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Jaenisch, Holger (2009) Data modeling applications in space science and astronomy. PhD thesis, James Cook University.

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Data Modeling Applications in Space Science and Astronomy

Holger Marcel Jaenisch

A portfolio mini-thesis submitted in partial fulfillment for the degree of

Doctor of Astronomy

James Cook University

ii

STATEMENT OF ACCESS

I, the undersigned, author of this work, understand that James Cook University will

make this thesis available for use within the University Library and, via the Australian

Digital Theses network, for use elsewhere.

I understand that, as an unpublished work, a thesis has significant protection under the

Copyright Act and;

I wish the following restrictions to be placed on this work:

1. Copyright (All Rights Reserved)

2. Permission to copy portions of this work for academic or research purposes is

granted, provided appropriate refere nce and citations to the author and this work are

made.

3. This work may not be reproduced and sold for profit without express permission

from the author.

Holger Marcel Jaenisch, PhD January 2009

Signature Date

iii

STATEMENT OF SOURCES

DECLARATION

I declare that this thesis is my own work and has not been submitted in any form for

another degree or diploma at any university or other institution of tertiary education.

Information derived from the published or unpublished work of others has been

acknowledged in the text and a list of references is given.

Holger Marcel Jaenisch, PhD January 2009

Signature Date

iv

STATEMENT ON THE CONTRIBUTION OF OTHERS

Supervision: Dr. Andrew Walsh

Dr. Graeme White

Dr. David Blank

Editorial Assistance: Mr. James Handley

Sponsorship: The following individuals and companies sponsored the presentation

and publication of several papers derived from this work (which are

noted in the References).

Dr. Nathaniel Albritton Amtec Corporation (USA)

Mr. Michael Hicklen dtech Systems, Inc. (USA)

Mr. Richard Esslinger Axiom Corporation (USA)

Mr. John Deacon Sparta, Inc. (USA)

Mr. Jeffrey Faucheux Sparta, Inc. (USA)

Dr. Carl Case Sparta, Inc. (USA)

Dr. Marvin Carroll Tec-Masters, Inc. (USA)

v

ACKNOWLEDGEMENTS

I would like to express my deep appreciation to the following individuals for technical

discussions and the development of creative test cases and examples demonstrating the

validity, utility, and appropriate application of Data Modeling as formulated in this thesis:

Marvin Barnett, James Handley, John Pooley, Dr. Marcel Thuerk, Ronald Juergens, Dr.

Miroslav Filipovic, Alexander Hons, William McDonald, Claude Songy, Richard Snow,

William Dionne, Michael McGary, Lee Ray, Dr. Kajesh Patek, Robert Morgan, James Myers,

Tim Aden, Scott McPheeters, Louis Bonham, Burt Vierge, Stoney Massey, Michael Conn,

Richard Gregg, Dr. David White, Marty Krizian, Stephanie Berry, Thomas Kotsch, Dr.

William Pickle, Dr. Mehmet Erengil, Dr. Stephen Kornguth, Dr. Matthew Edwards, James

Hunt, Todd Alexander, Emilio DiGiorgio, Thomas Martin, Joseph King, Dr. William

Douglas, Chester Rowe, Dr. Robert Carruth, Sieglinde Jaenisch, Kristi Jaenisch, David

Vineyard, Dr. Barry Johnson, Dr. John Caufield, Dr. Michael Curley, Dr. Nickolai Kukhtarev,

and Mr. Wintley Phipps.

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Data Modeling Applications in Space Science and Astronomy

Abstract

This Portfolio Thesis introduces seventeen scientific and technical papers that present

unique numerical and analytical algorithms that can be applied to any type of data and

imagery, independent of scene content, but are developed and illustrated here in the

context of astronomy and space sciences. Specifically, application to autonomous

remote vehicles for exploration is developed as a unique enabling technology. For

each paper, I present its significance to a hypothetical autonomous rover mission as an

example of how the new algorithms might find utility.

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Table of Contents

CHAPTER 1

PORTFOLIO THESIS OVERVIEW

1.1 Introduction. Page 1

1.2 Background. 3

1.3 Portfolio Introduction. 7

1.4 Robotic Rover Background –

An Example of the Need for Autonomous Software. 8

1.4.1 Autonomous Vehicles. 11

1.4.2 Path-Planning. 11

1.4.3 Reactive Navigation. 13

1.4.4 Data Modeling – The Key to Reactive Navigation. 14

1.5 Survey of Current Work. 15

1.6 Portfolio Thesis Papers – An Overview. 18

Paper 1-

Data Modeling For Radar Applications.

(Jaenisch & H andley, 2003). 18

Paper 2 -

Data Modeling of 1/f Noise Sets.

(Jaenisch & Handley, 2003). 19

Paper 3 -

Automatic Differential Equation Derivation From Sampled

Data For Simulation Parameter Sensitivity Analysis.

(Jaenisch, 2006). 19

Paper 4 -

Automatic Differential Equation Data Modeling for

UAV Situational Awareness.


viii

Paper 5 -

Data Model Predictive Control as a New

Mathematical Framework for Simulation and VV&A.

(Jaenisch, Handley & Hicklen, 2006). 21

Paper 6 -

Analytical Data Modeling for Equivalence

Proofing and Anchoring Simulations with Measured Data.

(Jaenisch, Handley & Hicklen, 2006). 21

Paper 7 -

Data Modeling Change Detection of Inventory

Flow and Instrument Calibration.

(Jaenisch, Handley & Bonham, 2003). 22

Paper 8 –

Data Modeling for Fault Detection.

(Jaenisch, Handley, Pooley & Murray, 2003). 22

Paper 9 -

Data Modeling for Virtual Observatory Data Mining.

(Jaenisch et al., 2004). 22

Paper 10 -

Virtual Instrument Prototyping with Data Modeling.


Paper 11 -

Data Modeling of Deep Sky Images.

(Handley, Jaenisch et al., 2004). 24

Paper 12 -

Image Resolution and Performance Analysis of

Webcams for Ground Based Astronomy.

(Lim, Jaenisch, et al., 2004). 24

Paper 13 -

Classification of Jacoby Stellar Spectra Data

using Data Modeling.

(Jaenisch and Filipovic, 2002). 25

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Paper 14 -

Enabling Unattended Data Logging and Publication

by Data Model Change Detection and

Environmental Awareness.


Paper 15 -

Data Driven Differential Equation Modeling of fBm Processes.


Paper 16 -

Data Modeling Augmentation of JPEG for Real-time

Streaming Video.

(Jaenisch and Handley, 2005). 27

Paper 17 -

Shai-Hulud: The Quest for Worm Sign.

(Jaenisch et al., 2005) 27

1.7 Conclusion. 28

x

CHAPTER 2

ADDITIONAL APPLICATIONS 30

2.1 Introduction. 30

2.2 Application to Data Ensemble Synthesis. 30

2.3 Application to Critical, Sensitive, and Key Parameter

Estimation for Analytical Modeling. 31

2.4 Application to State Estimation From Trajectory Estimates

for Tracking and Automatic Controller Design. 31

2.5 Application to Formal Proofing and Validation and Verification

of Software Simulations. 32

2.6 Application to Health Usage Management Systems (HUMS)

and System Health Diagnostics and Mean Time to Failure Prognostics. 32

2.7 Application to Data Mining. 33

2.8 Application to Virtual Instrument Prototyping. 33

2.9 Application to Image Processing, Analysis and Modeling. 34

2.10 Application to Image Compression and Synthesis. 34

2.11 Application to Spectral Analysis and Modeling. 35

2.12 Application to Unmanned Vehicles. 35

2.13 Application to Information Assurance and Network Security. 36

2.15 Conclusion. 37

References. 38

Portfolio Thesis Papers. 44

1

CHAPTER 1

PORTFOLIO THESIS OVERVIEW

1.1 Introduction.

This thesis addresses how requirements for planned deep space exploration missions

with autonomous sensor platforms can be met using current available mathematical

techniques. Current exploration missions rely on sending remote viewing cameras

and instruments to neighboring planets. As distances of these missions extend further

into space, message delays to and from Earth prohibit interactive decision-making and

mission execution.

Enabling remote exploration beyond the interactive delay boundary requires remote

sensor platform cognition, mentation (Blackmore 2004) and self -awareness. Onboard

algorithms must quickly process and understand their local changing environment to

enable in-situ and real-time coarse-of-action modification (the simplest form of which

is reactive navigation). Mission planners cannot be certain what will be encountered

on distant worlds. Without prior examples, knowing when something out of the

ordinary or a novelty is encountered requires cognitive awareness (because you

cannot know in advance what you do not know and therefore cannot prepare a

classifier to deal with it in advance). The intelligent sensor must discern novelties and

off-nominal events as though an investigator were standing planet-side on the distant

world. Infrequent data transmission and limited energy sources restrict onboard

computing power for image collection, processing, transmission, and in-situ science

analysis. Although the current use of digital based computing and processing has

become the main tool for remote control vehicles true efficient real-time computing

will require hardware based analog computing that is tolerant of harsh environments

and resistant to noise.

2

The shortcoming of analog computers (Korn and Korn 1964) has been touted to be the

limited processing of simple differential equations. Complex modern algorithms are

rarely in simple differential form. Hence, the real-time power of the analog computer

has not been brought forward because of the mathematical inability to recast complex

decision making architectures and algorithms into a series of differential equations

that can be adequately solved on analog computers that would enable miniaturization

and embedding on small frame autonomous vehicles. If this mathematical magic

bridge could be built (convert digital complex algorithms into differential equation

form), then the analog computational approach once again becomes a very reasonable

and very powerful tool to advance cognitive methods onto fully autonomous

platforms. The inability for an intelligent agent such as the analyst to reside station

side for remote missions limits the complexity of tasks that can be accomplished with

remote control based systems.

This limitation is partly compensated for by building knowledge models of the

environment and transmitting those models and any deviations from them as

“interesting” to mission analysts (Jacoby 1991). An intelligent and self-aware sensor

must be able to decide between photographing, analyzing and sampling rocks

encountered on a strewn field. The mathematical framework and bridge for modeling

changing information encountered during autonomous exploration with in-situ

derived differential equations and using them to detect novelties is called

Entropyology (en’tro-pe’-ah-lo-gee) (from entropy-ology, term coined in 1996 along

with Data Modeling by the author - the science of changing-information). This

portfolio thesis introduces the methods of Data Modeling and a myriad of successful

applications of these fundamental mathematical tools to demonstrate the feasibility of

developing autonomous probe missions.

Data Modeling is a fledgling theory. Full proofs of the general utility and limitations

of its methods are still being developed. The fact that a consistent set of methods can

be successfully applied to a diverse set of applications is itself not a proof of

universality. However, lacking such proofs does not preclude its utility. The power

of this new Data Modeling theory is the simplicity of the techniques involved once a

subtle but profound shift in the perspective of its use and interpretation of results has

occurred. The methods comprising Data Modeling are not themselves new. Many

3

will appear simple and previously tried-but abandoned. This is indeed the case.

Many simple but powerful techniques have fallen by the way side because of innate

limitations of the techniques themselves. But what if these limitations could be

addressed and solved or removed altogether? What if limitations that could not be

removed could be found to be exploitable and useful if viewed from a different

perspective and used in a novel way? Then, the powerful techniques resurrect and

become attractive again because the poison of the techniques limitations has been

transformed into a powerful cure. It is with a keen eye for recognizing the familiarity

of methods yet recognizing and discerning the subtlety of the way in which they are

used and applied that makes the difference between sulfur, salt-peter, charcoal and

gun powder. Simply recognizing the fundamental ingredients and knowing they must

be mixed or being aware of their general properties does not lead to the magical

formula for gunpowder or the more elusive technique for producing it. In the same

vein, the power of Data Modeling is not in the methods themselves but how they are

applied and how the results are used.

1.2 Background.

Cognition and mentation are aspects of self-awareness and do not fall under the

machine intelligence and artificial intelligence (AI) paradigm (Winkler 1972). AI is

the ability to program machines by example rather than by heuristic (text based rules

of thumb) or explicit program instructions. Programming by example is commonly

referred to as pattern matching or pattern recognition and emphasizes learning

differences between at least two distinct classes or sets. It is not possible to apply

pattern recognition methods to single class cases.

Machine learning is the ability to implement adaptive algorithms onto machines to

enable decisions to be modified in response to changing stimuli. Neither paradigm

encompasses the transition to biomechanics or mechanistic -bioinformatics which

embraces cognition (mental awareness and mentation), the human thought process of

extracting relationships and salient features, formulating hypothesis and making

predictions while updating views.

4

In order to achieve autonomy, the first principal of cognition which must be achieved

is self-awareness. Self-awareness is the ability to recognize that which is me from that

which is not me without anyone or anything explicitly showing the difference. This is

the process of forming expertise in a subject, just as an art specialist learns the details

of the technique of an individual master painter so that the specialist can then easily

discern forgeries when encountered in the future. This ability was attained without

having been trained on learning differences between the original and forgeries, since

forgeries are infinite in number but the originals are finite. Instead, sufficient detail is

discerned and learned that makes the specialist hypersensitive to deviations from such

details when encountered. The specialist is thus able to distinguish things that are the

same from a myriad of things that are similar but distinct without having to rely on

classification.

The expert may then attempt to teach such expertise to others by presenting examples

of forgeries and originals to give an example of what is being keyed off of. This

process is teaching classification by pattern recognition and is markedly inferior to

gaining expertise by experience alone. Students of the pattern recognition method

intrinsically learn to emphasize the differences between the presented examples more

than gaining expertise in details of the originals. This leads to an inability to recognize

good forgeries that the original expert can still easily distinguish. This original form

of gaining expertise is not pattern matching or pattern recognition it is a product of

cognition and mentation on aspects of similarities from a single class alone. This is

the ability that must be embodied as a computational process in a vehicle if it is to

achieve self-awareness or cognition.

The process of mentation is understanding and projecting variations of the learned

style or painting style that makes it possible for the expert who can recognize

forgeries to also recognize a newly discovered painting by the artist of interest even

when that image has never been seen before. This is a result of mentation which

followed cognition. This projecting of expectations ba sed on perceived experience

enables recognition of the sameness even when the object is vastly different from the

gross appearance of the original set. How can such elusive magical qualities be

embodied in a cybernetic process stored in an autonomous vehicle? This is the

question that is being addressed with the proper application of Data Models and the

5

Change Detectors they empower. This ability to discern from a single class can be

achieved with mathematical structures if used in profoundly subtle but powerful ways.

The Data Model is a formal differential equation representation of numerical data

(Christ, et al 1999) (Wang and Zheng 2005) that can be obtained from sensors that

observe the world around a rover. To make the processing power of the rover small

and able to respond to real-world environmental changes as they occur in real-time

requires true real-time computational speeds. Traditionally, differential equation

models are derived by humans after years of data collection and simplification with

abstraction of results into trends and can be represented with relatively simple smooth

and continuous differential equations. Such equations are few in number and their

solving is taught and readily available to the scientist or engineer, but deriving the

equations themselves is a holy grail in the symbolic computational arena and

represents a quest for true machine cognition (Cohen 2002) (Cohen 2003).

The techniques of Data Modeling allow new differential equations that describe the

relationships between observed data sets to be derived autonomously and independent

of human intervention. Once derived these equations can be solved using classical

mathematical techniques as required for analysis. The equations form a corpus of

knowledge or a descriptive hypothesis of the information relationships encountered.

This synthesis and abstraction of information relationships is the goal of data mining

and is vastly different from simple relational searching, keyword matching, or pattern

recognition applied to data base structures.

Database information structures (Duncan 1975) (Das 1992) have been labeled as

“data models”, though this is very misleading since these data models are simply flow

charts on the database contents showing relationships between information flow in an

organization. Nowhere is a model created or presented of the “data” itself.

Conversely, a Data Model is an explicit mathematical model of the data corpus itself

that can not only reconstruct the data from which it was derived, but also interpolate

as well as predict via extrapolation both forward and backward indefinitely. Hence, a

derived model of a data set is infinitely more powerful and useful than storing a data

set itself. Further, the information flow structures embodied by the database

community term of “data model” can in fact also be converted into a true Data Model

6

which would yield differential equation models of the information within the structure

itself, yielding formal proofing (Jaenisch 2006) and predictive modeling capability to

such models! This allows data models for databases to be converted into Data Models

and enables databases to be converted into true Knowledge bases. Information

processing is different from data processing. To be useful for cognition the former is

required.

In insects, such processing is referred to as hardwiring or “instinct” (Amos 2004). We

can make the analogy that instinct programming is to insects what mental thoughts are

to humans, in the same way analog computers based on operational amplifiers work in

true real-time but are specialized while digital computers are more general but far less

efficient and much slower in obtaining results. The ability to embody advanced

information processing algorithms into hardware in the form of analog computers has

heretofore been impossible. Since analog computers typically process only differential

equations, it would require all forms of information to be mapped into the form of a

forcing function feeding dynamic models or coupled differential equations.

Where would these system level model differential equations come from? These

differential equations are the autonomously derived Data Models that the author

created that extend the early Group Method of Data Handling (GMDH) (Madala and

Ivankhnenko 1994) and polynomial network developments far beyond simple pattern

matching. By generalizing and extending the GMDH work into a form of deriving

multi-variable analytical equations rather than simple nested second order

polynomials, true partial differential equation models can be derived and implemented

in analog computer hardware. This means the world can be sampled and understood

mathematically, enabling hardware to be created that processes in real-time these

perceptions only because they are in differential equation form.

Bayesian methods (Gustafson 1994) (Andrews and Stafford 2000) (Grzymala-Busey

1991) (Bolstad 2004) try to circumvent the inability to obtain the actual differential

equations of motion or dynamics by characterizing the average statistical behavior

and variance of observed dynamics. This covariance based description throws away

all the detailed information inherent in the relationships between the observed data by

7

representing only average occurrence as probabilities. The average observed behavior

never provides a mathematical model of the actual dynamics.

Without the actual equations, true forecasting forward and backward in time is not

possible. The crude GMDH-type methods, although applied to many areas, are

notorious for failing spectacularly to generalize or extrapolate and are very unstable

even for interpolating. By combining fractal and chaos algorithm methods with Data

Modeling differential equations, it is possible to convert streams of data directly into

differential equation form for ease of storing, comparing, inverse parameter

estimation, and future comparison as well as forecasting and backcasting. The tools of

Data Modeling make it possible to achieve a complete robust world-view paradigm

for a robotic sensor in a common framework tha t is independent of the sensor source

or type. This ability seems almost magical, but it is the goal of my thesis to show that

the fundamental mathematical foundation underpinnings already exist and are well

established. It simply requires a subtle but critical paradigm shift regarding how the

methods are viewed and applied. This shift enables a tremendous leap forward in

autonomous cognition. In fact, the change detector created from Data Models is the

first true example of a “Self-Aware” algorithm that is ready to be explored and

exploited for its potential to enable cognitive automatons (Collins and Horn 1991).

1.3 Portfolio Introduction.

In this Portfolio Thesis, I introduce seventeen scientific and technical papers that

present unique algorithms for information processing. These papers are the basis of

my submission for the degree of Doctor of Astronomy (DoA) at James Cook

University.

These algorithms can be applied to any type of data and imagery, independent of

scene content, but are developed and illustrated here in the context of astronomy and

space sciences. The computational processes could be as readily applied to

underwater reef studies, mammograms, portraits, microscopic images, radar images,

synthetic computer generated imagery, or simply pseudo-colored data arrays

comprised of code vectors, text, or any other embodiment of data.

8

The attached Portfolio Papers have been published over the period 2002 to 2006.

1.4 Robotic Rover Background - An Example of the Need for Autonomous

Software.

This section of the Portfolio Thesis is specifically to illustrate the purpose and need

for the algorithms that are presented below and in the attached papers. As stated

above, these algorithms have use in very many (possibly all) information processing

environments, but they are presented here in the context of astronomy as they were

developed for that use.

The use of robotic rovers (Bond and Allman 1996) is both an attractive and necessary

option for the exploration of planetary and or other remote objects. Although state-of-

the-art technology for remote control are presently used, fundamental limitations in

this approach exist preventing the realization of fully autonomous missions.

The principal issue with space exploration is the distance between the rover and the

Earth data acquisition and command station. The separation between the rover and its

base results in delayed communications and slower data streams.

For example, radio signals between Mars and Earth take from between 6 and 41

minutes transit time. Mars distance from Earth ranges from a minimum of 0.5 AU to

a maximum of 2.5 AU (when the Earth and Mars are on opposite sides of the Sun). 1

AU is about 150 million km and light travels at approximately 300,000 km/s. Light

travels 1 AU in approximately 500 seconds. Therefore, transmitting a radio signal to

Mars would take a minimum of 250 s (approximately 4 minutes) to a maximum of

1250 s (approximately 21 minutes). Round trip would be double these numbers, or 8

minutes to 42 minutes for a response to be received after transmitting the original

message. This does not take into account message drop out, handshaking for message

confirmation and message processing prior to response. This long distance itself also

imposes low communication bandwidth limits for information assurance and signal

quality.

9

This tyranny of distance precludes the use of teleoperation (every individual

movement controlled by human intervention) for controlling the vehicle and unveils

the requirement for vehicle autonomy.

Major improvements have been made in deep space telecommunications, including

the implementation of high-bandwidth communication systems operating at Ka-band

(Kurtz-above radio frequency of 32 GHz, which is off-limits to commercial users).

Ka-band is a factor of four higher in frequency than the current X-band technology

used in deep space communications, and will provide a data rate from Mars of more

than 2 megabits per second (today the maximum data rate transmitted to Earth by a

spacecraft at Mars is about 128 kilobits per second). Current technological

developments are also directed toward improving the performance of radio

communications through the use of large deployable spacecraft antennas and ground-

based antenna arrays.

In addition, optical transmission, which relies on laser light instead of radio waves, is

being explored to enable video-rate communications from Mars, and large gains in

data rate for outer solar system exploration. Laser communications from deep space

will be received by optica l telescopes operating either on the ground or on platforms

above the Earth's atmosphere. In the case of orbiting receivers and relay transducers,

real-time access can be anticipated in the future using a "trunk line" from the Earth to

a relay spacecraft in orbit around the distant planet, and proximity links between that

spacecraft and landers, rovers, and other vehicles. These exciting activities provide

intellectual and career opportunities for many systems engineers and experts in optical

communication.

However, the present communication delay makes Earth-bound remote control of

exploration vehicles at distances beyond the Earth's moon impractical – and

autonomous technology is a key to meeting this challenge. The Mars technology

program, for example , is developing technologies that will enable a rover to travel to

and sample a rock 10 meters away with a single command (instead of the five to 10

commands required by the present rovers) (Boden and Larson 1996).

10

Future planetary exploration missions may include the study of Mars by a series of

robotic missions, and the surface exploration of Venus and Titan. Plans currently call

for vehicles to carry samples back from distant bodies as far away as Venus or a

comet's surface, and for an investigation of the Moon as a source of minerals for

future space manufacturing, and even for earthly use. Deciding which rock to bring

back from a strewn field remains an elusive problem that will have to be addressed.

The subsurface exploration of solar system bodies , such as Mars and Jupiter’s moon

Europa, will require yet another dimension of robotic mobility, and more powerful

systems still will be needed for aerial vehicles operating in the atmosphere of Titan.

For the rover to function properly as an autonomous agent (Kott n.d.), it is essential

that it can sense, model, and discern its environment, enabling it to understand what it

perceives and allowing it to respond and react to these perceptions. For example, the

rover must know if it is going uphill or downhill, if there is an obstacle in front of it

and so on. This is achieved by the perception system of the rover. The perception

system senses the environment by using physical and virtual sensors.

Physical sensors could include wheel encoders, inclinometers, cameras, and laser

rangefinders which detect the immediate terrain environment of the robot. Virtual

sensors are mathematical functions defined by the values of some physical sensors.

For example, virtual sensors can give the absolute spatial location of the rover in

Cartesian coordinates.

Crude forms of virtual sensors are called smart sensors (Hall and McMullen 2004)

(Klein 2004). Smart sensors preprocess their own data prior to dissemination while a

virtual sensor may combine information from several sources including its own to

yield a data stream of information that is not directly observable by any sensor alone.

An example might be detecting networks of electrically connected objects such as

Improvised Explosive Devices (IED) or roadside bombs, where the entire IED is the

object of interest but no actual sensor can find such IEDs. Instead, a collection of

sensors is used to sense orthogonal and independent components of the IED with

varying levels of confidence, and the aggregate information from the various sensors

that alone only sees pieces of the IED infers the existence of the IED itself. This fused

suite of sensors forms a “virtual sensor” of IEDs and requires sensor fusion and

11

information fusion at all levels to be realized using a form of spatial voting and

reasoning to combine disparate sensor information.

1.4.1 Autonomous Vehicles

To meet the increasing requirements, a totally autonomous vehicle that can travel for

extended periods carrying out its assigned tasks is necessary. This is beyond the

present state-of-the-art of artificial intelligence, and methods currently under

investigation within the space community will provide only limited vehicle autonomy

at best.

This thesis sets out to demonstrate how Data Modeling overcomes many of these

limitations.

There are various degrees of autonomy and various approaches in the way autonomy

is granted to a rover. These factors determine the specific navigation technique used,

but all fall under two broad categories:

§ Path-Planning navigation

§ Reactive navigation (real-time obstacle avoidance navigation)

1.4.2 Path-Planning

In Path-Planning (Taguchi and Jugulum 2002), some form of terrain analysis is

performed and a safe route is decided before the vehicle is commanded to start

moving. In all cases, it is assumed that there is not previous knowledge of the terrain.

The route therefore requires reactive navigation as the rover moves towards a goal

and avoids obstacles or untraversable territory as it encounters them (Kim and Lewis

1998).

The core process invoked here is path planning, which suggests the superposition of a

global low-resolution height/elevation (topographic) map from the orbiter to high-

resolution map from the rover. The process of matching local terrain to global

12

overviews is called terrain matching and is achieved by an algorithm where the

translation that best matches the local and the global height map is sought.

An example of Path-Planning is Computer Aided Remote Driving (CARD). CARD

relies for its operation on stereo images sent from the rover to the ground-station.

Stereo imaging produces 3D pictures with depth perception, which enables the

operator to estimate a safe path for the vehicle, and designate this path using a 3D

cursor. The ground station computer calculates the control sequences for the vehicle

according to the designated path and then sends these to the vehicle. The vehicle

moves to the new location, sends a new picture and the process repeated. Depending

on the terrain and visibility conditions, the rover can move about 20 meters on each of

these iterations. Taking into consideration the delays involved (round-trip signal

delay, operator delay and computation delay), this approach results in an average

speed of somewhat less than a centimeter per second.

The major advantages of CARD compared to teleoperation is the relatively less

information transmitted. In addition, since the major computation is done on Earth,

the computers used can be powerful, saving the rover from carrying and powering

significant computers. However, no matter how fast the path planning and

computations are performed, the round-trip signal delay cannot be significantly

reduced and the average speed is unlikely to improve dramatically. Therefore, CARD

will be more suite d to short distance travels, such as traversing a difficult area or

performing a number of experiments in one location.

In semi-autonomous Path Planning navigation, the rover is given approximate routes

from Earth, but plans its local routes autonomously. Some operations are performed

on Earth while others are performed onboard the vehicle. In this scenario, a satellite

orbiting Mars sends stereo images of the areas of interest to the ground-station on

Earth. These images have a resolution of about a meter and enable operators to plan a

safe route for the vehicle for distances of, say, a few kilometers. In addition to path

planning, an elevation map is produced by computers. Both the elevation map and the

planned path are sent to the rover.

13

Onboard the rover, laser rangefinders and stereo cameras are used to obtain images of

the immediate environment. These images are used to compute a local topographic

map. This map is then matched to the local portion of the global map sent from Earth,

so that the rover can position itself on the global map and follow the designated route.

By comparison with the global map sent from Earth and the local map obtained from

the rover's sensors, a new detailed high-resolution map is produced by computers

onboard the rover. This map is eventually analyzed on the rover to determine the safe

areas over which to drive, while at the same time adhering to the route sent from

Earth.

1.4.3 Reactive Navigation.

Reactive Navigation (Maciejowski 1989) differs from path planning in that, while a

goal location is known, the rover does not plan its path but rather moves towards the

location by reacting to its immediate environment. There are various approaches to

Reactive Navigation, but all stem from the design belief that robust autonomous

performance can be achieved using minimal computational capabilities, as opposed to

the enormous computational requirements of path planning techniques.

Another modern approach to Reactive Navigation uses the adaptive capability of

artificial neural networks (Diamantaras and Kung 1996) (Jensen 1996), i.e., using

cumbersome back-propagation methods to tune reactive controllers or other more

specific methods such as support vector machines or radial basis function algorithms.

This approach maps perceptual situations into locomotion commands. These

mappings occur in distinct stages: (i) input data preprocessing, (ii) perceptual situation

classification, (iii) action association, and (iv) action validation. This series of

mappings leads to control architectures that are complicated, layered, brittle and

hierarchical. Existing applications of neural network techniques are limited by slow

training times and recent attempts to develop on-line reactive navigation capable of

on-line reflexive locomotion, using a trial-and-error feedback, have met with limited

success.

14

1.4.4 Data Modeling – The Key to Reactive Navigation.

Data Modeling has demonstrated autonomous performance – i.e., Data Modeling has

obtained Reactive Navigation (by deriving goals from mathematical predictions based

on local observed conditions) using minimal computational firmware (analog

computer). Analog computer implementation (Miola 1993) allows true real time

implementation. Analog computers are radiation hardened, temperature stable,

efficient in package size and power usage, and provide a method for closed loop servo

control to be achieved for the rover. Data Modeling gives the rover and its associated

sensor suite a higher reasoning capability (or a crude form of intuition) to supply

global and local navigation cues while enabling course-of-action autonomy (Widrow

and Walach 1996).

Entropyology combines, in novel and unique ways, mathematical theory and

techniques from data theory, similitude theory (Gelb and Vander Velde 1968),

analogy theory, cybernetics (Goldberg 1989), fractal geometry (Barnsley 2006), chaos

theory, catastrophe theory, complexity theory (self-organization), consciousness

theory, formal analysis (Jones 2002), parameter estimation (Raol, et al. 2004) (Doyle,

et al. 2002) (Mendel 1973), inverse modeling (Chalmond 2003), dimensional analysis

(Lipka 2003) (Palmer 2008), GMDH (ModelQuest 1996), Model Predictive Control

(MPC) theory (Lemke 2006), virtual sensor theory, and Data Modeling theory (Ladde

and Samba ndham 2004) into a series of analogical as opposed to simple numerical

and analytical methods for converting data into equations and exploiting them using

the rules of mathematics.

Even though the tools of Data Modeling were developed under the influence of the

various cited fields, there is no direct mathematical derivation or lineage from any of

them. The tools developed by the author are unique and were created after being

familiar with work in the various fields cited. The author created a commercial

software environment called MAJQANDA that is a collection of the classical and

Data Modeling based algorithms for decision-making and information processing.

15

1.5 Survey of Current Work.

A summary of the literature in this field is presented here in chronological order.

Only key works are presented, and the intent is to bring the reader up to speed in this

rapidly changing and exciting field. A more comprehensive survey of AI and machine

learning methods is outside the scope of this thesis and is readily available by simple

Google searching (http://www.google.com) on Internet resources. Repeating the

wealth of existing surveys here (none of which compare Data Modeling) is distracting

to the focus of this thesis. Further, a survey of machine learning and AI itself has also

been exhaustively documented on-line and will not be repeated here. The algorithms

in this thesis demonstrate the methods the author has personally applied to many real-

world applications in the past and where Data Modeling has proven superior.

Though this does not prove universal optimality, it does demonstrate diverse

applicability. The cited references for Data Modeling are by necessity sparse since the

author is the originator of the theory and it is too new to have wide audience. Several

examples of third parties using Data Modeling on their own projects is included in

this thesis to demonstrate impartial critical review and support of these methods.

Further, the application for a Data Modelin g patent as recent as April 2008 indicates

that these methods are indeed very new and therefore the review literature pertinent to

these methods cannot be expected to be extensive at this time.

This review starts with a paper describing a fully integrated modular, and object

oriented mobile vehicle simulator, named Expectations, that was published by Yung

and Ye (1997). In that work, all software was written in C/C++. Because this

software was modular, it allowed different algorithms for collision avoida nce,

navigation, path planning, and behavior learning to be easily swapped in and out

through a “plug and play” environment for algorithmic study and development. This

was a test-bed evaluation platform rather than a set of novel algorithms; however, it

did provide a variety of interesting benchmarks for later work, including that

presented in this thesis.

In the same year, Yung and Ye, this time joined with Fong, (Yung, Ye and Fong,

1997) presented an integrated virtual-world simulator for road vehicles that was

16

modular, object oriented, and hierarchical in its construction. This work now

provided a test-bed for automated driving, collision avoidance, navigation strategies,

tactics, and driving assistance. Again this work was a test-bed for evaluating the

proposed algorithms, and, again, it only presents a heuristic algorithm (explicitly

defined text based rules of thumb, and therefore cannot be learned from the

environment without human intervention).

Ye and Wang (2001), in a paper entitled “A Novel Na vigation Method for

Autonomous Mobile Vehicles” proposed software for a “navigator” that consisted of

software for obstacle avoidance, a goal seeker, a navigation supervisor, and an

environment evaluator. This methodology was specifically intended for use in

unknown and partially known environments, but relied on vehicles using traditional

fuzzy-logic operations (fuzzy-logic operations are a priori assigned binning of

numerical values into fixed increment meanings). Fuzzy rules cannot be discovered

autonomously but are specified by a human. These can be avoided.

The problems involved in navigation of autonomous vehicles using images from a

mounted camera are illustrated by Kumar, Jawahar and Narayanan (2004) . In their

work, items such as signboards were imaged by the camera and the vehicle position

was estimated with respect to the fixed signs. The methods employed were based on

contour correspondence between the current view and a reference view. This was an

elegant vision-system approach, and if sufficient onboard processing power were

available, it could have formed an interesting autonomous vision system. The paper

does not lead to analytical models, nor does it try to derive any such models.

In a paper published in 2005, Curtin et al., discusses the current state of autonomous

underwater vehicles (AUVs) and the steps involved to transform individual AUVs

into adaptive networked systems. Vehicle types are suggested based on their function

and operation environment (surface, interior, or bottom layer). In this work, various

platform navigation and control systems (software) were explored including methods

of creating autonomy for the vehicle. The laws that govern intelligent maritime

navigation are reviewed and an autonomous controller with conventional collision

avoidance behavior is described. Performance and behavior of various systems is

17

compared. The algorithms presented by Curtin et al., (2005) are, however, heuristic

rather than analytical.

Srini (2006), in web paper hosted by the IEEE Computer Society, presents a method

for using autonomous navigation principles in urban environments. In this

contribution, it was shown that current mobile sensor networks for autonomous

systems will also support autonomous modern vehicles equipped with actuators for

steering, throttling, and braking. Older vehicles are also supported by the system if

they are equipped with basic sensors and a smart phone. This paper documented

system interplay and mechanical fusion rather than autonomous environmental

awareness.

Fuzzy logic control, combined with behavior-based control, was introduced into the

science by Lei and Li (2007). Behavior-based control is changing the control

threshold based on the operational mode the device is in. This mimics but is not true

behavior based adaptation, since the various understood operational modes are pre-

programmed and the executed course of action simply depends on the mode the

device finds itself in.

Lei and Li presented a new method using behavior-based control of mobile robots,

specifically for path planning in unknown environments. A heading for the robot was

formed by connecting the target steering behavior and the obstacle avoidance

behavior that took the robot towards the target while avoiding obstacles. These

algorithms were statistical in their nature and not analytical.

Because these algorithms were statistical in nature, all the detailed information

inherent in the relationships between the observed data is represented as only average

occurrence using probabilities. The average observed behavior never provides a

mathematical model of the actual dynamics, which is crucial for use by a real-time

analog computer to eliminate stimulus A/D conversion and system latency.

A US patent (Brandon 2007) presents a method for gathering data with a Unmanned

Vehicle (UMV), storing that information into one or more Radio Frequency

Identification Units (RFID) storage devices (or some other data storage device), and

18

then releasing these devices for later collection and analysis of the stored data. This

technique was interesting because of the concept of allowing a small rover, with

limited on-board storage space, to offload data for future pick up. Limitations of the

number of off-loaded canisters are not addressed nor are the best methods for storing

data for later retrieval.

Again in 2007, a web published conference report by Kofod-Petersen et al. (2007),

gave several philosophical approaches to applying reasoning algorithms for

autonomous navigation. Most of the methods are heuristic in nature or statistical, and

there is no mention of analytical methods.

In 2008, several on-line articles appeared touting the military interest in swarm

technology and insect size autonomous vehicles for not only intelligence gathering

missions but also as offensive/defensive weapons with attack capability and

sophisticated ad hoc network centric information processing capabilities. It is these

later devices that most closely resemble the autonomous vehicles for remote sensing

that Data Modeling seeks to empower and enable.

1.6 Portfolio Thesis Papers – An Overview.

Here I list my papers presented for the degree of Doctor of Astronomy. Publication

details are given in the reference list. The papers are attached.

For each paper, I present the significance of the paper to a hypothetical autonomous

rover mission, which is, as explained above, here used as an example of where the

new algorithms are used.

Paper 1- Jaenisch, H., and Handley, J., “Data Modeling for Radar

Applications”. Radar Conference, 2003. Proceedings of 2003 IEEE,

pp. 379-386 (2003).

This is a general Data Modeling paper that shows the applicability and utilization of

the common core numerical algorithms across a large array of remote sensing

problems. This paper shows how, by converting numerical data into equation form,

19

universal data sharing and decision-making algorithms become computationally

efficient and autonomous. This work sets the stage for addressing in detail various

requirement of a rover mission that can be addressed using the Data Model methods,

thus alleviating the need for exotic Artificial Intelligence (AI) algorithms that are

unstable and computationally inefficient.

Paper 2 - Jaenisch, H., and Handley, J. “Data Modeling of 1/f noise sets”.

Noise in Complex Systems and Stochastic Dynamics. Edited by

Schimansky-Geier, Lutz; Abbot, Derek; Neiman, Alexander; Van

den Broeck, Christian. Proceedings of SPIE, Volume 5114, pp. 519-

532 (2003).

This paper presents a numerical method (that has the potential for use in an

autonomous planetary rover-type mission) for converting sensor data into a numerical

form suitable for modeling with analytical equations. This paper illustrates a general

purpose method for converting traditional measurements (such as temperature or

pressures as observed by a rover), imagery and internal status measurements, into a

single consistent algorithm for automatically deriving differential equations. These

derived differential equations are then used by the rover to make course-of-action

decisions.

Paper 3 - Jaenisch, H., “Automatic Differential Equation Derivation From

Sampled Data For Simulation Parameter Sensitivity Analysis”.

Intelligent Computing: Theory and Applications IV. Edited by

Priddy, Kevin L.; Ertin, Emre. Proceedings of SPIE, Volume 6229,

pp. 62290N (2006).

This paper presents a method by which the vehicle can process differential equations

obtained from sensor measurements (or status measurements) using the algorithm

described in the previous paper (Paper 2). The processing enables the identification

of the important parameters of the sensor measurement. This means, for example,

that the rover may be observing interesting changes in temperature or pressure as a

function of albedo or irradiance, and from these changes, for m a differential equation

model of the relationships. Once formed, parameter sensitivity analysis (as described

20

in this paper) enables the model to be simplified, and the crucial parameters to be

identified from among the available ensemble. Once identified, further detailed

monitoring by the vehicle can be undertaken to determine if unique or anomalous

events are occurring.

The algorithm presented in this paper enable the derivation of differential equation

models to be saved and checked with stored equations for similarity or uniqueness.

Instead of data basing new raw data, and trying to compare distributions yielding

ambiguous answers, this approach derives differential equation models and compares

the equations for similarity. Since these equations capture complex information

and/or relationships, the ability to compare such equations achieves far more than

pattern or template matching techniques allow, and is very computationally efficient,

and minimalist in storage size. Once the differential equations are built and

compared, they can be analytically inverted to obtain estimates and predictions of

input conditions expected for a variety of responses that may be encountered in the

future, or for simple function optimization. Novel in this approach is the ability of the

vehicle to vary the eigenmodes from a Fourier series model in real-time during the

mission.

Paper 4 - Jaenisch, H., and Handley, J., “Automatic Differential Equation

Data Modeling for UAV Situational Awareness”. Unmanned

Systems. Edited by McDaniel, Mark. Proceedings of the Huntsville

Simulation Conference (The Society for Modeling and Simulation

International), (2003).

This paper presents an algorithm for use in the machine reasoning center to achieve

inverse modeling and parameter estimation. This algorithm enables physics-based

differential equations to be derived from noisy sampled measurement data alone.

Once these physics-based dynamic equations are derived, this algorithm enables

automatic and non-human-guided inversion of the equation model to obtain estimates

of physical parameters which cannot be directly measured by the sensors. An

example would be to infer wind speed aloft, or atmospheric turbulence, from the

model using measurements of star scintillation.

21

Paper 5 - Jaenisch, H., Handley, J., Hicklen, M., “Data Model Predictive

Control as a New Mathematical Framework for Simulation and

VV&A”. Intelligent Computing: Theory and Applications IV. Edited

by Priddy, Kevin L.; Ertin, Emre. Proceedings of SPIE, Volume

6229, pp. 62290U (2006).

This paper gives an algorithm that enables the autonomous reasoning capability of the

vehicle to compare two different (and independently derived) differential equations

that come from a single set of sensor observations. The purpose of this comparison is

to identify important underlying changes that may have occurred over time (an

example might be the comparison of two independently derived models of

atmospheric turbulence, made over two different time intervals, and at two different

locations). Being able to analytically compare the models eliminates the need for

cumbersome “heuristic expert rules” that otherwise would have to be pre-programmed

into the rover. Rather, this algorithm compares the structure and form of the

equations and the scale of the coefficients, and is able to see if the differences are

important and if they require further sampling or noting by the science package.

Paper 6 - Jaenisch, H., Handley, J., Hicklen, M., “Analytical Data Modeling

for Equivalence Proofing and Anchoring Simulations with Measured

Data”. Modeling, Simulation, and Verification of Space-based

Systems III. Edited by Motaghedi, Pejmun. Proceedings of SPIE,

Volume 6221, pp. 62210H (2006).

This paper presents an algorithm which will give the ability to make two independent

predictions of measurement. It is thus possible to form a set of “expected”

observation. This ability forms the basis of a “goal” for decision-making and route

planning by enabling the rover to navigate to areas where the sensor measurement

most resembles the “expected” data based on the derived models. This ability is

achieved by giving the vehicle the power to convert differential equation models into

a Transfer Function form that makes mathematical “prognostication” possible.

22

Paper 7 - Jaenisch, H., Handley, J., Bonham, L., “Data Modeling Change

Detection of Inventory Flow and Instrument Calibration”,

Proceedings of SOLE Conference, Huntsville, AL, Aug 12-14, 2003 .

This paper presents an algorithm that enables an autonom ous vehicle to monitor its

own internal system status for mission readiness and health. This is accomplished by

a model of “nominal” performance derived throughout the mission made from

internal sensors monitoring operations such as system power, temperature, vibrations,

energy usage, fuel availability, etc. This ability enables the calibration of the sensors

and systems to be checked automatically, and internally, by comparison with previous

equation models of nominal performance and system health (using the modeling

procedures presented in the earlier papers). This paper introduces situational

awareness that allows decisions to be dealt with during the mission.

Paper 8 – Jaenisch, H., Handley, J., Pooley J., Murray S., “Data Modeling for

Fault Detection”, Society for Machinery Failure Prevention

Technology (MFPT), Volume 57, (2003).

This paper presents an algorithm that models internal system status to predict when

early engine or power system fault will occur. This will then enable a corrective and

preemptive course of action to be taken to mitigate system failure. It is critical for a

vehicle to efficiently manage its own critical resources whilst working autonomously

in an alien environment.

Paper 9 - Jaenisch, H., Handley, J., Lim, A., Filipovic, M., White, G., Hons,

A., Deragopian, G., Schneider, M., Edwards, M., “Data Modeling for

Virtual Observatory Data Mining”. Optimizing Scientific Return for

Astronomy through Information Technologies. Edited by Quinn,

Peter J.; Bridger, Alan. Proceedings of SPIE, Volume 5493, pp. 198-

220, (2004).

This paper presents a core Data Modeling algorithm that enables a large variety of

different types of equation models derived during a mission to be sorted and stored for

very fast retrieval and processing as needed. Efficient database management of

23

derived models enables the rover’s computational resources to be optimized, and

conserved, by using the minimum number of models that are required for the current

mission goals. Fast swapping, and storing, of new models is important to minimize

the delay in response time to changes in mission status, and an efficient method of

coding and storing derived models is necessary and presented here.

Data Modeling enables prediction of similarly but distinct imagery from previously

seen images and allows mathematical combining of information content into

analytical form, enabling fast mathematical analysis on the image structure. Unlike

bootstrapping or statistical methods that either resample the original image or use the

average statistics of the image to generate pixel values, the Data Modeling approach

provides an analytical expression where the input parameters are varied to create new

images that are similar to the original yet distinct.

These analytical equation Data Models are stored as a knowledge base and are

retrieved using index tags such as the Heptor or similarity (Reference Paper 6). Once

retrieved, the coefficients of the Data Model are perturbed to generate predictions of

the rover’s surrounding environment and establish a model of familiarity.

Paper 10 – Jaenisch, H., Handley, J., Pooley, J., Murray, S., “Virtual

Instrument Prototyping With Data Modeling”, JANNAF 39th

Combustion, 27th Airbreathing Propulsion, 21st Propulsion Systems

Hazards, and 3rd Modeling and Simulation Subcommittees Joint

Meeting, Colorado Springs, CO, (December 1 -5, 2003).

This paper presents a method to create custom “virtual sensors” as it encounters new

surroundings or observations. Using the previously defined algorithms, the algorithm

presented here enables the determination that some observed or predicted parameters

are critical. The important parameters can now be grouped automatically into a met-

model or combined-important-parameter model that enables observation of a new

inferred parameter. An example might be that a change in atmospheric turbulence

was found to be linked to changes in latitude and strongly sensitive to altitude. The

result would be a new equation model that couples altitude and latitude as a predictive

model for variation of weather patterns. This would have been unexpected, but would

24

have been crafted as the important variation encountered during its mission. The

algorithms presented here enable such a fusion of model information.

Paper 11 - Handley, J., Jaenisch, H., Lim, A., White, G., Hons, A., Filipovic,

M., Edwards, M., “Data Modeling of Deep Sky Images”. Modeling

and Systems Engineering for Astronomy. Edited by Craig, Simon C.;

Cullum, Martin J. Proceedings of SPIE, Volume.5497, pp. 449 -460

(2004).

This paper presents an algorithm for converting image data into a mathematical

equation. The equation takes up much less room than the raw data and preserves

sufficient fidelity for decision-making algorithms to work. Unlike current approaches

that require humans to decipher imagery, low-resolution image models are sufficient

for rovers because their decision criteria is simplified and based on scene roll-up

statistics. They do not impose the artificial pattern perception on imagery that

humans do and are able to work satisfactorily with perception models rather than

large memory expensive images. Moving beyond storing images is critical and

conversion into scene models for analytical comparison is necessary. These image

models may now be analytically compared and proofed with techniques described in

other papers (such as image compression). I demonstrated this algorithm on the M51

galaxy as shown in Data Modeling Enabled Real Time Image Processing for Target

Discrimination and assigned this to Handley so that he could demonstrate as a school

project the use of this algorithm on the entire Messier list which is published in this

work.

Paper 12 - Lim, A., Jaenisch, H., Handley, J., Filipovic, M., White, G., Hons,

A., Berrevoets, C., Deragopian, G., Payne, J., Schneider, M.,

Edwards, M., “Image resolution and performance analysis of

webcams for ground based astronomy”. Ground -based Telescopes.

Edited by Oschmann, Jacobus M., Jr. Proceedings of SPIE, Volume

5489, pp. 1152-1163 (2004).

This is an algorithm that enables small footprint and low profile optics to be used to

acquire high quality images. Space in rover vehicles is critical and large high density

25

CCD array cameras are slow to process. However, they are not necessary if sufficient

image and resolution enhancing software is available to enable smaller cameras with

higher frame readout speeds to be used. The algorithms I developed and published in

this paper demonstrate that a great deal of imaging efficiency can be gained by image

tracking and enhancing using low-resolution cameras. Such systems can rivals slower

high-resolution systems.

Paper 13 - Jaenisch, H., and Filipovic, M., “Classification of Jacoby Stellar

Spectra Using Data Modeling”. Imaging Spectrometry VIII. Edited

by Shen, Sylvia S. Proceedings of SPIE, Volume 4816, pp. 296 -307

(2002).

This paper presents an algorithm that creates a complicated classification scheme. As

large amounts of data are observed, the ability to sort and organize, and be able to

make decision about the significance of the data is critical for data management. The

algorithm provided here enables autonomous information ontology.

This paper solved what was considered in the literature to be an unsolvable problem:

using a neural network or other method to build a classifier to distinguish the Jacoby

stellar classes based on stellar spectra. The literature reported only achieving 70-80%

classification success on the baseline data with no reasonable explanation why the

baseline data itself could not be learne d 100% correctly. It was not even possible to

overfit the data to learn it 100% correctly in the published literature. My contribution

was to show with Data Model Change Detectors I was able to achieve 100% correct

classification and was the first to do so, but I also found the explanation for why the

other methods failed. I found which stellar classes were populated with contradictory

examples of spectra, indicating that some of the classes specified by human observers

were contradictory, overlapping, and not numerically distinct. This is very

fundamental and a very important result for the astronomy community because it

points to which star classes called different by astronomers are actually not distinct

from an information theory point of view and suggests the areas where reevaluation

by astronomers should be done. So the finding that 100% correct classification on the

training data was achieved was very unique and a breakthrough report.

26

Paper 14 - Jaenisch, H., “Enabling Unattended Data Logging and Publication

by Data Model Change Detection and Environmental Awareness”.

Unattended Ground, Sea, and Air Sensor Technologies and

Applications VIII. Edited by Carapezza, Edward M. Proceedings of

SPIE, Volume 6231, pp. 62310R (2006).

This paper describes the fully autonomous rover. The paper examines the use of the

core algorithms to enables an orbiter to select a suitable planetary landing site and to

deploy surface rovers.

The paper outlines the processes where surface rovers are deployed and allowed to

acclimatize themselves without prior knowledge. Objects are sensed and

autonomously recognized as being interesting and worthy of closer scrutiny. If closer

inspection with change detectors to guide decisions yields evidence of prior examples

then no further interest is taken. If the detection is novel then science is performed

and the novelty studied. The core knowledge gained by rovers can be embodied as a

coupled and sorted set of transfer function models capable of being interrogated to

reconstruct visited environments, or interrogated to predicted unseen environments.

This ability enables full transfer of experience to a new pick-up mission in a

completely transparent mathematical form. Legacy information is preserved and the

sensors are able to exhibit self-organizing behavior as they learn their environment.

Paper 15 - Jaenisch, H., Handley, J., Faucheux, J., “Data Driven Differential

Equation Modeling of fBm processes”. Signal and Data Processing

of Small Targets 2003. Edited by Drummond, Oliver E. Proceedings

of SPIE, Volume 5204, pp. 502 -514 (2003).

This paper presents a refined algorithm for automatic differential equation derivation

by the rover and enables the equation to be solved both numerically, and more

importantly, analytically. Analytical solution enables very small amounts of memory

to be used for storage and enables very fast execution of the derived models in the

memory. Numerical forms enable the equations to be converted into look-up-table

(LUT) format if that is a mission requirement for fast processing. Some system

27

architectures planned may require vector processors that require LUTs and integer

math operations in place of real math packages as mandated by standard differential

equation modeling. This algorithm facilitates conversion into integer processing

format.

Paper 16 - Jaenisch, H., and Handley, J., “Data Modeling Augmentation of

JPEG for Real-Time Streaming Video”. Visual Information

Processing XIV. Edited by Rahman, Zia -ur; Schowengerdt, Robert

A.; Reinbach, Stephen E. Proceedings of SPIE, Volume 5817, pp.

258-269 (2005).

This paper presents a novel algorithm for determining the quality of images being

modeled and is used in this thesis to score the results of scenery and to rank predicted

images with actual observed ones.

Paper 17 - Jaenisch, H., Handley, J., Faucheux, J., Lamkin, K., “Shai-Hulud:

The Quest for Worm Sign”. Data Mining, Intrusion Detection,

Information Assurance, and Data Networks Security 2005. Edited by

Dasarathy, Belur V. Proceedings of SPIE, Volume 5812, pp. 321 -329

(2005).

This paper introduces an algorithm for converting all binary data collected into a

special “fractal” format that can be concisely described with a small number of

descriptive statistics. These statistics are then used to derive change detectors and

useful equation models for decision-making during the mission. The descriptive

features required to adequately describe complicated data is limited to seven, simple

to calculate features, thereby forming a seven element vector called a “heptor”. This

heptor is used as the discriminating information to reduce the dimensionality of large

number sets when describing equation models in the vehicles memory.

28

1.7 Conclusion.

This is a mini Portfolio Thesis designed to introduce seventeen papers that constitute

this Doctor of Astronomy Portfolio.

In this chapter, I introduce a series of algorithms that have been developed and which

are the core of the Portfolio Papers. In Chapter 1 Section 4, I illustrate their use in the

context of an autonomous planetary rover; however, this work has many uses outside

this area. These other areas will be looked at in Chapter 2. In Chapter 1 Section 5, I

look briefly at work in the literature that is convergent with the work in my seventeen

papers, and in Chapter 1 Section 6 I introduce each of the papers of the Portfolio with

a small summary of their content. Here I lean heavily on the context of an

autonomous planetary roving vehicle.

30

CHAPTER 2

ADDITIONAL APPLICATIONS

2.1 Introduction.

I examine here in Chapter 2, a series of diverse applications that incorporate the core

algorithms that are presented in the seventeen Portfolio Papers and which have been

illustrated in Chapters 1.2 and 1.4 in the context of an intelligent and autonomous

planetary rover. The various core algorithms can also be used for a wide range of

astronomy and other disciplines; for example, for data mining, signal processing,

image processing, decision-making aides, and as a general replacement for

applications that traditionally have used classical neural networks.

To illustrate these applications, I now examine a brief cross-section of applications

that have or could be benefited from the rover-inspired algorithms. These are

demonstrated in the attached portfolio papers and details are given illustrating how the

core techniques have been used for various applications. Data Modeling has been

successfully applied to missing data reconstruction, signature synthesis, interpolation,

data compression, classifier performance modeling and adaptive feature selection.

2.2 Application to Data Ensemble Synthesis.

A novel application of these methods is for solving the inverse fractal problem for 1/f

noise sets. This simply means the ability to predict the missing va lues of a

complicated data set, while conserving the dynamics and structure of the original data.

The performance of these methods has been compared with classical data modeling

methods (Portfolio Paper 2, Data Modeling of 1/f Noise Sets, (Jaenisch and Handley,

2003)) and the applicability to different distributions of noise has been presented,

along with an overview of important applications including data and image

compression.

31

2.3 Application to Critical, Sensitive, and Key Parameter Estimation for

Analytical Modeling.

The portfolio algorithms enable an analytical differential-equation model to be

derived from a single simulation input and output data vector. The derived model was

analytically varied (real versus imaginary) to determine critical, sensitive, and key

parameters without the use of Design of Experiments (DOE). This ability is very

useful for formal analysis, data modeling, Validation Verification and Accreditation

(VV&A), and automatic software correctness proofing.

This work is described in Portfolio Papers 3 and 5, Automatic Differential Equation

Derivation From Sampled Data For Simulation Parameter Sensitivity Analysis

(Jaenisch, 2006) and Data Model Predictive Control as a New Mathematical

Framework for Simulation and VV&A, (Jaenisch, Handley and Hicklen, 2006).

2.4 Application to State Estimation From Trajectory Estimates for Tracking

and Automatic Controller Design.

Another important, and demonstrated application of the portfolio algorithms has been

the ability to autonomously make assessments of the physical characteristics of a

monitored object from its motion in the x-y plane. Traditionally, the process of state

estimation from a trajectory has been in the realm of the Kalman filter.

Unfortunately, the Kalman filter and Extended Kalman filter require a-priori defined

models of the dynamics of the object being observed. Predictions can only be made if

there exists a physics-based partial-differential equation that describes the equations

of motion. The published techniques for deriving the differential equation from

measured data enables the Kalman filter to be used for tracking very nonlinear

dynamics, and trajectories, and is applicable across many disciplines (see Portfolio

Paper Data Modeling Enabled Guidance, Navigation, and Control to Enhance the

Lethality of Interceptors Against Maneuvering Targets, (Jaenisch, Handley,

Faucheux, and Lamkin, 2005).

32

2.5 Application to Formal Proofing and Validation and Verification of

Software Simulations.

These analysis algorithms provide an analytical method for comparing simulated data

with measured data; the goal is to prove Equivalence and Consistency between the

model and the real data. The model output for a sensor and “real-world” measured

data are collected for comparison, and this method derives analytical Data Models

that are analyzed in frequency space, yielding a quantitative assessment of the model

relative to the measured data.

For logistics research, these methods enable reference points to be established for

complex dynamic processes such as inventory control and instrument calibration.

These mathematical models are a basis of comparison (and for the detection of

significant change) in inventory flow or instrument calibration.

The work on the formal proofing and Validation and Verification is covered in

Portfolio Papers 6 and 7; Analytical Data Modeling for Equivalence Proofing and

Anchoring Simulations with Measured Data, (Jaenisch, Handley and Hicklen, 2006)

and Data Modeling Change Detection of Inventory Flow and Instrument Calibration,

(Jaenisch, Handley and Bonham, 2003).

2.6 Application to Health Usage Management Systems (HUMS) and System

Health Diagnostics and Mean Time to Failure Prognostics.

The Data Modeling methods used by the rover of Chapter 2 can also be generalized

for use with other vehicles and systems (both manned and unmanned). Such

algorithms can be used to build a real-time diagnosis of operational conditions and

prognostic, i.e. a database of fault classification and time -to-failure estimates. Data

Modeling can achieve anomalous detection while only requiring nominal (no-fault)

conditions for training. This makes Data Modeling an attractive tool for Novelty

Detection and ambiguity resolution between nominal/anomalous; i.e. Data Modeling

can resolve ambiguities in diagnostic calls and manage risk uncertainty in prognosis

estimates of time-to-fail.

33

Data Modeling was successfully applied to Novelty Detection, and it can be applied to

Diagnostic Ambiguity Resolution and Prognostic Risk Uncertainty Management (see

Portfolio Paper 8; Data Modeling for Fault Detection, (Jaenisch, Handley, Pooley and

Murray, 2003)). Classified methods, such as adaptive multi-dimensional distance

measure neural networks, and Divergence Classifiers, were unsuccessful when

applied to a set of vibration diagnostics vectors with contain only slightly anomalous

conditions. When Data Modeling was applied to the same feature vector sets, 100%

correct classification was achieved (Jaenisch, Handley, Pooley and Murray, 2003).

2.7 Application to Data Mining.

The information storing and characterization capability intrinsic in Data Modeling (as

illustrated in Chapter 1) makes it possible for index-tagging of Virtual Observatory

data files with descriptive statistics. This is achieved by calculating 6 standard

moments at the time of data collection which are attached to the data file as

descriptive file tags. Data Change Detection Models are derived from these tags and

used to filter databases for similar or dissimilar information. This information could

be in the form of stellar spectra, photometric data, images, and text.

This application is covered in Data Modeling for Virtual Observatory Data Mining,

(Jaenisch et al., 2004).

Currently, no consistent or reliable method exists for searching, collating, and

comparing two-dimensional images. Traditionally, methods used to address these

data problems are disparate and unrelated to text data mining and extraction. Data

Modeling provides a unifying tool to enable data mining across all data classes.

2.8 Application to Virtual Instrument Prototyping.

Virtual instruments and sensors derived from mathematical models onboard an

autonomous rover (as outlined in Chapter 1) have utility in far more general settings

(Portfolio Paper 10, Virtual Instrument Prototyping with Data Modeling, (Jaenisch et

al., 2003)). They are mathematical embodiments of physically unrealizable devices.

The mathematical theory for prototyping virtual sensors using Data Modeling can

34

now be defined for many intelligent agent and robotic applications. The theory of

Change Detection using Data Modeling is presented. Examples of prototype virtual

sensors include: physical parameter sensor using trajectory, THz anthrax detector,

gear tooth health sensor, and fault detector, (Jaenisch et al., 2003).

2.9 Application to Image Processing, Analysis and Modeling.

These “information modeling”, and specifically imagery modeling techniques, can

also be used as a method for simulating CCD Focal Plane Array (FPA) images of

extended deep sky objects. These tools are used to model FPA fixed pattern noise,

shot noise, non-uniformity, and the extended objects themselves. The mathematical

model of the extended object is useful for correlation analysis and other image

understanding algorithms used in Virtual Observatory Data Mining. Application to

the objects in the Messier list resulted in a classifier that achieves 100% correct

classification (More details are given in Portfolio Paper 11, Data Modeling of Deep

Sky Images, (Handley, Jaenisch et al., 2004)).

Here another important image processing application was demonstrated and published

(Portfolio Paper 12: Image Resolution and Performance Analysis of Webcams for

Ground Based Astronomy, (Lim, Jaenisch, et al., 2004). This paper shows that it is

possible to achieving real-time super-resolution from ground-based images using for

small aperture telescopes.

2.10 Application to Image Compression and Synthesis.

Another application to image processing and storage is the augmentation of JPEG

image compression with fractal modeling techniques. The method enables sub-

sampling in conjunction with JPEG compression algorithms. Rather than directly

compressing large high-resolution images, we propose decimation to thumbnails

followed by compression (Portfolio Paper 16; Data Modeling Augmentation of JPEG

for Real-time Streaming Video, (Jaenisch and Handley, 2005). This enables

Redundant Array of Independent Disks (RAID) compression and facilitates real-time

streaming video with small bandwidth requirements. Image reconstruction occurs on

35

demand at the receiver to any resolution required using Data Modeling based fractal

interpolation.

This device -independent resolution capability is useful for real-time sharing of images

across virtual networks where each node has a different resolution capability. The

same image is constructed to whatever limitations exist at each individual node,

keeping the image “device independent” and image resolution scalable up or down as

hardware/bandwidth limitations and options evolve. For this work, a novel figure of

merit was created and demonstrated. It was useful for comparing image data that

mimics the human perception with a numerical score better than any previous metric

identified in the litera ture. It is used in this thesis to evaluate predicted imagery with

existing true image data.

2.11 Application to Spectral Analysis and Modeling.

An unusual application for the decision-making algorithms is its applicability to

Expert systems. By deriving Data Model polynomials in the place of neural network

and expert-system classifier rules, it was possible to achieve 100% correct

classification of the Jacoby stellar spectra (Portfolio Paper 13; Classification of

Jacoby Stellar Spectra Data using Data Modeling, (Jaenisch and Filipovic, 2002)).

The Jacoby set is a challenging group of 161 spectra spanning the full range of

temperature, sub-temperature and luminosity groupings of standard star types. To

achieve full learning, the development of a cascaded decision architecture linking an

extensive network of polynomial decision equations was required. The two dominant

features were extracted, and complex decision maps generated. Also, the sensitivity

of the equation architecture to misclassification due to measurement noise was

analyzed.

2.12 Application to Unmanned Vehicles.

The principles which are applied to the “fully autonomous planetary rover of Chapter

1” are also applicable to any form of unmanned vehicle or robotic system. Unmanned

vehicles can be deployed and allowed to form environmental models of familiarity.

The vehicle is then capable of deciding if further inspection is warranted based on

36

change detectors. The core knowledge gained can be embodied as a coupled and

sorted set of transfer-function models capable of being interrogated to reconstruct

visited environments. Legacy information is preserved and the vehicle is able to

exhibit learned self -organizing behavior. These issues are covered in the Portfolio

Paper 14, Enabling Unatte nded Data Logging and Publication by Data Model

Change Detection and Environmental Awareness, (Jaenisch, 2006).

2.13 Application to Information Assurance and Network Security.

A final, and extremely important, application of the algorithms is in cyber security

and detection of attacks or anomalies in network information flow. Successful worm

detection at real-time OC-48 and OC-192 speed requires hardware to extract web-

based binary sequences at faster than these speeds, and software to process the

incoming sequences to identify worms. Computer hardware advancement in the form

of Field Programmable Gate Arrays (FPGAs) makes real-time extraction of these

sequences possible. Lacking are mathematical algorithms for worm detection in the

real-time data sequence, and the ability to convert these algorithms into LUTs that can

be compiled into FPGAs.

Data Modeling provides the theory and algorithms for an effective mathematical

framework for real-time worm detection and conversion of algorithms into LUTs.

Detection methods currently available, such as pattern recognition algorithms, are

limited both by the amount of time needed to compare the current data sequence with

a historical database of potential candidates, and by the inability to accurately classify

information that was unseen in the training process. Data Modeling eliminates these

limitations by training only on examples of nominal behavior. This results in a highly

tuned and fast running equation model that is compiled in a FPGA as a LUT and used

at real-time OC-48 and OC-192 speeds to detect worms and other anomalies. A proof

of concept has been demonstrated using binary data from a WEBDAV, SLAMMER

packet, and RED PROBE attack (Portfolio Paper 17, Shai-Hulud: The Quest for

Worm Sign (Jaenisch et al., 2005)).

37

2.14 Conclusion.

This Doctor of Astronomy Portfolio Thesis is in two chapters. In both chapters I

introduce the Portfolio Papers that have developed a series of core algorithms. In

Chapter 1 Sections 4 and 6, the algorithms are illustrated in the context of an

autonomous planetary rover vehicle. In Chapter 1 Section 5, I look briefly at work in

the literature that is relevant to the work in my seventeen papers, and in Chapter 1

Section 6, I introduce each of the papers of the Portfolio.

In this second chapter, I show how the mathematical basis of this work is applicable

to many areas of data collection and imagery.

The mini-thesis supports the portfolio of papers bound into this volume. The author

recommends this portfolio to the reader.

38

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44

Portfolio Thesis Papers

Data Modeling For Radar Applications

Holger M. Jaenisch and James W. Handley Sparta, Inc.

4901 Corporate Drive, Suite 102

Huntsville, Alabama 35805 USA

Abstract - This paper presents the process of Data Modeling

for generating functional {[f(x)]n} models of algorithms or data

set relationships in simple polynomial form. Data Modeling has

been applied to missing data reconstruction, signature synthesis,

interpolation, data compression, classifier performance

modeling, and adaptive feature selection. This paper highlights

the mathematical process and successful applications.

I. DATA MODELING

A. Overview

The Data Modeling process can be summarized as a fusion

of numerical methods. In general terms, the process isolates

support vectors which are then cast into Turlington

polynomial form [1] for single variables and reconstructed at

index locations using Deslauriers-Dubuc dyadic interpolation

[2]. For the multivariate case, the vector of support vectors

are transformed into Peel-Willis-Tham basis functions [3]

across the support vectors that are used as input into

Kolmogorov-Gabor polynomials [4]. This forms the forward

model solution shown in (1).

),,()( 21 nxxxfzf K= (1)

The inverse solution shown in (2) is obtained using the

method of partial derivatives as defined by Darwiche [5] and

is readily approximated by using Turlington polynomial

approximation to which we apply Deslauriers-Dubuc dyadic

interpolation, but this time comprised of the Data Model

output as the predictive functional input.

[ )()()()( 21 zfxfxfxfT

n =L ] (2)

Jaenisch fused these numerical methods into the process of

Data Modeling, an enabling technology consisting of 1)

Deslauriers-Dubuc dyadic interpolation generalized to

Barnsley’s fractal based affine geometry transforms to

synthesize many different classes of data sets; and 2) the

ability to generate functional equation based models of order

3n of any algorithm or data set relationship. Equation based

models use both parametric and non-parametric statistics as

descriptive features for inputs and outputs, and yields discrete

equations even where no closed form previously exist. An

additional benefit of the equation abstraction process is the

robust determination of the contribution of each input feature

and their subsequent ranking [6] in terms of a) power to

which each variable is raised; and b) frequency of occurrence

of each variable in the final Data Model equation.

B. Fractional Brownian Motion fBm (1/f)

The most common type of noise found in nature is fBm as

shown in (3).

( )∫ ∞−

−−+Γ

=t H

H tdBttH

tB )'()'()(

1)( 2

1

21

(3)

The term 1/f noise is applied to any fluctuating quantity

V(t) whose spectral density SV(f) in log-log space varies as

1/fȕ over many decades with 0.5 < ȕ < 1.5 such as fBm. ȕ characterizes the steepness of the slope of the spectral

density, with larger ȕ values representing steeper slopes.

Gaussian or white noise would be characterized with ȕ = 0,

Brownian motion with ȕ = 2, and 1/f noise for ȕ values in

between [7].

The baseline methods of Data Modeling work best on those

data sets that are characterized as being 1/f; however, Data

Modeling incorporates numerical integration of data sets as

shown in (3) that are uncorrelated noise to convert them into

1/f data sets before generating the model of the data. This

works because the numerical integration of Gaussian noise

yields a data set that is a 1/f process. To recover the original

from the integrated model simply requires numerical

differentiation of the reconstructed data. Other methods for

achieving 1/f type data includes converting binary sequences

to –½ to ½ prior to integration and performing an inverse

cosine transform on the support vector, thereby generating a

waveform.

C. Dyadic Fractal Reconstruction

Data Modeling combines Deslauriers-Dubuc dyadic

interpolation as an operator on support vectors [8] (4) cast as

Turlington polynomials (5)

[ ] 01),( =−+ bwxy iiiα (4)

∑∑⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

++

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

+

++=

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

kd

dt

d

k

i

c

dt

d

c

dt

d

ik

i

i

DCBtAty1

10

1

10 101log

101

10log)(

(5)

to regenerate the original data. According to Deslauriers-

Dubuc, the dyadic extension of the fundamental interpolant is

given as

)3()1()1()3()()2/( ++++−+−+= tdFtcFtbFtaFtFtF (6)

where a, b, c, and d are scaling coefficients, and satisfies the

identity given in (7). In (7), [t] denotes the integer portion of

the number t.

∑+

−=

−=3][

2][

)()()(t

tn

ntFnyty (7)

Since the solution of the coefficients in (6) cannot be

obtained directly, regression methods may be used.

However, recasting Deslauriers-Dubuc dyadic interpolation

in a geometric form enables Barnsley’s Iterated Function

Systems (IFS) defined as

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛

n

n

nn

n

nf

e

y

t

dc

a

y

tw

0 (8)

to be used instead. Elton [9] proved that this process will

converge for all continuous functions in (9), which we have

achieved using Turlington polynomials.

∑ ∫=

∞→=

+

n

kX

kn

xdxfxfn 0

)()()(1

1lim µ (9)

Unlike Deslauriers-Dubuc dyadic interpolation and

Barnsley’s IFS interpolation where only 4 or 5 control points

and the scaling coefficients are used and supplied artificially

and separate from the data process, Data Modeling uses

straight-line segment transforms of all available data points

(if sub-sampled data, these support vectors are the available

data) and derives all the coefficients from the support vector

data itself, including an estimate of dn which is obtained from

Jaenisch [10] and is related by

( ).

1

1

log

1

NJN

NJ

N

J

dJ

n

≤

>−

=−

(10)

The sign of the scaling coefficient dn is assigned to be the

same as the sign of the derivative between support vector n

and support vector n - 1.

This innovation solves the inverse fractal problem by

enabling fractal interpolation and image regeneration to be a

robust and autonomous process readily implemented in real-

time.

Support vector identification is simplified to equi-spaced

partitioning of the original data vector. The number of points

N to sub-sample [11,12] in building the model is bounded by

(11) where R is the range of the data and ı is the standard

deviation of the data.

5.01

log

log

lim)Re(

1log

log

limmin)Re(

01

01

=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎟⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎟⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

=

→∆+

→∆+

N

NJ

R

J

N

NJ

R

J

i

Ji

i

Ji

σ

σ

(11)

The number of sub-sample points N can be solved for

between these bounds minimizing the variance of the residual

between reconstructed and original. The authors have found

that naively using either 10% of the original data set or using

N from the second equation in (11) only as the sub-sample

rate yields excellent results.

This process can be used to increase the resolution of any

data set to any user specified resolution by treating the

current in-hand data set as a set of support vectors. Data

Modeling preserves the fine structure content of the data, is

computationally efficient, and enables optimal interpolation

for ensemble generation.

D. Equation Modeling

The equation modeling technology of the Data Modeling

process generalizes neural networks by truncated Taylor

series expansion and uses multivariate regression to derive

high order polynomials using simple low order (< 3) building

blocks. This technique is amenable to the generation of

multi-class decision classifiers. Advantages of using Data

Modeling include: 1) Feature selection process is fully

autonomous; 2) Generates a polynomial decision equation; 3)

Input variables may call external simulations and executables

for their value(s); 4) Output variables may call external

simulations and executables with value(s); 5) Output may

become input to new equation models; 6) Expert systems

may be linked with polynomials as either input or output; 7)

Totally open-ended and scaleable; 8) Executes in polynomial

time; 9) No nested FOR-NEXT loops; and 10) Can be

derived without transcendental functions.

Any decision transfer function that can be cast as a neural

network, Dempster-Shafer probability table, or Bayesian

Belief Network (BBN) architecture can be recast in

polynomial form and used to generate an objective transfer

function directly. This new simple algebraic model can be

optimized using other methods developed by Jaenisch in Data

Modeling to provide fast and efficient inverse models as well.

Equation modeling builds high order (3n) polynomials

using polynomial building blocks that never exceed third

order, thereby preventing precision problems associated with

numerical implementation. These third order building block

forms are derived from generating a prototype neural network

comprised of 3 input nodes, 2 hidden nodes, and 1 output

node. These basis functions process data from input to output

layer through a hidden layer. The processing of input data

with a bias into the hidden layer is modeled according to

Peel-Willis-Tham as in (12).

...1

1)(

3

3

2

210)(++++=

+= +− jjjbxb xaxaxaa

exf

j

(12)

The solution is obtained in a single pass using linear

regression, thereby avoiding back propagation.

The transformation of the hidden layer data into the output

layer is done using a sigmoidal processing function. With

this approximation, the transformation of the input data

through the hidden layer to the output layer is given in (13):

∑∑= =

=N

h

k

j

j

hhj tusty1 0

, )()( (13)

where N is the number of neurons in the hidden layer; k is the

neuron polynomial order; sj,h is the combined coefficients

representing the jth coefficient of the hth polynomial neuron

in the hidden layer multiplied by the weight from the hth

neuron to the output layer; uh(t) is the total input to the hth

neuron in the hidden layer at time t; and y(t) is the output

data. The equation modeling technology in Data Modeling

uses a maximum polynomial order of k = 3 and solves for the

resulting coefficients (sj,h) using multivariable linear

regression to map the input data points (uh(t)) into the output

(y(t)). This equation can be written in matrix form (14)-(17)

to allow for the solution of the system of equations using

multivariable linear regression:

Θ= Xy (14)

T

jjj ptytytyy )](),...,1(),([ ++= (15)

T

NkNNk ssssss ],...,,,...,,...,,[ ,,1,01,1,11,0=Θ (16)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

++++=

k

NN

k

k

NN

k

ptuptuptuptu

tutututu

X

)(...)(1...)(...)(1

)(...)(1...)(...)(1

11

11

MOMMMOMM (17)

The final Data Model is derived from appropriate solutions

of the Kolmogorov-Gabor polynomial shown in (18).

L++++= ∑∑∑∑∑∑i j k

kjiijk

i j

jiij

i

ii xxxaxxaxaa0φ (18)

The final form of the Data Modeling equations and the

polynomial order shown in (19) are nested following

Ivakhnenko with n representing the number of layers in the

final Data Model and x(bi(t)) the inputs mapped from the

previous layer.

[ ] n

n

txO

tbxtbxbxtftx

3)(

)))))))))((((((((,()( 21

=

= K (19)

If a Data Model is needed for data sets whose

dimensionality is greater than one such as images (2-D) or

image sequences (3-D), the Hilbert sequence shown in (20)

and (21) is used to transform the two dimensional data into a

one dimensional data sequence for application of the Data

Modeling techniques [13].

)()()()( 43211 nnnnn PwPwPwPwH ∪∪∪=+ (20)

)()()()( 43211 nnnnn PwPwPwHwP ∪∪∪=+ (21)

Equations (20) and (21) are subject to Lindenmayer’s L-

system grammars represented in (22) and defined by (23).

°=−→−+→+

→−++−→+−−+→

90δ

FF

FLRFRLFR

FRLFLRFL

(22)

),,(),,(

),,(),,(

),sin,cos(),,(

δααδαα

αααα

+→⇒−−→⇒+

++→⇒

yxyx

yxyx

lylxyxF (23)

Using Data Modeling, polynomials as large as order 65,000

(~ [O(3)]10 from (19)) have been generated in real-time.

For completeness, since Data Modeling as described thus

far is contractive and therefore assumes interpolation,

expansive projection of support vectors and associated

scaling coefficients is possible. This extrapolation is

accomplished by invoking the Ruelle-Takens Theorem [14]

and treating the Data Model as the attractor for the underlying

process. Then using support tuples as defined by (24) and

linear prediction (25), reasonable forecasts have been

achieved within the Cramer-Rao bound (26).

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−+

+=Χ

))1((

)(

)(

)(

τ

τ

Ntx

tx

tx

tM

(24)

n

N

j

jnjn xydy += ∑=

−1

(25)

N

original

error

σσ ≥ (26)

Discussions regarding the non-violation of the Nyquist-

Shannon sampling theorems due to application driven

sampling criteria can be found in [15].

The focus of this work thus far has been on the generation

of Data Models in a forward direction by mapping multiple

functional inputs into the output objective function. The

inverse Data Modeling problem solution uses the output

objective function as the only input into a set of Data Models

estimating each functional input independently. This can be

achieved using Bayesian Belief Networks as a low-resolution

approach, or the compilation of a resultant BBN into a

canonical polynomial form given by

∑ ∏Θ∏=Θx

xxx

fxf

fx ii

ii

iiF .),(

~~

λλ (27)

The inverse solution is obtained by applying the method of

partial derivatives [5] to (27). The partial derivative equation

can be represented as

.)()(

)()(

)(

1)|Pr( 2

2 ⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

Θ∂∂

−∂Θ∂

∂=

Θ∂∂

yxuyxuxu

eFeFeF

eF

eF

ey

λλ(28)

An estimate to these partial derivatives can be obtained

from Turlington polynomials (5), and in fact the entire Data

Modeling process (1)-(26). The authors have examined each

of these methods and prefer the full Data Modeling approach

for solving the inverse problem.

E. Decision Architectures

Although the process of passing information through a

completed decision architecture is top down (Fig. 2), the Data

Modeling approach for constructing a decision architecture is

unlike other approaches by working from the bottom up.

Data Modeling can characterize new data sets (multiple

partitions if necessary) and simply build an architecture

consisting of classifier and controller transfer functions. Data

Modeling begins at the lowest decision level and breaks the

process up into a series of bimodal decisions. This is possible

because if two non-coincident points exist in N-space, then an

N-dimensional analog to y = mx + b passing between the two

points can always be found, limiting the maximum number of

classifiers to converge to N - 1.

This process is learned in a cascade fashion guaranteeing

100% learning (Fig. 1) and results in a classifier (shown as A,

B, and C in Fig. 2). If more than one classifier is created, a

decision has to be made at a higher level which classifier to

execute and under what conditions. The sole function of this

controller equation model would be to analyze the conditions

and determine which classifier under its control to execute.

These are shown as 1 and 2 in Fig. 2.

100

Examples

A

83% of total

learned

168 Unlearned

251 Siphoned

F21, F32, F47B

99% of total

learned

11 Unlearned

35 Siphoned

F33, F55, F77, F69

99.99% of total

learned

1 Unlearned

4 Siphoned

F21, F65, F87

C

100% of total

learned

0 Unlearned

F20, F43, F74

D

Training

FinishedAvailable Features (F21- F90) Outcomes For A, B, C, D – Fault/No Fault

100

Examples

A

83% of total

learned

168 Unlearned

251 Siphoned

F21, F32, F47B

99% of total

learned

11 Unlearned

35 Siphoned

F33, F55, F77, F69

99.99% of total

learned

1 Unlearned

4 Siphoned

F21, F65, F87

C

100% of total

learned

0 Unlearned

F20, F43, F74

D

Training

FinishedAvailable Features (F21- F90) Outcomes For A, B, C, D – Fault/No Fault

Fig. 1 Data Model construction using cascade approach.

1

F1, F2, F5

ARCHITECTURE

• Common feature subset

• Multiple NETWORKS

with different TASKS

NETWORK


• Multiple CLASSIFIERS with

different DECISIONS

=I

3

1 2

A B C D

II3

1 2

A B C D

α3

1 2

I II III IV

=

CLASSIFIER


• Multiple DECISION

TRANSFER FUNCTIONS

F1, F5, F7

A

F2, F3, F4

B

F2, F3, F7, F9

1

C1? C2? C3?

Step 1

(Classifiers)

Step 2

(Apex)

F2, F4, F9

C

F1, F7, F9

2

Step 3

(Branch Addition)

C1? C2? C3?C1? C2? C3?

1

F1, F2, F5

ARCHITECTURE


• Multiple NETWORKS

with different TASKS

NETWORK


• Multiple CLASSIFIERS with

different DECISIONS

=I

3

1 2

A B C D

II3

1 2

A B C D

α3

1 2

I II III IV

=

CLASSIFIER


• Multiple DECISION

TRANSFER FUNCTIONS

F1, F5, F7

A

F2, F3, F4

B

F2, F3, F7, F9

1

C1? C2? C3?

Step 1

(Classifiers)

Step 2

(Apex)

F2, F4, F9

C

F1, F7, F9

2

Step 3

(Branch Addition)

C1? C2? C3?C1? C2? C3?

Fig. 2. Completed decision architecture representation.

This allows the Data Modeling approach to be totally open-

ended and scalable, and allows for entire branches to be

added to the tree if necessary without retraining the entire tree

[16]. The tree structure can continue to expand if the

complexity of the decision architecture increases. This

allows for entire branches to be added to the entire tree if

necessary without retraining the entire tree. If necessary,

each individual equation in the architecture can be

represented by an entire network of polynomial based

equation models, thereby allowing for the nesting of decision

architectures within other decision architectures. This

method is fully adaptive, and only chooses and uses those

features that are necessary to accurately make the correct

decision at each decision transfer node.

This is akin to discovering the optimal Bayesian Belief

Network structure from the data autonomously, where the

grouping of terms in the polynomial directly defines the

connectivity and strengths between decision nodes in a

Bayesian Belief Network. Each individual equation in the

architecture can be represented by an entire network of

polynomial based equation models, thereby allowing for the

nesting of decision architectures within other decision

architectures. The best features do not need to be identified a

priori because they will be identified by the decision

architecture when anomalous conditions are encountered.

Traditionally, if new data becomes available, decision

architectures must be entirely retrained by evaluating where

the new data should be placed and new classifiers constructed

using both the old exemplar data and the new exemplars. By

converting Bayesian Belief networks into a polynomial form,

new branches and nodes can be added without the need to

retrain the entire network.

Data Modeling is used to characterize the data feature

vector to determine which original feature vector the new

data is the most like (worst case to distinguish between).

Transfer functions are derived to classify the new feature

vector. This constructs a new layer in the architecture

consisting of the newly generated transfer functions and the

transfer functions related to the most alike feature vector

selected above. In the place of the original transfer functions

in the architecture, we place new controller transfer functions

that simply distinguish when to use each of the two transfer

functions below it [16].

F. Generation of Lookup Tables (LUT)

Once a Data Model is built, it can then be used to generate

a series of simple analog nomographs. The nomograph is a

tool that provides quick graphical answers for applications

where numerical data are substituted into formulas [17]. The

nomograph shown on the left in Fig. 3 is a graphical

“program” that solves the simple equation x + y = w. This

concept is generalized to more complicated functional forms

and to many more variables with the Data Modeling equation

generation approach. Yet the use of the final decision look

up table will remain just as simple and easy to use in real

time.

Through the creation of a transfer function equation and the

use of the method of partial derivatives (23), a decision map

is generated that captures the equation output for various

values of the input features. These decision maps become a

lookup table.

An example of a nomograph (lookup table) is shown on the

right in Fig. 3, where each axis represents a different physics

based feature input into the process and the output decision is

read directly based on the x and y location and associated

lookup table values. By varying two features at a time across

the valid range while holding the other variables constant, the

polynomial output is plotted at each increment location

creating the decision map.

Fig. 3. Lookup table generation with Data Modeling.

Features 1 and 2 are used directly to determine the same

decision as the original Data Model equation without running

the equation. The use of lookup tables that are generated by

these equations as nomographs is amenable in applications

that have only a limited amount of processing power and

storage capability, as well as in applications where real time

implementation is required.

Field updates of new information in the lookup table also

occur directly. As new information comes in during a

mission or as the mission evolves, feature values in the LUT

are changed at locations defined by the intercept of two

feature values. This eliminates the need to generate an entire

new decision map whenever new information becomes

available. This method provides updates to the decision

landscape lookup table in real-time and in situ. Post-mission

or whenever necessary, these changes can be used to derive a

new decision map that encompasses the new information.

These decision lookup tables form a basis of graphically

encoding complex decision making algorithms into real time.

Non-computational forms for in-flight updates (IFTU) or

situational awareness network sharing is possible by

minimizing the data and optimizing the information form that

can be shared across a limited bandwidth channel.

G. Calibration on Demand (COD)

Calibration On Demand (COD) can be used in real-time to

indicate when incoming data sets are of a form not previously

accounted for or anticipated during training on either actual

or simulated data sets. COD allows sensing of off-nominal

conditions while being trained exclusively on nominal

conditions. This is a key theoretical enabling concept for

Situational Awareness via algorithms.

A use for COD is in monitoring individual sensor

measurements and overall system snapshots for change

detection. Individual sensor measurements provide a

characterization of system level conditions. This yields an

estimate of nominal or off-nominal condition.

Off-nominal detection can be used to initiate man in the

loop intervention or the automatic generation of new or

additional features, new or additional sensor looks to generate

more detailed data, spawn the generation of a new classifier

to deal with the new off-nominal information, or simply yield

a tip-off condition.

Rather than use the actual input feature values, we use

parametric and non-parametric descriptions of the entire

feature value set. This can include the conversion of features

to standardized Z-scores. This treats the feature value set as a

time series and models the distribution by overfitting to a

minimized regression function. By setting the target value of

the regression to minimize and then characterizing the

polynomial fluctuation around a minimum, in practice the

output feature value used for training is ½.

An overfitted Data Model can be generated that uses

Shewhart’s Control Chart theory [18] to monitor if the

modeled process is under Statistical Process Control. In

Shewhart’s theory, trial limits are generated for X using

RAXLCL

RAXUCL

X

X

2

2

−=

+= (29)

where A2 is a table lookup value and R is the average range.

In Data Modeling, these trial limits are set using the

polynomial output from the training data. The upper and

lower boundaries are assigned to be one-half of a standard

deviation above and below the mean value of the polynomial

output from the training data.

It is very important to note that even though the ranges of

input features may be the same as seen during training, which

is easily tested without resorting to Data Modeling; we can

also detect instantly combination of values for incoming

features that are not anticipated or seen before. This off-

nominal detection is a tip-off condition that can be used to

predict additional regions that need to be populated by

decision making classifiers or decision transfer nodes.

COD can evaluate the operational ranges and feature

regimes of proposed classifiers. This method shows regions

of nominal behavior that indicate adequate coverage by the

proposed classifier.

II. APPLICATIONS

A. Adaptive Wavelet Basis Function Selection

The first problem with using wavelet transforms is

determining when to stop sampling and apply the transform.

The second problem is a priori selection of the proper basis

function.

The non-constant location of minimum and maximum

scale-space values when using orthogonal basis functions

necessitates the need for a computer intensive sort algorithm

to insure their location at the beginning of the data array. No

current method for picking the basis function a priori and in

an autonomous fashion that insures constant location of the

minimum and maximum scale space values has been reported

in the literature.

The wavelet transform cannot be applied to a single point,

it requires that a group of measurements be taken and

characterized together. To overcome this, a fixed number of

samples, called a data window, are measured and the wavelet

transform applied to them. This technique requires the a

priori selection of either a dyadic or non-dyadic window size,

and the transform is applied only when enough measurements

are taken to fill the window.

Window selection can be performed autonomously and

robustly using Data Modeling. The entire Data Modeling

process of defining the optimal data window size, along with

the ideal information based sub-sampling to derive the basis

function is called the Self Defining Basis Function Adaptive

Window (SDBFAW) [19]. The total number of points N to

sample insures stationarity across the window and satisfies

the second equation in (11), thereby providing a solution to

the first wavelet transform application problem listed above.

The actual SDBFAW is determined by sub-sampling with the

window found in the first equation in (11). Sub-sampling

works because of ergodicity within the stationary window.

This provides a solution to the second wavelet transform

problem listed above.

In (11), R is the range and ı the standard deviation. Note

that in order to calculate (11) in real time, Jaenisch developed

an approximation to the standard deviation that can be

calculated recursively on the data in real time as it comes in.

The mean (30) and standard deviation approximation (31) are

1

1

1

1

+

+

++ +⎟⎟

⎠

⎞⎜⎜⎝

⎛=

i

i

i

i

iiN

y

N

Nµµ (30)

( ) ( ).

1

2

11

1

2

1

+

++

++

−+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

i

ii

i

iii

N

y

N

N µσσ (31)

Once the optimal basis function is derived, the wavelet

transform can be generated using standard wavelet transforms

routines available in the literature. It should be noted that this

process does not require that the basis function be orthogonal

or compactly supported. Both orthogonal and non-orthogonal

basis functions can be used to characterize sensor

measurements.

B. Data Synthesis

Data Modeling has been successfully compared to classical

methods including polynomial curve fitting, Markov chains,

wavelet transforms, mean and covariance methods, and

Kalman filters (Fig. 4). Column 1 of Fig. 4 shows the

original data sequence of 1024 points and a histogram

showing the distribution of the original. In Column 2, results

from using each of the classical techniques mentioned above

are shown along with a histogram of the distribution of each

resultant. Note that only the Markov chain process captures

the underlying distribution of the original, but fails to capture

the temporal and spatial characteristics of the original.

Column 3 of Fig. 4 shows the fractal Data Modeling

approach using five different sub-sampling rates varying

from 1% to 25% of the original. For each of these sub-

sampling rates, both the temporal, spatial, and underlying

distribution characteristics of the original are captured.

Columns 4 and 5 show the spatial error and power spectra

differences between the original, classical method, and the

fractal reconstruction. Using the 5% Data Model (50 points)

as a baseline, only the wavelet transform method provided a

slightly better RSS error, but with a substantial difference in

both the power spectra and in temporal and spatial

characteristic retention. Two peaks occurred in the log power

spectra that were captured and amplified by the fractal

method albeit with a shift in their locations.

Fig. 4. Comparison of fractal Markov modeling and classical methods.

The wavelet power spectra showed no peaks at all in the

lower trace, indicating a significant distortion in the original

signal and loss of frequency content. It should also be noted

that all of the fractal Data Models generated power spectra

more like the original than any of the classical methods, and

that using 50 points of the original data outperformed using

50 wavelet transform coefficients.

C. Adaptive Feature Selection

Data Modeling provides the ability to do adaptive feature

sub-selection in a robust fashion where the contribution of

each feature to the final solution is calculated. The process of

generating the Data Models generates a ranked list of each of

the inputs in terms of the number of times each input appears

in the final solution Data Model. Input features that provided

only redundant information are removed from consideration

during the equation generation process.

This process was successfully demonstrated on the

THAAD missile program, where IV&V was performed on

both a Bayesian quadratic classifier and neural network.

Using this adaptive feature ranking and selection process, the

authors were able to suggest the use of different features from

the current set that increased the performance of both the

original Bayesian quadratic classifier and the original neural

network. This boosted the performance of the Bayesian

quadratic classifier equal to the original neural network

performance, and enabled the neural network to equal the

performance of the Data Model in achieving perfect

classification. The Data Model form still surpasses the neural

network form in terms of computational efficiency and in

providing the actual functional form represented by the neural

network. This robust feature selection algorithm can be

applied generically to other feature sets in an attempt to

increase performance.

D. Classifier Modeling

Fig. 5 shows an example of a Data Modeling output that

yielded 100% classification a difficult simulated THAAD-

like data set. Shown in the upper right corner is the actual

equation used to make the decision. To the lower left of this

group is a ranking of the input features (bar chart) derived

from Data Modeling and a graph of the overlap of each of the

two best input features (above bar chart).

Fig. 5. Application of Data Modeling to radar feature data.

It should be noted that a good deal of overlap exists

between the features selected by Data Modeling, and yet Data

Modeling was able to take these features and generate perfect

classification. Such is not the case with the quadratic

classifier (example shown in the lower right quadrant using

the best features available in terms of separability). Also

shown in the lower right of Fig. 5 is an example of a decision

map that was generated using an actual Data Modeling based

equation classifier, which shows the complex nature of the

decision space as compared to the quadratic classifier

decision space shown in the upper left. Training support

vectors are denoted as white dots.

Successful construction of a Phased Derived Range (PDR)

physics based feature classifier polynomial using Data

Modeling was demonstrated on PDR type data sets.

Beginning with data that existed in hardcopy format only,

these were digitized and scaled to the correct range. Once

this was done, Data Modeling was used to synthesize

ensembles of PDR type data sets in order to create

characterizing statistics. These statistics were then used to

derive equation-based classifier like the one discussed

previously for the THAAD work.

Data Modeling has also been used to successfully capture a

Dempster-Shafer classifier process. Using only hardcopy

data, the data representing Dempster-Shafer classifier output

versus range and quality of the intelligence gathered was

scanned into the computer and digitized and scaled to the

correct range in the same manner as the PDR data mentioned

earlier. This equation-based classifier yields the probability

of an object being a Class 1 type given the range and the

quality of the intelligence gathered, and can be embedded

into testbeds and simulations for real time use. This negates

the need for Dempster-Shafer source code or executables.

The process listed above was done blindly without any

knowledge of the Dempster-Shafer algorithm other than the

hardcopy data that was used.

E. Discovery of Bayesian Belief Network Architecture

Data Modeling has also been used to successfully capture

the decision architecture process of a Bayesian Belief

network. Using a 12 node BBN generated in Netica™, 100

examples of decisions from the BBN (conflicted versus

deconflicted scenarios) and the values for the input variables

for each of these 100 cases was learned by equation based

Data Modeling technology. Data Modeling determined how

significant each feature was to the final result. Data

Modeling found that only 3 of the variables (upper tier

intercept time after launch, and lower tier commit time after

launch, and kill assessment time) were required for the

equation-based model to adequately capture the decision

process. In order to demonstrate that Data Modeling can in

fact use all variables if necessary, the training process was

allowed to continue past this much simpler solution until all

variables were used.

Data Models were also generated that solved for each of

the inputs given the output conflicted/deconflicted value. This

provides a mechanism for solving the inverse problem of

predicting the input values for a given output.

III. DATA MODELING PERFORMANCE

In analyzing different algorithms for conversion to Data

Models, it was found that many of these complex algorithms

cannot run in real-time due to the length of code (~ 5000

SLOC), memory usage (> 1 MB), and execution time (~

seconds). Using the Data Modeling approach, equation-based

models were built that used 75 Kbytes of memory and took

approximately 2 milliseconds to execute, and yielded

performance comparable to the original algorithms. In

addition, when the results were encoded into a lookup table

format, the execution time was reduced to 0.7 milliseconds

with an increase in memory required to approximately 300

Kbytes of memory to execute. A final icon encoding scheme

was employed that stored the same amount of information as

the lookup table in a 6 Kbyte JPEG compressed image.

REFERENCES

[1] T. Turlington, Behavioral Modeling of Nonlinear RF and Microwave

Devices. Boston, MA: Artech House, 2000.

[2] G. Cherbit, Fractals: Non-integral Dimensions and Applications. New

York: Wiley, 1991.

[3] G. Page, J. Gomm, and D. Wilson, Applicaton of Neural Networks to

Modeling and Control. London: Chapman and Hall, 1993.

[4] A. Ivakhnenko and V. Lapa, Cybernetics and Forecasting Techniques.

New York: Elsevier, 1967.

[5] A. Darwiche, “A differential approach to inference in Bayesian

Networks,” University of California (Los Angeles), 2000.

[6] H. Jaenisch and M. Filipovic, “Classification of Jacoby spectra using

Data Modeling,” Proceedings of SPIE: Imaging Spectrometry VIII, vol.

4816, pp. 296-307, July 2002.

[7] J. Feder, Fractals. New York: Plenum Press, 1988.

[8] B. Scholkopf and A. Smola, Learning with Kernels. Cambridge, MA:

MIT Press, 2002.

[9] M. Barnsley, Fractals Everywhere, Second Edition. Boston, MA:

Academic Press, 1993.

[10] H. Jaenisch, S. Taylor, J. Handley, and M. Carroll, “Optimal fractal

image and data compression,” Proceedings of the Southeastern

Simulation Conference, pp. 67-77, October 1995.

[11] J. Handley, H. Jaenisch, and M. Carroll, “Identification of the behavior

transition point in data using a robust fractal method,” Proceedings of

the Southeastern Simulation Conference, October 1995.

[12] J. Handley, On The Existence of a Transition Point in Sampled Data

using Fractal Methods. Ann Arbor, MI: UMI, 1995.

[13] H. Peitgen, H. Jurgens, and D. Saupe, Chaos and Fractals, New York:

Springer-Verlag, 1992.

[14] D. Ruelle, Chaotic evolution and strange attractors. Cambridge:

Cambridge University Press, 1989.

[15] T. Lago, “Digital sampling according to Nyquist and Shannon,” Sound

and Vibration, vol. 36:2, pp. 20-22, February 2002.

[16] H. Jaenisch, J. Handley, S. Massey, C. Case, and C. Songy, “Network

centric decision architecture using for financial or 1/f data models,”

Proceedings of SPIE: App. and Science of Neural Networks, Fuzzy

Systems, and Evolutionary Comp, V, vol. 4787, pp. 86-97, July 2002.

[17] A. Levens, Nomography. New York: Wiley, 1948.

[18] E. Grant, Statistical Quality Control. New York: McGraw-Hill, 1952.

[19] H. Jaenisch, J. Handley, C. Songy, and C. Case, “Adaptive self-

defining basis functions for wavelet transforms specified with Data

Modeling,” Proceedings of SPIE: Algorithms and Systems for Optical

Information Processing VI, vol. 4789, pp. 123-133, July 2002.

Data Modeling of 1/f noise sets

Holger M. Jaenisch*a

and James W. Handley a

aSparta Inc., 4901 Corporate Drive, Suite 102, Huntsville, AL 35805

ABSTRACT

A novel method is presented for solving the inverse fractal problem for 1/f noise sets. The performance of this method is

compared with classical data modeling methods. Applicability to different distributions of noise is presented, along with

an overview of important applications including data and image compression.

Keywords: Data Modeling, fractal, noise, 1/f, inverse fractal modeling

1. BACKGROUND

Beginning with a random data set (N points) from an unknown stochastic process, we wish to identify the system

governing the process (system identification) given only discrete sampled estimates. The process of estimating the

stochastic process model parameters from estimates of the measured data is termed inverse modeling. If the final inverse

model can be cast into a functional form subject to certain constraints, it is termed a Data Model.

1.1 Data Model definition

A Data Model is defined as a function [f(x)]n subject to the following constraints:

1. Must be a continuous function or an equivalent discrete representation.

2. Must be an interpolation function.

3. Number of coefficients should not exceed 2

1−N and must not exceed N-1.

Further, a more desirable Data Model is achieved by adhering to the following guidelines:

1. Should contain no Boolean or programming statements, only algebraic expressions.

2. Should not contain looping structure or recursion.

3. Should contain no random number generators.

4. Should contain no probability based variables, conditionals, or output.

Because of the first constraint, discontinuous or piecewise linear solutions such as high order spline functions are not

valid Data Models. The second constraint eliminates regression based fits to data such as low order polynomials unless

the fit of the polynomial is sufficient to use as an interpolation function. For the third constraint, if an interpolation

function has greater than 2

1−N coefficients, such as a Lagrangian polynomial that requires approximately N-1

coefficients, it is termed a trivial Data Model.

1.2 Lagrange polynomials

An example of a trivial Data Model is an inverse model represented as a Lagrange polynomial. Lagrange polynomials

are of degree N-1 and are derived from a measurement set of N points. This polynomial is obtained using Lagrangian

interpolation and is defined as in (1). For data sets greater than 10 points, the Lagrange polynomial exhibits unstable

solutions, especially near end points. For this reason, it is of limited utility as a basis for Data Modeling1.

∑ ∏−

=

−

≠= −

−=

1

0

1

0

)(N

j

N

jii ij

ij

xx

xxyxP (1)

*[email protected]; phone 1 256 337-3768; fax 1 256 830-0287; sparta.com

mailto:[email protected]

1.3 Chebyshev polynomials

Another inverse model form is based on the use of orthogonal basis functions such as Chebyshev polynomials of the

explicit form T that can also be combined with trigonometric identities to yield the explicit

expression form

))(coscos()( 1 xnxn

−=

1)(0 =xT xxT =)(1 )()(2)( 11 xTxxTxT nnn −+ −= (2) .1≥n

These polynomials are then used to approximate a function using the form

121

1

1 )()( cxTcxfN

k

kk −≈ ∑=

− .11 ≤≤− x (3)

Functions derived from the Chebyshev approximation technique or from other polynomial methods such as Legendre

polynomials, Laguerre polynomials, Hermite polynomials, or Radial Basis Functions do not capture the desired fine

resolution detail (fractal characteristics) of the original stochastic process2.

1.4 Multi-variable Data Modeling

Data Modeling can be generalized to the multi-variable case by invoking the constraints listed in Section 1.1. The multi-

variable form must be continuous, be an interpolation function, and result in fewer coefficients than the factorial

combinations of variables or the number of realizations (all possible combinations representation is a trivial case). In this

work, we only look at the single variable case. The multi-variable case is treated by the authors elsewhere3.

2. FRACTIONAL BROWNIAN MOTION

1/f or fBm noise is the most commonly found type of noise in nature. Formally, fBm is the integral of white noise.

However, the term 1/f noise is applied to any fluctuating quantity V(t) whose spectral density in log-log space varies as

1/fȕ, where ȕ characterizes the steepness of the slope of the spectral density. Gaussian or white noise would be

characterized with ȕ = 0, Brownian motion with ȕ = 2, and 1/f noise for ȕ values in between4. Mathematically, the fBm

process is represented by

( ).)'()'(

)(

1)( 2

1

21 ∫ ∞−

−−+Γ

=t H

H tdBttH

tB (4)

2.1 fBm Analogies

The direct integration method (5) uses a random number generator in order to capture the spatial characteristics of (4),

while FFT filtering (6) uses a random number generator to capture the frequency characteristics of (4). The Weierstrass-

Mandelbrot method5 (7) is a functional representation that eliminates the need for a random number generator, but does

not represent the spatial or frequency characteristics of (4) as well as (5) or (6). On the other hand, the Sxe method (8)

combines the benefits of (5) and (7) to represent both characteristics in a functional form. The discrete Sxe method (9) is

a discrete implementation of (8).

)(21

1 −+= − rndBB nn Direct Integration (5)

( ) ( )[ ]∑−

=

−+−

=

12

0

2

21

21

2

1

N

m

tif

nnmerndirnd

fB

πβ

FFT-filtering (6)

((∑∞

=

−=0

cos1)(i

ii nabnB π )) Weierstrass-Mandelbrot (7)

∑=

=stepn

x

exnB/

0

)sin()( Sxe Function (8)

)sin(1

e

nn nBB += − Discrete Sxe Function (9)

2.2 fBm Modeling

For the Chebyshev polynomial approximation method in (2) and (3), sampling of the continuous interval [-1, 1] is

achieved for an arbitrary f(x) by transforming the dependent variable x that occurs on the interval [a, b] by the

relationship

)(

)(2

ab

abxy

−+−

= .11 ≤≤− y (10)

We propose that a model of a specific fBm realization data set can be created from sub-sampled points (called the

stencil) of that realization as

].)1([)(

)()(

0 tktBkF

FwnB

H

n

∆−+==

(11)

We refer to this archetypal kernel as the Data Model of the fBm realization. This is possible because the fBm realization

defines the attractor for an unknown stochastic process. According to the Ruelle-Takens theorem6, addresses on this

attractor harbor information about the entire attractor. This should enable attractor reconstruction via interpolation from

a sparse sampling of address points. These address points define support vectors (12) or fiducial points for this process

because the support vectors are the points that define the critical decision boundary as a hyperplane in n-dimensional

space7.

[ ] 01),( =−+ bwxy iiiα (12)

If we think of a complex-spaced line in n-dimensional space as defining this boundary, we see its projection into 1-D

takes on the form of a 1/f type curve. Therefore, the fiducial points necessary to delineate and reconstruct this boundary

function are the support vectors defined as the sub-sampled points in (11) and can be represented in functional form

using Turlington polynomials8 of the form

.101log

101

10log)(

1

10

1

10 ∑∑

++

+

++=

md

dk

d

m

i

c

dk

d

c

dk

d

im

i

i

DCBkAkF (13)

We use the generic term 1/f when representing the entire class of 1/fβ type data sets. Because these processes are short-

time stationary and ergodic over finite non-zero length intervals, it should be possible to adequately capture the

statistical self-similarity within discrete intervals by sparse sampling of the interval to capture the local dynamics of the

attractor.

3. DATA MODELING

3.1 Dyadic fractal reconstruction

One theoretical approach for achieving functional Data Modeling from the sub-sampled 1/f realization is to use the

method of Deslauriers-Dubuc dyadic interpolation9. According to Deslauriers-Dubuc, the dyadic extension of the

fundamental interpolant is given as


where a, b, c, and d are scaling coefficients, and satisfies the identity given in (15). In (15), [t] denotes the integer

portion of the number t.

∑+

−=

−=3][

2][

)()()(t

tn

ntFnyty (15)

Since the solution of the coefficients in (14) cannot be obtained directly, regression methods may be used. However,

recasting Deslauriers-Dubuc dyadic interpolation using Barnsley’s Hidden Variable Iterated Function Systems (IFS)

recasts the interpolation into geometric form using affine transformations defined as

+

=

n

n

n

nnn

nnn

n

n

g

f

e

z

y

x

mlk

hdc

a

z

y

x

w

00 (16)

to be used instead. This affine transformation can be further simplified for our application by

.0

+

=

n

n

nn

n

nf

e

y

x

dc

a

y

xw (17)

Elton10 proved that this process will converge for all continuous functions in (17), which we have achieved using

Turlington polynomials.

∑ ∫=

∞→=

+

n

kX

kn

xdxfxfn 0

)()()(1

1lim µ (18)

3.2 Iterated Function Systems (IFS) reconstruction

IFS reconstruction is defined by Barnsley11 in (17) as an affine transform subject to the constraints

=

−

−

1

1

0

0

n

n

ny

x

y

xw and for n=1,2,…, N. (19)

=

n

n

N

N

ny

x

y

xw

Therefore, the five real numbers a, c, d, e, and f that specify the transformation must obey the equation relationships

.

,

,

,

100

10

nnNnNn

nnnn

nnNn

nnn

yfydxc

yfydxc

xexa

xexa

=++=++

=+=+

−

−

(20)

This is a system of four equations and five unknowns, resulting in one free parameter in each transformation. This free

parameter is chosen to be dn and is called the vertical scaling factor. This choice allows the other four parameters to be

written in terms of the data and the parameter dn as

.)(

)(

)(

)(

,)(

)(

)(

)(

,)(

)(

,)(

)(

0

00

0

01

0

0

0

1

0

01

0

1

xx

yxyxd

xx

yxyxf

xx

yyd

xx

yyc

xx

xxxxe

xx

xxa

N

NNn

N

nnNn

N

Nn

N

nnn

N

nnNn

N

nnn

−−

−−−

=

−−

−−−

=

−−

=

−−

=

−

−

−

−

(21)

Barnsley gives no method for choosing the value for dn, and instead leaves this as a free parameter that is only

constrained to be on the interval [0, 1]. The development of this method is based on Deslauriers-Dubuc dyadic

interpolation using only 4 or 5 control points. The vertical scaling parameter along with the 4 or 5 control points are left

as user input. Barnsley stores only the IFS coefficients derived from the 4 or 5 control points and the dn scaling

parameters in order to reconstruct a crude statistical representation of the data.

IFS reconstruction is performed by starting with an initial point (usually one of the points given in (19)) and performing

one of the transforms given in (17) and defined by the coefficients in (21) to the point to generate a new point that lies on

the attractor for the function. This new point is then used as a starting point and another transform applied to it to find

another new point on the attractor. This is repeated iteratively until the attractor for the function fills in to within a

specified tolerance given as

.ˆ ε≤− xx (22)

The IFS coefficients in (21) determine the distance the current x location is from the starting point of the data set (as a

fractional percentage of the length of the entire data set) and then maps the new y value to an x location the same

fractional percentage into the transform interval. The y value is determined by the coefficients in (21) by determining

how far the vertical scaling coefficient raises the previous y value above or below the sloped line connecting the

beginning y and ending y of the transform interval.

Barnsley presents two different methods for performing this iterative calculation to reconstruct the attractor: 1) Random

Iteration Algorithm and 2) Deterministic Algorithm. The Random Iteration Algorithm is a Markov process. It uses a

single starting point and the transform to be applied is chosen at random. The resulting point is then remapped with

another transform chosen at random, and the process repeated iteratively until all desired points have met the condition

in (22).

The Deterministic Algorithm assumes the data set to be on a grid and performs all transforms against all available points

(control points). The result is the synthesis of the same number of interpolated points as the number of control points. As

before, each of these points are then mapped iteratively until the condition in (22) is met. Neither the Random Iteration

Algorithm or the Deterministic Algorithm yield a true functional f(x) form that can be directly evaluated by x.

Innovations by the authors to solve problems with this method include: 1) Modification of the algorithm to interpolate on

all available data; 2) Selection of support vectors (control points) directly from data set; 3) Estimation of dn; 4)

Reduction of number of iterations required to evaluate y for a given x; and 5) Replacement of random number generator

for Random Iteration Algorithm by an equivalent function are covered in the next section.

3.3 Data Modeling innovations for IFS

3.3.1 Interpolation on all available data

IFS interpolation routines as published in the literature use only 4 or 5 control points from the original data in order to

capture the dynamics of the process and derive part of the scaling coefficients. The authors have found that instead of

arbitrarily selecting 4 or 5 points from the data set, all available data points can instead be used to reconstruct any 1/f

type noise data set. This required a paradigm shift from the approach of Barnsley, where the goal was to select the

appropriate 4 or 5 points that yielded the best reconstruction results. By using more data points, better reconstruction

results was achieved without the necessity to search for combinations of control points.

3.3.2 Support vector selection

Support vector identification is simplified to equi-spaced partitioning of the original data vector. The number of points N

to sub-sample12,13 in building the model is bounded by (23) where R is the range of the data and ı is the standard

deviation of the data.

=→∆+

N

NJ

R

Ji

Ji

1log

log

limmin)Re(0

1

σ 5.01

log

log

lim)Re(0

1 =

=→∆+

N

NJ

R

Ji

Ji

σ (23)

The number of sub-sample points N can be solved for between these bounds in order to minimize the variance of the

residual between reconstructed and original. We have found excellent results by solving the second equation in (23) for

N (the number of points to sub-sample), and have determined that a good rule of thumb to go by in the absence of the

information contained in (23) is to sub-sample 10% of the original data set. The process of sub-sampling itself occurs as

shown in (11), and is related to the [-1, 1] interval sampling shown previously in (10).

3.3.3 Estimation of dn

3.3.3.1 Method 1

The morphological fractal dimension Df of an IFS interpolation function can be related to dn, by the relationship

)log(

)log(1 1

N

dD

N

n n

f

∑ =+= (24)

where Df is bounded between 1 and 2 and the interpolation function is not a straight line. The authors have noted that if

the fractal dimension Df of the desired interpolation function in (24) is known, its vertical scaling coefficients dn, can be

represented as a function of the fractal statistics of the data, and hence can be estimated from a characterization of the

data itself.

Using the morphological fractal dimension relationship in (24), it naturally follows that a first estimate of the dn

coefficients from the data can be obtained by inverting (24) and solving for dn. By isolating dn in (24), we obtain

.1

1

−

==∑ fDN

n n Nd (25)

If we assume that the magnitude of each of the scaling coefficient is equal and uniformly distributed over all intervals of

the data, the summation (Σ) can be removed from (25) by dividing both sides of the equation by N. This yields the

relationship

.2−= fD

n Nd (26)

The sign of the scaling coefficient dn is assigned to be the same as the sign of the derivative between support vector n

and support vector n – 1 given as

.

1

1

<−

≥=

+

+

nnn

nnn

n

yyd

yyd

d (27)

The authors have found that in practice, the use of the relationship in (26) yields scaling coefficients that are too large

and cause the interpolation function to not reconstruct the original data well. Care must also be taken when choosing

which fractal dimension characterization method to use when mapping Df into dn. The relationship in (26) is developed

based on morphological fractal statistic methods. In order to make the relationship in (26) useful for entropy based

fractal characterization methods, the parameter β was added yielding

.2 β−−= fD

n Nd (28)

with the value of β selected based on the underlying distribution of the original data and the signs of the scaling

coefficients set using (27).

3.3.3.2 Method 2

Because fractal statistics imply multi-resolution scaling, we propose that the estimate of the fractal dimension Df derived

from the sub-sampled points (11) is sufficient to enable reconstruction. Data Modeling uses these sub-sampled points

(support vectors) and derives all the coefficients from the support vector data itself. This includes an estimate of the

vertical scaling coefficient dn obtained from Jaenisch14 as

)(2

13 −+= JJJ β (29)

.

1

1

log

)1(

≤

>−

=−

NJN

NJ

N

J

d

J

n (30)

where β=0.25 for the Gamma distribution, 0.5 for the Weibull and Poisson distributions, -0.25 for the Exponential

distribution, –0.5 for the Triangular distribution, and zero for Gaussian and Uniform distributions. The sign of the

scaling coefficient dn is assigned to using the relationship in (26). This technique, as in the case of (26) and (28) also

assumes that the fractal dimension Df is distributed uniformly among all intervals of the data. If this is not the case,

multi-fractal analysis may be applied and the data partitioned into subintervals so that a spectrum of fractal dimensions

can be obtained15 or a functional representation of Df provided to allow the fractal dimension to vary across the data set.

Techniques based on multi-fractal analysis work very well, but are not covered in the scope of this work due to space

constraints.

3.3.3.3 Method 3

As a further simplification in the determination of dn, we propose that these scaling coefficients can be estimated directly

from the data itself without any preprocessing of the data set for fractal characterization. This estimate uses the

differences of means (∆µ) of neighboring segments connecting support vector points and is given by

2

)(

2

)(

2

)( 2112 nnnnnnn

yyyyyyd

−=

+−

+= ++++ (31)

This scaling coefficient estimate method can be done in real-time as the data is being measured.

3.3.4 Iteration reduction

In applications where the IFS algorithms used for Data Modeling yield an acceptable solution in sufficient time, no

further enhancement is needed. Real time versions must reduce the number of iterations necessary for convergence. We

propose the use of zooming as shown in Fig. 1. Using the Ruelle-Takens theorem, we can use the interpolated values that

define the new bounded segments near the desired x interval. These new points are used to restart the interpolation

sequence on the newly constrained domain.

B

C

D

E

F

f(3.5)?

ε

C

D

2

3

5 6

8

911

12

1

4

7

10

13

1

4

7

10

13C

D

1

4

7

10

13C

D

3.5

A

1 2 3 4 5 6

f(3.5)=3.721

ε

1

2

3

4

B

C

D

E

F

f(3.5)?

ε

C

D

2

3

5 6

8

911

12

1

4

7

10

13

1

4

7

10

13

1

4

7

10

13C

D1

4

7

10

13C

D

1

4

7

10

13C

D

1

4

7

10

13C

D

3.5

A

1 2 3 4 5 6

f(3.5)=3.721

ε

1

2

3

4

Fig. 1. Iteration reduction using zooming.

For the Random Iteration algorithm, once the number of points generated on the desired x interval equals the original

number of sub-sampled points, the IFS transform coefficients in (21) are calculated from these newly generated points

and the iteration process restarted on the new domain. This process is repeated until a solution is found within the

tolerance given in (22).

Using the Deterministic algorithm, only the transform from the segment that the desired data point lies on is used along

with the sub-sampled points. All of the sub-sampled points are mapped using this single transform, and the results stored

in a temporary buffer. The sub-sampled points are replaced with the newly generated points, and the process is repeated

until a solution is found that meets the criteria given in (22). The number of calculations required to find a solution

meeting the criteria in (22) for the Deterministic algorithm is bounded by Nx, where N is the number of points in the

Data Model, and the number of zooms x is directly proportional to the order of the tolerance (x ~ |log(ε)|).

3.3.5 Replacement of random number generator

In Monte Carlo applications where a uniform random number generator is used (such as the Random Iteration

Algorithm), we propose instead to use a stochastic function to derive approximate random digits. We have found that a

good functional representation for approximating random number digits on the interval [-1,1] is the derivative of the Sxe

function (8) and is given as

)sin( exR = (32)

where e is the exponential function for value one (~ 2.7183) , and x is generally used as a non-repeating increasing index

variable. If the desired deviates are to be on an interval other than [-1,1], the results can be renormalized to any

minimum (a) and maximum (b) value using

).)](1([ˆ21 abRaR −++= (33)

Use of this stochastic function for the generation of random deviates in the Data Modeling process eliminates one major

impediment towards a functional representation f(x).

4. APPLICATIONS OF DATA MODELING

4.1 Data hole patching

One application of Data Modeling is in hole patching for data sets16. Although many classical data and signal processing

methods assume that a data set is evenly sampled and contains no holes, this is not always the case in measured data

because of sampling constraints and data dropout. We have found that we can achieve excellent results in patching one

or more holes in data sets using the Data Modeling approach. This is done using the same sampling constraints shown

above in (23) or by naively sampling 10% of the available data evenly. If an evenly space point falls in a data hole

region, we simply omit it from the set of support vectors. The IFS transforms listed above are performed on the available

support vectors to reconstruct the missing data in the dropout area. This new data is simply placed into the original data

sequence where the missing data would have occurred. Results are shown in Fig. 6 in Section 5 of this work.

4.2 Data compression

The authors generally use Data Modeling to perform compression through decimation of the original data and saving

only the support vectors. This facilitates reconstruction on demand to any desired resolution using the Data Modeling

IFS transforms discussed previously. Unlike Barnsley’s method where all of the coefficients in (21) and the vertical

scaling coefficient dn are stored, we store only the sub-sampled points and generate all of the necessary coefficients upon

demand. This method yields an approximately 80% savings in storage from Barnsley’s techniques, and since the number

of points that are to be saved is known ahead of time, the compression ratio is also known in advance. This yields

excellent reconstruction results as demonstrated in Fig. 5 in Section 5.

4.3 Image compression

Data Modeling compresses images by first converting them to a one dimensional data sequence and decimating as

shown previously in (23). This generates a thumbnail representation of the original image. These thumbnails are low-

resolution images that capture the salient features of the original, can be stored in much less space than the original, and

can be reconstructed to any desired resolution upon demand. Two dimensional images are transformed into a one

dimensional data sequence using the Hilbert sequence15 given in (34) subject to Lindenmayer’s L-system grammars

defined by (35).

)()()()(

)()()()(

43211

43211

nnnnn

nnnnn

PwPwPwHwP

PwPwPwPwH

∪∪∪=∪∪∪=

+

+ (34)

°=−→−+→+

→−++−→+−−+→

90δ

FF

FLRFRLFR

FRLFLRFL

).,,(),,(

),,(),,(

),sin,cos(),,(

δααδαα

αααα

+→⇒−−→⇒+

++→⇒

yxyx

yxyx

lylxyxF (35)

The decimation methods contained earlier in this work can then be used to sub-sample the sequence, and Data Modeling

used to reconstruct the sequence to any resolution desired. The inverse Hilbert sequence (reverse of (34)) is then used to

reconstruct the two dimensional image derived from the Data Modeling process. It has been found that this process

yields excellent results as demonstrated in Fig. 2.

Fig. 2. Data Modeling technique applied to an image.

Once the thumbnail has been generated, it can then be further compressed using techniques such as JPEG-2000 and

transmitted in an efficient manner. On the receive end, the thumbnail would be uncompressed with JPEG-2000 and Data

Modeling used to reconstruct the image to the original size or larger on demand, thereby enabling streaming real time

video application.

4.4 Data extrapolation

In order to extrapolate data points, it is necessary to first introduce at least one new support vector that is outside of the

original transform intervals. This is accomplished by appending the first point of the data set to the end of the data set

according to

)( 11 iiNN xxxx −+= ++

iN yy =+1 Ni ,...,2,1= (36)

to form an extrapolation interval. New transforms using (21) are calculated, and the data sequence is reconstructed. Only

the points generated in the first ½ of the extrapolation interval are kept as extrapolated points. This process is repeated

by increasing i in (36) and appending the next point in the original data set to the end of the saved extrapolated points to

create a new extrapolation interval. The reconstruction process is continued until all of the original data points have been

used, and can be repeated as many times as necessary to generate the desired length forecast.

Historical

Fractal Forecast

Linear Prediction

256 5121

Fig. 3. Screenshot of Data Modeling compared to Linear Prediction for forecasting.

4.5 Quantum computing

Using the Data Modeling approach, mathematical operations can be performed on the sub-sampled portion of a larger

data set, reducing the number of calculations required by as much as one third (1/3). The resultant is nearly equal to

performing the same mathematical operation on the full data set. To prove this, we do the following:

Given a set of 256 numbers A, and an equal length filter B of 256 numbers • • • • • • •

Sub-sample every 10th number from A to yield 26 numbers in C

Sub-sample every 10th number from B to yield 26 numbers in D

Convolve A and B yielding 256 numbers in E

Convolve C and D yielding 26 numbers in F

Use fractal reconstruction to inflate 26 numbers in E to 256 numbers in G

Compute Differences in E and G yielding 256 numbers in H

This is shown in Fig. 4 below. The maximum difference value of H is 11%, near the approximate 10% reconstruction

tolerance. This approach can also be applied to an image using Data Modeling, because we treat the 2-D image as a 1-D

data set using the Hilbert sequence.

E

G

Maximum Error ~ 11%

A

C

Thumbnail

Original

A

C

Thumbnail

Original

B

Original

Filter

Thumbnail Filter

D

B

Original

Filter

Thumbnail Filter

D

⊗ =H

Fig. 4. Complex Process Applied to Sub-Sampled Data Yields Equivalent Result.

5. RESULTS

5.1 Data synthesis

Data Modeling has been successfully compared to classical methods including polynomial curve fitting, Markov chains,

wavelet transforms, mean and covariance methods, and Kalman filters (Fig. 5). Column 1 of Fig. 5 shows the original

data sequence of 1024 points and a histogram showing the distribution of the original. In Column 2, results from using

each of the classical techniques mentioned above are shown along with a histogram of the distribution of each resultant.

Note that only the Markov chain process captures the underlying distribution of the original, but fails to capture the

temporal and spatial characteristics of the original.

Column 3 of Fig. 5 shows the fractal Data Modeling approach using five different sub-sampling rates varying from 1%

to 25% of the original. For each of these sub-sampling rates, both the temporal, spatial, and underlying distribution

characteristics of the original are captured.

Columns 4 and 5 show the spatial error and power spectra differences between the original, classical method, and the

fractal reconstruction. Using the 5% Data Model (50 points) as a baseline, only the wavelet transform method provided a

slightly better RSS error, but with a substantial difference in both the power spectra and in temporal and spatial

characteristic retention. Two peaks occurred in the log power spectra that were captured and amplified by the fractal

method albeit with a shift in their locations.

The wavelet power spectra showed no peaks at all in the lower trace, indicating a significant distortion in the original

signal and loss of frequency content. It should also be noted that all of the fractal Data Models generated power spectra

more like the original than any of the classical methods, and that using 50 points of the original data outperformed using

50 wavelet transform coefficients.

Kalman Filter

Mean & Covariance

50 pt. Wavelet

256 pt. Fractal Model1/f Random Signal 1024 pts.

100 pt. Fractal Model




Markov Chain

Poly RSS 2.19

FMM RSS 0.94

Mark RSS 10.16

FMM RSS 1.48

Wave RSS 1.76

FMM RSS 1.99

M&C RSS 8.61

FMM RSS 3.23

Kalm RSS 2.31

FMM RSS 4.76

Polynomial

Poly RSS 33.42

FMM RSS 3.82

Mark RSS 70.23

FMM RSS 5.07

Wave RSS 26.55

FMM RSS 6.66

M&C RSS 53.63

FMM RSS 11.48

Kalm RSS 31.95

FMM RSS 8.91

0 max

0 max

0 max

0 max

Spatial Error Distribution Power Spectra (dB)

0 maxFig. 5. Comparison of fractal Data Modeling and classical methods.

5.2 Noise model comparison

Fig. 6 shows examples of Data Modeling for reconstruction and hole patching for three different 1/f type data sets. The

first two data sets (Rows 1 and 2) were generated using a Gamma distribution as shown, and the third data set was

generated using the Sxe function proposed earlier. We naively sampled the originals in Column 1 by 10% and

reconstructed as discussed earlier in order to obtain Column 2, and we introduced the dropouts shown in Column 3 and

patched them as discussed in Section 4.1. The Cramer-Rao bound of the process is the a priori theoretical best that we

can reconstruct each of the data sets and is given as

)/(100 NCR σ= (37)

where σ is the standard derivation of the N sub-sampled points. The error for the reconstructed curve is given in (38),

and the hole patching error is calculated in the same manner as (38) using only the dropped out points.

))ˆ((1

100 ∑=

−=npts

i

iinptsyyErr (38)

Original

Decimated and

Reconstructed Original with Dropouts Dropouts Patched

Cramer-Rao = 1.72% Recon Error = 2.03% Hole Error = 2.29%

Gamma

(A=2)

Gamma

(A=1)

Sxe

C-R = 2.07% Error = 3.01% Error = 2.95%

C-R = 2.36% Error = 3.98% Error = 3.67%

66 dropouts

53 dropouts

47 dropouts

1816 pts.

1253 pts.

Sub-sample

175 pts.

Sub-sample

119 pts.

1146 pts.Sub-sample

61 pts.

Original

Decimated and

Reconstructed Original with Dropouts Dropouts Patched

Cramer-Rao = 1.72% Recon Error = 2.03% Hole Error = 2.29%

Gamma

(A=2)

Gamma

(A=1)

Sxe

C-R = 2.07% Error = 3.01% Error = 2.95%

C-R = 2.36% Error = 3.98% Error = 3.67%

66 dropouts

53 dropouts

47 dropouts

1816 pts.

1253 pts.

Sub-sample

175 pts.

Sub-sample

119 pts.

1146 pts.Sub-sample

61 pts.

Fig. 6. Examples of Data Modeling for reconstruction and hole patching.

6. CONCLUSIONS

Data Modeling has solved the inverse fractal problem for 1/f noise sets. Innovations developed by the authors during the

course of this work include modifying the IFS algorithm to use all available data instead of 4 or 5 control points,

autonomous selection of the support vectors or control points directly from the data set, the ability to estimate dn from

the data itself, reducing the number of iterations required to find solutions, and the replacement of the random number

generator in the Random Iteration algorithm with a stochastic function. Applicability of these techniques has also been

demonstrated for data and image compression and reconstruction, hole patching, data forecasting, and applicability to

the field of quantum computing for various noise sets. It is possible to process your own data sets through the Data

Modeling algorithms we have provided in the Appendix in the form of a MathCAD function and work document or

QBASIC source code. The author’s permission is granted to use this algorithm provided the author’s are acknowledged

and referenced in any and all derived implementations. If the investigator prefers to use a uniform random number

generator, the Sxe function can be replaced at the appropriate locations.

ACKNOWLEDGMENTS

The authors would like to thank Marvin Carroll, Tec-Masters, Level-13, Licht Strahl Engineering INC, and Kristi

Jaenisch and Technical Dive College for the use of the MAJQANDA algorithm suite during the course of this work.

REFERENCES

1. Moin, P. Fundamentals of Engineering Numerical Analysis, Cambridge: Cambridge University Press, c2001.

2. Press, W.H., Flannery, B.P., Teukolsky, S.A., and W.T. Vetterling, Numerical Recipes in FORTRAN, Cambridge:

Cambridge University Press, c1989.

3. Jaenisch, H.M. and J.W. Handley, “Data Modeling for Radar Applications”, To be published in Proceedings of the

IEEE Radar Conference 2003, Huntsville, AL.

4. Feder, J. Fractals. New York: Plenum Press, 1988.

5. Edgar, G.A. Classics on Fractals, Reading, MA: Addison-Wesley, c1993.

6. Ruelle, D. Chaotic evolution and strange attractors. Cambridge: Cambridge University Press, 1989.

7. Scholkopf, B. and A. Smola, Learning with Kernels. Cambridge, MA: MIT Press, 2002.

8. Turlington, T. Behavioral Modeling of Nonlinear RF and Microwave Devices. Boston, MA: Artech House, 2000.

9. Cherbit, G. Fractals: Non-integral Dimensions and Applications. New York: Wiley, 1991.

10. Elton, J. “An Ergodic Theorem for Iterated Maps,” Journal of Ergodic Theory and Dynamical Systems, 7:481-488

(1987).

11. Barnsley, M. Fractals Everywhere, Second Edition. Boston, MA: Academic Press, 1993.

12. Handley, J., Jaenisch, H. and M. Carroll, “Identification of the behavior transition point in data using a robust fractal

method,” Proceedings of the Southeastern Simulation Conference, October 1995.

13. Handley, J. On The Existence of a Transition Point in Sampled Data using Fractal Methods. Ann Arbor, MI: UMI,

1995.

14. Jaenisch, H.M., Taylor, S.C., Handley, J.W., Carroll, M.P., “Optimal Fractal Image and Data Compression”,

Proceedings Of The Southeastern Simulation Conference '95, pp. 67-77, Orlando, FL 1995.

15. Jaenisch, H., Barton, P., and R. Carruth, “Determining the Fractal Dimension of Scenes and Digital Signals using

ROSETA and Other Novel Approaches,” Proceedings of SPIE Conference, Session: Signal Processing, Sensor

Fusion, and Target Recognition II, Vol. 1955, pp.298-315, Orlando, FL 1993.

16. Jaenisch, H.M. “Fractal Interpolation for Patching Holes in Data Sets”, Proceedings of the Southeastern Simulation

Conference ’92, Pensacola, FL 1992.

APPENDIX

0 50 1000

0.5

1

yy i

xxi

0 500 10000

0.5

1

original i

model ii

i ii,

yyi

ck i( ) xx

i 1−⋅ dk i 1−( ) yy

i 1−⋅ ffk i( )+

+:=xxi

ak i( ) xx

i 1−⋅ ek i( )+:=

yy0

model0

:=xx0

0:=ki

floor Ki( ):=K

insegs

sin iexp 1( )( ) 1+

2

⋅ 1+:=

ffn

xnnsegs

modeln 1−⋅ xn

0model

n⋅− d

nxn

nsegs( ) model0

⋅ xn0

− modelnsegs

− ⋅−

b:=

cn

modeln

modeln 1−− d

n 1− modelnsegs

model0

−( )⋅−

b:=

dn

if n nsegs<model

n 1+ modeln 1−−

2,

model0

modeln

−

2,

:=

en

xnnsegs

xnn 1−⋅ xn

0xn

n⋅−

b:=a

n

xnn

xnn 1−−

b:=

b xnnsegs

xn0

−:=xnn

n:=

n 1 nsegs..:=nsegs 99=nsegs newn 1−:=

modelii

original

floor iinpts 1−

newn 1−

⋅

:=ii 0 1, newn 1−..:=original

i

originali

min original( )−

max original( ) min original( )−:=

newn ceilnpts

10

:=original

ioriginal

i 1− sin ie( )+:=original

00:=i 1 npts..:=npts 1000:=

DM fx itol, x, y,( ) newn length y( )←

nsegs newn 1−←

Nk

k 1+←

k 0 newn 1−..∈for

b xnsegs

x0

−←

dn

yn 1+ y

n 1−−

2←

n 1 nsegs 1−..∈for

dnsegs

y0

ynsegs 1−−

2←

an

xn

xn 1−−

b←

en

xnsegs

xn 1−⋅ x

0x

n⋅−

b←

cn

yn

yn 1−− d

ny

nsegsy

0−( )⋅−

b←

fn

xnsegs

yn 1−⋅ x

0y

n⋅− d

nx

nsegsy

0⋅ x

0y

nsegs⋅−( )⋅−

b←

n 1 nsegs..∈for

xprev x0

←

yprev y0

←

bestx x0

←

besty y0

←

k floor rnd nsegs( ) 1+( )←

xcurrent ak

xprev⋅ ek

+←

ycurrent ck

xprev⋅ dk

yprev⋅ fk

+( )+←

bestx xcurrent←

besty ycurrent←

xcurrent fx− bestx fx−<if

xprev xcurrent←

yprev ycurrent←

bestx fx− itol>while

besty

:=

'Fractal Data Modeling - QBASIC

'Holger Jaenisch, Ph.D. and Jamie Handley, c1992-2003.

nameagain: INPUT "Data file name => ", zz$

IF zz$ = "" THEN GOTO nameagain

percagain: INPUT "Subsample percentage (1-100) => ", perc

IF perc <= 0 THEN GOTO percagain

IF perc > 100 THEN GOTO percagain

perc = perc / 100!

tolagain: INPUT "Reconstruction tolerance (.1) => ", tol

IF tol <= 0 THEN GOTO tolagain

OPEN zz$ FOR INPUT AS #1

ntotal = 0

DO UNTIL EOF(1)

REDIM PRESERVE yorig(ntotal)

INPUT #1, yorig(ntotal)

ntotal = ntotal + 1

LOOP

CLOSE #1

ymin = yorig(0)

ymax = yorig(0)

FOR i = 1 TO ntotal - 1

IF yorig(i) > ymax THEN ymax = yorig(i)

IF yorig(i) < ymin THEN ymin = yorig(i)

NEXT i


yorig(i) = (yorig(i) - ymin) / (ymax - ymin)

NEXT i

n = CINT(perc * ntotal)

REDIM xx(n - 1), yy(n)

FOR i = 0 TO n - 1

xx(i) = CINT(((ntotal - 1) * i) / (n - 1))

yy(i) = yorig(CINT(((ntotal - 1) * i) / (n - 1)))

NEXT i

yy(n) = yy(0)

ERASE yorig

ntrans = n - 1

bb = xx(n - 1) - xx(0)

REDIM a(ntrans), b(ntrans), c(ntrans), d(ntrans), e(ntrans), f(ntrans)

FOR i = 1 TO ntrans

d(i) = ((yy(i + 1) + yy(i)) / 2) - ((yy(i) + yy(i - 1)) / 2)

a(i) = (xx(i) - xx(i - 1)) / bb

b(i) = 0!

c(i) = (yy(i) - yy(i - 1) - d(i) * (yy(n - 1) - yy(0))) / bb

e(i) = (xx(n - 1) * xx(i - 1) - xx(0) * xx(i)) / bb

f(i) = (xx(n - 1) * yy(i - 1) - xx(0) * yy(i) - d(i) * (xx(n - 1) * yy(0) - xx(0) * yy(n - 1))) / bb

NEXT i

OPEN "input.dat" FOR OUTPUT AS #1

FOR i = 0 TO n - 1

PRINT #1, xx(i), yy(i)

NEXT

CLOSE #1

REDIM yyout(ntotal - 1)


yyout(i) = -999

NEXT i

icnt = 0

again1:

OPEN "input.dat" FOR INPUT AS #1

OPEN "output.dat" FOR OUTPUT AS #2

DO UNTIL EOF(1)

INPUT #1, x, y

FOR k = 1 TO ntrans

IF INKEY$ <> "" THEN END

xnew = a(k) * x + b(k) * y + e(k)

ynew = c(k) * x + d(k) * y + f(k)

PRINT #2, xnew, ynew

IF ABS(xnew - CINT(xnew)) <= tol THEN yyout(CINT(xnew)) = ynew

icnt = icnt + 1

NEXT k

ileft = 0


IF yyout(i) = -999 THEN ileft = ileft + 1

NEXT i

IF ileft = 0 THEN

GOTO done1

END IF

LOOP

CLOSE #1

CLOSE #2

KILL "input.dat"

NAME "output.dat" AS "input.dat"

GOTO again1

done1:

CLOSE #1

CLOSE #2

KILL "input.dat"

KILL "output.dat"

CLS

SCREEN 1

WINDOW (-1, 1.1)-(ntotal, -.1)

PRINT "Total calcs = "; icnt

OPEN "final.out" FOR OUTPUT AS #1


PSET (i, yyout(i))

yyout(i) = yyout(i) * (ymax - ymin) + ymin

PRINT #1, yyout(i)

NEXT i

CLOSE #1

END

Automatic Differential Equation Derivation From Sampled Data For

Simulation Parameter Sensitivity Analysis

Holger M. Jaenisch*

James Cook University, Townsville QLD 4811, Australia

ABSTRACT

An analytical differential equation model from a single simulation input and output data vector is derived. The derived

model is analytically varied (real versus imaginary) to determine Critical, Sensitive, and Key parameters without the use

of Design of Experiments (DOE).

Keywords: Formal Analysis, Data Modeling, VV&A, Equivalence, Consistency, Transfer Function Modeling, Automatic Proofing

1. INTRODUCTION

When a large number of parameters exist for a simulation (greater than 40 input and/or output parameters), the

application of Design of Experiments1 (DOE) becomes unwieldy. Coupling of variables is lost when 2 way or 3 way

variation of parameters is performed, since there may be more than 2 or three parameters directly coupled. Instead, the

simulation is modeled analytically by a transfer function derivation using only input vector and associated output

vectors, and without access to source code or the compiled program. If more than one example is available, they may be

used as well. By not adjusting all coupled parameters simultaneously, a true estimate of the parameter sensitivity on the

output is not obtained. Instead, frequency space is used as an environment that limits the number of parameters into

dominant eigenmode terms and varies them one at one time by varying the real and imaginary terms on the analytical

model of the input vector and passing it through the analytical transfer function model of the simulation.

2. DATA MODELING

Data Modeling2 is a unique term coined to describe the process of deriving mathematical equations as models from

simulated or measured data. What follows is a specific embodiment that achieves Data Modeling using classical control

and Fourier theory. The final differential equations generated by the process are generically termed Data Models.

Classic transfer functions3-5 are defined using the raw input and output data from a plant (in this case simulation of

interest). The Z transform of the input and output data is defined as

n

n zazazazX −−− ++++= K2

2

1

11)( (1) n

n zbzbzbbzY −−− ++++= K2

2

1

10)(Single simulation input variable multiple realizations Single associated simulation output variable multiple realizations

In this form, the classic transfer function is formed by

Simulation

n

n

n

n

zazaza

zbzbzbb

zX

zY

InputZ

OutputZzH −−−

−−−

++++++++

===K

K2

2

1

1

2

2

1

10

1)(

)(

)(

)()( (2)

where the input vector X(z) defines the characteristic polynomial of the simulation, and whose roots (poles) and their

placement on a root-locus plot define the sensitivity of the simulation. This represents the TF for a single input variable

with multiple realizations resulting in a single output variable of multiple associated realizations. For multiple simulation

input/output variable cases, the transfer function is

[ ]

=

)()()(

)()()(

)()()(

)(

21

22221

11211

zHzHzH

zHzHzH

zHzHzH

zH

KKKK

K

K

L

MOLM

L

L

(3)

*[email protected]; phone 1 256 337-3768


where each resultant output variable is a linear combination of all input variables. The transfer function from input

variable j to output variable i is Hij(z). This matrix captures the influence of each individual input variable element to all

output variables.

This process can be greatly simplified by adopting a unique and novel interpretation or perspective of definitions. By

taking a vector of multiple simultaneous input variables and an equal length vector of multiple simultaneous output

variables (MIMO) and transposing them (transposing column vectors to rows), the result is simplified to an equivalent

single input/output system (SISO) where the multiple realizations of the single input variable are actually the single

simultaneous input variables. This MISO variable transfiguration (transpose followed by new interpretation of resulting

vector) is comprised of 8 simulation describing variable inputs and 2 computed output variables (from a kinetic impact

debris simulation6). The vector of simulation variable values is formed as follows:

=

=

69.71

14.337

11.0

51.9

27.155

41.0

36.7

38.0

)(

)(

)(

(deg)

(sec)

)( 2

ssScaleIntToTgtMa

kgMass

mAperture

mWavelength

FOV

RelSigVal

Time

mArea

ai µ

=

=

1

1

1

1

1

1

11.7

345

)(

Padding

Padding

Padding

Padding

Padding

Padding

mmMinSize

NFrags

bi

N inputs, single realization N outputs, single realization

i = [ 1 2 3 4 5 6 7 8 ]

[ ]ssScaleIntToTgtMaMassApertureWavelengthFOVRelSigValTimeAreaa =ℑ

[ ]69.7114.33711.051.927.15541.036.738.0=ℑa Single virtual input, N realizations

[ ]PaddingPaddingPaddingPaddingPaddingPaddingMinSizeNFragsb =ℑ

[ ]11111111.7345=ℑb Single virtual output, N realizations (4)

The justification for using a single realization or Monte Carlo run of data for developing a Simulation model from a TF

is obtained when measuring information content or entropy of the variables. The TF in (4) can be used to model the

simulation. Entropy is calculated by converting all digits into gray codes, and processing the codes as though they were

binary digits according to Shannon. The single realization in (4) has an entropy measure of 184; increasing entropy by

1%, requires another 40 cases. The equivalent mean and covariance matrix model (2x2 covariance plus 2 means for the

output), has entropy 40, require 532 Monte Carlos to equal the entropy in a single case derived TF. Figure 1 graphs

entropy as a function of variable number for Kolmogorov-Gabor polynomial, covariance,, and TF type simulation

models.

From (2), the differential coefficients a0, a1, b0, b1…,bn, are

)()()1)(( 2

2

1

10

2

2

1

1 zXzbzbzbbzazazazY n

n

n

n

−−−−−− ++++=++++ KK (5)

where a are original input variable values (the order of the z coefficient becomes the order of the derivative and is equal

to the number of input points) and b are original output variable values (again, order of the z coefficient is the equal to

the order of the derivative and equal to the number of input points).

)()( 011

1

1011

1

1 txbtdt

dxbt

dt

xdbt

dt

xdbtyat

dt

dyat

dt

ydat

dt

yda

n

n

nn

n

nn

n

nn

n

n ++++=++++ −

−

−−

−

− LL (6)

Kolmogorov-Gabor

Polynomial

Kolmogorov-Gabor

Polynomial

Transfer

Function

Covariance

Model

10

105

Log(Entropy)

1 100Log(Nvars)

Transfer

Function

Covariance

Model

10

105

Log(Entropy)

1 100Log(Nvars)

Fig. 1. Log of Entropy versus Log of number of variables.

By substitution, the detailed debris differential Data Model is obtained. Note the special numerical meaning of the

coefficients in front of the derivatives as being the various initial variable values.

=+++++++ )(38.036.741.027.15551.911.014.33769.712

2

3

3

4

4

5

5

6

6

7

7

tytdt

dy

dt

yd

dt

yd

dt

yd

dt

yd

dt

ydt

dt

yd

)(34511.71111112

2

3

3

4

4

5

5

6

6

7

7

txtdt

dx

dt

xd

dt

xd

dt

xd

dt

xd

dt

xd

dt

xd+++++++ (7)

Note that derivative terms, now by substitution, display the variable names they represent from the a and b arrays in (4).

dt

dMinSize

dt

1Pd

dt

2Pd

dt

3Pd

dt

4Pd

dt

5Pd

dt

6Pd36.741.027.15551.911.014.33769.71

2

2

3

3

4

4

5

5

6

6

7

7

++++

3

3

4

4

5

5

6

6

7

7

1111138.0dt

FOVd

dt

Wavelengthd

dt

Apertured

dt

Massd

dt

ScaleCoeffdNFrags ++++=+

Areadt

dTime

dt

lSigValRed34511.71

2

2

+++ (8)

Next, variable names for the derivative terms are substituted, emphasizing the substantial cross coupling captured.

3

3

04

4

05

5

06

6

00dt

2PdFOV

dt

3PdWavelength

dt

4PdAperture

dt

5PdMass

dt

P6dScaleCoeff

7

7

+++

7

7

002

2

0dt

ScaleCoeffd6PNFragsArea

dt

dMinSizeTime

dt

1PdlSigValRe 0=+++

2

2

3

3

4

4

5

5

6

6

dt

lSigValRed1P

dt

FOVd2P

dt

WavelengthdP3

dt

Apertured4P

dt

Massd5P 00000 +++++

AreaNFragsdt

dTimeMinSize 00 ++ (9)

To demonstrate the powerful cross coupling in the model, expand only the second order derivative of the inputs as an

example. Equation (10) shows a subset of the derivatives in (9) in order to demonstrate complexity and small variable

influence.

AreaNFragsdt

dTimeMinSize

dt

lSigValRed1P 0 002

2

++ =

AreaNFragsTimelSigValReMinSizeTimeRelSigValAreaTime1P 0 00 )())()(( +−+−−− (10)

As the derivative order of (10) grows (meaning more simulation variables), it becomes unwieldy. This is mitigated by

using a cascade of bi-quadratic term transfer functions1 and using the resulting coupled differential equations for each

transfer function in the cascade. Because the time index represents different variables, when the differentiation across

different variables occurs, high order variable coupling occurs. This functional (function whose variables themselves are

functions) differential equation is analytically solvable using the well known Volterra series. Further, the well know

Kolmogorov-Gabor (KG) discrete form of the Volterra Series is the solution to our functional differential equation. This

becomes a nested functional multi-variable polynomial shown as

LK ++++= ∑∑∑∑∑∑i j k

kjiijk

i j

jiij

i

iiL xxxaxxaxaaxxxy 021 ),,,( (11)

))))))))),,(((((((((),,,( 2121 kjinL xxxybybybyfxxxy KK = [ ] n

LxxxyO 3),,,( 21 =K (12)

whose direct solution gives the integral to (9) while mitigating the use of numerical integration solvers. The KG

polynomial uses the Group Method of Data Handling (GMDH) and self organizing methods to parsimoniously solve for

its coefficients using 3rd order building blocks and regression in a nested fashion. This yields a multi-input variable

polynomial for a single simulation output variable that provides a direct simulation output without solving by integration.

It should be noted that the number of examples required for generating a KG polynomial must be greater than number of

terms. Alternatively, a single variable GMDH version of the KG polynomial is obtained from all inputs going into a

single variable being tuned to the associated padded output.

This KG Data Model formulation is still somewhat cumbersome and does not lend itself well to simultaneous multiple

output variables or to periodic behavior modeling. Therefore, trigonometric basis functions in the form of Fourier series

are used as shown below, making Data Modeling in frequency space possible using the Fast Fourier Transform (FFT).

This is easily accomplished using the classic z transform defined for z = eiω , which is the same as the discrete Fourier

transform (DFT) defined by

∑=k

j kn

ji

nC )

2exp(

1 π (13)

Therefore, the TF is formed in frequency space using the fast Fourier transform (FFT) to determine the Fourier series of

the input and output sequences. Classically, these two Fourier series would be combined into a TF

∑=

+=n

j

jjjj xBxAxy1

sincos)( ωω )(

)(

InputFFT

OutputFFTTF = (14)

The Fourier series is a lengthy representation that includes many alias terms due to frequency folding. A simplified form

identifies the dominant frequency components or eigenmodes, labeling them as lambda terms after zeroing all other

frequency locations.

otherwise0Ȝdominantif1Ȝ

BAfxyN

ij

j

j

N

ij

jk ==

+=≅ ∑ )sin()cos(),,,()(22

21

ππ λλλλλ K

xBxA jjjjj ωωλ sincos += (15)

Eigenfunction extraction uses a variation of Wiener filtering to identify significant peaks in the dB power spectrum.

These peaks correspond to dominant sine and cosine terms occurring in the power spectrum. Once the FFT is generated,

the power spectrum of the data is calculated, which is the sum of the squares of the real and imaginary terms in (1), and

is then transformed into decibels (dB).

22 )Im()Re( iii BAP += ))max(

(log10)( 10

i

ii

P

PdBP = (16)

Least squares regression is used to fit a straight line to the entire dB power spectrum of the form

bmxy += (17)

where m is the slope of the line and b is the y-intercept. It should be noted that the zero order term is not used in the

fitting and detrending process; however, it is maintained in the final result. The power spectrum is detrended and squared

(or raised to higher powers to reduce the number of modes passing our detrend; or decreased to 1 to pass more nodes). A

second fit using linear regression is performed, where the y-axis of the data is the detrended power spectrum and the x-

axis is simply an index variable. This y-intercept is the noise threshold. Locations where the detrended power spectrum

peaks are greater than the threshold pass the filter unchanged in value from the original spectrum. Adjacent peaks are

combined together, leaving only the maximum value (peak) of the group. The truncated components are normalized to

the minimum and maximum of the full spectrum, thereby conserving entropy lost during peak pruning. This process is

repeated until convergence and avoids the matrix conditioning requirements associated with Principal Component

Analysis (PCA) The full process is described graphically in Figure 2.

Now that an eigenmode model exists, an important excursion can be made. If such simulation models are derived from

I/O data sets representing 2 different simulations to be validated or a simulation and measured data to be calibrated, both

can be accomplished by calculating Equivalence and Consistency between the structure of the 2 eigenmodels and their

coefficient values.

Fig. 2. Modified Wiener filtering process for identifying values above noise floor in power spectrum. Left graph illustrates log-log

power spectrum with linear regression fit used for detrending. Center graph shows the detrended power spectrum (order=2) and its fit.

Right graph shows final peaks retained (zero order maintained at original value throughout process and removed only for graphing).

TF models are now generated for both simulation and test flight or measured data, and their eigenfunction

representations (if chosen) determined. The eigenfunction TF models are compared for Equivalence and Consistency to

characterize the similarities and differences between the simulation and test flight data, with eigenfunction mode

locations and amplitudes compared to determine how alike or different the two models are. Equivalence is determined by

calculating the correlation between the magnitude of each of the TF models, and given by

( )( )( ) ( )∑

= −−

−−=

N

jjj

jj

Model2Model2Model1Model1

Model2Model2Model1Model1eEquivalenc

122

100 (18)

where Model1 and Model2 are the magnitudes of the TF models for the simulation and measured data given by

22

jjj BAModel1 += (19)

and Model1 and Model2 in (18) are the means of the result of (19).

Consistency between Model1 and Model2 is calculated by first subtracting the mean from each frequency vector exactly

as shown in (18). Next, each vector is multiplied by a Mask of zeros and ones, with ones in the mask vector at the

location of an eigenfunction mode, and a zero at all other points. This mask converts the full model into the

eigenfunction model, and is given by

≠=

=0Model11

0Model10Mask1

j

j

j (20)

≠=

=0Model21

0Model20Mask2

j

j

j

Mean subtraction and mask multiplication are given in equation form as

jjjj Mask2Mask1Model1Model1Model1 )( −=

jjjj Mask2Mask1Model2Model2Model2 )( −= (21)

Next, each frequency terms is converted into a Z-score using the standard deviation of the 2 (Model1 and Model2)

frequency terms at the jth location from (21) appended with a zero to make a total of three points, resulting in

j

j

j

Model1Model1

σ=

j

j

j

Model2Model2

σ= (22)

Mask2Mask1

))Norm(d)(1)(Mask22(Mask1

yConsistenc

N

1j

jjj

o

∑=

−= jjj Model1Model2d −= (23)

where Norm in (23) returns a normal probability on a distribution of mean zero and standard deviation of one, and the

argument dj for the probability calculation is given as shown. Figure 3 shows Equivalency and Consistency analyses. The

number of inputs for the first data set is 10 and for the second set 4. Three different scenarios are given. In the 1st,

input/output variable examples from two different Monte Carlo runs of the same simulation are shown; in the 2nd two

different fidelity versions of the same simulation; and in the 3rd the same simulation is running unrelated scenarios.

The Eigenmode Transfer Function (TF) is formed from lambda terms

=))((

))(()(

InputFFT

OutputFFTTF

λλλ (24)

))))))))),,((((((((( 21 kjin xxxybybybyf K = ),,,( 21 kf λλλ K (25)

where the KG polynomial model in (12) is replaced with an equivalent eigenfunction Data Model. If derivative variable

data is modeled with (15), it is simple to integrate analytically forming an integral equation. The integral of (15) is

∑∫ ∑==

−=+n

j

j

j

j

j

j

jn

j

jjjj xB

xA

dx xBxA11

cossin)sincos( ωω

ωω

ωω (26)

while the derivative of (15) is simply

0cossin1

=+−−∑=

n

j

jjjjjj xBxAdx

dy ωωωω (27)

0 5 100

2

j

p

0 10 20 30 400

500

1000

1500

0 2 40

2

jj

5.33 10.67 160

50

100

1

1

Equivalency = 100%Consistency = 100%

Input 1 (10 inputs)

Input 2 (4 inputs)

Outputs 1&2 Common Eigenfunction Modes

0 5 100

2

j


Input 1 (10 inputs)

Input 2 (4 inputs)


0 2 40

2

jj

0 10 20 30 40500

0

500

1000

1500

5.33 10.67 160

20

40

0 5 100

2

j


Input 1 (10 inputs)

Input 2 (4 inputs)

Outputs 1&2 Com m on Eigenfunction Modes

0 2 40

2

jj

0 10 20 30 401000

0

1000

2000

3000

5.33 10.67 161

0

1

Fig. 3. Comparison of Equivalency and Consistency.

Because the Fourier series is orthogonal, extraction of only dominant terms to form the differential equation model does

not destroy the underlying shape of the variable value curve. However, this is not the case in the time domain, where the

removal of a single power term of a polynomial gives a new polynomial, thereby completely changing the shape of the

function . Hence, simulation Data Modeling is performed in the frequency domain.

A transfer function model was derived using the input variable vector elements in (4), resulting in E coefficients of the

λTF.

)(

)(

aFFT

bFFTTF =λ (28)

E

0.666

0.692−

0.726−

0.196

38.213

0.196

0.726−

0.692−

0

1.654

0.647

0.946−

0

0.946

0.647−

1.654−

0

1

2

3

4

5

6

7

=

E is the matrix given in (28). The first column contains the real part of each frequency term, the second column the

imaginary part including its conjugate, and frequency index in column 3. This is used with (29) to create the derivative of

the TF given by the form in (29). Substituting (28) into (29) and removing the conjugate terms results in (30) (for

simplicity of display; they must be replaced to use the expression with an inverse FFT).

dTF

dx

4π

8 81

nterms 1−

j

Ej 0,− sin 2π E

j 2,i

8⋅

⋅ Ej 1, cos 2π E

j 2,i

8⋅

+

Ej 2,⋅

∑

=

⋅− 0

(29)

which is simply

(dTF/dx) – ((0.77 sin(0.7854 x)) + (1.61 sin(1.5708 x)) + (-0.65 sin(2.3562 x))+ (-152.85sin(3.1416x))

+(1.84 cos(0.7854 x)) + (1.44 cos(1.5708 x)) + (-3.15 cos(2.3562 x)))=0 (30)

Next, the simulation equivalent Data Model is used for Critical and Sensitive Parameter analysis.

3. CRITICAL/SENSITIVE/KEY PARAMETER IDENTIFICATION

• Critical parameters are simulation input variable vector elements for which smallest change in value

causes the largest change in range of one or more simulation output variable vector elements.

• Sensitive parameters are simulation input variable vector elements for which smallest change in value

cause the largest change in output simulation variable variance across all output variable vector

elements, regardless of their range.

• Key parameters are the intersection of the truncated critical and sensitive parameter lists.

This unique method for critical, sensitive, and key parameter estimation can mitigate the need for Design of Experiments

(DOE) with large numbers of (*.exe) inputs. This is only possible with analytical models. Variations of variables are

applied to the input model and changes in output variables measured when convolved with the simulation model,

represented in frequency space by the transfer function. This variation is always 2 way, regardless of number of

variables. The benefit of variable variation in frequency space is that the nonlinear coupling between variables in the

simulation is maintained without the explicit need to account for it using statistical methods, because modifying one

frequency space mode of the input model affects all input and output values simultaneously. Variable variation is not

applied to the transfer function, because any change in the modes of this function is a change in the system model, rather

than variations of the system model. This is consistent with control theory, which indicates that the input z transformed

denominator of the system transfer function defines the simulation characteristic polynomial, whose roots on the root-

locus plot indicate system stability, and hence parameter sensitivity.

One can use eigenmodels or the full system model TF. For this debris model, the TF consists of a total of 8 modes as

shown in (4) and (28), and is short enough to not require eigenmodeling. The TF for the baseline (example 67) is

)75.0cos(20.)5.0sin(65.0)5.0cos(73.)25.0sin(65.1)25.0cos(69.66.0 iiiiiTF πππππ ++−+−=5.1sin(65.0)5.1cos(73.)25.1sin(95.0)25.1cos(2.)cos(2.38)75.0sin(95.0 iiiiii )ππππππ −−+++−

)2sin(65.1)2cos(69. ii ππ −− (31)

The input model is equal in length and terms to the TF, but is changed into a notch or single band-pass filter applied to

the TF. This is done by zeroing all terms in the input model except the zero order, a single mode using 2 way variation,

and the complex conjugate of the term (3 non-zero terms total). For even number of points, the final mode before the

complex conjugate is real only and is also included and held constant along with the zero order. For odd number of

points, this mode does not exist. The real and imaginary values for the single mode are allowed to vary from –2 times to

+2 times their original mode value when only one example is available. When multiple examples are available, the initial

boundaries are set using the global minimum and maximum for each mode calculated from all the cases, and is modified

so that the maximum distance from current case to boundary is maintained on both sides of the current case. This ensures

that the current case is centered for all modes in the map. This variation establishes minimum and maximum margins for

constructing a parameter map for further analysis, and ensures that the original mode values are recoverable from the

center of each map. These margins are only a starting point for the analysis. The parameter maps must be observed to

determine if all of the structure in the parameter space is being observed. If not, the automatic scale is increased by the

analyst until no new structure is being introduced into the map at the boundaries, or until identified boundary conditions

yield invalid equivalent inputs.

Also, when multiple cases are available, the magnitude of the dot product between the real and imaginary components of

the modes is calculated. The dot products are then sorted and the example corresponding to the highest dot product

extracted as the baseline example from which the TF models will be built. For the debris model, this was example 67.

At each grid location, change in input and output variable values from baseline (due to single mode from input causing

changes in model output values in response) are obtained by summing the difference between the varied and original

input and variable output values given by

∑−

=

−=1

0

N

i

ii BaseInNewIndiffIn ∑−

=

−=1

0

N

i

ii BaseOutNewOutdiffOut (32)

These values are rollups across all 8 input and output variable vector indexes for the current complex point. Once these

changes are calculated, the ratio of the maximum output variable change to minimum input variable change is calculated

diffIn

diffOut

eInputChang

geOutputChan= (33)

Once the parameter map (µ map) is generated for the first mode, the process is repeated by allowing the next single

mode of the input model and its complex conjugate to change values. This is done for modes (in our example 3 total) by

excluding the final real valued only mode and the zero order. As before, the value of the real and imaginary components

of the current mode are varied from either –2 times to +2 times their original mode value or between the boundaries set

from the global minimum and maximum of the available cases or those increased by the analyst. These maps are

generated in equal size increments, enabling them to be combined independent of scale in their x and y axes. Figure 4

(left) shows the result of multiplying the magnitude of the logarithm of each of the 3 single mode based independent

maps created processing the baseline example from the debris model.

∏=

=3

1

,, )log(i

jiji MapFinalMap (34)

Real Mode 1

Mode 2

Mode 3

Mod

e 1

Mo

de

2M

od

e 3

Imag

Log(∑−=

−= 1

0

N

i

ii BaseOutNewOutLog(diffOut) )

-660.5

-265.8

-300.7

608.9

81.6

348.5

-467.5

-404.5

-564.5

711.9

235.3

437.5

-263.8

-157.2

-97.8

122.2

-84.6

-63.5

2 1 43

5

Real Mode 1

Mode 2

Mode 3

Mod

e 1

Mo

de

2M

od

e 3

Imag

Log(∑−=

−= 1

0

N

i


-660.5

-265.8

-300.7

608.9

81.6

348.5

-467.5

-404.5

-564.5

711.9

235.3

437.5

-263.8

-157.2

-97.8

122.2

-84.6

-63.5

Real Mode 1

Mode 2

Mode 3

Mod

e 1

Mo

de

2M

od

e 3

Imag

Log(∑−=

−= 1

0

N

i


-660.5

-265.8

-300.7

608.9

81.6

348.5

-467.5

-404.5

-564.5

711.9

235.3

437.5

-263.8

-157.2

-97.8

122.2

-84.6

-63.5

Imag

Log(∑−=

−= 1

0

N

i


-660.5

-265.8

-300.7

608.9

81.6

348.5

-467.5

-404.5

-564.5

711.9

235.3

437.5

-263.8

-157.2

-97.8

122.2

-84.6

-63.5

2 1 43

5

Log(∑−

=

−=1

0

N

i

ii BaseVarNewVarLog(diffOut) )

Real Mode 1

Imag

-660.5

-265.8

-300.7

608.9

81.6

348.5

-467.5

-404.5

-564.5

711.9

235.3

437.5

-501.8

-222.4

-219.5

85.3

-104.6

-94.8

Mode 2

Mode 3

Mode

1

Mo

de

2M

od

e 3

1

2

3

4

5

Log(∑−

=

−=1

0

N

i


Real Mode 1

Imag

-660.5

-265.8

-300.7

608.9

81.6

348.5

-467.5

-404.5

-564.5

711.9

235.3

437.5

-501.8

-222.4

-219.5

85.3

-104.6

-94.8

Mode 2

Mode 3

Mode

1

Mo

de

2M

od

e 3

Log(∑−

=

−=1

0

N

i


Real Mode 1

Imag

-660.5

-265.8

-300.7

608.9

81.6

348.5

-467.5

-404.5

-564.5

711.9

235.3

437.5

-501.8

-222.4

-219.5

85.3

-104.6

-94.8

Mode 2

Mode 3

Mode

1

Mo

de

2M

od

e 3

1

2

3

4

5

Fig. 4. Critical parameter location identification using Maximum Output Change over Minimum Input Change (34) (left), and

sensitive parameter location identification using variance instead of difference (right). White dot in center is baseline, lines mark the

maximum valued red dot, with 5 maximum cases numbered. Coordinates are marked for each mode with mode 1 closest to the graph.

For sensitive parameter identification, the approach is the same as for critical parameters, except that the difference in

standard deviation of the output to baseline and the standard deviation of the input are used instead of difference in input

and output value directly. This is shown in (35) as

)( ii BaseInNewInStdDevdiffIn −= )()( BaseOutStdDevNewOutStdDevdiffOut −= (35)

and results in the map on the right in Figure 4 by again multiplying the log magnitude of each of the 3 single mode based

maps. For each of these maps, the maximum value and its corresponding location (x,y) or (real, imaginary) in each is

determined. This maximum point is the location of the maximum output change that corresponds to the minimum input

change (34), and defines the critical parameter test case (left) and sensitive parameter test case (right).

Once the maximum grid location is found, a new full mode input model is constructed using the amplitudes

corresponding to each mode of fixed grid location of maximum value. All terms are recovered using the minimum and

maximum range value for each map stacked to yield the ones in Figure 4 and knowledge of the maximum location. This

allows for direct calculation of the values. The zero order and final real only value mode are copied over directly from

the original input model. The identified varied input model is inverse FFT to return it from frequency space to its original

space.

Critical, sensitive, and key parameter analysis is applied to both single and multiple example case from the debris model,

varying each mode in the input model one at a time and generating a ranked critical, sensitive, and key parameter lists of

the debris model inputs. Table 1 shows the results obtained using example 67 (found to have the maximum dot product

as described previously in this work) as the baseline.

Table 1. Critical, Sensitive, and Key parameter ranking of inputs.

Rank

Critical

(Single Max

Output Value)

Effect

Sensitive

(Max Output

Variance Change)

Effect

Key

(Max Magnitude

Output Change)

Effect

1 Interceptor Target

Mass Ratio 25.9 x Target Area 295.8 x Time to Impact 18.6 x

2 Relative Signature

Value 6.6 x Wavelength 206.5 x Sensor Aperture 13.8 x

3 Time to Impact 6.3 x Target Mass 173.0 x Relative Signature

Value 12.2 x

4 Sensor Aperture 5.3 x Field of View 135.3 x Wavelength 11.8 x

5 Wavelength 1.5 x Time to Impact 76.5 x Target Area 11.7 x

6 Field of View 1.3 x Sensor Aperture 67.1 x Field of View 6.8 x

7 Target Area 1.0 x Relative Signature

Value 47.5 x Target Mass 6.7 x

8 Target Mass 1.0 x Interceptor Target

Mass Ratio 1.0 x Interceptor Target

Mass Ratio 1.0 x

For identifying the ranked critical parameter list, the ranking is done relative to the maximum output variable change.

Therefore, only knowledge of the input change is required off of the baseline vector (which is assumed to be zero). These

inputs are sorted into ascending order (from most critical to least critical). The effect is calculated by taking the

reciprocal of the ranking divided by the reciprocal of the least critical ranking. For identifying the ranked sensitive

parameter list, each input vector element is tested one at a time by zeroing the other input variables. The variance of the

resulting output is calculated, and the inputs ranked by sorting the variances into descending order. The effect is

calculated by dividing the ranking of each by the ranking of the least sensitive parameter. For identifying the ranked key

parameter list, the critical effect is multiplied by the sensitive effect for each variable, and then sorted into descending

order. A MathCAD document of this technique is provided in the Appendix, along with a short description.

When designing an experiment, first choose the number of variables that can be tested. Next, specify the total number of

tests that can be done. As an example, specify 2 variables to test per case and only 5 test cases. Therefore, the top 2 key

parameters from the analysis are the experiment test case parameters. To identify calibration or anchor test cases, map

values and corresponding locations are raster scanned out and sorted by value into descending order given by

)()( ,ijjNi MapSortRasterOutSort =+ N=Number of Columns (36)

These are the ranked validation cases (traditionally representing Latin hypercube axes for DOE). These locations are

used to reconstruct the full eigenmodes for the test cases, which are then inverse Fourier transformed as previously

described. The list is processed until 5 valid anchor test cases of unique key parameters are identified. This is done by

performing the full critical/sensitive/key parameter analysis above for each case in descending order. Figure 4 identifies

by number 1-5 on each map the 5 anchor test cases.

REFERENCES

1. Jones, R. B., Symbolic Simulation Methods for Industrial Formal Verification, Boston, MA: Kluwer Academic

Publications, 2002.

2. Jaenisch, H.M., Algorithms for Autonomous Unmanned Deep Space Probes, D.Sc. Thesis, James Cook Univ., 2006.

3. Rosko, Joseph S., Digital Simulation of Physical Systems, Reading, MA: Addison-Wesley, 1972.

4. Barnett, S. & Cameron, R.G., Introduction to Mathematical Control Theory, 2nd ed., Oxford: Clarendon, 1993.

5. Doyle, F.J., Pearson, R.K., Ogunnaike, B. A., Identification and Control Using Volterra Models, London: Springer-

Verlag, 2002.

6. Jaenisch, H.M., Handley, J.W,. Hicklen, M.L., “Data Model Predictive Control as a New Mathematical Framework

for Simulation and VV&A”, SPIE Defense and Security Symposium, Orlando, FL, April 2006.

APPENDIX (Critical/Sensitive/Key Parameter Estimation (MathCAD 12))

Description

The MathCAD 12 document contained in the Appendix takes an n input and 2 output data file. This file is read into the

variable z at the top of the first page of the document. Variables are in columns in the file, with the 2 output columns

appended as the final 2 columns. Monte Carlos (multiple examples) are accommodated in the input file, with each on an

individual line. This program can also accommodate simulation runs where the inputs do not vary, but the outputs vary

due to Monte Carlos. This program analyzes the dot product from the examples and chooses one as the baseline (ic),

processes that example using the single mode technique described in this work, and returns rankings both in the ratio

given in (33) and in numerical score (1, 2, 3, …). In order to increase the size of the parameter map space, multiply both

occurrences of jval on the left hand side of the equation by a number greater than 1 (for our examples, this was 6 and 24).

This document is currently set up to extract the maximum point and perform the full critical/sensitive/key parameter

identification described in this paper. In order to extract multiple key parameters per test case (cases 1 to 5 as shown in

Figure 4), an outer loop can be supplied around this document. Otherwise, it must be run manually once for each of the

5 numbered locations in Figure 4.

Automatic Differential Equation Data Modeling for UAV Situational Awareness

Holger Jaenisch, Ph.D. and James Handley

Sparta, Inc.

4901 Corporate Drive, Suite 102

Huntsville, AL 35805

email: [email protected]

KEYWORDS

Data Modeling, UAV, differential equations, situational

awareness, parameter estimation

ABSTRACT

This paper demonstrates the ability to

autonomously make assessments of the physical

characteristics of a monitored object only from

knowledge of its motion in the x-y plane.

Traditionally, the process of state estimation from a

trajectory estimate has been in the realm of the Kalman

filter. Unfortunately, the Kalman filter and Extended

Kalman filter require a priori defined models of the

dynamics of the object being observed. This entails the

existence of a physics based partial differential equation

that describes the equations of motion in the filter to

enable predictions of state for comparison to be made.

INTRODUCTION

With the advent of Data Modeling theory (Jaenisch

et al. 2003d), it becomes possible to introduce a

significant amount of intelligence into very small

footprint UAVs. This results in the ability for a UAV to

report salient information across an ad hoc network

rather than overwhelming available bandwidth with

unprocessed data.

As an alternative to reporting full up data streams, it

is possible to create real-time Data Models

(mathematical equations) of sensor measurements by

transmitting only sparsely sub-sampled sensor

values, and reconstructing via Data Modeling fractal

interpolation the full length high fidelity time series on

demand at the receiver. This algorithm and results have

been published in previous work given in the

References (Jaenisch and Handley 2003).

It is also possible to monitor the status of onboard

sensors to determine calibration and operational

characteristics using the Data Modeling theory of

change detection for situational awareness (Jaenisch

et al. 2003c). This algorithm and approach has also

been published in previous work given in the

References (Jaenisch et al. 2003b).

Data Modeling has also been used to compress

images via sub-sampling to a thumbnail size, and the

thumbnail then readily transmitted and reconstructed on

the receive end. Applications include IDM enabled

modems transmitting streaming live video at slow 56-

kilobyte rates.

The use of Data Driven Differential Equation Data

Modeling enables a differential equation to be

constructed from measured data alone enabling physics

based aerodynamics modeling with limited

computational resources. This enables estimates of the

object’s mass, equivalent area, and probable control

feedback loop parameters to be determined in a least

mean squared error (LMSE) fashion.

DATA MODELING THEORY

The Data Model is the smallest amount of

information required to functionally model empirical

dynamics. This model can take the form of a group of

real numbers, a single equation, or a network of

equations. The general philosophy of Data Modeling is

Data “Blank” Modeling, where the “Blank” takes on

the descriptive term Sequence, Relations, or Change.

The “Blank” depends on the type of construct modeled.

Data Sequence Modeling uses fractal methods to

capture the self-similarity and scaling dynamics of

sequences of data within a data series. Data Relations

Modeling are equations that capture the nonlinear

relationships between different data series. Data

Relations Models are tuned to be sensitive to changes

of input conditions enabling system change detection.

This is called Data Change Modeling. Other constructs

may be Data Modeled as well; this list is illustrative

rather than exhaustive.

DATA SEQUENCE MODELING

Data sets (time series, image, or other vectors of

information) can be reconstructed from a sparse sub-

sampling (Jaenisch and Handley 2003). Sparse sub-

sampling is achieved by naive decimation, and the

sparsely sub-sampled points are the Data Model of the

original data set. Because they are demonstrated to be

the support vector

[ 01),( =−+ bwxy iiiα ] (1)

of the set, the sample size need never exceed 10% of

the original number of points, akin to reconstructing a

musical score by saving only every 10th

note. Data

Sequence Modeling is based on a unique fractal

operator defined as

{ } 10111 ))(()(

−−+−

−−

−−

≡ iNnnnn

if xN

yyyy

N

yyyD

)()( 1101111 −+−−−+ −−+−+ nnninn yyyyyyy (2)

DATA RELATIONS MODELING

Data Relations Modeling is the process of finding a

multivariate expression that maps given finite sample

values of the independent variables into associated

values of the dependent values of the process. This

process involves approximating the Kolmogorov-

Gabor polynomial

L++++= ∑∑∑∑∑∑i j k

kjiijk

i j

jiij

i


using nested functions of the form

))))))))).((((((((,()( 21 tbxtbxbxtftx nK= (4)

Data Modeling differs from conventional methods such

as linear, quadratic, and polynomial regression

techniques that only solve for the numerical coefficients

for a user specified function. Data Models discover the

form of the best function adaptively and autonomously.

The Data Modeling process of discovering automatic

functions is an evolutionary/genetic directed

approach where the set elements consist entirely of

simple polynomial building blocks. Data Modeling

also differs from genetic algorithms and symbolic

regression because it incorporates fractal

interpolation functions for achieving mathematical

modeling of complex multivariable processes (Jaenisch

et al. 2003a).

IMAGE COMPRESSION

In Fig. 1, the image on the left is that collected and

telemetered by the UAV. If the whole image was only

sampled at every 10th point (10:1 compression), the

thumbnail superimposed at the bottom center between

the two images is all that would be telemetered down.

This thumbnail was inflated on the receive side back up

to its original size using a fractal operator as shown by

the image on the right of Fig. 1. It has sufficient

resolution to be considered “lossless” by image feature

extraction and decision algorithms.

CALIBRATION ON DEMAND

Change detection type Data Models have been

created and reported in the literature for many

applications including network monitoring for

information based attacks. Unique to these change

detection Data Models is the ability to discern nominal

and abnormal behavior without ever having been

exposed to the latter, thereby achieving external

situation al awareness without a priori assumptions.

A particularly exciting example was reported in

recent literature showing where only 400 examples of

nominal information flow were needed out of a

possible 4 million training records to accurately

discriminate between nominal and 14 different types of

attack behavior. Yet not a single example of attack

was used for training. This unique ability to detect that

which is different makes change detection models

suitable for many roles in the UAV mission (Jaenisch et

al. 2003e).

DYNAMIC DIFFERENTIAL EQUATION

GENERATION ALGORITHM

One future requirement for UAV systems will be to

autonomously observe a moving target and form an

opinion as to its type, size, shape, likely trajectory, and

intent. To enable such an assessment, a critical

intelligence gathering and processing capability is the

system identification and parameter estimation

process. To achieve such estimates autonomously

requires the formulation of the dynamic equations of

motion governing the observed target.

Distinguishing an aircraft from a landcraft is done

using a 100 mph threshold, and landcraft can be further

divided into land and water based craft using a

threshold of 40 mph. Aircraft can be divided into

propeller (100-150 mph), turboprop (150-250 mph), jet

(200-500 mph), and missile (500+ mph).

An object’s maneuvering is characterized by

calculating the change in slope of the trajectory

anchored at the initial point. This change is

characterized in terms of higher order moments

i

N

j

i

j

Ni

xxM µ

σ−

−= ∑

=1

1 (5)

Once the object type has been determined, various

subsets of a priori physics based derivative forcing

functions are available for fitting. If this type of bulk

filter determination is not available, then all classes of

forcing functions are used. Because each class of

forcing function contains many of the same types of

forcing functions, it is not necessary to determine the

class to a great degree of accuracy.

2r

GMaGravity =

m

pAvCa D

Drag

25.0= (7)

Fig. 2 shows the measured example of a

maneuvering object trajectory. The true trajectory is

biased and distorted by the sensor measurement process

and therefore is not known. A Kalman filter cannot be

applied because this object is maneuvering and the

equations of motion cannot be known in advance. No a

priori model exists for the sensor measurement bias

and noise. This leads to a double confounded problem

of blind system identification and parameter

estimation in the form of modeling and predicting the

true trajectory and modeling and predicting the sensor

noise and bias.

Using the filtering criteria above, it was determined

that the object whose trajectory is given in Fig. 2 is

most likely a maneuvering missile, which activated the

following group of forcing functions as candidates

1

2

1

2

1 KvdA ρ−= (Axial drag)

21

2

2 KvdB ρ= (Cross force due to

cross velocity)

31

3

2 KdC ωρ−= (Magnus cross force

due to cross velocity)

41

4

2 KdD ωρ= (Magnus cross force

due to cross spin)

51

3

2 KvdE ρ−= (Cross force due to

cross spin)

The idea is to estimate and model the second

derivative of the moving object. This is accomplished

by autonomously selecting related physics based

derivative forcing functions. The control parameters

from these forcing functions are estimated using a

genetic algorithm to gain close initial guesses for final

optimization using Levenberg-Marquardt nonlinear

least squares fitting.

611

4

81 KvdP ωρπ−= (Spin-decelerating

moment)

71

4

2 KdQ ωρ= (Magnus cross torque

due to cross velocity)

81

3

2 KvdR ρ−= (Cross torque due to

cross velocity)

91

4

2 KvdS ρ= (Cross torque due to

cross spin)

The genetic algorithm is a directed random search

parameter estimator defined as follows: 1) Select a gene

pool size of nn, where n is the number of coefficients in

the forcing function; 2) Populate pool with random

values (apply prior probabilities if known); 3) Rank

according to selected cost function (determine points

falling outside ¼ standard deviation boundary of

objective function, and sum their Euclidean distances);

4) Select frequency of occurrence of mutation and its

form (negation or random perturbation); 5) Select

number of offspring to propagate forward as nn/2; 6)

Populate offspring pool with top ½ of current pool and

spliced offspring (splicing selects a random location in

the coefficient array and splices the left side of one

parent and the right side of another into a new

offspring); 7) Pool is reranked as in 3); 8) Stopping

criteria is all points are within ¼ standard deviation

boundary or user specified maximum number of

iterations achieved. The best candidate solution is then

passed through Levenberg-Marquardt for a final

optimization.

101

5

2 KdT ωρ−= (Magnus cross torque

due to cross spin)

2r

GMmFg = (Gravity force) (6)

As many as 72 other smaller perturbing forces

describing body shape, fin orientation, fin shape, fin

placement, as well as other stability derivatives could

also be included in the list; however, the above 11 are

the major contributors. The desired form of the final second order

differential equation is Because the local body coordinate system is

unknown and changes orientation with respect to a

fixed earth centered inertial coordinate system, only the

magnitudes of the forcing functions was used.

Performing correlation on the 11 forces in (6) ranked

the axial drag force and gravity were the most

significant and were modeled as

),,,,,,,( 112

2

mn uuccdt

dyyf

dt

ydKK= (8)

where f is a continuous function,

dt

dy and 2

2

dt

yd are the

first and second derivatives of y with respect to time, c1

to cn are constants, and u1 to um are forcing functions.

Step B – Physics based forcing function derivatives

A genetic algorithm is used as a part of Data

Modeling to determine the optimal mass, drag

coefficient, and effective area of the object. For the

trajectory in Fig. 2, the genetic algorithm predicted:

In order to ensure that smooth data curves and

derivatives were used in the differential equation

discovery process, data sets were sub-sampled and

Turlington polynomials and derivatives of the form

2037.0 mA = (Equivalent body area) "5.8217.0 == mD (Diameter equivalent body)

( )

∑

+++=

−m

d

xx

mm

m

DBxAxT1

10 101log)( (9) 69.0=DC

lbskgM 7318.33 == (12)

( )

( )∑ −

−

+

+=m

d

xx

d

xx

m

m

m

m

m

m

d

DB

dx

dT

1

101

10 (10) As discussed previously, two physics based forcing

functions were used and resulted in a Data Model of the

form

).5.0

,,(2

2

2

2

2

m

pAvC

r

GMtf

dt

yd D=

(13) were used. Turlington polynomials are continuous

orthogonal functions that are built in a piecewise

fashion but are everywhere differentiable.

Orthogonality enables construction of the polynomial

and its derivatives in real-time as data is being

received by a sensor such as a UAV. Data Modeling

theory has shown previously that sub-sampling of data

processes has a negligible impact on the results, and

was therefore used to decrease the number of

calculations and thereby speed up the process.

Step C – Thrust derivative

A Data Model for thrust of the form

),(),(2

2

3

2

2

T

T

dt

ydtf

m

Ftf

dt

yd

==

(14)

The modeling steps in the process used to determine

the differential equation are labeled as:

was built using a Turlington polynomial function

derived from a nominal thrust acceleration profile based

on the STAR, LOKI, Viper IIIA class targets. Fig. 2

shows the acceleration profile used and a comparison of

this Data Model to the sub-sampled Turlington 2nd

derivative curve. If the thrust and mass profiles are

known, they can be used to calculate the

acceleration due to thrust directly and this then in

turn used in the Data Model.

1. Target physics model

a. Position and velocity derivative

b. Physics based forcing function

derivatives

c. Thrust derivative

d. Lumped parameter stability derivatives

2. Sensor bias model

a. Position bias

b. Orientation error Step D – Lumped parameter derivatives

c. Measurement error

d. Boresight error Lumped parameters represent fin number,

placement, aspect, and other perturbing forces lumped

into a single group and modeled using eigenfunctions.

Generation of these eigenfunctions is done with a novel

method published previously using a modified Weiner

filtering technique on the Turlington polynomial of the

2nd derivative of the data in Fig. 6. This results in a Data

Model of the form

DERIVING THE TARGET PHYSICS MODEL

Step A – Position and velocity derivative

The 1200-point data set in Fig. 6 was 10% sub-

sampled to 120 points and Turlington polynomials

constructed of the sub-sampled trajectory, and its 1st

and 2nd derivatives. A Data Model was then generated.

).,,,( 321

4

2

2

λλλtfdt

yd=

(15)

),,(

1

2

2

dt

dyytf

dt

yd=

(11)

Target physics model

The derivatives from (11), (13), (14), and (15) are

treated as metavariables and fused into a Data Model

of the form

).,,,,(

4

2

2

3

2

2

2

2

2

1

2

2

5

2

2

=

dt

yd

dt

yd

dt

yd

dt

ydtf

dt

yd (16)

DERIVING THE SENSOR BIAS MODEL

Step A – Position bias

First, (16) is double integrated (using either a 4th

order Runge Kutta method or direct numerical

integration) at the proper step size to return the same

number of points (1200) as the original. The result is

compared with the original trajectory in Fig. 6 in order

to determine the sensor position bias, which is based on

the range ratio of the original and the target physics

model. For this data set, the bias (B) was 1.4 and when

applied to (16) results in

5

2

2

6

2

2

=

dt

ydB

dt

yd (17)

Step B – Orientation error

Sensor orientation error is modeled as a first

order differential equation of the form

))))))(((((,()(

1 tyytfdt

ydn

i

λλ ∆∆=

∆

K (18)

where ∆y is the difference between (17) and the original

trajectory, and λι are the lumped parameter dominant

eigenfunctions of dt

yd )(∆. The results of (18) can be

fused directly into the Data Model in (17) as a velocity

change after the first integration, or (18) can be

integrated and added to the resultant trajectory

after double integration of (17).

Step C – Measurement error

Sensor measurement error is modeled as a first

order differential equation of the form

))))))(((((,()(

1 tyytfdt

ydn

i

λλ ∆∆=

∆

K (19)

where ∆y is the difference between (17) corrected for

orientation error and the original trajectory, and λι are

the lumped parameter dominant eigenfunctions of

dt

yd )(∆. The results of (19) can be fused directly into

the Data Model in (17) as a velocity change after the

first integration, or (19) can be integrated and added

to the resultant trajectory after double integration

of (17).

Step D – Boresight error

Boresight error is modeled as a first order

differential equation of the form

))))))(((((,()(

1 tyytfdt

ydn

i

λλ ∆∆=

∆

K (20)

where ∆y is the difference between (17) corrected for

orientation and measurement error and the original

trajectory, and λι are the lumped parameter dominant

eigenfunctions of dt

yd )(∆. The results of (20) can be

fused directly into the Data Model in (17) as a velocity

change after the first integration, or (20) can be

integrated and added to the resultant trajectory

after double integration of (17).

Results

The resultant Data Modeling derived differential

equation output is shown in Fig. 2 along with the

residual error and its histogram. The proof of concept

mode lies in the 8th

bin of the histogram which

corresponds to a most frequently occurring error

bound of negative one (-1.0) to ½ (0.5) meters,

matching the original Cramer-Rao bound ambiguity.

REFERENCES

Jaenisch, H. and J. Handley, “Data Modeling of 1/f

noise sets”, Proceedings of SPIE Fluctuations and

Noise 2003, June 1-4, 2003. Santa Fe, NM.

Jaenisch, H., J. Handley, and L. Bonham, “Data

Modeling for Unmanned Vehicle Situational

Awareness,” Proceedings of the AIAA, 2003 Unmanned

Vehicle Conference, September 14-18, 2003. San

Diego, CA.

Jaenisch, H., J. Handley, and L. Bonham, “Data

Modeling Change Detection of Inventory Flow and

Instrument Calibration”, Proceedings of SOLE 2003,

August 10-14, 2003. Huntsville, AL.

Jaenisch, H., J. Handley, L. Bonham, “Enabling

Calibration on Demand for Situational Awareness”,

Army Aviation Association of America (AAAA)

Tennessee Valley Chapter Huntsville Alabama, Feb 12,

2003.

Jaenisch, H., J. Handley, and J. Faucheux, “Data Driven

Differential Equation Modeling of fBm processes”,

Proceedings of SPIE, August 3-8, 2003. San Diego,

CA.

Jaenisch, H., J. Handley, and J. Faucheux, “Data

Modeling of network dynamics”, Proceedings of SPIE,

August 3-8, 2003. San Diego, CA.

FIGURES

Fig 1. Complex Process Applied to Sub-Sampled Data Yields Equivalent Result.

Residual

Error

Original and

Model Overlay

0.0

-6151.4

0.0 1200.0

5.16

-13.18

Thrust

0.0 1200.0

Gravity

Drag

Lumped Stability Derivatives

0.0 1200.0

0.0 1200.0

Boresight

Measurement

Orientation

Fig. 2. Measured trajectory data compared with model overlay (top left), physics based and lumped stability

derivative forcing functions (top right), user input thrust acceleration profile (bottom left), and orientation,

measurement, and boresight error derivatives (bottom right).

Data Model Predictive Control as a New Mathematical Framework for

Simulation and VV&A

Holger M. Jaenisch*a,b

, James W. Handleya, and Michael L. Hicklen

b

aJames Cook University, Townsville QLD 4811, Australia

bdtech Systems Inc., P.O. Box 18924, Huntsville, AL 35804

ABSTRACT

This paper presents the mathematical theory and procedure for comparing two simulations analytically. The result is the

derivation of two equation models; one for each respective simulation. The derived models are analytically compared to

determine: equivalence, consistency, linearity, similarity, and degree of overlap. This yields a unique analytical tool for

comparing simulation versions or scenarios for VV&A. Methods as simple as regression can then be used to determine if

accreditation is maintained on new simulations or models. The derived analytical functions can themselves be

appropriately combined into an adaptive intelligent lookup table (LUT) equivalent model for real-time simulation

purposes.

Keywords: Formal Analysis, Data Modeling, VV&A, Equivalence, Consistency, Transfer Function Modeling, Automatic Proofing

1. FORMAL VV&A

Traditional validation, verification, and accreditation (VV&A) methods use exhaustive variable instantiating and testing.

This includes closing of shall statements and matching requirements to expected performance values. True validation and

verification follows a sequence of steps leading from requirements to source code inspection and analysis, and finally

collection of numerical data from the various subroutines and functions of operational code and comparing these

numbers to expected values. This is a very cumbersome time consuming process and can never be shown to be fully

completed. It remains a process of refining the V&V process and always leaves doubt as to the validity of the code for

new scenarios. This limits accreditation for intended uses only rather than global accreditation enabling excursions on a

validated simulation across all possible input scenarios.

Formal VV&A attempts to apply the tools of mathematics in the form of lambda calculus and formal logic to treat

programs as proof-able entities. The promise of being able to automatically solve for and prove the validity and

correctness of programs is an elusive but worthy goal. If we can demonstrate that programs are in fact equivalent to

mathematical equations, then transformation of the equations into other forms can be accomplished with simple algebraic

manipulation. Further, once these equations in the form of differential equations are derived from programs, then the

tools of formal mathematical proofing can in theory be applied to formally prove the correctness of the programs.

Towards this goal we present a novel numerical method for doing that which bypasses the lambda calculus and formal

logic in favor of dimensional and structural analysis of the derived equation models themselves, equivalence and

consistency become quantifiable values that have meaning and can formally be used to prove two different versions of a

program to be the same or different with a simple to understand number. Finally, the program source code itself no

longer needs to be evaluated. Rather, equations are derived that map the input into the resulting output. By comparing the

behavior of two or more independently derived equations, their similarity can again be formally evaluated and used for

VV&A. First, we will develop an informal proof of the equivalence of programs and equations.

EQUAL means equal over infinite domain, Equivalency is equal over a bounded domain. Consistency means

correlated linear changes in output between two models when driven by proportional similar perturbations.

Consistency quantifies implied causality. When both Equivalence and Consistency are 100%, the models are

congruent.

Given: Functions f1 (x,y,z) and f2 (x,y,z) and assuming Consistency and Equivalency Between f1 and f2 is proof-

able using methods of FORMAL ANALYSIS.



Let

y1 = sin(x) y2 = x – x3 / 3! + x5 / 5! – x7 / 7! (Truncated Taylor Series)

Consistency and Equivalency are proof-able over a bounded range, Therefore

1) y1 = sin(x) Is Equivalent to y2 = x – x3 / 3! + x5 / 5! – x7 / 7!

2) y = f(x) Is a Black Box Description of the Equations

3) y1 and y2 Are Equivalent to f1(x) and f2(x)

4) Changes of variables in f1(x) and f2(x) yield consistent change in output.

5) f(sin(x)) = (sin.exe) and f(Taylor(x)) = (Taylor.exe)

6) Since an arbitrary f(x,y,z) is transformed into Fourier series, FFT is an invertible mapping

A Program (*.EXE) is a black box f(x) which takes as input numerical value for variables and returns as output

numerical values. Since a causal mapping between these I/O exist the program is a function or LUT table

generator. Since a Formal FUNCTION is also a causal mapping between I/O and a LUT generator (*.EXE) and

f(x…) are equivalent. Since any function can be encoded as a program (*.EXE) then the function and program

are equivalent:

Given f1(x) ~ f(*.EXE) f2(x) ~ f(*.EXE)

Then f1(*.EXE) and f2(*.EXE) are generalized fA(*.EXE) and fB(*.EXE)

Therefore fA(*.EXE) = fA(x,y,z) fB(*.EXE) = fB(x,y,z)

Assuming System Inputs FAin = fAin(x,y,z) and Outputs FAout = fAout(x,y,z), Then

x1 = FFT(FAin) and y1 = FFT(FAout) yields the FORMAL TRANSFER FUNCTION MODEL

TF1=y1/x1 And TF2=y2/x2

From Which New Output is Generated Using New Input x1’ and x2’. Since Fourier series are convergent, and

can be converted into differential equation form, they are proof-able. QED

Analytical comparison of simulations is accomplished using the method of classical control theory transfer functions

(TFs) derived from input/output data from the two simulations. The methods analyze the form of the two independently

derived TF models by comparing their mathematical structure and the value of their coefficients. Equivalence and

Consistency between models is measured, and therefore between the simulations from which they are derived.

Specifically, for comparison of a simulation and its equivalent Data Model, we are interested in the structure and form of

the TF model constructed from the simulation compared with the TF model constructed from the Data Model. Since the

TF models are comprised of unique modes mathematically expressed as cosine and sine functions, they are readily

transformed into analytical derivative terms, and the number of unique modes can be reduced by performing

eigenfunction modeling using modified Wiener filtering.

2. DATA MODELING

2.1 Transfer function theory

Data Modeling1 with transfer functions can be used to model discrete input and output (I/O) systems. The transfer

function2-3 (TF) provides an algebraic representation of a linear, time-invariant filter in the frequency domain. Because

Data Modeling transfer function models occur in frequency space using complex variables, they are called analytical

representations. The frequency domain transfer function is determined using the fast Fourier transform (FFT). The

frequency space representation is separated into the real and imaginary components

∑==j

N

ij

iii yyFFTA )cos())(Re( 2

21 ππ ∑==

j

N

ij

iii yyFFTB )sin())(Im( 2

21 ππ (1)

where A are the real terms, B the imaginary terms, N the number of points, and both i and j are counters ranging from

zero to N-1. Each frequency term in (1) results in a cosine-sine pair in the analytical convergent Fourier series

∑=

+=n

j

jjjj xBxAxy1

sincos)( ωω (2)

The transfer function is defined as the frequency domain representation of the output divided by the frequency domain

representation of the input, given in equation form by

ionDeconvolutInputFFT

OutputFFT

zX

zYzH ===

)(

)(

)(

)()( (3)

where H(z) is the transfer function, Y(z) the frequency domain representation of the output, and X(z) the frequency

domain representation of the input. The division in (3) is a deconvolution, which removes the effects of the input data set

from the output data set. The resulting ratio of FFT(output) to FFT(input) is the TF model of the process. Once

generated, the TF model is used to transform (convolve) any input data set into its related output data set, given by

))()(

)(( NewInputFFT

InputFFT

OutputFFTInvFFTNewOutput = (4)

If an inverse Fast Fourier Transform (InvFFT) is performed on the TF model in (3), the result is the system response

function (SRF)

)sin()cos()(22

N

ij

j

j

N

ij

jii BATFInvFFTSRFππ +== ∑ (5)

which is in the time domain and used with spatial convolution of a new input. Conversely, in the frequency domain

convolution is a simple complex multiply of the TF with a new input as shown in (4). Using eigenfunction Data

Modeling or modified Weiner filtering, the number of frequency terms can be reduced, yielding a compact function of

dominant eigenmodes (λ) remaining after pruning. This is shown in


BAfxyN

ij

j

j

N

ij

jk ==

+=≅ ∑ )sin()cos(),,,()(22

21

ππ λλλλλ K (6)

2.2 Eigenfunction modeling

Eigenfunction extraction uses a variation of Wiener filtering to identify significant peaks in the dB power spectrum.

These peaks correspond to dominant sine and cosine terms occurring in the power spectrum. Once the FFT is generated,

the power spectrum of the data is calculated, which is the sum of the squares of the real and imaginary terms in (1), and

is then transformed into decibels (dB).


(log10)( 10

i

ii

P

PdBP = (7)


bmxy += (8)

where m is the slope of the line and b is the y-intercept. It should be noted that the zero order term is not used in the

fitting and detrending process; however, it is maintained in the final result. The power spectrum is detrended and squared

(or raised to higher powers to reduce the number of modes passing our detrend; or decreased to 1 to pass more nodes). A

second fit using linear regression is performed, where the y-axis of the data is the detrended power spectrum and the x-

axis is simply an index variable. This y-intercept is the noise threshold. Locations where the detrended power spectrum

peaks are greater than the threshold pass the filter unchanged in value from the original frequency space representation.

Adjacent peaks are combined together, leaving only the maximum value (peak) of the group. The truncated components

are multiplied by a scale factor to maintain equal power in the truncated and full models, thereby conserving entropy lost

during peak pruning. This process is repeated until convergence and avoids the matrix conditioning requirements

associated with Principal Component Analysis (PCA) The full process is shown graphically in Figure 1.

Fig. 1. Modified Wiener filtering process for identifying values above noise floor in power spectrum. Left graph illustrates log-log

power spectrum with linear regression fit used for detrending. Center graph shows the detrended power spectrum (order=2) and its fit.

Right graph shows final peaks retained (zero order maintained at original value throughout process and removed only for graphing).

2.3 Comparison metrics

TF models are now generated for both simulations and their eigenfunction representations determined (λTF). The two

models are compared using Equivalency, Consistency, Probability of Being Random, Probability of Being a Linear

System Model, Sensitivity, Relative Model Sensitivity, Correlation, Percentage of Variation in One Model Captured by

the Second., Root Sum Squared Difference Between Models, Variance Similarity Between Models, Number of Unique

Eigenmodes, Entropy, Model Sensitivity Ratio, Percent Overlap in Model Eigenmodes, and Analysis Confidence4,5.

The λTF models are compared for Equivalence and Consistency to characterize the similarity or degree of congruence

between simulation models. This is done by comparing the structure of the two equations. In frequency space, this is

simply comparing model amplitudes. Equivalence is determined by calculating the correlation between the magnitude of

each TF (PSD) given by

( )( )

( ) ( )∑∑

∑

==

=

−−

−−=

N

j

i

N

j

j

N

j

jj



eEquivalenc

1

2

1

2

1100 (9)

where Model1 and Model2 are the magnitudes of the TF (PSD) models for the simulation and measured data given by

22


and Model1 and 2Model in (10) are the respective means of the result of (10).

Consistency between Model1 and Model2 is calculated by first subtracting the mean from each PSD as shown in (9).

Next, each respective result is multiplied by a Mask of zeros and ones, with ones in the mask corresponding to the

locations of an eigenfunction modes, and zeros at all other points. This mask converts the full model into the


≠=

=0Model11

0Model10Mask1

j

j

j (11)

≠=

=0Model21

0Model20Mask2

j

j

j


jjjj MaskMaskModel1Model1Model 21)(1 −=

jjjj MaskMaskModelModel2Model 21)2(2 −= (12)

Next, each frequency term is converted into a Z-score using the standard deviation of the 2 (Model1 and Model2)


j

j

j

ModelodelM

σ1

1 =

j

j

j

ModelodelM

σ2

2 = (13)

21

))(1)(2)(1(21

MaskMask

dNormMaskMask

yConsistenc

N

j

jjj

o

∑=

−= jjj ModelModeld 12 −= (14)


argument dj for the probability calculation is given as shown. Equivalence and Consistency examples are displayed

below in Fig. 2.

0 5 100

2

j

p

0 10 20 30 400

500

1000

1500

0 2 40

2

jj

5.33 10.67 160

50

100

E q u i v a l e n c y = 1 0 0 %C o n s i s t e n c y = 1 0 0 %

I n p u t 1 ( 1 0 in p u t s )

In p u t 2 ( 4 in p u t s )

O u t p u t s 1 & 2 C o m m o n E ig e n f u n c t io n M o d e s

0 5 100

2

j

E q u i v a l e n c y = 5 0 %C o n s is t e n c y = 4 2 %

I n p u t 1 ( 1 0 in p u t s )

In p u t 2 ( 4 i n p u t s )

O u t p u t s 1 & 2 C o m m o n E ig e n f u n c t i o n M o d e s

0 2 40

2

jj

0 10 20 30 40500

0

500

1000

1500

5.33 10.67 160

20

40

0 5 100

2

j

E q u i v a l e n c y = 0 %C o n s is t e n c y = 0 %

I n p u t 1 ( 1 0 i n p u t s )

I n p u t 2 ( 4 i n p u t s )

O u t p u t s 1 & 2 C o m m o n E i g e n f u n c t i o n M o d e s

0 2 40

2

jj

0 10 20 30 401000

0

1000

2000

3000

5.33 10.67 161

0

1

E X E 1

E X E 2

E X E 1

E X E 2

E X E 1

E X E 2

0 5 100

2

j

p

0 10 20 30 400

500

1000

1500

0 2 40

2

jj

5.33 10.67 160

50

100

E q u i v a l e n c y = 1 0 0 %C o n s i s t e n c y = 1 0 0 %

I n p u t 1 ( 1 0 in p u t s )

In p u t 2 ( 4 in p u t s )

O u t p u t s 1 & 2 C o m m o n E ig e n f u n c t io n M o d e s

0 5 100

2

j

E q u i v a l e n c y = 5 0 %C o n s is t e n c y = 4 2 %

I n p u t 1 ( 1 0 in p u t s )

In p u t 2 ( 4 i n p u t s )

O u t p u t s 1 & 2 C o m m o n E ig e n f u n c t i o n M o d e s

0 2 40

2

jj

0 10 20 30 40500

0

500

1000

1500

5.33 10.67 160

20

40

0 5 100

2

j

E q u i v a l e n c y = 0 %C o n s is t e n c y = 0 %

I n p u t 1 ( 1 0 i n p u t s )

I n p u t 2 ( 4 i n p u t s )

O u t p u t s 1 & 2 C o m m o n E i g e n f u n c t i o n M o d e s

0 2 40

2

jj

0 10 20 30 401000

0

1000

2000

3000

5.33 10.67 161

0

1

E X E 1

E X E 2

E X E 1

E X E 2

E X E 1

E X E 2

Fig. 2. Comparison of Equivalency and Consistency with input and output sizes between models different.

If known inputs are available, the input vector is created for each model. If the input vector is only available for one

simulation, then it is assumed to be the same for both. If one simulation assumes input in different dimensions (number

of parameters and units of those parameters) than the other simulation, then the input vector can be transformed into a

common space by using a transfer function. The TFs in (15) yield the transformation of one set of units into another,

many inputs into a few outputs, and a few inputs into many outputs respectively.

)(

)()(

MetricFFT

EnglishFFTunitsTF =

)100,,2,1(

)2,1()(

inininFFT

outoutFFTManyToFewTF

K=

)2,1(

)100,2,1()(

ininFFT

outoutoutFFTFewToManyTF

K= (15)

Another method for handling unequal numbers of input vector elements is to pad the shorter of the two with zeros to be

equal length with the longer one. An example of this is given in Figure 2, which shows a comparison using a notional

data set where the number of inputs for the first data set is 10 and for the second set the number of inputs is 4. The inputs

and outputs are varied to show that the totally overlapping case (outputs closely match and inputs in the same locations

in different sized input vectors) flags at 100% Equivalence and Consistency, and degrades gracefully to a non-matching

case of 0% Equivalence and Consistency.

For each model, the probability of being random is given by

N

sNEigenModeProbRnd

2100= (16)

where N is the number of points (and equal to 2X the maximum possible number of eigenmodes), NEigenModes the

number of eigenmodes in the TF, and multiplied by 100 to convert the ratio into a percentage.

The probability of the resultant model being a linear system model is given by

)21max(

)21min(

1001

0

1

1

0

1

1

0

1

1

0

1

∑∑

∑∑−

=

−

=

−

=

−

==n

i

n

i

n

i

n

i

Linear

OutorOut

OutorOut

Prob [ ]2)1)()(( += Input1FFTTFIFFTOUT1 (17) ( 2

Output1OUT2 = )

and answers the question Is squaring the input the same as squaring the output? In (17), OUT1 is generated by taking

the FFT of the input, squaring it, multiplying it by the TF, and then taking the inverse FFT. This is then compared

directly in (17) with the square of the original output given as OUT2. In this work, all input vectors have 1 added to them

in the frequency space to insure that no division by zero errors are encountered; however, in practice it has been found

that this is not always necessary.

For multiple input cases, each model can be examined independently to determine the input element accounting for the

most model variance. This is done by setting one input vector element to its original value while setting all remaining

elements to zero and is given by

++

=∑∑

ModelTF

Input

MaxElemBA

BAModVar i

)(2

)(2100max

22

22

(18)

where 2(ΣA2+B2) defines the variance from the frequency space representation and A and B defined as in (1). Once all

input vector elements have been turned on one at a time and their variance ratio with the variance of the TF calculated,

the input vector element corresponding to the maximum value shown in (18) is the element that accounts for the most

model variance.

Relative Model Sensitivity (higher values indicate more sensitivity) is calculated by turning all input elements on

(defined as the baseline), and the variance compared to the variance of the TF by

ModelTF

BaseInput

BA

BARelModSens

)(2

)(2100

22

22

∑∑

++

= (19)

By first multiplying the TF by the FFT of a new input and then performing an inverse FFT, a new output is created,

where classical time space comparisons can be applied. Three metrics used by Data Modeling to compare the model with

the original are correlation, the percentage of the variance of one model captured by the second, and the root sum

squared difference. In equation form, these three metrics are given by

( )( )( ) ( )∑∑

∑−−

−−=

i

i

i

i

i

ii



Correl22

100

( )( )∑

∑−

−=

i

i

i

i

Model1Model1

Model2Model2

VarRatio 100

−

−=

∑∑

∑−

=

−

=

−

=

),max(

)(

11001

0

1

0

1

0

2

n

i

i

n

i

i

n

i

ii

Sim

Model2Model1

Model2Model1

RSS (20)

where Model1 and Model2 here are in time space and result from the TF multiplication and inverse FFT process, and

Model1 and 2Model are the means of each resultant. These 3 metrics can be calculated between the model and the

original data as a direct comparison of how well they match, or it can be performed between two different models to

score the degree of match..

The variance similarity between 2 TF models is given as a percentage by

))(2)(2max(

))(2)(2min(100

2222

2222

Model2Model1

Model2Model1

SimBAorBA

BAorBAVar

∑∑∑∑

++++

=

(21)

The unique eigenmode information captured by the model is also given as a percentage by

−=)

2

1(

)2

(

1100

erf

zerf

Unique EigenModes

lnModesModeNumberEige

onnModesCommNumberEigez = (22)

The entropy preserved in the model from the original data is given as a percentage by

22222

2222

))(2)(2max(

)(2)(2100

Model2TFModel1TF

Model2TFModel1TF

BAorBA

BABAEntropy

∑∑∑∑

++

++= (23)

The model sensitivity ratio is the ratio of the relative model sensitivity given in (19), and is given as a percentage by

(Model1)RelModSens

(Model2)RelModSensioModSensRat 100= (24)

The percentage of terms overlapping between models is the ratio of the number of eigenmodes common to Model 1 and

Model 2 to the maximum between the number of eigenmodes in Model 1 or Model 2. This percentage is given by

100),( Model2Model1

nModesCommonEige

sNEigenModesNEigenModeMax

NrlapTermsPercentOve = (25)

Finally, the analysis confidence is defined as the average of the consistency, variance similarity, correlation, and the

RSS similarity. In equation form, this is given as

4

SimSim RSSCorrelVARyConsistencnfidenceAnalysisCo

+++= (26)

Calculation of these metrics for two equation models independently derived from two independent I/O data sets

quantifies their degree of equality for VV&A. High equivalency validates the models, consistency verifies their similar

behavior, and accreditation is possible by checking that threshold conditions for these metrics are maintained.

Congruence is proven when all required thresholds are met or exceeded for a given application or intended use. Also,

theoretical predictability is a function both of being non-random and being linear, but is not pursued further at this time.

2.4 Metrics example

The authors chose to demonstrate the metrics in Section 2.3 using an interceptor debris model and its derived Data

Model5. Eight (8) input variables (wavelength, relative signature value, target area, field of view, scale coefficient, sensor

aperture, time to impact, and target mass) control the input to both the original model and the Data Model, with both

returning the number of fragments and the minimum fragment size. Transfer function models were constructed for each

by running the same 64 sets of 8 input vector elements through both the original model and the original Data Model

(extracting 64 examples). The 64x8=512 input vector elements were placed into a single input vector, and the 2 output

elements (number of fragments and minimum fragment size) were first padded up from 2 elements to 8 elements, and

then each set of 8 concatenated together in the same manner as the input vector elements. This process is the same used

for Monte Carlo type simulations, where multiple inputs sets are concatenated together into a single input vector, and

when convolved with the SRF or TF yields the outputs for the Monte Carlo cases en masse. For cases where 1 or a

subset of Monte Carlo runs is required, the input vector for the TF is filled to the maximum number of cases (here, 64

cases) by padding with repetitions of the Monte Carlo input cases.

Single TF models for original debris model and original Data Model were generated and analyzed. Results are given in

Table 1. TF output and the original output are close matches both in terms of correlation (88-90%) and RSS difference

(96%). In terms of the frequency space TF models constructed for the original debris model and its original polynomial

Data Model representation, the Equivalence between the model scored 77%, Consistency 45%, and percentage overlap in

model frequency terms 49%. Inclusion of more terms raised the Equivalence, Consistency, and Overlap scores, but not

the final Correlation or RSS scores. The authors chose to use shorter models with lower Equivalency and Consistency.

Model 1

(Original)

Model 2

(Data Model)

TF Model Correlation With Original Data (%) 88.7 89.7

TF Model Variance % of Original Data Variance (%) 291.8 214.6

RSS Similarity Between TF Model and Original Data (%) 96.2 96.7

Unique Modes Between Models (%) 45.1 37.4

Entropy Preserved From Original (%) 95.5 95.0

Probability of Being Random (%) 18.0 15.6

Probability of Being a Linear System Model (%) 12.3 86.6

Most Sensitive (Min Input Element Change Causing Max Output Variance) Tgt. Mass (6) Tgt. Mass (6)

Relative Model Sensitivity (Higher is More Sensitive) 787.1 862.3

Most Critical (Min Input Element Change Causing Max Output Range) Wavelength (4) Wavelength (4)

Model 1st Derivative 1.7 1.5

Model 2nd Derivative 1.9 1.6

Model 1 and Model 2 Compared

Correlation Between Models (%) 99.0

Model Variance Ratio (Model 2 / Model 1) (%) 73.5

RSS Similarity Between Models (%) 98.7

Variance Similarity Between Models (%) 91.3

Model Sensitivity Ratio (Model 2 / Model 1) (%) 109.5

Terms Overlapping Between Models (%) 48.9

Equivalence Between Models (%) 77.0

Consistency Between Models (%) 45.3

Analysis Confidence (%) 83.6 Table 1. Summary of Analysis between Full Debris Model and Data Model Representation.

2.5 Lookup Table (LUT) generation

Finally, for applications where even simple equation models cannot be executed in real-time, the equation models are

converted into lookup table (LUT) format. For the lookup tables in this work, the TF was used for initial LUT

population. This population was performed by creating new input vectors consisting of combinations of uniform random

input variables that are automatically normalized between zero and one. These new input vectors are then fast Fourier

transformed, multiplied by the TF, and inverse fast Fourier transformed. This LUT derived from successive repetition of

this process still contained areas for which no values were obtained. For this reason, we construct a multi-variable

polynomial as an equation Data Model that both adequately reproduces the data output from the TF, and also gives the

ability to predictably populate the remaining regions of the LUT not reached by processing through the TF. The

Kolmogorov-Gabor polynomial of the form

LK ++++= ∑∑∑∑∑∑i j k

kjiijk

i j

jiij

i


represents a multiple input, single output transfer function that maps the inputs x1, x2,…, xL into the output y. Data

Modeling approximates this Kolmogorov-Gabor polynomial using a hierarchy of functional models as


LxxxyO 3),,,( 21 =K (28)

Polynomials of the form in (28) are Data Relations Models, which are multi-variable polynomials built using building

blocks no larger than 3rd order and comprised of combinations of up to 3 input variables. These polynomial building

block solutions are nested into higher order polynomial fits as shown in (27), which makes the Data Model a functional

(an equation or function that can be embedded into another equation or function as an input variable). This process is

illustrated in Figure A.1 of the Appendix.

LUTs for number of fragments and minimum fragment size (mm) were created using a multi-variable polynomial

derived from the TF for each output vector element (Figures A.2 and A.3 in the Appendix). This polynomial creation

process was provided the input list and allowed to perform its own ranking and choice of 3 inputs to use. Significance of

these 3 inputs was based on frequency of usage of each input in the final resultant polynomial. This resulted in the

selection of wavelength (defines sensor type and resolution), target mass, and target area for number of fragments; and

relative signature value, wavelength, and target mass for minimum particle size.

LUT construction is illustrated in Figure A.1 in the Appendix. Input/output pairs are passed through the multi-variable

linear regression routine using only at most 3 variables at a time, with multiple 3rd order polynomials nested together and

used as described above as functionals. Figure A.1 illustrates the code structure that includes the looping across nested

layers, building block types, and input variable combinations Once the equation Data Model is constructed, new inputs

corresponding to the grid locations on the LUT are passed through the final equation. Also illustrated in Figure A.1 in

the Appendix is the mapping of the output into the 16 individual 16x16 smaller LUTs contained in the two LUTs that

follow in the Appendix.

For convenience, 16 step increments were chosen for interrogating the equation and constructing the LUTs. In Figure 3

(number of fragments), wavelength was used the most number of times in the polynomial, and given no further

information is used to determine which of the 16x16 blocks contains the resultant. On the right side of the LUT is the

mean and standard deviation of each block for use as the answer. If wavelength and target mass are known, the

individual row in the block on the LUT is found, and the row averaged or its median or mode calculated. If all 3 inputs

are known, then the value is read off the grid directly by first going to the 16x16 block for wavelength, the row inside the

16x16 for target mass, and finally the intersecting column value in the row for target area. This process is described

directly on the LUT itself, and is repeated for the LUT in Figure A.3 in the Appendix for minimum fragment size.

REFERENCES

1. Jaenisch, H.M. and Handley, J.W., “Data Modeling for Radar Applications”, Proceedings of the IEEE Radar

Conference 2003, (May 2003).


3. Barnett, S. & Cameron, R.G., Introduction to Mathematical Control Theory, 2nd ed., Oxford: Clarendon Press,

1993.

4. Jaenisch, H.M., Algorithms for Autonomous Unmanned Deep Space Probes, D.Sc. Thesis, James Cook University.

(Submitted to University for Approval), 2005.

5. Handley, J.W., Jaenisch, H.M., Barnett, M.H., Esslinger, R.A., Grover, D.A, Hunt, J.R., “Data Modeling Predictive

Control Theory for Deriving Hyper-Real-Time War Game Models from Real-Time Models”, SPIE Defense and

Security Symposium, Orlando, FL, April 2006.

APPENDIX

Combinations of

Inputs (3 at a time)

RSS Error

DeriveDM

(Nesting Loop)

Nest

(Building

Block Loop)

Variable

Loop

Generate

Inputs for LUT

Generate

Output in LUT

Format

Combinations of


RSS Error

Combinations of


RSS Error

DeriveDM

(Nesting Loop)

DeriveDM

(Nesting Loop)

Nest

(Building

Block Loop)

Nest

(Building

Block Loop)

Variable

Loop

Generate

Inputs for LUT

Generate

Inputs for LUT

Generate

Output in LUT

Format

Generate

Output in LUT

Format

Triple = a0 + a1x1 + a2x2 + a3x3 + a4x12 + a5x2

2 + a6x3

2 + a7x1

3 + a8x2

3 +

a9x33+ a10x1x2 + a11x1x3 + a12x2x3 + a13x1x2x3 + a14x1x2

2 +

a15x1x32 + a16x1

2x2 + a17x1

2x3 + a18x2x3

2 + a19x2

2x3

Fig. A.1. Lookup Table (LUT) creation routines. Shown is the outer nesting loop (layers), the individual building block calls inside of

nesting loop (AllVars, Single, Double, Triple, Quad, Quint), Combinations of Variables loop inside the Triple building block, LUT

input generation routine, LUT output formatting routine, and the equation form of the Triple building block.

Fig. A.2. Number of Fragments (x 10) LUT using wavelength (0.3 to 11.5 microns), target mass (255 to 345 kg), and target area (0.1

to 0.5 sq. meters).

Fig. A.3. Minimum Fragment Size LUT using Rel. Sig. Val. (0.04 to 0.98), wavelength (0.3 to 11.5 microns), and target mass (255

to 344 kg).

Analytical Data Modeling for Equivalence Proofing and Anchoring

Simulations with Measured Data


, James W. Handleya, and Michael L. Hicklen

b

aJames Cook University, Townsville QLD 4811, Australia

bdtech Systems Inc., P.O. Box 18924, Huntsville, AL 35804

ABSTRACT

This paper presents a novel analytical method for comparing simulation data with measured data, with the goal of

proving Equivalence and Consistency between the model and the real data. Our method overcomes the problems of

disparity in the inputs to the simulation and varying number of parameters between the simulation and the measured

flight data. Our method derives analytical Data Models that are analyzed in frequency space, yielding quantitative

assessment values for the model performance relative to the measured data. The model output for a sensor and its “real-

world” measured data are collected for comparison.

Keywords: Formal Analysis, Data Modeling, V&V, Equivalence, Consistency, Transfer Function Modeling, Automatic Proofing

1. INTRODUCTION

Traditional software development focuses upon code verification at a relatively low level adequate for small to medium

scale sub-system and system level development. Technology that evaluates “higher level models” has been developed

and is effective for large software intensive systems; however, anchoring and calibration of these models must still be

done to measured data. We propose the use of Data Modeling of simulation input/output (I/O) data and measured data as

the basis for this comparison.

In this work, we show that simulation to measured data anchoring can be done without access to source code or even the

simulation itself. All that is required is the output of the simulation and the measured data it represents. The input data is

not required, because it is assumed to be common to the simulation and the measured data process. By setting the sum of

the inputs equal to one (1) for both the measured process and the simulated process, a common denominator for Transfer

Function (TF) modeling is established. This leaves the independent numerators to be mathematically compared.

2. TRANSFER FUNCTION DATA MODELING

2.1 Transfer function theory

Data Modeling1-4 using transfer functions can be used to model both input and output systems. The transfer function5,6

(TF) provides an algebraic representation of a linear, time-invariant filter in the frequency domain. The frequency

domain for the transfer function is determined using the Fourier transform (FFT). The frequency space representation is

separated into the real and the complex or imaginary component

∑==j

N

ij

iii yyFFTA )cos())(Re( 2

21 ππ ∑==

j

N

ij

iii yyFFTB )sin())(Im( 2

21 ππ (1)

where A are the real terms, B the complex (imaginary) terms, N the number of points, and both i and j are counters

ranging from 0 to N-1. Each frequency term in (2) results in a cosine-sine pair in the analytical equation

∑=

+=n

j

jjjj xBxAxy1

sincos)( ωω (2)

which is the solution to the differential equation given by the derivative of (2) and given as the kth derivative in (4)



=+

=−

=−−

=+−

=

∑

∑

∑

∑

=

=

=

=

04modsincos

34modcossin

24modsincos

14modcossin

)(

1

1

1

1

kxBxA

kxBxA

kxBxA

kxBxA

xy

n

j

jj

k

jjj

k

j

n

j

jj

k

jjj

k

j

n

j

jj

k

jjj

k

j

n

j

jj

k

jjj

k

j

k

ωωωω

ωωωω

ωωωω

ωωωω

(3)

If derivative data is modeled with (2), it is necessary to integrate the result in order to obtain the analytical solution to the

differential equation. The kth integral of (2) is given by

=+

=+−

=−−

=−

=

∑

∑

∑

∑

∫

=

−−

=

−−

=

−−

=

−−

04modsincos

34modcossin

24modsincos

14modcossin

))(()(

1

1

1

1

kxBxA

kxBxA

kxBxA

kxBxA

dxxy

n

j

jj

k

jjj

k

j

n

j

jj

k

jjj

k

j

n

j

jj

k

jjj

k

j

n

j

jj

k

jjj

k

j

k

ωωωω

ωωωω

ωωωω

ωωωω

(4)

The transfer function is defined as the frequency domain representation of the output divided by the frequency domain

representation of the input. In equation form, this is given by

)(

)()(

zX

zYzH = (5)

where H(z) is the transfer function, Y(z) the frequency domain representation of the output, X(z) the frequency domain

representation of the input, and both Y(z) and X(z) can be represented in the polynomial form

L++++= 3

3

2

210)( zazazaazY (6)

which is written in terms of the single variable z and is analogous to the Kolmogorov-Gabor polynomial used extensively

in Data Modeling. The input is transformed directly along with the output and the ratio calculated. If required, Data

Modeling techniques described in previous work and included in the References can be used to interpolate or pad the

input and output number of points to be equal.

Once the ratio in (5) is calculated, it can be inverse transformed back into the original space as a system response

function (SRF) given by

)sin()cos()(22

N

ij

j

j

N

ij

jii BATFInvFFTSRFππ +== ∑ (7)

or it can be maintained in the frequency domain as a TF using A, B, and the frequencies (2πj/N) shown in (7). Using

eigenfunction Data Modeling and modified Weiner filtering, the number of frequency terms kept can be reduced,

yielding a more compact function form, and the transfer function is written in terms of the dominant eigenfunctions.

2.2 Eigenfunction modeling

Eigenfunction extraction uses a variation of Wiener filtering to identify the peaks in the dB power spectrum. These peaks

correspond to sine and cosine terms when using a discrete Fourier transform for spectral decomposition. To facilitate

peak identification and extraction, a model of the data set’s dB power spectrum is built. This is done using the Fourier

transform defined by (1). Once the Fourier transform is generated, the power spectrum of the data is calculated, which is

the sum of the squares of the real and imaginary terms in (1), and is then transformed into decibels (dB).


(log10)( 10

i

ii

P

PdBP = (8)


bmxy += (9)

where m is the slope of the line and b is the y-intercept. Once the power spectrum is detrended and squared, a second fit

using linear regression is performed, where the y-axis of the data to fit is the detrended power spectrum and the x-axis is

simply an index variable. This y-intercept is used as a threshold, with locations where the detrended power spectrum

points greater than the y-intercept are allowed to pass the filter unchanged in value from the original spectrum, while

those locations that are less than the y-intercept are set to zero. Once a threshold is applied to the frequency terms and the

non-peak values zeroed, each remaining dominant frequency term represents a single term in the final eigenfunction of

the data set, which is given in (2). Adjacent peaks are combined together by leaving only the central (middle) peak of the

group. Finally, the truncated components are normalized to the minimum and maximum of the full spectrum, thereby

conserving entropy lost during peak removal.

2.3 Construction and use of transfer functions

When only an executable or examples of simulation runs are available to derive a Data Model, a slightly different

approach is used. If the number of example scenarios (Input/Output) is still fairly substantial, then the regression based

approach is still appropriate; however, in the case where very few cases are possible or even limited to just one test case

example, a completely different approach is necessary. For our example below (IRSim), we have only one example;

therefore we apply predictive control theory and use the transfer function as tool of choice.

Analytical comparison of measurement or flight data with test data is accomplished using the method of classical control

theory transfer functions (TFs) described in the previous section. Because transfer function models occur in frequency

space using complex variables, they are formally termed analytical methods. Simple TF models are derived from

input/output pairs of data. Independent TF models are built for both the simulation and for the test data using one data set

from the simulation and one set from the test data. The resultant TFs are Data Models that are compared using formal

mathematical methods of analysis.

A TF model is built by a simple ratio of Fast Fourier Transforms (FFTs) as shown in (5). The transfer function is

constructed by dividing the FFT of the output data set by the FFT of the input data set

ionDeconvolutInputFFT

OutputFFTTF ==

)(

)( (10)

This division is called a deconvolution. Deconvolution removes the input data set from the output data set. The resulting

ratio of output FFT to input FFT is the TF model of the process. The TF model is used to transform (convolve) any input

data set into its related output data set, given by

))()(

)(( NewInputFFT

InputFFT

OutputFFTInvFFTNewOutput = (11)

If an inverse Fast Fourier Transform (InvFFT) is performed on the TF model in (5) or (10), the result is the creation of

the system response function (SRF) in (7). The system response function is a time domain model can then be spatially

convolved with a new input to yield a prediction of the response output, or an FFT can be performed to transform it back

into frequency space for convolution.

The methods here analyze the form of the two independently derived differential equation models by comparing their

mathematical structure and the value of their coefficients to measure the Equivalence and Consistency between models,

and therefore between simulation and flight test data. Specifically, for comparison of simulation with flight test data, we

are interested in the structure and form of the TF model constructed from the simulation compared with the TF model

constructed from the flight test data. Since the TF models are comprised of unique modes mathematically expressed as

cosine and sine functions as shown in (2), they are readily transformed into analytical derivative terms a shown in (3).

The number of unique modes can be reduced by performing eigenfunction modeling and modified Wiener filtering, but

with or without eigenfunction modeling, the method yields an analytical differential equation form model of the transfer

function.

2.4 Application to measured flight test data

TF models are now generated for both simulation and test flight data, and their eigenfunction representations (if chosen)

determined. The eigenfunction TF models are compared for Equivalence and Consistency to characterize the similarities

and differences between the simulation and test flight data, with eigenfunction mode locations and amplitudes compared

to determine how alike or different the two models are. Equivalence is determined by calculating the correlation between

the magnitude of each of the TF models, and given by

( )( )( ) ( )∑

= −−

−−=

N

jjj

jj


Model2Model2Model1Model1eEquivalenc

122

100 (12)

where Model1 and Model2 are the magnitudes of the TF models for the simulation and measured data given by

22


and Model1 and Model2 in (12) are the means of the result of (13).

Consistency between Model1 and Model2 is calculated by first subtracting the mean from each frequency vector exactly

as shown in (12). Next, each vector is multiplied by a Mask of zeros and ones, with ones in the mask vector at the

location of an eigenfunction mode, and a zero at all other points. This mask converts the full model into the


≠=

=0Model11

0Model10Mask1

j

j

j (14)

≠=

=0Model21

0Model20Mask2

j

j

j


jjjj Mask2Mask1Model1Model1Model1 )( −=

jjjj Mask2Mask1Model2Model2Model2 )( −= (15)

Next, each frequency terms is converted into a Z-score using the standard deviation of the 2 (Model1 and Model2)


j

j

j

Model1Model1

σ=

j

j

j

Model2Model2

σ= (16)

Mask2Mask1

))Norm(d)(1)(Mask22(Mask1

yConsistenc

N

1j

jjj

o

∑=

−= jjj Model1Model2d −= (17)


argument dj for the probability calculation is given as shown.

If known inputs are available, the input vector is created for each model. If the input vector is only available from the

simulation, then it is assumed to be the same for both the simulation and the measured data. If the measured data

assumes input in different dimensions (number of parameters and units of those parameters), then the input vector can be

transformed into a common space by using a transfer function to do so. Also, this method works well even in cases

where the number of inputs in Model 1 and Model 2 are different. An example of this is given in Figure 1, which shows

a comparison using a notional data set where the number of inputs for the first data set is 10 and for the second set the

number of inputs is 4. The inputs and outputs are varied to show that the totally overlapping case (outputs closely match

and inputs in the same locations in different sized input vectors) flags at 100% Equivalence and Consistency, and

degrades gracefully to a non-matching case of 0% Equivalence and Consistency.

0 5 100

2

j

p

0 10 20 30 400

500

1000

1500

2

0 2 40

2

jj

5.33 10.67 160

50

100

1

1


Input 1 (10 inputs)

Input 2 (4 inputs)


0 5 100

2

j


Input 1 (10 inputs)

Input 2 (4 inputs)


0 2 40

2

jj

0 10 20 30 40500

0

500

1000

1500

5.33 10.67 160

20

40

0 5 100

2

j


Input 1 (10 inputs)

Input 2 (4 inputs)


0 2 40

2

jj

0 10 20 30 401000

0

1000

2000

3000

5.33 10.67 161

0

1

Fig. 1. Comparison of Equivalency and Consistency.

3. EXAMPLES

3.1 Simulation data compared to measurements

In our case, the input to both IRSim and the real-world data are assumed to be the same, but unknown to us. We used an

index array for the denominator ranging from zero to the number of measurements minus 1. We now proceed to derive

transfer function models as Fourier series representations for IRSim and for the measured data. Once created, the two TF

models are then analyzed to see how alike the two Fourier series are. As shown in Section 2, Equivalence is the

correlation between the two Fourier transforms, which measures if the peaks go up and down at the same time.

Consistency measures actual ratios of the value of the peaks to measure equivalency of the Fourier series coefficients.

Once these models are derived, analytical methods such as the Method of Derivatives can be applied to isolate critical

parameters or setting and perform sensitivity analysis. Likewise, the difference between the simulation and the real data

can be resolved by deconvolution of the two models in frequency space. This isolates the high frequency component not

captured by IRSim and also provides an analytical model for the difference that could now be included in IRSim as a

correctional factor if desired.

This simple method for achieving critical parameter estimation without resorting to Design of Experiments7-10 (DOE) is

only possible because we have an analytical model for the system and can thereby exploit the Method of Derivatives and

Jacobian analysis. Jaenisch has proposed an abbreviated form of this method for estimating Critical, Sensitive, and Key

parameters. This method uses an abbreviated TF consisting of the first order (mean) term plus the largest magnitude

eigenmode and its complex conjugate. When multiplying the FFT of the input by this abbreviated TF, the TF acts as a

notch or single band-pass filter on the input, which reduces the input variation space to the real and complex frequency

terms of the corresponding single eigenmode of the input. This reduction provides a method for then constructing a 2

parameter plot using the real frequency term on the x-axis and the complex part of the frequency term on the y-axis.

Minimum and maximum for the axes are determined by applying frequency space perturbations to the single eigenmode

in the original input.

Parameter plots are then constructed by varying the real and complex part of the input eigenmode between minimum and

maximum, constructing the corresponding controlled single eigenmode input along with its mean and corresponding

complex conjugate. The input is then convolved with the TF and a new output obtained. Baseline input and output

vectors are calculated by applying an inverse FFT to the abbreviated form of the original input vector and its

corresponding output. For one parameter plot, change in input and output from baseline are captured by summing the

difference between the derived and original input or output.

The authors were provided with two measured infrared (IR) signature histories and their one corresponding IRSim

generated signature. Figure 2 shows the two measurement signature histories: one for the notional solid fuel rocket

booster and the other for the notional liquid fuel rocket booster.

0 2 4 6 8 10 12 14 162

0

2

4

S o lid

L iq u id

IR

S ig n a tu re

In te n s ity

Fig. 2. Graph of measured data for solid fuel rock and liquid fuel rocket.

3.2 Solid Rocket Model versus Measured Signature

Using the methods described in Section 2, the TF model between IRSim and the measured data for the solid rock motor

(17 outputs) was created and compared for Equivalence and Consistency. The left graph in Figure 3 shows graphically

the output vector for both the measured and simulated data, and on the right shows the differences between them using

I×χ (chi-by-eye), since in Figure 2 they appear to be equal.

0 2 4 6 8 10 12 14 162

0

2

4

2.6

0.882−

sig1meask

sig1simk

0 5 10 150.05

0

0.05

3

8

ask

IR

Signature

Intensity

Difference

IR

Signature

Intensity

Fig. 3. Graph of IRSim data and measured data for the solid rocket motor (left), and their differences (right)

Once TFs are built for both the measured data and IRSim, they are then graphically displayed using the magnitude of the

frequency terms in each model as bars on a bar chart as shown in Figure 4.

0 50

5

10

m3 k

0 50

5

10

m4 k

Fig. 4. Case 1 TF model comparison of IRSim data and measured data.

These TF models are then compared using Equivalence and Consistency as described in Section 2 to yield an

Equivalence of 99.98% and a Consistency of 85.47%, which is consistent with the data shown in Figure 3. This results in

the differential equations for IRSim TF model given in (18) and the measurement TF model given in (19).

)6sin(107.4)4cos(724.4)4sin(038.1)2cos(203.)2sin(005.2(1717

4)1(xxxxx

dx

Modeld ππππππ−−++=

)12sin(964.2)10cos(14.)10sin(595.4)8cos(06.)8sin(348.)6cos(02.1 xxxxxx ππππππ +−−++−

))16cos(2.3)16sin(48.4)14cos(551.5)14sin(334.5)12cos(684.6 xxxxx πππππ −−−++ (18)

)6sin(963.3)4cos(686.4)4sin(014.1)2cos(215.)2sin(013.2(1717

4)2(xxxxx

dx

Modeld ππππππ−−++=

)12sin(294.3)10cos(055.)10sin(715.4)8cos(14.)8sin(328.)6cos(984.0 xxxxxx ππππππ ++−−+−))16cos(48.2)16sin(72.4)14cos(062.6)14sin(362.5)12cos(714.6 xxxxx πππππ −−−++ (19)

To verify the differential equation results, each one is passed through a Fourth Order Runge-Kutta (RK4) integrator to

solve the differential equation. The results for these two cases is shown in Figure 5 (left and center), where the curves lie

on top of each other and are indistinguishable as before. Figure 5 (right) is provided to show the small differences

between the two differential equation results.

Fig. 5. Graph of integrated results for the differential equation model in (18) overlaying simulated data (left); the differential equation

model in (19) overlaying measured data (center); and difference graph (right).

3.3 Determination of Extra Significant Frequency Components in Measurement Data

While comparing the IRSim TF model and the measured data TF model for the solid rocket, it was found by taking the

eigenfunction of the difference that two significant frequency components existed in the measured data that were not

present in the simulation data as shown in Figure 6. Two methods can be used to correct for this difference. One is to

build a TF of the solid rocket simulated data into the solid rocket measurement data that can then be used with new

simulated data to yield the equivalent measurement data. The second method is to construct an additive model of the

extra terms that can then be added to new simulated data runs to convert them into equivalent measurement data cases.

0

0.05

0.1

E ig en fu n c tio n

O f D iffe ren ce

(M ag n itu d e )

C y c le s

E ig en m o d es F o rm

C o rre c tio n F ac to r

0 1 2 3 4 5 6 7 8

Fig. 6. High frequency term (right most bar) shows frequency difference that existed in the measured data that was not found in the

IRSim data. Plot is of the magnitude of the differences.

To correct for the difference between the models, we first build a TF model that is the ratio of the measured data TF

model to the IRSim TF. This is given in equation form in (20) and in differential equation form in (21).

SBIRSSim

Measured

TF

TFController = (20)

)4cos(006.0)4sin(982.1)2cos(006.)2sin(005.1(1717

4)(xxxx

dx

Controllerd πππππ++−= (21)

)10cos(215.)10sin(125.5)8cos(192.2)8sin(392.3)6cos(0002.)6sin(895.2 xxxxxx ππππππ ++++++))16cos(056.1)16sin(672.7)14cos(3.)14sin(35.7)12cos(24.)12sin(132.6 xxxxxx ππππππ ++++++

By inputting the current IRSim output into this TF model, the result is the transformation of the output simulation data

into representative measurement data. This model can be used either to post-process IRSim output, or can be made

available for integration directly into IRSim.

As a second correction method, we build an additive model that can be used instead of the TF method given in (20) and

(21), and whose result is added to the simulated data. This model captures the two frequency terms given in Figure 6 and

is given in differential equation form as

))16cos(536.)16sin(176.)8cos(148.)8sin(016(.1717

4)(xxxx

dx

Correctord πππππ++−= (22)

3.4 Solid Rocket Measured Data versus Liquid Rocket Measured Data

To demonstrate how the Equivalence and Consistency metrics behave for data sets that are not alike, we compare the

measured data from the solid rocket with the measured data from liquid rocket (10 outputs), both of which are shown in

Figure 2 graphically, and whose TF models are given in Figure 7. In order to perform the comparison, the shorter 10

output liquid rocket case was zero padded to be equal length with the 17 output solid rocket case.

0 50

5

10

m3 k

0 5

0

5

10

m4 k

Fig. 7. TF models for measured data sets 1 and 2. Note differences occur after the first two bars.

The Equivalence and Consistency measures for this comparison were 12.46% and 4.58%, respectively, which shows a

low but non-zero Equivalence and Consistency between the two rocket measurement sets as expected. Table 1 is a rollup

summary of Equivalence and Consistency for the like-data comparison (solid rocket model versus measurement) and the

different data comparison (solid rocket measurement versus liquid rocket measurement).

Comparison of Solid Rocket

Model and Measurement

Comparison of Solid and Liquid

Rocket Measurement Sets

Equivalence 99.98% 12.46%

Consistency 85.47% 4.58% Table 1. Equivalence and Consistency Summary for Case 1 and Case 3

3.5 IRSim Data Model

Using the signature data provided in the two examples, a full Data Model of IRSim was generated. This TF DM uses the

two IR data signatures shown previously to generate a composite or prototype (average of the two in frequency

space).Once the prototype is created, equivalent inputs are solved for using the mathematical relationship

))()(

1( OutputFFT

prototypeFFTInvFFTInputEquivalent = (23)

Equivalent inputs are derived for both original signatures (solid and liquid rocket), and the FFT of the prototype is now

used as the TF. Since the FFT of the prototype was used to solve for the equivalent inputs, multiplying the new TF by the

FFT of the inputs will yield the original boundary cases. Additionally, by passing input vectors where each element is

constrained to be within the minimum and maximum of the element values from the solid rocket and liquid rocket

(boundary cases), new valid cases are created. The differential equation for this TF model is given by

)4cos(16.2)4sin(274.1)2cos(4.1)2sin(046.1(1717

4)(xxxx

dx

TFModeld πππππ+++−= (24)

)10cos(77.)10sin(52.1)8cos(452.)8sin(564.1)6cos(423.)6sin(269.1 xxxxxx ππππππ −+++++16cos(348.)16(12.)14cos(336.)14sin(595.)12cos(626.1)12sin(048. xxinxxxx ))ππππππ −++−−+

and an example of output is shown graphically in Figure 8. This was constructed using 17 notional inputs, whose

minimum and maximum values are given in Table 2. Also, if actual input variables are known, regardless of the number

they can be transformed into this equivalent set of 17 notional inputs, again using a TF of the form

)).()(

)17((17 nputsNewActualIFFT

tsActualInpuFFT

utsDerivedInpFFTInvFFTInputsNew = (25)

0 2 4 6 8 10 12 14 162

0

2

4

IR

Signature

IntensityCase 1

Case 2

Proto

New Output

Fig. 8. Sample output from the IRSim Data Model.

Minimum Maximum Minimum Maximum

Input 1 3.435 4.812 Input 10 -2.945 2.945

Input 2 -1.532 1.532 Input 11 -1.86 1.86

Input 3 -0.46 0.46 Input 12 -3.028 3.028

Input 4 -1.225 1.225 Input 13 -2.035 2.035

Input 5 -1.307 1.307 Input 14 -1.876 1.876

Input 6 -1.166 1.166 Input 15 -0.087 0.087

Input 7 -0.486 0.486 Input 16 -0.175 0.175

Input 8 -0.532 0.532 Input 17 -1.444 1.444

Input 9 -1.177 1.177 Table 2. Ranges for 17 notional inputs for the IRSim Data Model.

REFERENCES

1. Jaenisch, H.M., Algorithms Enabling Unmanned Deep Space Probes, D.Sc. Thesis, James Cook University (2005-

2006).

2. Jaenisch, H.M., and Handley, J.W., “Data Modeling for Radar Applications”, Proceedings of the IEEE Radar


3. Jaenisch, H.M., and Handley, J.W., “Automatic Differential Equation Data Modeling for UAV Situational

Awareness”, Society for Computer Simulation, Huntsville Simulation Conference 2003, (October 2003).

4. Jaenisch, H.M., Handley, J.W., Faucheux, J.P. “Data Driven Differential Equation Modeling of fBm processes”,

Proceedings of SPIE, San Diego, CA, August 4, 2003.


6. Barnett, S. & Cameron, R.G., Introduction to Mathematical Control Theory, 2nd ed., Oxford: Clarendon Press,

1993.

7. Doyle, F.J., Pearson, R.K., Ogunnaike, B. A., Identification and Control Using Volterra Models, London: Springer-

Verlag, 2002.

8. Jones, R. B., Symbolic Simulation Methods for Industrial Formal Verification, Boston, MA: Kluwer Academic

Publications, 2002.

9. Taguchi, G., and Jugulum, R., The Mahalanobis-Taguchi Strategy: A Pattern Technology System, New York: John

Wiley and Sons, 2002.

10. Murphy, G., Similitude in Engineering, New York, Ronald Press Company, 1950.

Data Modeling Change Detection of Inventory Flow and Instrument

Calibration

Holger M. Jaenisch, James W. Handley, and Louis A. Bonham

Sparta Inc., 4901 Corporate Drive, Suite 102, Huntsville, AL 35805

ABSTRACT Data Modeling is a new mathematical tool for logistics research. It enables reference points to be

established for complex dynamic processes such as inventory control and instrument calibration. This paper presents an overview of Data Modeling and how it can be used to achieve

mathematical models that are a basis of comparison for detecting significant changes either in inventory flow or instrument calibration.

1. INTRODUCTION

1.1 Inventory Flow Data Modeling for inventory flow and control involves the creation of a mathematical equation such as

),,,(' PDPOVVfy ∆= (1)

that describes the relationship between inventory levels and their usage as measured by volume

(V), change in volume (∆V), point of origin (PO), or point of departure (PD). Simple metrics

that are part of traditional logistics analys is of inventory levels can be monitored autonomously to give a new level of self-awareness to business as usual for complex operations. For example,

keeping munitions and supplies at the proper levels between storage points and required field or transition points requires careful orchestration of delivery schedules, volumes, and queuing time

arrival events. If the intricacies contained in the complex inventory flow system or process can be adequately

modeled, the model can be used as a diagnostic and predictor for assessing when inventory flow has changed in an off-nominal fashion. Such change detection (novelty detection) would enable

monitoring and detection of inappropriate allocations of resources and potentially generates derivative indicators of impending critical events and disasters.

1.2 Inventory Abnormality The time directly preceding possible escalations of hostilities frequently results in an increase in

the requisition of certain materials or the unusual storage of quantities of certain items. This change in the flow of business as usual is very difficult to detect while it is happening, because this form of anomalous behavior is usually spread out over time and across many nodes in the

inventory flow structure. If an accurate mathematical model of the nominal flow characteristics of the inventory site were available for comparison, then changes in the underlying dynamics

would be easily seen and would draw the logistic analyst’s attention.

1.3 Instrument Response Modeling Aircraft use sophisticated ins truments in the form of Improved Data Modems (IDM), flight

recorders, transponders, radar, communication links, leveling instrumentation, and flight and attitude controllers. Whenever one of these instruments is suspected of being out of calibration, it

is removed from the aircraft and returned to the manufacturer for recalibration and testing. This process is both costly and lengthy, and is done many times unnecessarily. Data Modeling can be used to eliminate the unnecessary removal and recalibration of an instrument. Using Data

Modeling, the response of a newly calibrated instrument to a known stimulus for different instrument settings can be monitored and recorded during the early history of its useful life or

whenever the instrument is known to be functioning adequately. These stimuli need not be the same as those prescribed by the manufacturer for use during test bench calibration. Instead, the stimulus can be something that is readily available for use in the field.

These settings and responses are used to generate a tuned equation Data Model that associates

these instrument readings and nominal variations to a nominal output. This Data Model captures the performance curve of the instrument, and is sensitive to any changes or variations. Once generated, the Data Model can then be used to determine at any point and time in the future if the

current instrument is within nominal bounds, or if the instrument is now out of calibration and must be removed from the device and returned to the manufacturer.

2. DATA MODELING

2.1 Introduction Data Modeling is the process of finding a mathematical expression that provides a good fit

between given finite sample values of the independent variables and the associated values of the dependent variables of the process. The mathematical expression that fits the given sample of data is called a Data Model. This process involves finding both the functional form of the Data

Model and the numeric coefficients for the Data Model. Data Modeling differs from conventional methods such as linear, quadratic, and polynomial regression techniques that only

solve for the numerical coefficients for a function whose form is specified ahead of time. Data Modeling also differs from genetic algorithms and symbolic regression because it incorporates fractal interpolation functions for achieving mathematical modeling of complex multivariable

processes. The Data Modeling process of discovering automatic functions is an evolutionary and genetic directed approach where the set elements consist entirely of simple polynomial building

blocks. The final form of the Data Modeling equations and the polynomial order shown in (2) are

solutions to the Kolmogorov-Gabor polynomial with n representing the number of layers in the final Data Model and x(bi(t)) the inputs mapped from the previous layer.

[ ] n

n

txO

tbxtbxbxtftx

3)(

)))))))))((((((((,()( 21

=

= K

(2)

Data Model polynomials on the order of ~ O[310] have been generated within minutes which adequately capture high degrees of nonlinearity but still execute in real-time.

2.2 Change Detection The power of statistical quality control charts is the ability to flag when a dynamic process such

as inventory flow or instrument response is not in statistical control. One commonly used type of control chart is the X-Bar Chart defined by the upper control limit (UCL) and the lower control

limit (LCL)

.2

2

RAXLCL

RAXUCL

X

X

−=

+= (3)

The X-Bar chart plots how far away from the average value of a process the current measurement falls. According to Shewhart, values that repeatedly fall outside of three standard

deviations of the average have assignable causes and are labeled as being out of statistical control (off-nominal).

2.3 Calibration on Demand (COD) A Data Model of a complex process in statistical control can be used to generate a lookup table

(LUT), which is analogous to the X-Bar chart. Such a Data Model is built using only nominal cases. Tuning makes the model sensitive to feature values that fall outside the training

boundaries. Whenever these new features fall outside of nominal boundaries, tip-off is flagged. Combinations of incoming feature values that are not anticipated or that are within valid ranges but in combinations not seen before can be instantly detected with this approach. This enables

calibration to be specified and monitored dynamically or on demand (COD). The lookup table trial limits are set using the polynomial output from the training data. The upper and lower

boundaries are assigned to be one-half (½) of a standard deviation (ε) above and below the mean value of the polynomial output from the training data.

Fig. 1 illustrates how either inventory flow sensor measurements or instrument responses are converted into features and analyzed to detect if a change has occurred in system status from

nominal to yield tip-off conditions from the model.

Tip-Off

Tip-Off

Nominal

Tip-Off

Tip-Off

Nominal

Equation

Output

σ

T = w0+w1x1+w2x2+w3x3 +w4x12

+w5x

22 +w

6x

32 +w

7x

13 +w

8x

23

+w9x32 +w10x1x2 +w11x1x3

+w12x2x3

Skewness Standard

Deviation

KurtosisRollup Stats

Data Model

MeasurementData

MeasurementDataInstrument Response

ZScores

ZScores

Inventory Flow

[1 3 –2 4 –2 1 4]

DataSeriesData

SeriesPDFPDF

Fig. 1. Calibration on Demand using Data Modeling.

COD can monitor individual sensor measurements, feature vector elements, and overall system snapshots for change detection. Individual sensor measurements provide a characterization of system level calibration states.

3. APPLICATIONS

3.1 Inventory Flow and Monitoring

An inventory flow scenario was created using the Dynamic Inventory Model (DIM) and associated inputs given in the Appendix that simulates an inventory problem on a period-by-period basis. This inventory model was run using two different products stored at a single

location (e.g., warehouse). Two different customers that were the end users of these two products were also modeled ordering inventory from this storage location. These orders were placed and

filled along with inventory reordering and backordering by the storage location for a total of 256 time periods. During each of the 256 time periods, the number of orders received, inventory, quantity of sales lost, and quantity of units sold was recorded. These quantities were recorded on

a customer by product basis, and summed in order to determine the overall inventory flow, number of orders, number of sales lost, and number of units sold for the storage location. The

goal was to construct a LUT picture of nominal inventory flow conditions and to predict what off-nominal conditions might be.

Our approach does not directly use the inventory feature values as inputs into the Data Model. Instead, each of the 256 nominal feature examples was rolled up, thereby characterizing the

feature distribution independently. This was done for each nominal example using the following parametric and non-parametric statistics where N=4, x1 is inventory flow, x2 is the number of orders, x3 is the sales lost, and x4 the quantity of units sold, and the Range is the difference

between the highest and lowest xj.

∑=

=N

jjN xx

1

1 (4) ∑=

− −=N

j

jNxx

1

2

112 )(σ (8)

∑=

−=

N

j

j

N

xxSkew

1

3

1

σ (5) 3

1

4

1 −

−= ∑

=

N

j

j

N

xxKurt

σ (9)

151

6

16

−

−= ∑

=

N

j

j

N

xxM

σ (6) 105

1

8

18 −

−= ∑

=

N

j

j

N

xxM

σ (10)

=→∆+

N

NJ

Range

Ji

Ji

1log

log

limmin)Re(0

1

σ (7)

++= ∑

−

=

−+1

1

21 )(1log1 1

N

j

Range

xx

Nf

jjD (11)

Once the rollup features in Eq. (4) through Eq. (11) were calculated, a Data Model was tuned to

εσ ±=2

186

2 ),,,,,,,(f

DJMMKurtSkewxf (12)

where ½ is the target regression value and ε is the variation of the regression fit about ½. Once

the equation is trained and tuned, boundaries are placed on either side of the target regression

value that characterizes the fluctuation ε of the polynomial about ½. The Data Model generated

is shown in Fig. 2, and the boundaries that characterize the polynomial fluctuation were set at 0.4992 and 0.5010.

Fig. 2. Data Model of the Dynamic Inventory Model (DIM).

Fig. 2 also contains a Decision Map (LUT) generated using the Data Model. The white box represents the training range of the nominal cases. From the Decision Map, only the features shown on the x-axis need to be monitored in order to determine if the inputs will produce

nominal or off-nominal inventory flow, since the nominal region is bounded to a narrow band along the x-axis but extends along the full range of values for the y-axis. This boundary is

marked on the Decision Map and represents the values 0.33 < J < 0.37, 1.43 < Df < 1.45, 12.3 <

x < 16.3, and 16.1 < σ < 22.7. This model now predicts what inventory statistics would fall outside of the boundary, indicating unusual inventory dynamics that demand scrutiny by the analyst.

3.2 Instrument Calibration To demonstrate instrument calibration, the MiniOx oxygen analyzer with an accuracy of plus or

minus 1% was used. Since this type of oxygen sensor is based on measuring oxygen partial pressures (PPO2), the manufacturer specifies that concentrations of 100% oxygen (EAN100) and

20.9% oxygen (air or EAN21) be used for calibrating these instruments at a constant flow rate. The instrument is first connected to the EAN100 source, and once the sensor response has come to equilibrium, the instrument is then adjusted until the reading is 100%. If the instrument cannot

be adjusted so that the reading is 100%, then the sensor should be replaced. Once the reading is adjusted, the instrument is disconnected from the EAN100 source, exposed to EAN21, and the

reading noted. If the sensor reading does not return to indicating between 20-22%, then the sensor should be replaced. As the sensor nears the end of its useful life, it is possible to calibrate the sensor to 20.9% in air, but it may read only as much as 85% in 100% oxygen. This makes it

necessary to check the sensor periodically at both EAN100 and EAN21. However, in field applications EAN100 is not always available for calibration of oxygen sensors and analyzers. For

this reason, it is not always possible to explicitly follow the manufacturer’s procedures to check

instrument calibration in the field. However, EAN21 is generally readily available at varying pressures or flow rates in the form of SCUBA cylinders or tire fill stations.

3.2.1 Performance Curve Modeling

Using Data Modeling, the performance curve of a new or in-calibration oxygen analyzer instrument can be modeled using varying flow rates of air as a stimulus. Using a flow meter, known flow rates of air can be applied to the calibrated sensor and the oxygen percentage or

voltage output from the sensor recorded. From this, a tuned equation based Data Model is constructed that can be used in the field using the same flow rates. All that is then required in the

field to determine if the instrument or sensor is still calibrated is a flowing air source to yield the constant flow rates. Alternatively, a flow meter to regulate the flow of air across the sensor in a quantified manner may be used.

The sensor exhibits a small hysteresis effect that slightly elevates the reading of oxygen and then

rebounds to a low value. We propose to characterize this hysteresis curve for a calibrated sensor and then compare to this model in the field to determine instrument calibration based solely on airflow induced perturbations to shift the hysteresis curve.

To do this, an oxygen analyzer sensor that was in calibration was characterized. Ten different

measurement realizations were recorded as shown in Fig. 3, consisting of 1) Flow rate in liters per minute (L/min), 2) the power on equilibrium oxygen sensor percentage readings, 3) instrument oxygen percentage reading with air flowing across the sensor, and 4) final oxygen

sensor percentage reading (including any hysteresis effects) once the airflow was removed from the sensor. Using the power on instrument readings as a baseline, the amount of change in the

instrument readings while air was flowing across the sensor from the initial power on value and the change in the final readings from the power on values (thereby capturing the sensor hysteresis) were calculated.

0.1 -0.05 0.2 -0.2 -0.35 -0.5 -0.2 -0.4 0.1 0.0 -0.1 -0.8 0.2 -0.1 0.2 0.15 0.1 -0.1 -0.55 -0.8

6.0 22.2 22.3 22.156.0 22.3 22.5 22.15.75 22.0 21.65 21.55.75 21.6 21.4 21.25.75 22.3 22.4 22.34.8 22.0 21.9 21.24.8 21.4 21.6 21.34.75 21.75 21.95 21.94.7 22.3 22.4 22.24.5 22.4 21.85 21.6

-0.2 -0.5 0.1 -0.3 -1.2 -0.9 0.2 0.3 0.1 -0.1 0.05 -0.2 0.1 -0.1 0.1 -0.1 0.1 0.0 0.1 -0.1

6.0 20.4 20.2 19.96.0 20.9 21.0 20.65.9 20.5 19.3 19.65.75 20.0 20.2 20.35.75 20.1 20.2 20.04.6 20.5 20.55 20.34.5 20.4 20.5 20.34.5 20.5 20.6 20.44.5 20.5 20.6 20.54.5 20.5 20.6 20.4

Flow

Rate

(L/min)

Flow

Rate

(L/min)

Calibrated +6% Out of Calibration

Mean = 20.3

Standard Deviation = 0.34

Mean = 21.9

Standard Deviation = 0.40

% O 2

(Power ON)

(P)

% O2

(AirFlow)

(A)

% O 2

(Final)

(R)

A-P R-P

(Dr)

% O 2

(Power ON)

(P)

% O 2

(AirFlow)

(A)

% O 2

(Final)

(R)

A-P R-P

(Dr)

Rollup of All Calibrated

% O 2 Measurements

Rollup of All Out of Calibration

% O 2 Measurements

Fig. 3. Tabulated sensor measurements.

An additional ten measurements (also shown in Fig. 3) from a +6% calibration biased sensor

were also measured using the same conditions and flow rates as the nominal cases. Both the calibrated and biased measurements were within the manufacturer’s accuracy specifications for

acceptability for EAN21 (between ~ 20-22% as stated above), making it impossible to conclude that the sensor was out of calibration based on the measurements alone.

The Data Modeling process ranks these inputs and builds a tuned equation, which characterizes

the performance curve using only knowledge derived from the nominal sensor measurements. The tuned Data Model only required the final instrument reading (R) and the sensor hysteresis

(Dr) to model the performance curve. This yielded 100% correct classification. The equation for this Data Model is shown at the top of Fig. 4, labeled as nom1. If nom1 gives a value between 0.499 and 0.501, the sensor is calibrated. Any other values indicate the sensor is out of

calibration. The lookup table graphic can be used directly in the field by simply taking the power on and final oxygen analyzer percentage readings to determine the values for R and Dr and the

calibration status. Alternatively, the equation for nom1 itself can be evaluated if a calculator is at hand. Additionally, the known off-nominal cases were run against the tuned Data Model and all ten (100%) were flagged as off-nominal, indicating that the sensor needed to be either

recalibrated or replaced.

nom1 111.632Dr⋅ 383.752 R⋅+ 10.818 Dr2.⋅ 18.9819R

2.⋅− 10.86 Dr⋅ R⋅−+ 7.736 10-2⋅ Dr

3.⋅+ .312949 R3.⋅ .5324 R⋅ Dr

2.⋅−+ .2641 Dr⋅ R2.⋅ 2585.369−+:=

-1.2 Final – Initial Readings (Dr) 0.619.3

Fin

al R

eadin

g (

R)

2

0.9

17

.5 F

ina

l R

ea

din

g (

R)

2

2.7

-3.1 Final – Initial Readings (Dr) 2.5

3 SigmaTestPoint

Boundary

TrainingBoundary

3 SigmaTraining

DataBoundary

Fig. 4. Lookup table generated using Data Model for oxygen sensor monitoring and calibration.

The LUT shown in Fig. 4 can be used to see if the oxygen sensor is out of calibration. The

current values for R and Dr are determined and the intersection for the two values can be read directly from the chart. If the value falls in a white region, the sensor is calibrated. If the value

falls into black, the instrument is out of calibration. The tuned equation Data Model is able to predict that regions of calibration and out of calibration exist outside of the training box region.

On the left side of Fig. 4, the training region and boundary for the tuned equation based Data Model is enlarged and shown in detail. The box on the interior represents the training boundary and encompasses the region where the training examples (shown as points on the graph) reside.

Shown on the right side of Fig. 4 is a larger interrogation region derived from the tuned equation based Data Model, the same training boundary box, and the off-nominal points used for testing

the model.

4. CONCLUSIONS

Data Modeling has been successfully applied to inventory flow and instrument calibration

monitoring. A Data Model was created that defined the boundary for nominal inventory

dynamics. Additionally, Data Modeling was used to demonstrate in field Miniox calibration monitoring. This was accomplished by modeling the response hysteresis of the sensor to various

flow conditions. This created an equation and lookup table that can be used in the field to determine if the Miniox is still in calibration without the use of a pure O2 source.

ACKNOWLEDGMENTS

The authors would like to thank Marvin Carroll, Tec-Masters, Level-13, Licht Strahl Engineering INC, and Kristi Jaenisch and Technical Dive College for the use of the

MAJQANDA algorithm suite during the course of this work, and Greg Ogle of Southeastern Divers Inc. for his assistance in the oxygen sensor data collection process.

BIBLIOGRAPHY

1. Blanchard, B. Logistics Engineering and Management. Englewood Cliffs, NJ: Prentice-Hall, 1981.

2. Buzacott, J.A. and J.G. Shanthikumar. Stochastic Models of Manufacturing Systems.

Englewood Cliffs, NJ: Prentice-Hall, 1993. 3. Fraden, J. Handbook of Modern Sensors. New York: AIP Press, 1996.

4. Grant, E. Statistical Quality Control. New York: McGraw-Hill, 1952. 5. Jaenisch, H.M. and J.W. Handley, “Data Modeling for Radar Applications,” Proceedings of

the IEEE Radar Conference 2003, May 5-8, 2003. Huntsville, AL.

6. Jaenisch, H.M., J.W. Handley, L. Bonham, “Enabling Calibration on Demand for Situational Awareness”, Army Aviation Association of America (AAAA), Tennessee Valley Chapter,

Huntsville, AL, February 12, 2003. 7. Jaenisch, H.M., J.W. Handley, J.C. Pooley III, S.R. Murray. “Data Modeling For Fault

Detection,” 58th MFPT, Virginia Beach, VA, 2003.

8. Jaenisch, H.M., J.W. Handley, K.H. White, J.W. Watson Jr., C.T. Case, C.G. Songy, “Virtual Prototyping With Data Modeling”, Proc. SPIE. Seattle, WA July 7-11 2002.

9. Lamb, J.S. The Practice of Oxygen Measurement for Divers. Flagstaff, AZ: Best, 1999. 10. Lotfi, V. and C. Pegels. Decision Support Systems for Operations Management & Science.

Chicago, IL: Irwin, 1996.

11. McNitt, L.L. BASIC Computer Simulation. Blue Ridge Summit, PA: TAB Books, 1983. 12. MiniOx I Oxygen Analyzer Instruction Manual. Pittsburg, PA: MSA Company, 1995.

13. Webster, J.G. Medical Instrumentation: Application and Design. Boston, MA: Houghton Mifflin, 1978.

BIOGRAPHY

Dr. Holger Jaenisch has over 20 years experience working in the field of applied mathematics and algorithm creation. Dr. Jaenisch’s experience has included the application of Data Modeling to advanced discrimination algorithms and concepts, data, message, and image compression

techniques and simulation modeling. His expertise covers the areas of signal and data processing, image processing, and the modeling of processes, simulations, and systems with neural and

equation based architectures. His concepts have been successfully applied in the development of two commercially sold software packages for nonlinear data analysis.

APPENDIX

Dynamic Inventory Model Inputs

Number of customers 2 Number of products 2 #######################################Customer 1 Product 1# Demand levels 2 Level 1 # Items 1 Prob of Demand Level .8 Level 2 # Items 10 Prob of Demand Level .2 # Lead times 2 Lead Time 1 # Periods 5 Prob of lead time level .8 Lead Time 2 # Periods 15 Prob of lead time level .2 Cost per order placed ($) 100 Annual per unit- holding cost ($) 10Stockout cost per unit ($) 100 Number of periods per year 52 Order quantity 2 Reorder point 2 Beginning inventory 10 Number of periods to simulate 256 #######################################Customer 1 Product 2# Demand levels 1 Level 1 # Items 3 Prob of Demand Level .5 # Lead times 2 Lead Time 1 # Periods 2 Prob of lead time level .5 Lead Time 2 # Periods 4 Prob of lead time level .5 Cost per order placed ($) 10 Annual per unit- holding cost ($) 1Stockout cost per unit ($) 5 Number of periods per year 52 Order quantity 20 Reorder point 10 Beginning inventory 20 Number of periods to simulate 256 #######################################

Customer 2 Product 1# Demand levels 2 Level 1 # Items 3 Prob of Demand Level .7 Level 2 # Items 4 Prob of Demand Level .3 # Lead times 1 Lead Time 1 # Periods 3 Prob of lead time level 1 Cost per order placed ($) 100 Annual per unit -holding cost ($) 20Stockout cost per unit ($) 30 Number of periods per year 52 Order quantity 10 Reorder point 25 Beginning inventory 5 Number of periods to simulate 256 #######################################Customer 2 Product 2# Demand levels 3 Level 1 # Items 1 Prob of Demand Level .5 Level 2 # Items 5 Prob of Demand Level .25 Level 3 # Items 10 Prob of Demand Level .25 # Lead times 3 Lead Time 1 # Periods 1 Prob of lead time level .25 Lead Time 2 # Periods 5 Prob of lead time level .25 Lead Time 3 # Periods 10 Prob of lead time level .5 Cost per order placed ($) 125 Annual per unit -holding cost ($) 25Stockout cost per unit ($) 10 Number of periods per year 52 Order quantity 15 Reorder point 20 Beginning inventory 20 Number of periods to simulate 256 #######################################

Dynamic Inventory Model Source Listing 'Dynamic Inventory Model (DIM) 'Holger M. Jaenisch, Ph.D. and Jamie Handley CLS DIM l(50, 2), d(50, 2), y(50) OPEN "input.log" FOR OUTPUT AS #1 INPUT "Number of customers" ; ncust PRINT #1, "Number of customers"; ncust INPUT "Number of products"; nprod PRINT #1, "Number of products"; nprod PRINT #1, "#######################################" REDIM max1(ncust) mx1 = 0 FOR nc = 1 TO ncust REDIM nrow(nprod) maxval = 0 FOR np = 1 TO nprod PRINT PRINT "Currently on Customer"; nc; "and Product"; np PRINT #1, "Customer"; nc; "Product"; np 916 GOSUB 1106 920 GOSUB 1206 924 GOSUB 1306 928 GOSUB 1406 932 GOSUB 1506 nrow(np) = n2 IF n2 > maxval THEN maxval = n2 IF U$ = "1" THEN 924 IF U$ = "2" THEN 920 IF U$ = "3" THEN 916 NEXT np 'rollup by customer REDIM w(maxval, 8) FOR np = 1 TO nprod zz$ = "c" + FORMAT$(nc, "0") + "p" + FORMAT$(np, "0") + ".dat" OPEN zz$ FOR INPUT AS #2 FOR i2 = 1 TO nrow(np) FOR kz = 1 TO 8 INPUT #2, h w(i2, kz) = w(i2, kz) + h NEXT kz NEXT i2 CLOSE #2

NEXT np zz1$ = "c" + FORMAT$(nc, "0") + ".dat" OPEN zz1$ FOR OUTPUT AS #2 FOR i2 = 1 TO maxval FOR kz = 1 TO 8 PRINT #2, w(i2, kz); NEXT kz PRINT #2, NEXT i2 max1(nc) = maxval IF maxval > mx1 THEN mx1 = maxval CLOSE #2 NEXT nc 'final rollup REDIM w(mx1, 8) FOR nc = 1 TO ncust zz1$ = "c" + FORMAT$(nc, "0") + ".dat" OPEN zz1$ FOR INPUT AS #2 FOR i2 = 1 TO max1(nc) FOR kz = 1 TO 8 INPUT #2, h w(i2, kz) = w(i2, kz) + h NEXT kz NEXT i2 CLOSE #2 NEXT nc zz1$ = "rollup.dat" OPEN zz1$ FOR OUTPUT AS #2 FOR i2 = 1 TO mx1 FOR kz = 1 TO 8 PRINT #2, w(i2, kz); NEXT kz PRINT #2, NEXT i2 CLOSE #2 PRINT CLOSE END 1106 PRINT PRINT "Probability distribution for demand period" PRINT PRINT "3 Demand levels = Low, Med, High"

PRINT "Input Demand levels and probabilities in ascending order" PRINT "to build histogram" PRINT INPUT "Number of levels of demand"; k1 PRINT #1, "# Demand levels"; k1 PRINT "demand, probability" p = 0 e1 = 0 FOR i = 1 TO k1 PRINT "Level"; i INPUT "# Items"; d(i, 1) INPUT "Prob of Demand Level"; d(i, 2) PRINT #1, "Level"; i; "# Items"; d(i, 1); PRINT #1, "Prob of Demand Level"; d(i, 2) e1 = e1 + d(i, 1) * d(i, 2) p = p + d(i, 2) d(i, 2) = p NEXT i PRINT PRINT "Expected demand for period"; e1 PRINT PRINT "Probability distribution for lead time" PRINT PRINT "Number of different lead times"; INPUT k2 PRINT #1, "# Lead times"; k2 PRINT PRINT "3 Lead time levels = Low, Med, High" PRINT "Input Lead time levels and probabilities in" PRINT "ascending order to build histogram" PRINT PRINT "If period is defined as month” PRINT "# Periods/yr=52 below), 3 weeks" PRINT "would be 0.75" PRINT PRINT "lead time (# time periods), probability" p = 0 e2 = 0 FOR i = 1 TO k2 PRINT "Lead Time"; i; INPUT "# Periods"; l(i, 1) INPUT "Prob of lead time level"; l(i, 2) PRINT #1, "Lead Time"; i; "# Periods"; l(i, 1); PRINT #1, "Prob of lead time level"; l(i, 2) e2 = e2 + l(i, 1) * l(i, 2) p = p + l(i, 2) l(i, 2) = p NEXT i PRINT PRINT "Expected lead time", e2 RETURN 1206 PRINT PRINT "Cost per order placed ($)"; INPUT c1 PRINT #1, "Cost per order placed ($)"; c1 PRINT "Annual per unit-holding cost ($)"; INPUT c2 PRINT #1, "Annual per unit -holding cost ($)"; c2 PRINT "Stockout cost per unit ($)"; INPUT c3 PRINT #1, "Stockout cost per unit ($)"; c3 PRINT PRINT "52 periods/year = week, 12 periods/year = month" PRINT "Number of periods per year"; INPUT n1 PRINT #1, "Number of periods per year"; n1 1299 RETURN 1306 PRINT PRINT "Order quantity"; INPUT q PRINT #1, "Order quantity"; q PRINT "Reorder point (Quantity threshold for reorder)"; INPUT r PRINT #1, "Reorder point"; r PRINT "Beginning inventory"; INPUT b PRINT #1, "Beginning inventory"; b PRINT "Number of periods to simulate"; INPUT n2 PRINT #1, "Number of periods to simulate"; n2 PRINT #1, "#######################################" FOR i = 1 TO k3 y(i) = 0 NEXT i RETURN 1406 t1 = 0 t2 = 0 t3 = 0 t4 = 0 k3 = l(k2, 1) zz$ = "c" + FORMAT$(nc, "0") + "p" + FORMAT$(np, "0") + ".dat" OPEN zz$ FOR OUTPUT AS #2 FOR i2 = 1 TO n2 'UPDATE PIPELINE GOSUB 2006

'GENERATE DEMAND GOSUB 2106 'PLACE ORDER GOSUB 2206 PRINT #2, t1; t2; t3; t4; jwh; b; s2; s1 t2 = t2 + b t3 = t3 + s2 t4 = t4 + s1 NEXT i2 CLOSE #2 'DISPLAY RESULT GOSUB 2306 RETURN 1506 PRINT PRINT "MENU OF OPTIONS" PRINT "1 TRY ANOTHER SET OF DECISIONS" PRINT "2 TRY ANOTHER SET OF COSTS " PRINT "3 TRY ANOTHER SET OF PROBABILITIES" PRINT "4 NEXT/TERMINATE " PRINT PRINT "ENTER OPTION"; INPUT U$ IF U$ = "1" THEN 1599 IF U$ = "2" THEN 1599 IF U$ = "3" THEN 1599 IF U$ = "4" THEN 1599 PRINT "INVALID RESPONSE" GOTO 1506 1599 RETURN 'UPDATE PIPELINE 2006 b = b + y(1) FOR i = 1 TO k3 - 1 y(i) = y(i + 1) NEXT i y(k3) = 0 RETURN 'GENERATE DEMAND 2106 p = RND FOR i = 1 TO k1 IF p > d(i, 2) THEN 2116 z = d(i, 1) i = k1 2116 NEXT i IF z > b THEN 2126 s1 = z s2 = 0 GOTO 2130 2126 s1 = b s2 = z - b 2130 b = b - s1 RETURN 'PLACE ORDER 2206 v = b FOR i = 1 TO k3 v = v + y(i) NEXT i jwh = 0 IF v > r THEN 2299 p = RND FOR i = 1 TO k2 IF p > l(i, 2) THEN 2226 l1 = l(i, 1) i = k2 2226 NEXT i t1 = t1 + 1 jwh = jwh + 1 IF l1 > 0 THEN 2236 b = b + q GOTO 1299 2236 y(l1) = y(l1) + q 2299 RETURN 'DISPLAY RESULT 2306 PRINT PRINT "RESULT OF SIMULATION" PRINT "DEMAND", t3 + t4 PRINT "UNITS SOLD", t4 PRINT "SALES LOST", t3 PRINT "ORDERS", t1 PRINT "AVE RAGE INVENTORY LEVEL ", t2 / n2 t5 = (n1 / n2) * c1 * t1 t6 = c2 * t2 / n2 t7 = (n1 / n2) * c3 * t3 t = t5 + t6 + t7 PRINT PRINT "COSTS ADJUSTED TO ANNUAL BASIS" PRINT PRINT "ORDERING", t5 PRINT "HOLDING", t6 PRINT "STOCKOUT", t7 PRINT "TOTAL", t PRINT PRINT "TYPE THE VALUE 1 TO CONTINUE"; INPUT U$ RETURN

DATA MODELING FOR FAULT DETECTION

H. M. Jaenischa, J. W. Handley

a, J. C. Pooley III

b, and S. R. Murray

b

aSparta, Inc., 4901 Corporate Drive, Suite 102, Huntsville, AL 35805

bAmtec, Inc., 500 Wynn Drive, Suite 314, Huntsville, AL 35816

Abstract: Data Models are high order [O(3n)] multi-variable characterizations derived

from simple third order polynomial building blocks. Such functions can be used to build

real-time operational condition diagnostic and prognostic, i.e. fault classification and time

to failure estimates. Data Modeling can achieve anomalous detection while only requiring

nominal (no-fault) conditions for training. This makes Data Modeling an attractive tool

for Novelty Detection and ambiguity resolution between nominal/anomalous; also Data

Modeling can resolve ambiguities in diagnostic calls and manage risk uncertainty in

prognosis estimates of time to fail. This paper presents the theory of how Data Modeling

was successfully applied to Novelty Detection, and how it can be applied to Diagnostic

Ambiguity Resolution and Prognostic Risk Uncertainty Management. Classifier methods

such as adaptive multi-dimensional distance measure neural networks and Divergence

classifiers were unsuccessful when applied to a set of diagnostic vibration feature vectors

with only slightly anomalous conditions. When Data Modeling was applied to the same

feature vector sets, 100% correct classification was achieved.

Key Words: Change detection; Data Modeling; diagnostics; health status; off-nominal;

prognostics; situational awareness; transmission vibration

Introduction: Fault diagnostics of rotordynamic devices such as the gearboxes of

helicopters (BlackHawk, AH64, Chinook, Kiowa, Huey, etc.) require the extraction of

diagnostic features from the spectral signals measured within the gearbox. These features

characterize the operational condition of the vital parts of the gearbox and are derived

from spectral signals comprised of a non-Gaussian signal plus white noise. Because of

this noisy environment, fault diagnosis can be a difficult challenge for most standard fault

diagnosis methods, leading to misdiagnosis of fault conditions. Techniques such as Phase

Coherent Filtering (PCF) can be used to boost the signal-to-noise (SNR) ratio and better

estimate the signal before feature extraction. Once this is done, features are extracted and

a Data Modeling equation based classifier algorithm built to perform fault detection [1].

Feature Vectorization: Hybrid time and spectral analysis techniques are used for feature

selection. The presence of severe noise makes regular diagnostic evaluation difficult. In

this environment, the underlying cause of an anomaly can be difficult to determine.

Anomalies are analyzed using hybrid techniques in order to determine the relevant

causality relationship to the shaft synchronous frequency or its harmonics that are

produced by the gearing and drive train reducers and multipliers. A causality relationship

determines when anomalies may indicate the presence of distress due to a defect. The

presence of these defect distresses can be seen as sidebands originating from modulation

of the component defect with the shaft synchronous frequency. In order to achieve

effective fault diagnosis, it is necessary to look at the combined effect of Amplitude

Modulation (AM) and Frequency Modulation (FM) because AM and FM rarely exist in

isolation. The combined effect of AM and FM increases the number and/or amplitude of

the sidebands and makes their pattern asymmetrical due to reinforcement and

cancellation. The result of AM and FM modulations has different phase relationships,

making it necessary to use higher order spectral analysis techniques in the form of the

Hilbert Transform to filter and isolate contributing effects. Once filtering is done, a

statistical acceptance criterion that is derived from the analysis process is used as a

fault/false alarm discriminator. Traverses outside of this region constitute severity of the

fault and the rate of the traversal is the fault degradation rate. This type of hybrid analysis

fault determination criterion also allows for the resolution of ambiguous cases.

Data Modeling: Data Modeling is the process of finding a mathematical expression that

provides a good fit between given finite sample values of the independent variables and

the associated values of the dependent values of the process. Data Modeling finds a Data

Model that fits the given sample of data. This process involves finding both the

functional form of the Data Model and the numeric coefficients for the Data Model. Data

Modeling differs from conventional methods such as linear, quadratic, and polynomial

regression techniques that only solve for the numerical coefficients for a function whose

form is specified ahead of time. Data Modeling also differs from genetic algorithms and

symbolic regression because it incorporates fractal interpolation functions for achieving

mathematical modeling of complex multivariable processes. The Data Modeling process

of discovering automatic functions is an evolutionary/genetic directed approach where

the set elements consist entirely of simple polynomial building blocks [2].

Change Detection: The power of statistical quality control and specifically control charts

is the ability to recognize when a process is not in statistical control. One commonly used

types of control chart is the X-Bar Chart defined by the upper control limit (UCL) and the

lower control limit (LCL)

.2

2

RAXLCL

RAXUCL

X

X

−=

+= (1)

The X-Bar chart plots how far away from the average value of a process the current

measurement falls. According to Shewhart [3], values that fall outside of three standard

deviations of the average have assignable causes and are labeled as being out of statistical

control (off-nominal).

Analogous to the X-Bar chart, a model of the complex process that is in statistical control

is called a Data Model [4]. A Data Model can be built by training all known nominal

cases against a constant output value. Overfitting makes the model more sensitive to

feature values or groups of feature values that fall outside the training boundaries.

Whenever these new features or groups of features begin to depart from within the valid

training boundaries, off-nominal or tip-off conditions are flagged.

Calibration On Demand (COD) is the ability to modify the boundaries between nominal

and off-nominal conditions as the evolving situation requires. This boundary is defined

by a set of control parameters. COD allows restricted pockets or segments within

previously declared nominal or off-nominal regions to be changed to the other condition

through the use of a lookup table (LUT). COD can be used in real-time to indicate when

incoming data sets are of a form not anticipated during simulations and not accounted for

previously. Off-nominal detection can then be used to initiate the generation of additional

feature sources to deal with the new off-nominal information in the form of a new

classifier. Fig. 1 illustrates how new gearbox accelerometer measurements are converted

into features and analyzed to detect if a change has occurred in system status from

nominal to yield tip-off conditions from the model.

Tip-Off

Tip-Off

Nominal

Equation

Output

σ

T = w0+w1x1+w2x2+w3x3 +w4x12

+w5x22 +w6x3

2 +w7x13 +w8x2

3

+w9x32 +w10x1x2 +w11x1x3

+w12x2x3

Skewness Standard

Deviation

KurtosisRollup

Stats

Data Model

Measurement

DataAccelerometer

Z

Scores

Gear

Box

[1 3 –2 4 –2 1 4]

Data

SeriesPDF

Tip-Off

Tip-Off

Nominal

Tip-Off

Tip-Off

Nominal

Equation

Output

σ

T = w0+w1x1+w2x2+w3x3 +w4x12

+w5x22 +w6x3

2 +w7x13 +w8x2

3

+w9x32 +w10x1x2 +w11x1x3

+w12x2x3

Skewness Standard

Deviation

KurtosisRollup

Stats

Data Model

Measurement

Data

Measurement

DataAccelerometerAccelerometer

Z

Scores

Z

Scores

Gear

Box

Gear

Box

[1 3 –2 4 –2 1 4]

Data

Series

Data

SeriesPDFPDF

Fig. 1. Calibration on Demand using Data Modeling.

COD can be used in real-time to indicate when incoming data sets are of a form not

previously accounted for or anticipated during training on either actual or simulated data

sets. COD allows sensing of off-nominal conditions while being trained exclusively on

nominal conditions. This is a key theoretical enabling concept for Situational Awareness

via algorithms.

A use for COD is in monitoring individual sensor measurements, feature vector elements,

and overall system snapshots for change detection. Individual sensor measurements

provide a characterization of system level conditions. This yields an estimate of nominal

or off-nominal condition.

Rather than use the actual input feature values, we use parametric and non-parametric

descriptions of the entire feature value set. This can include the conversion of features to

standardized Z-scores. This treats the feature value set as a time series and models the

distribution by overfitting to a minimized regression function. By setting the target value

of the regression to minimize and then characterizing the polynomial fluctuation around a

minimum, in practice the output feature value used for training is ½.

An overfitted Data Model can be generated that uses Shewhart’s Control Chart theory to

monitor if the modeled process is under Statistical Process Control. In Data Modeling,

the trial limits are set using the polynomial output from the training data. The upper and

lower boundaries are assigned to be one-half of a standard deviation above and below the

mean value of the polynomial output from the training data.

COD can evaluate the operational ranges and feature regimes of proposed classifiers.

This method shows regions of nominal behavior that indicate adequate coverage by the

proposed classifier. It is also very important to note that even though the ranges of input

features may be the same as seen during training, which is easily tested without resorting

to Data Modeling, we can also detect instantly combinations of incoming feature values

that are not anticipated or that are within valid ranges but not seen before. This off-

nominal detection is a tip-off condition that can be used to predict additional regions that

need to be populated by decision-making classifiers or decision transfer nodes.

Feature Vector Description: Feature vector data sets were synthetically generated for

both nominal and varying degrees of anomalous conditions as recorded by a single

gearbox accelerometer [5]. Generally, the accelerometer data is used to generate a feature

vector set or matrix denoted by

.

100,2,1,

100,22,21,2

100,12,11,1

2

1

,

=

=

mmmm

nm

aaa

aaa

aaa

FV

FV

FV

FV

L

MMMM

L

L

M

(2)

The vibration data recorded by this accelerometer goes through a process called

segmenting before the feature vector is generated. Segmenting is used for reducing noise

and is also known as synchronous averaging. The segmentation process generates a total

of 100 pieces of data that are then in turn used to generate a total of 100 feature vectors.

Each feature vector FVm (row vector) contains 100 elements that are generated using

various nominal and anomalous diagnostic discrimination techniques. The results of these

diagnostic discrimination techniques are converted into Z-scores [6] using

NZ

i

i

i

i

α

αα σ

µα −=

( )ii

i

M

M

2=α for even i. (3)

These Z-score values ranged in values from –4 to 4. The Z-scores from all 100 pieces of

data segmented above and from all 100 different diagnostic discrimination techniques

make up a feature vector matrix that is a total size of 100x100.

The synthetically generated data used is representative of vibration data recorded using

0.1 second intervals for durations of up to 12 seconds. A total of 4 different anomaly

percentages were used (1%, 5%, 10% and 15%) corresponding to 1, 4, 7, and 10

anomalous feature vector elements (matrix columns). The single anomalous feature

vector element case (1%) represents a bearing fault, and the increasing percentages of

anomalous feature vector elements up to 15% represent the propagation of this bearing

fault and its effects on the other critical components of the gearbox (such as deterioration

of the bearing, races, gears, and cage). These 4 different anomaly percentages were

varied across 4 different severities of anomaly value (plus or minus 1.5, 2, 3, and 4) for a

total of 16 anomalous feature vector matrices. In addition, a feature vector matrix of

nominal behavior was also generated (values between –1 and 1), for a total of 17 feature

vector matrices.

Model Construction: This nominal feature vector matrix was used to build an equation

based Data Model for nominal/off-nominal detection. The feature data consisted of 100

different Z-score values described above that were derived from different diagnostic

discrimination techniques. Only Features 21-90 were used as input features for the Data

Model. Features 1-20 were ignored because their values did not change across all of the

examples provided, and Features 91-100 were ignored because they were rollup Z-scores

(91-94) or zeros (95-100). For the 70 Z-scores used, there were a total of 100 examples of

nominal behavior generated as described above that were used in constructing the Data

Model. It should also be noted that although the Z-score values for the data used were in

the range –4 to 4, values used in actual Data Model training were left in raw from ranging

from –4096 to 4096. This was done to save the processing step of converting all of the

synthetically generated data from raw Z-scores to final Z-scores.

The conventional approach for performing novelty detection would be to take examples

of all 70 Z-scores for both nominal and anomalous cases and construct a high

dimensional (70 dimensional) classifier that would be used to predict if unknown sets of

Z-scores represent a nominal or anomalous case, and if anomalous, the type of anomaly.

The approach taken by the authors differs from the conventional approach. First, our

approach uses only nominal behavior examples to construct the Data Model. This Data

Model is then used to determine if a new unseen set of Z-scores is nominal/anomalous

using the principles of Shewhart’s Control Chart theory to determine when the process is

out of statistical control.

Second, our approach does not directly use any set or subset of the 70 Z-scores as input

into the Data Model. Instead, each of the 100 nominal examples of 70 Z-scores provided

was rolled up by characterizing the Z-score distribution independently. This was done for

each nominal example using the following parametric and non-parametric statistics:

∑=

=N

j

jNxx

1

1 (4) )min( xMin = (5)

∑=

− −=N

j

jNxx

1

2

112 )(σ (6) )max( xaxM = (7)

∑=

−=

N

j

j

N

xxSkew

1

3

1

σ (8) MinMaxRange −= (9)

31

4

1 −

−= ∑

=

N

j

j

N

xxKurt

σ (10) (11) 22 )1( σχ −−= NN

=→∆+

N

NJ

Range

Ji

Ji

1log

log

limmin)Re(0

1

σ (12)

++= ∑

−

=

−+1

1

21 )(1log1 1

N

j

Range

xx

Nf

jjD (13)

}))]((max[))(({10

1

∑=

−=j

i fPDFfPDFFOM αα (14)

The f(α) term in Eq. (14) is a multi-fractal spectrum [7]. Using this approach, each of the

70 Z-scores that represented a single example were reduced to 11 descriptive statistical

features that both adequately characterized the Z-score distribution for each of the 100

examples and reduced the quantity of inputs to be processed.

Once the rollup features in Eq. (4) through Eq. (14) were calculated, a Data Model was

generated using a target regression value of ½ as the nominal output. Once the equation is

trained, boundaries are placed on either side of the target regression value that

characterizes the fluctuation of the polynomial about ½. This is shown in Fig. 1, with

values outside of the boundary representing anomalous conditions. The Data Model

generated is given in the Appendix, and the boundaries that characterize the polynomial

fluctuation were set at 0.499 and 0.501.

The contribution of each rollup feature is given in Table I, and is the number of

occurrences of each feature in the final Data Model. The correlation of each rollup

feature with the target regression value is also given along with eigenranking values. The

eigenranking values are calculated by first determining the most highly correlated input

with the target regression value and removing it from the list. Next, the most highly

correlated feature remaining in the list is compared with the features that have been

removed from the list to determine cross-correlation. Eigenranking is then calculated by

subtracting the highest cross-correlation from the correlation with the target regression

value, thereby determining its new information contribution. This feature is then removed

from the list and the process repeated until all features have been processed.

Table I.

118.685.598.643.780.885.527.511.035.398.668.

118.100.035.016.005.110.103.022.054.064.002.

22171717171732324747

)11()9()5()10()12()6()7()13()14()4()8(

−−−−−−−−−−−−−

ngEigenRanki

nCorrelatio

Occurrence

Equation

Model Performance: Using these threshold values, the original nominal feature vector

matrix rollup stats were then passed through the Data Model and all 100 (100%) were

correctly classified as nominal. We chose from the anomalous cases the two cases that

were felt to be the most stressing. These cases were the single anomalous feature vector

element with the closest value to nominal (1% with values near plus or minus 1.5) and

the 10 anomalous element case with the closest value to nominal (15% with values near

plus or minus 1.5). These same cases had been looked at in previous work using both a

neural network approach and a divergence classifier to perform novelty detection. Neither

of these methods was able to achieve 100% correct classification as in the Data Modeling

approach.

These two cases provide the support vectors for the cases, and if the Data Model can

correctly identify these cases, all other nominal or anomalous cases can also be correctly

typed. Support vectors are the points that lie on the boundaries of the solution space and

define the complex decision boundary between nominal and anomalous behavior. The

support vectors are all of the non-redundant and non-coupled examples that are necessary

to define a model of this boundary. Examples that recast the same information contained

by other examples, but in different terms such as derivative terms or highly cross-

correlated features, would be redundant and would not be support vectors.

Each of these feature vector matrices was rolled up with the same characterizing statistics

given in Eq. (4) through Eq. (14), and these rollup statistics were passed through the Data

Model. For both cases, 100% of the cases were correctly classified as anomalous. These

results are shown graphically in Fig. 2.

The classical chi-squared test was applied as a discriminant for most stringent anomalous

cases and required a significance level of 0.25 ( to discriminate nominal from

anomalous. For the same cases, Novelty Detection using Data Modeling could

discriminate nominal from anomalous at significance level 0.44 ( . Therefore, the

Data Modeling approach is a superior method for preventing False Safes, and

significantly reduces misdiagnosis and associated consequent maintenance cost.

)75.02χ

)56.02χ

Nominal

1 Anomaly 10 Anomalies

26.0

0.0

Nominal

Nominal

Off-Nominal

Nominal

Nominal

Off-Nominal

11.7

.501

.499 Fig. 2. Results of Data Model used for nominal and anomalous test cases.

On the right side of Fig. 2 is a Decision Map generated using the Data Model. Along the

X-axis of the Decision Map is the feature given in Eq. (12) and on the Y-axis the feature

given in Eq. (14). These two features were allowed to range within their original training

boundaries plus or minus one standard deviation (σ). All other features were held at their

median value. The white box represents the valid training ranges of the original nominal

cases. The Decision Map displays the complex nature of the decision space that is

captured by the equation model, and that is not captured by other statistical

characterizations such as quadratic classifiers or Bayesian networks. The Decision Map

can be used as a direct lookup table (LUT) for real time implementations. This allows

restricted pockets or segments within previously declared nominal or off-nominal regions

to be changed to the other condition as the need arises by modifying the values in the

LUT directly and bypassing the original equation model. The Decision Map shown in

Fig. 2 can be reproduced using the code and the associated values listed in the Appendix.

Application to Diagnostic Ambiguity Resolution and Prognostic Risk Uncertainty

Management: Performing Diagnostic Ambiguity Resolution results in the use of two

separate networks: 1) Novelty Detection Network and 2) Fault Classification Network.

The Novelty Detection Network discussed thus far serves as a general monitor of all

feature vectors simultaneously by examining their Z-score distributions. This indicates if

feature vector Z-scores across all elements are occurring as expected. Once the novelty

detection network determines the current status to be either nominal or anomalous, this

information is then passed to the fault classification network. Data Modeling was used in

the construction of the Novelty Detection Network shown in this work; however, it can

also be used in the construction of both a Fault Classification Network and a Risk

Uncertainty Management (RUM) Network.

The high-resolution Fault Classification Network (Fig. 3) is comprised of four separate

embedded networks: fault detection, fault identification, fault severity ranking, and fault

degradation rate networks. These networks monitor individual feature vector elements to

verify the findings of the Novelty Detection Network, thereby mitigating false positives

and negatives.

Fault classification requires individual feature vector element and feature vector element

grouping information to be extracted and used. In this fashion, individual Data Models

are created for each individual sensor or feature vector element. Each one of the networks

can rely on its own independent features to achieve modeling and tip-off monitoring, and

no requirement is made that all available features be used. The feature driver sub-

selection is evolved by the decision architecture discovery process algorithm and is

entirely adaptive to the nature of the individual feature vector elements.

Rollup

Stats

Novelty

Detection

Network

CAR/COA

Network

Fault Classification Network

Detection

Identification

Severity Ranking

Degradation Rate

Classification Ambiguity Resolver (CAR)

Optimal

Classifier

Fault ID

Severity

Degradation

Rate

Diagnostic

Z-ScoresRollup

Stats

Novelty

Detection

Network

CAR/COA

Network

Fault Classification Network

Detection

Identification

Severity Ranking

Degradation Rate

Classification Ambiguity Resolver (CAR)

Optimal

Classifier

Fault ID

Severity

Degradation

Rate

Diagnostic

Z-Scores

Fig. 3. Classification Ambiguity Resolver (CAR) Network.

The Risk Uncertainty Manager (RUM) (Fig. 4) uses standardized Z-scores as input to

determine the theoretical failure time, probability of failure at the theoretical failure time,

and will directly give probability of failure for Resource Managers to anticipate logistic

and parts inventory management.

Prognosis

Uncertainty

Z-Scores

RUM/COA

Network

Risk

Prediction

Network

Risk Uncertainty Manager (RUM)

Action

Triage

Model

Prognosis

Uncertainty

Z-Scores

RUM/COA

Network

Risk

Prediction

Network

Risk Uncertainty Manager (RUM)

Action

Triage

Model

Fig. 4. Risk Uncertainty Manager (RUM) Network.

Conclusions: Data Modeling has been demonstrated as a viable algorithm for novelty

detection. Data Modeling has enabled anomalous behavior to be detected without a priori

examples of anomalous behavior. In addition, the anomalies detected were of a very

small magnitude and not perceptible to other advanced adaptive algorithms. This initial

success sets the stage for Data Modeling to be used in Diagnostic Ambiguity Resolution

and Prognostic Risk Uncertainty Management.

References

[1] Pooley, J.C. and S.R. Murray, “Rotordynamic Fault Diagnostics Using Phase

Coherent Filtering,” To be published in the Proceedings of the Society for

Machinery Failure Prevention Technology (MFPT), Virginia Beach, VA, April

2003.

[2] Jaenisch, H.M. and J.W. Handley, “Data Modeling for Radar Applications,” To be

published in Proceedings of the IEEE Radar Conference 2003, Huntsville, AL.

[3] Grant, E. Statistical Quality Control. New York: McGraw-Hill, 1952.

[4] Jaenisch, H.M., J.W. Handley, L. Bonham, “Enabling Calibration on Demand for

Situational Awareness”, Army Aviation Association of America (AAAA),

Tennessee Valley Chapter, Huntsville Alabama, Feb 12, 2003.

[5] Lee, C. and J.C. Pooley, “Feature Vector Generation and Fusion Techniques for SH-

60 Helicopter Gearbox Diagnostics,” Final Report to the JAHUMS Project Office,

February 2003.

[6] Nikias, C. L. and A. P. Petropulu. Higher-Order Spectra Analysis: A Nonlinear

Signal Processing Framework. Englewood Cliffs, NJ: Prentice-Hall, 1993.

[7] Jaenisch, H.M., C.J. Scoggins, M.P. Carroll, and J.W. Handley, “Entropy Fractal

Analysis of Medical Images Using ROSETA,” Proceedings of SPIE, Los Angeles,

CA January 24, 1994. Session: Medical Applications of Modern Imaging

Technology.

Appendix

DEFCUR A-ZOPEN "input" FOR INPUT AS #1INPUT #1, maxval, minval, range, mean, stdevINPUT #1, skew, kurt, chisq, Jaen, Hand, rosfomCLOSE #1maxval = (maxval - 3027.78@) / 268.4416@minval = (minval - -3107.3@) / 235.948@range = (range - 6134.3@) / 346.41@mean = (mean - -1.6663@) / 100.0386@stdev = (stdev - 1079.6204@) / 64.3872@skew = (skew - -.0622@) / .3681@kurt = (kurt - 1.9463@) / .4347@chisq = (chisq - -80957000@) / 9532140@Jaen = (Jaen - .32@) / .0527@Hand = (Hand - 1.3741@) / .0071@rosfom = (rosfom - 4.5092@) / .8543@GOSUB COEFF1iflag1 = 0IF l6o3 < .499 THEN iflag1 = 1IF l6o3 > .501 THEN iflag1 = 1l6o3 = LOG(ABS(l6o3))/LOG(10@)OPEN "output" FOR OUTPUT AS #1PRINT #1, l6o3, iflag1PRINT l6o3, iflag1CLOSE #1END

Use the following values to

reconstruct the Decision Map

in Fig. 2:

maxval = 2029

minval = -2545

range = 4710

mean = 16.3072

stdev = 964.8604

skew = 0.0874

kurt = 2.1116

chisq = -70450000

Hand = 1.5248

Jaen = Vary From .1057 to .5859

rosfom = Vary From .0059 to 7.2724

COEFF1:l1o2 = .1779 + .0294*rosfom + .4022*mean -.2041*maxval -.2074*rosfom*rosfoml1o2 = l1o2 + .0407*mean*mean + .0834*maxval*maxval -.033*rosfom*meanl1o2 = l1o2 + .0162*rosfom*maxval -.0081*mean*maxval -.2349*rosfom*mean*maxvall1o2 = l1o2 - .0479*rosfom*rosfom*rosfom -.0132*mean*mean*meanl1o2 = l1o2 -.2092*mean*rosfom*rosfom -.0854*rosfom*mean*meanl1o2 = l1o2 + .117*rosfom*maxval*maxval -.0203*maxval*rosfom*rosfoml1o2 = l1o2 + .0071*maxval*mean*mean + .0362*mean*maxval*maxvall1o3 = -.0549 + .1685*SIN(6.2832*chisq) + .5903*COS(6.2832*chisq)l1o3 = l1o3 -.0716*SIN(6.2832*rosfom) -.2275*COS(6.2832*rosfom)l1o3 = l1o3 + .134*SIN(6.2832*mean) -.1696*COS(6.2832*mean)l1o3 = l1o3 -.0673*SIN(6.2832*Hand) -.4242*COS(6.2832*Hand)l1o3 = l1o3 + .1103*SIN(6.2832*maxval) -.0127*COS(6.2832*maxval)l1o3 = l1o3 -.2275*SIN(6.2832*minval) -.3622*COS(6.2832*minval)l1o3 = l1o3 + .1424*SIN(6.2832*kurt) + .0648*COS(6.2832*kurt)l1o3 = l1o3 + .1099*SIN(6.2832*skew) + .1226*COS(6.2832*skew)l1o3 = l1o3 -.1476*SIN(6.2832*range) -.0191*COS(6.2832*range)l1o3 = l1o3 -.0014*SIN(6.2832*Jaen) -.1282*COS(6.2832*Jaen)l1o3 = l1o3 + .3289*SIN(6.2832*stdev) -.7289*COS(6.2832*stdev)l2o1 = .0244 + .6126*l1o2 -.1719*Hand + 1.4384*l1o3 -.2076*l1o2*l1o2l2o1 = l2o1 + .0004*Hand*Hand -.0239*l1o3*l1o3 + .3297*l1o2*Handl2o1 = l2o1 + .373*l1o2*l1o3 -.1078*Hand*l1o3 -.7229*l1o2*Hand*l1o3l2o1 = l2o1 + .3051*l1o2*l1o2*l1o2 + .0107*Hand*Hand*Handl2o1 = l2o1 -.5415*l1o3*l1o3*l1o3 -.0947*Hand*l1o2*l1o2l2o1 = l2o1 -.0662*l1o2*Hand*Hand + .6466*l1o2*l1o3*l1o3l2o1 = l2o1 -.8223*l1o3*l1o2*l1o2 -.022*l1o3*Hand*Hand + .3764*Hand*l1o3*l1o3l3o2 = .1214 + 1.3191*l2o1 -.1414*skew + .1093*stdev -.1803*l2o1*l2o1 l3o2 = l3o2 -.017*skew*skew -.0793*stdev*stdev + .2141*l2o1*skewl3o2 = l3o2 -.1368*l2o1*stdev + .1585*skew*stdev + .0057*l2o1*skew*stdevl3o2 = l3o2 -.1218*l2o1*l2o1*l2o1 -.0164*skew*skew*skew -.0342*stdev*stdev*stdevl3o2 = l3o2 + .0971*skew*l2o1*l2o1 -.0396*l2o1*skew*skew -.0441*l2o1*stdev*stdevl3o2 = l3o2 -.0611*stdev*l2o1*l2o1 -.0352*stdev*skew*skew + .0889*skew*stdev*stdevl2o2 = .0056 +.5233*l1o2 -.1429*skew + 1.6838*l1o3 + .0186*l1o2*l1o2l2o2 = l2o2 + .0026*skew*skew -.1106*l1o3*l1o3 -.1814*l1o2*skew l2o2 = l2o2 -.2555*l1o2*l1o3 + .5827*skew*l1o3 -.2691*l1o2*skew*l1o3l2o2 = l2o2 -.0629*l1o2*l1o2*l1o2 + .0083*skew*skew*skew -.5017*l1o3*l1o3*l1o3l2o2 = l2o2 + .1548*skew*l1o2*l1o2 + .1036*l1o2*skew*skew + .1486*l1o2*l1o3*l1o3l2o2 = l2o2 -.2098*l1o3*l1o2*l1o2 -.2228*l1o3*skew*skew + .0629*skew*l1o3*l1o3l3o1 = .0288 + 1.1318*l2o2 + .1871*mean -.2148*Hand + .0552*l2o2*l2o2l3o1 = l3o1 -.0372*mean*mean -.0443*Hand*Hand -.1808*l2o2*meanl3o1 = l3o1 + .1393*l2o2*Hand + .1863*mean*Hand + .0386*l2o2*mean*Handl3o1 = l3o1 + .0026*l2o2*l2o2*l2o2 -.038*mean*mean*mean + .0014*Hand*Hand*Handl3o1 = l3o1 -.0529*mean*l2o2*l2o2 -.0314*l2o2*mean*mean -.0274*l2o2*Hand*Handl3o1 = l3o1 -.0019*Hand*l2o2*l2o2 + .0524*Hand*mean*mean -.0337*mean*Hand*Handl4o3 = .1059 + .8827*l3o2 -.2142*Jaen + .0731*l3o1 + .0672*l3o2*l3o2l4o3 = l4o3 -.0812*Jaen*Jaen -.0185*l3o1*l3o1 + .3007*l3o2*Jaenl4o3 = l4o3 -.091*l3o2*l3o1 -.3331*Jaen*l3o1 -.3113*l3o2*Jaen*l3o1l4o3 = l4o3 -.1367*l3o2*l3o2*l3o2 + .037*Jaen*Jaen*Jaen + .5027*l3o1*l3o1*l3o1l4o3 = l4o3 + .3029*Jaen*l3o2*l3o2 -.2689*l3o2*Jaen*Jaen -1.2086*l3o2*l3o1*l3o1l4o3 = l4o3 + .8481*l3o1*l3o2*l3o2 + .3087*l3o1*Jaen*Jaen + .1116*Jaen*l3o1*l3o1l5o1 = -.0615 + .9876*l4o3 + .0708*rosfom -.0493*skew -.0079*l4o3*l4o3l5o1 = l5o1 + .1561*rosfom*rosfom -.0341*skew*skew + .1969*l4o3*rosfoml5o1 = l5o1 -.0968*l4o3*skew + .1803*rosfom*skew -.0531*l4o3*rosfom*skewl5o1 = l5o1 -.0085*l4o3*l4o3*l4o3 + .0471*rosfom*rosfom*rosfom -.0448*rosfom*skew*skewl5o1 = l5o1 + .0122*skew*skew*skew -.0228*rosfom*l4o3*l4o3 +.0763*l4o3*rosfom*rosfoml5o1 = l5o1 + .02*l4o3*skew*skew + .0022*skew*l4o3*l4o3 + .0329*skew*rosfom*rosfoml5o3 = .0213 + 1.0512*l4o3 + .3126*minval + .1807*kurt + .0269*l4o3*l4o3l5o3 = l5o3 + .0188*minval*minval + .0161*kurt*kurt -.0311*l4o3*minvall5o3 = l5o3 -.1121*l4o3*kurt + .1604*minval*kurt -.0744*l4o3*minval*kurtl5o3 = l5o3 + .0141*l4o3*l4o3*l4o3 -.0463*minval*minval*minval -.1443*minval*kurt*kurtl5o3 = l5o3 -.0237*kurt*kurt*kurt -.0743*minval*l4o3*l4o3 -.0773*l4o3*minval*minvall5o3 = l5o3 + .0144*l4o3*kurt*kurt -.0097*kurt*l4o3*l4o3 -.1706*kurt*minval*minvall6o3 = .0065 + .6065*l5o1 -.0362*mean + .3729*l5o3 -.3338*l5o1*l5o1 + .0161*mean*meanl6o3 = l6o3 -.1255*l5o3*l5o3 -.2353*l5o1*mean + .4507*l5o1*l5o3 + .1453*mean*l5o3l6o3 = l6o3 -1.1258*l5o1*mean*l5o3 + .0312*l5o1*l5o1*l5o1 + .0023*mean*mean*meanl6o3 = l6o3 + .3571*l5o3*l5o3*l5o3 + .528*mean*l5o1*l5o1 -.1531*l5o1*mean*meanl6o3 = l6o3 -.702*l5o1*l5o3*l5o3 + .3228*l5o3*l5o1*l5o1 + .1499*l5o3*mean*meanl6o3 = (l6o3 + .6534*mean*l5o3*l5o3)*.0005 + .5001RETURNSTOP

Data Modeling for Virtual Observatory data mining

Holger Jaenisch*a,b, James Handleya, Albert Lima, Miroslav Filipovicc,a, Graeme Whitea, Alex Honsa, Gary Deragopiana, Mark Schneidera, Matthew Edwardsb

aJames Cook University, Centre for Astronomy, Townsville QLD 4811, Australia bAlabama Agricultural and Mechanical University, Department of Physics, Huntsville, AL 35811

cUniversity of Western Sydney, Locked Bag 1797 PENRITH SOUTH DC NSW 1797, Australia

ABSTRACT

< 869.47 -3.27 41.37 602.25 10053.48 620.0042>

We propose a novel approach for index-tagging Virtual Observatory data files with descriptive statistics enabling rapid data mining and mathematical modeling. This is achieved by calculating at data collection time 6 standard moments as descriptive file tags. Data Change Detection Models are derived from these tags and used to filter databases for similar or dissimilar information such as stellar spectra, photometric data, images, and text. Currently, no consistent or reliable method for searching, collating, and comparing 2-D imagery exists. Traditionally, methods used to address these data problems are disparate and unrelated to text data mining and extraction. We explore the use of mathematical Data Models as a unifying tool set for enabling data mining across all data class domains.

Keywords: Data Modeling, data mining, Virtual Observatory, exo-planet, photometric, stellar spectra modeling,

eclipsing binary, video-based adaptive optics, fractal

1. INTRODUCTION

Search and processing time of legacy data1 can be improved using waveform characterization and modeling techniques. In order to decrease the time required to identify similar information in a database, it is necessary to bypass time-consuming sequential processing of complete files.

With files sizes of photometric and spectrographic databases in excess of terabytes of information collected per month, a method other than sequential access search and correlation must be developed and integrated in the early planning stages to insure legacy archive data sets already have hooks and tagging information embedded. The SLOAN Digital Sky Survey2 recently made its second release of stellar object data to the public. This second release consisted of 88 million objects taking up more than 6 terabytes of space. This equates to between 50 kilobytes and 100 kilobytes of information for each object in the SLOAN data release. The SLOAN survey is able to acquire data on 640 objects simultaneously, thereby pushing data acquisition rates into the terabytes per month class. For example, mining the Sloan database for exo-planets requires sequential processing of over 100,000 candidate files. Once candidates are identified, discriminating and resolving binary stars and exo-planets requires another sequential processing pass of 10-20,000 files selected in the first processing step. The flexible file structure of FITS has sufficient overhead room to accommodate several lines in the header of

descriptive numbers. These numbers accurately and uniquely identify the data sets. Because these features are located in the header, processing of the entire file is avoided. Using Data Modeling theory, dynamic descriptive variables given in Equations (8) – (12) and (23) characterize each file in the Virtual Observatory database. Once generated, these six numbers are stored directly in the FITS header of the file, and are extracted for further processing or characterization of the files. *[email protected]; phone 1 256 337 3768; James Cook University; Centre for Astronomy; Townsville, QLD 4811; AUSTRALIA.


We propose to add the following 6 numbers into the FITS header as tags:

1. Square root of 2nd order moment (Standard Deviation) 2. 3rd order moment (Skewness) 3. 4th order moment (Kurtosis) 4. 6th order moment (M6) 5. 8th order moment (M8) 6. Unique physics feature specific to data type such as the wavelength where maximum occurs, classification

of object, Dewey Decimal subject area describing object, file type, or any other physics based descriptor. These 6 features are described in Figure 1 and in Equations (8) – (12) and (23). For the 161 stellar spectra in the Jacoby database, the value of these 6 tags is given in Table 1. Throughout this paper, various applications of these 6 tag feature values are shown. Actual examples of these tag numbers were incorporated into the paper and are found in the abstract for the entire body of the paper, as well as in Figure 18 for the accompanying images in the paper. For the entire paper tag in the abstract, the Dewey Decimal system classification for this subject area is used for tag 6. Figure 2 shows this SPIE paper transformed into a fractal curve. Creation of this curve and the tags is discussed in Section 2.11.

2. EXAMPLES

2.1 Correlation

One classic method for comparing an unclassified spectrum is correlation. Correlation determines which object the current set of measurements is most like. Correlation between two equal length data sets x and y is given by

∑∑∑

−−

−−=

i

i

i

i

i

ii

yyxx

yyxx

yxnCorrelatio)()(

))((

),( (1)

The output value for the correlation ranges in value between 0 and 1, where a value of 1 signifies that the two data sets are identical, and a correlation value of 0 that no similarities exist between the two data sets. Using correlation, unclassified measured stellar spectrum must be compared point by point with each file in the database. Once this processing is completed, there are still several limitations that limit its usefulness. These limitations include:

1. Exhaustive searching – requires presence of actual legacy data files rather than header information 2. Time consuming 3. Lack of a well-defined decision boundary 4. Sensitivity of correlation

Models of extensive databases can be published and distributed easily to investigators by processing file names and tag data rather than full data files to perform local initial searches. Hot links to legacy data files enables data pull on request. Each point in the file and database must be processed for correlations. Next, it is compared against all other correlations for maximum. This requires exhaustive searching. The determination of the maximum correlation is without a well-defined criteria. Figure 3 shows the result of using correlation as a classifier for the Jacoby data set. Although the requested example compared favorably with the Jacoby database (correlated higher than 0.9 with the same spectra from which it was derived), it also scored equally high with 140 out of 161 other members of the same database. This ambiguity makes correct identification difficult without a priori knowledge of truth. This is shown on the graph in Figure 3 with most of the files exhibiting correlations between 0.9 and 1.0.

2.2 Graphics based search methods

Virtual Observatory databases can be studied using graphics based searching3. This discovers and displays spatial relationships that exist across data files without a priori knowledge or assumptions. Automated clustering of groupings within the database allows visual presentation of files as point clouds demarcated by ellipses encompassing clusters and labeled according to average, most frequently occurring, and range of content. From this initial search, groups are easily deleted, merged, post-processed, or Data Modeled. Once an ellipse or group is selected for Data Modeling, a rapid and real-time equation is derived that recognizes the tagged features associated with the selected groups being related. This highly tuned and sensitive Data Model filter now enables rapid search and similarity match or reject with higher precision than correlation or nearest neighbor methods. Figure 4 shows this type of graphics based search engine applied to the 6 tags for the data. The presence of 7 different groups in the clustering space denotes different characteristics exist in the data. 2.3 Stellar object change detection model

If a star is un-catalogued or if there is doubt about its accepted classification, then a new classification must be performed. This new classification is done by comparing its spectrum with an atlas of well-studied spectra of bright stars. Figure 5 shows 13 standard digital spectra from the principal MK spectral types measured with a spectrograph along with a histogram describing the distribution of values in the representative spectra4,5. The number of bins in the histogram is selected to be equal to the number of points in the spectral curve.

A single Data Change Model is generated that uses only the descriptive features of the single unclassified object given in Equations (8) – (12) and (23). The feature values for each of the objects in the database (previously stored in the header) are then passed through the Data Change Model. Stellar objects that are unlike the new set of measurements will tip-off and can be ignored, while objects whose statistics are like the new set of measurements will not tip-off. This simple Data Change Model provides the ability to condense the universe of terabytes of stellar spectra into a short list of candidate files for further processing. An example Data Change Model for this process is shown in Figure 6. This model contains no nested loops and is read directly down the page. Reading this equation down the page is the same as reading the terms in the equation across the page from left to right on a single line. The Data Change Model is orthogonal, meaning that each successive layer in the equation builds on the previous ones and refines the solution. Any intermediate solution layer can be used as the current best estimate of the output. It should be noted that to save space, lines in the Data Model were combined together. 2.4 Modeling of stellar database features

For cases where the stellar class is known but the measured spectrum for the object is unavailable, estimates of the tags are generated from a Data Model of the features themselves. Using the calculated tags for each Jacoby stellar spectra in Table 1, separate independent equation models are generated for each element comprising the tag. These equations are based on the Turlington polynomial model of the form

( )

∑−

=

−

−

+−+−+=

1

2

001.101111 101log)(001.)()(

N

j

xx

jj

j

mmxxmyxT jj

jj

jxx

yym

−

−=

+

+

1

1 (2)

where xi is the stellar classification and yi the individual tag element values in Table 1. Source code for generating these equation models is listed in Figure 7. This code uses the x and y data from Table 1 for each element to generate separate database feature programs. It should be noted that (2) consists of N-2 terms from the portion inside of the summation, or one term less than the number of coefficients generated using the source code in Figure 7. Stellar classification (temperature, sub-temperature, and luminosity) is encoded into a single number that

uniquely rolls up the three different classifications into one. These classifications are combined using

XY.ZZ (3)

where X is the luminosity class, Y is the temperature class, and ZZ is the sub-temperature class. The luminosity class is defined as I = 1, II = 2, III = 3, IV = 4, and V = 5. The temperature class is defined as O = 1, B = 2, A = 3, F = 4, G = 5, K = 6, and M = 7. The sub-temperature class maps directly into ZZ as 05 to 95. Six (6) equation models (1 for each tag element) are generated by transforming the stellar classification into a

single number using (3), and substituting the values from Table 1 for the tag element directly into (2). The tag values in Table 1 and this transformation are all that is required to generate the model. The starting condition (x1, y1) is read from Table 1 directly for each tag element. 2.5 Stellar database modeling using change detection

Each stellar spectrum in the database is characterized using the descriptive statistics described in the next section. Once these descriptive statistics are calculated, Data Change Models are constructed that allow alike database objects to pass through, while rejecting via tip-off those that are different from the training set. These Data Change Models can be run either independently or in a cascade fashion, with the output being the correct stellar classification. In previous work, the authors have demonstrated that Data Modeling can be used to generated a classifier that 100% correctly classifies the 161 stellar spectra in the Jacoby database. This Data Model based classifier was built in a cascade fashion using the values of the spectra at individual spectral lines as input. The resulting Data Model was an O(34) = 81st order polynomial model contained within approximately 200 software lines of code (SLOC). Using change detection, the resultant Data Model is O(33)=27, and is captured by 20 SLOC. Fewer Data Models were required, and each change detector was much shorter and less complicated than the original Data Models generated in previous work. Figure 8 gives an example of a Data Change Model for determining if a stellar spectrum is class O or one of the other temperature classes in the Jacoby database. 2.6 Change detection for brightness curve modeling

Eclipsing binary star light curves can also be header tagged using the same descriptive statistics except for wavelength. Once tagged, these curves can be readily data mined using the same concept demonstrated previously with the Jacoby stellar spectra database. Determining if a star is an eclipsing binary is dependent on the viewing geometry and in turn the orbital inclination of the binary star system. Because of this sensitivity, measurements of the light curve of the star look remarkably different for different orbital inclinations, and therefore viewing geometries. In order to show the applicability of Data Modeling for brightness curve modeling, we first must demonstrate that a change detection model can be built that will recognize light curve data from the same object, but from different viewing aspects and orbital inclinations, as being from the same object. This change detection model must also reject light curves from other objects as being different. To illustrate this, 63 synthetic light curves were generated, characterized, and a Data Change Model constructed. Table 2 lists the parameters used to generate 63 synthetic light curves, examples of which are shown in Figure 9. This model correctly classified all 33 of the original light curves of the object for varying orbital inclinations between 58 and 90 degrees as being the same object. The statistics from the remaining 30 light curves were then passed through the Data Model, and all 30 (100%) were correctly flagged as not being the binary star system of interest. 2.7 Change detection for exo-planet detection

Automatic exo-planet processing6 includes sorting collected photometric data of systems containing either eclipsing binary stars or exo-planets into two classes. Currently, the process of determining if exo-planets are present in a data set consists of taking measurements, correlating the measurements, and stacking them over fixed time periods to reduce the overall noise in the data. Once enough samples have been stacked and the noise reduced, classical pattern matching and search methods are employed in order to detect exo-planets. A simple change detection model such as the one shown in Figure 10 is constructed from the exo-planet data of interest. Once constructed, data from light curves including exo-planets and binary stars can be passed through the Data Model, and only the exo-planet data will be accepted by the Data Model. All other data types will be rejected as not matching the object.

Table 3 lists the parameters used to generate 100 synthetic light curves to demonstrate this concept (50 exo-planet light curves plus the 50 binary star cases) as shown in Figure 9. As before, 100% of the training data containing an exo-planet was flagged correctly. Next, 50 additional synthetic binary star data light curves were generated, characterized, and passed through the change detector. From this 50, 43 were correctly labeled as binary stars and the remaining 7 labeled as exo-planets. This corresponds to a correct classification of 86% of the binary stars and 100% of the exo-planets. This facilitates and demonstrates a filter that pre-qualifies light curves as exo-planetary. Tables 2 and 3 are provided in this paper to enable other investigators to use the same data set for comparison. 2.8 Image Data Modeling

Data Modeling can be used to construct short and concise Turlington and eigenfunction models for extended image morphology. These models can be generated in real-time as the image is acquired, and once generated, can be stored in a much smaller space than the pixel values of the original image. Only a limited number of eigenfunction terms compared to the number of points in the original image need to be saved in order to capture the spatial morphology of the object class. This lends itself to autonomous characterization, analysis, and data mining of images for class as stellar field, nebula, or galaxy. To demonstrate this modeling concept, the authors used the image of M51 given in Figure 11. This image was transformed from 2-D to 1-D using the Hilbert sequence, thereby maintaining 2-D correlation. An eigenfunction based Data Equation Model was then constructed. Results of this model are given in Figure 11, along with the eigenfunction model. Pseudo-code for generating eigenfunction based Data Equation Models is given in Figure 12. A detailed treatment of this method is contained in Reference 7.

2.9 Image selection for stacking and software speckle imaging

We assert as a fundamental premise that nominal occurs more frequently than off-nominal. If the reverse is true,

then the definition simply swaps, making it still hold. Based on this assumption, we can proceed with an unsupervised algorithm that will determine normal from abnormal without a priori knowledge or representation of either. For the application of image stacking and enhancement, we apply this principle to see if a Change Detection Model can be autonomously generated that can identify sharp images from blurry ones without a priori training on these image distinctions. In the same manner that stellar spectra and light curves can be coherently averaged and stacked to remove noise, images of stellar objects can also be stacked once they are registered (correlated with one another). To date, methods for selecting the best images as candidates for stacking have required man-in-the-loop to view the image and make a determination as to whether the candidate image is considered a sharp image. It is desired to rank images in terms of sharpness and only pick the very best images for stacking because it enables further blind deconvolution and image enhancement methods to bring out more fine detail8,9,10,11,12,13. Generation of the Data Model can be done in two different ways: 1) supervised, and 2) unsupervised. In supervised mode, legacy database images are analyzed with man-in-the-loop to determine if individual frames are sharp or non-sharp. Once a few candidate sharp frames are identified, the statistics are extracted from the images and a Data Change Model constructed. The sharp images are considered to be nominal with a value of 0.5, and if the equation Data Change Model yields tip-off, the image is different from the training sharp image and is discarded. Additionally, Data Models may be constructed using sub-block regions of the image, calculating their individual statistics, and averaging either the statistics or the Data Model output in order to make a decision whether the image is sharp or non-sharp. Pseudo code for constructing a Data Change Model is given in Figure 13. In unsupervised mode, no a priori knowledge of images being sharp or non-sharp is required. It is only assumed

that the majority of the images are background (non-sharp). A Data Change Model that flags the images as being either sharp or non-sharp is constructed as shown in Figure 14. Pseudo-code for constructing a Data Change Model unsupervised is also given in Figure 13. For the images analyzed for this paper, the final Data Change Model was determined by placing a tight boundary of 0.0012 on either side of the final sharp image class designation. All images that flagged inside of the two boundaries are considered candidate sharp images, while all others are considered non-sharp images and are discarded from further consideration.

2.10 Image enhancement

Once these images are stacked, they can be processed further for enhancement using Data Modeling. The authors have determined that the Van Cittert deconvolution nonlinear image processing method can be approximated through the application of a series of high pass and low pass kernel filters to an image. Additionally, using knowledge of the number of times the filters will be applied, the individual high pass and low pass filter can be combined together into an equivalent kernel. An equivalent kernel is one that when applied to the image gives the same answer as if a series of individual kernels had been applied to the image one at a time. To illustrate this, consider an image processing sequence consisting of applying a high pass filter, followed by a low pass filter, followed finally by a Gaussian filter to an image. Examples of these filters are

111

1151

111

−−−−−−−−

111

131

111

121

242

121

High Low Gaussian (4) Convolving each of these three kernels together yields a larger kernel that is called the equivalent kernel. This is shown below:

111

1151

111

−−−−−−−−

⊗111

131

111

⊗121

242

121

=

4810841

4210161024

8109415294108

10161522521521610

8109415294108

4210161024

14810841

−−−−−−−−−−−−−−−−−−−−−−

1

−−−−−−

(5)

When the equivalent kernel is applied to the image, the result is the same as convolving the image with each of the

three smaller kernels separately. Mathematically, the order that the kernels are applied in does not matter. The result of the convolution process that creates the resultant kernel is the same as long as all the kernels are applied the correct number of times. If a high pass filter of size 3x3 is applied to an image alternatively with a low pass filter that is also of size 3x3, the resultant for one application of these 2 filters is the same as having used a single equivalent kernel of 5x5. Taking this one step further, a series of 5 sets of 3x3 high pass and 3x3 low pass filters could be applied to the image so that

== )(5),,,,( HPLPHPLPHPLPHPLPHPLPHPLP 21 x 21 Equivalent Kernel (6)

For this example, an equivalent kernel of size 21x21 would yield the same answer in a single pass. To keep from having to spatially convolve such a large kernel or to invoke a Fast Fourier Transform (FFT), the authors have instead derived a Data Model of the kernel. We have found that the equivalent kernel process can be approximated through the use of a Data Modeling

equation that uses either the equivalent of tag features describing the individual pixel neighborhoods operated on

by the equivalent kernel, or a subset of the pixel neighborhood values directly as input into the Data Modeling

equation. The subset of the pixel neighborhood chosen corresponds to the center row and center column of the kernel as shown in Figure 15. Figures 16 and 17 show examples of a series of images that were selected, processed, and stacked interactively with Registax, a commercial freeware program for registering, processing, and stacking images. Also presented in Figures 16 and 17 is the stacked image that results from using the Data Modeling concept, and the scores determined for each. Available was a total pool of 66 images for Mars and 225 images for the moon. For Mars, a total of 18 of the 300

images were flagged as sharp by the Data Model equation, registered, and stacked. For the moon, Data Modeling determined that a total of 38 out of 225 images were sharp and good candidates for stacking. As noted under the figures, the Data Modeling flagged images provided an image whose resultant quality factor was better (larger) than using the Registax algorithm. Figure 18 contains the tags for each of the images in Figures 16 and 17. These tags were calculated by converting the bitmaps into ASCII files consisting of 4 columns of numbers (index, red, green, and blue color planes). The files were then read in and processed as text files as described in Section 2.11. The tag values were generated using the real-time versions of the standard moments given by

1

1

1

1

+

+

++ +

=

i

i

i

iii

N

y

N

Nµµ (7) ( ) ( )

1

2

11

1

2

1

+

++

++

−+

i

ii

i

ii

N

y

N

µσ=i

Nσ (8)

1

3

11

1

+

++

+

−+

=i

i

iiii

iN

yNSkew

Skewσ

µ

(9) 3

)3(

1

4

11

1 −

−++

=+

++

+i

i

iiii

iN

yNKurt

Kurtσ

µ

(10)

15

)15(

1

6

116

16 −

−++

=+

++

+i

i

iiii

iN

yNM

Mσ

µ

(11) 105

)105(

1

8

118

18 −

−++

=+

++

+i

i

iiii

iN

yNM

Mσ

µ

(12)

Equations (8)-(12) generate the first five elements of the tag, and the final tag value describes the image class. In this example, 3.40 represents an image (3) of the fourth planet Mars (0.4), and of the planet itself (0). For 3.31, this represents an image (3) of the third planet Earth (0.3) and of the 1st moon (0.01). Figure 19 contains the source code for generating the tag values. 2.11 Text mining

The final application for data mining is the application of the Change Detection principle to the querying of text-only articles. If the text documents are transformed in an appropriate manner to facilitate characterization with the same tag features already in use throughout the foregoing examples, then the same principles and techniques already outlined can be brought to bear. This would enable examples of text to be searched for in documents by deriving the numerical values for the aggregate features and then using change detection equations to pass or tip-off text documents that are either related or different from the search vector. If it is desired to flag only the unique word content of the document, it can be processed as in Reference 2, and then only the unique word content further processed as described in this work. The ability to avoid scanning and processing full text documents is identical to avoiding image file processing or light curve processing by simply reading the descriptive header tags. If a similar set of tags can be generated for standard text files, then only the headers need to be read to uniquely identify information content. The problem we now address is illustrating how such descriptive statistics as used in all previous examples (standard moments) can be meaningfully derived form text. We initiate the process by converting ASCII text into its equivalent binary representation. First, we determine the base 2 equivalent representations for each character in the ASCII character set (numbers 0-255). Once these unique 8 digit long binary sequences are determined, this new table of binary sequences is used to convert each character in the text document under consideration into binary. When in the binary form, it is a simple matter to convert the binary sequence wi into an equivalent fractional Brownian motion increment zi with unitary value by the transformation

12 −= ii wz (13)

Integration of the sequence of the form

∑=

=i

j

ji zy0

(14)

yields a Weiner curve or classic 1/f type data curve as a transformation of the entire text document into a fractal curve. This fractal curve is now readily and uniquely characterized by the tag features14. The tags are stored in the header of the document and change detection models built using these headers for unique identification. Figure 20 illustrates how text is converted into a fractal curve for characterization, culminating in the tags for this text document in the abstract.

3. DATA MODELING 3.1 Theory

Data Modeling15

is the process of deriving a mathematical expression from measured data. This inverse modeling and parameter estimation process finds both the functional form of the equation and its numeric coefficients. In the general sense, a Volterra series model is used to describe the linear and non-linear portions of a dynamic system as a continuous process

∫∑∫∞∞

=

∞

−−=0

111)(

0 0

)()(),,()( qqq

q

q

ddtutuhty ττττττ KKKL (15)

which can be more efficiently represented in the discrete case by replacing the integrals with summations to yield16

∑ ∑ ∑∞

=

∞

=

∞

=

−−∆=0 0 0

11)(

1

)()(),,()()(q

qq

qq

t

q

tutuhtyτ τ

ττττ KKL (16)

For multiple inputs, the Volterra series in (15) and (16) requires the use of kernels and convolutions rather than simple multiplications. For a simple second order two input case, the multiple input Volterra series will include a zero-order kernel, two first-order kernels, two second-order self-kernels (kernels working on a single input) and one second-order cross kernel (kernels that act on different inputs). As the system order and number of inputs is increased, the number of kernels and convolutions becomes combinatorial. For this reason, the multiple-input Volterra series approach is only

practical for low-order systems with two or three inputs. Kolmogorov-Gabor presented a discrete polynomial form of the Volterra series that is more amenable to computer implementation, especially when more than three inputs are necessary. Data Modeling approximates the Kolmogorov-Gabor polynomial of the form

LK ++++= ∑∑∑∑∑∑i j k

kjiijk

i j

jiij

i


Ivakhnenko demonstrated that the Group Method of Data Handling (GMDH) using nested functions of the form17


LxxxyO 3),,,( 21 =K (18)

may be used to converge to the Kolmogorov solution by approximation. However, the GMDH approach fails to yield a parsimonious form. The authors have combined GMDH with genetic evolutionary directed algorithms to obtain a

single directed solution and final expression18. This information flow is provided in the pseudo code in Figure 13 for

constructing Change Detection Models. This structured approach forms intermediate meta-variables from combinations of three inputs combined into a single new fused output. This fused meta-variable becomes a new input variable available at the next layer. Each meta-variable is a node and can be thought of as the decision node or common point for information flow that is highly correlated and

related to the output goal. This allows the information flow tree to be visualized as a directed graph and the information relationships to be discovered between the various inputs and the ultimate goal output. Since the algorithm only uses the inputs necessary to achieve convergence, pruning of inputs occurs automatically and requires no external intervention. To show how these polynomials can be used to derive differential equations, we note that polynomials can be used to model forcing functions and derivative terms directly. These polynomials can be thought of in equation form as

)(xfdx

dy= (19)

which represents an x or index only solution to the differential equation that is of the same form as a trajectory solution (a traditional polynomial uses an x index and associates with it a y value). If y happens to be a derivative, the derivative as a function of time is gained directly. To demonstrate how a dynamic system can be Data Modeled with the same polynomial, we use as input the previous or lagged derivative value and associate with it the current derivative value. Now the x index is no longer an input, and we have specified y completely dependent on previous values, yielding the typical form

)(yfdx

dy= (20)

If the analytic derivative is not directly available from the measured data, it can be derived from the Turlington polynomial given in Equation (2) following Jaenisch et.al.19, who showed its derivative is obtained directly from

( )

( )

( )∑∑−

=

−

= −

−

−+

+

−−+=

1

2,

1

1001.

001.

1

1

)10ln(

101

10

)001(.

))(001(.1)(

N

j

ik

kk

ii

xx

ixx

k

jj

i

k

k

j

j

mmkB

dx

Td ψ ∑≠≠−=

−=i

kjj

ij

jkik j

01

,1, ψψ (21)

The data can also be directly modeled using eigenfunctions, and although this process is slower, analytical derivatives and integrals are then available as well. 3.2 Change Detection

Figure 13 provides pseudo-code of the algorithm for creating change detection models. By training to the category value of ½, an orthogonal term model is produced whose sensitivity can be controlled using higher and lower tap points. Also, since the feature vector is rolled up using parametric and non-parametric statistics, the queriant is free to obtain when constructing the Change Detection model groups of descriptive statistics that are lumped into a large feature vector set. This feature vector set is labeled nominal and used to find matches or deviations, with matches scored as a zero flag and deviations scored as a one flag. Tuning20,21,22 makes the model more sensitive to feature values or groups of feature values that fall outside the training boundaries. Whenever new features or groups of features begin to depart from within the valid training boundaries, off-nominal or tip-off conditions are flagged23,24.

Rather than use the actual input feature values, we use parametric and non-parametric descriptions of the entire feature

value set. This treats the feature value set as a time series and models the distribution by tuning to a minimized

regression function. Tuning is achieved by setting the target value of the regression to minimize and characterizing the ensuing polynomial fluctuation around the converged minimum for the placement of decision thresholds. In practice the output feature value used for training is the categorical value of ½.

3.3 Change Model construction

Each Data Model is generated using only a portion (50% or less) of the available nominal data, with the

remainder held back for testing the Data Model after construction. This supports the Data Modeling re-sampling theory that only a portion of the available data is required to produce adequate change models. A feature vector data set or matrix is generated and denoted by

.

,2,1,

,22,21,2

,12,11,1

2

1

,

=

=

nmmm

n

n

m

nm

aaa

aaa

aaa

FV

FV

FV

FV

L

MMMM

L

L

M

(22)

Each row of feature vector FV contains elements that are generated using different descriptive statistics. For analyzing the Jacoby stellar spectra, six (6) descriptive statistics that constitute the FV are given by

)( jxMax=λ (23)

Once the features in Equations (8) - (12) and (23) are calculated for each nominal data set, a Data Change Model is generated using a target regression value of ½ as the nominal output. Once the equation model is trained, boundaries are placed on either side of the target regression value that characterizes the fluctuation of the polynomial about ½. Typically, these boundaries are set at 0.499 and 0.501 with a standard deviation of .0012.

4. FUTURE WORK

Model based archive searching enables a full compliment of mathematical tools to be used, including the method of derivatives, finite element analysis, and automatic differential equation modeling. The automatic generation of analytical function models of stellar and photometric data allows interpolation and modeling in the form of analytical derivatives for use in constructing differential equation models across related groups. A Data Modeling algorithm is needed to decompose an equivalent kernel into the convolution of a single column and a single row vector. Finally, we propose an alternative to adaptive optics and speckle imaging using real-time Data Models to enable unsupervised sharp image detection for stacking and image enhancement. Using the concepts of Data Modeling, a novel real-time video and software based artificial adaptive optical system is possible.

5. SUMMARY

In this work, the authors have successfully demonstrated the use of Data Modeling and the tag descriptors to construct Change Detection models of individual data files, models of stellar database features, models of the stellar database itself, brightness curve modeling, exo-planet detection, stellar object image modeling, image selection for stacking and speckle imaging, stellar image enhancement, and text mining. Index tagging of Virtual Observatory data files with descriptive statistics enables both rapid data mining and mathematical modeling. The inclusion of these tags directly into the FITS structure facilitates a substantial improvement in search and processing time of legacy databases. Data Change Detector Modeling uses the descriptive tags for each object to determine if matches to the file of interest exist in the database. The recording of all future observational measurements should include the calculation and storage of these tags. For currently existing databases, computer programs should be written and executed that post-process and add these features to the information stored in current databases. This would facilitate rapid dissemination and sharing of model based information.

ACKNOWLEDGMENTS

The authors would like to thank Scott McPheeters, Tim Aden, and John Deacon for their continued support during the course of this work.

REFERENCES

1. National Virtual Observatory, http://www.us-vo.org/summer-school/index.html, May 24, 2004. 2. Sloan Digital Sky Survey, http://www.sdss.org/, May 20, 2004. 3. Jaenisch, H.M., Handley, J.W., Case, C.T., Songy, C.G., “Graphics based intelligent search and abstracting using

Data Modeling”, Proceedings of SPIE, Seattle WA July 7-11 2002. 4. Jaenisch, H.M., Filipovic, M.D., “Classification of Jacoby Stellar Spectra Using Data Modeling”, Proceedings of

SPIE Vol.4816 (2002). 5. Jaenisch, H.M., Collins, W.J., Handley, J.W., Hons, A., Filipovic, M.D., Case, C.T., Songy, C.G., “Real-time visual

astronomy using image intensifiers and Data Modeling”, Proceedings of SPIE Vol. 4796 (2002). 6. Hudgins, D., “Catch an Extrasolar Planet”, Astronomy, pp. 76-79, May 2002. 7. Handley, J., Jaenisch, H., Lim, A., White, G., Pennypacker, C., Edwards, M., “Data Modeling of Deep Sky Images”,

Proceedings of SPIE Vol. 5497 (2004). 8. Cidadao, A. J., “Thoughts on Super-Resolution Planetary Imaging”, S&T, pp. 127-134, December, 2001. 9. Jaenisch, H.M., Handley, J.W., Scoggins, J., Carroll, M.P., "Atmospheric Optical Turbulence Model (ATOM) Based

On Fractal Theory, " Proceedings of SPIE, Los Angeles, CA January 24, 1994. 10. Roddier, F. Adaptive Optics in Astronomy. Cambridge, UK: Cambridge University Press, 1999. 11. Lim, A., Jaenisch, H., Handley, J., Berrevoets, C., White, G., Deragopian, G., Payne, J., Schneider, M., “Image

Resolution and Performance Analysis of Webcams for Ground Based Astronomy”, Proceedings of SPIE Vol. 5489 (2004).

12. Lambert, A., Fraser, D., Jahromi, M.R.S., Hunt, B.R., “Super-resolution in image restoration of wide area images viewed through atmospheric turbulence”, Proceedings of SPIE Vol. 4792 (2002).

13. Fraser, D., Lambert, A., Jahromi, M.R.S., Clyde, D., Donaldson, N., “Can broad-band image restoration rival speckle restoration?”, Proceedings of SPIE Vol. 4792 (2002).

14. Jaenisch, H. and Handley, J., “Data Modeling of 1/f noise sets”, Proceedings of SPIE Vol. 5114 (2003). 15. Jaenisch, H.M. and Handley, J.W., “Data Modeling for Radar Applications”, Proceedings of the IEEE Radar

Conference 2003, (May 2003). 16. Westwick, D.T. Kearney, R.E., Identification of Nonlinear Physiological Systems, Piscataway, NJ: IEEE, 2003. 17. Madala, H.R., Ivakhnenko, A.G., Inductive Learning Algorithms for Complex Systems Modeling, Boca Raton, FL:

CRC Press, 1994. 18. Jaenisch, H.M., and Handley, J.W., “Automatic Differential Equation Data Modeling for UAV Situational

Awareness”, Society for Computer Simulation, Huntsville Simulation Conference 2003, (October 2003). 19. Jaenisch, H.M., Handley, J.W. , Faucheux, J.P., “Data Driven Differential Equation Modeling of fBm processes”,

Proceedings of SPIE Vol. 5204(2003). 20. Jaenisch, H.M., Handley, J.W., Pooley J.C., Murray S.R., “Data Modeling for Fault Detection”, Society for

Machinery Failure Prevention Technology (MFPT), (April 2003), published in Proceedings of MFPT (2004). 21. Pooley, J.C. and Murray, S.R., “Rotordynamic Fault Diagnostics Using Phase Coherent Filtering”, Proceedings of

the Society for Machinery Failure Prevention Technology (MFPT), (April 2003). 22. Pooley, J.C., Murray, S.R., Jaenisch, H.M., and Handley, J.W., “Fault Detection via Complex Hybrid Signature

Analysis”, JANNAF 39th Combustion, 27th Airbreathing Propulsion, 21st Propulsion Systems Hazards, and 3rd

Modeling and Simulation Subcommittees Joint Meeting, Colorado Springs, CO, (December 2003). 23. Jaenisch, H.M., Handley, J.W., Bonham, L.A., “Enabling Calibration on Demand for Situational Awareness”, Army

Aviation Association of America (AAAA), Tennessee Valley Chapter, (February 2003). 24. Jaenisch, H.M., Handley, J.W., Faucheux, J.P., Harris, B., “Data Modeling of Network Dynamics”, Proceedings of

SPIE Vol. 5200 (2003).

http://www.us-vo.org/summer-school/index.html

http://www.sdss.org/

FIGURES

Tip-Off

Tip-Off

Nominal

Tip-Off

Tip-Off

Nominal

Equation

Output

σ

T = w0+w1x1+w2x2+w3x3 +w4x12

+w5x22 +w6x3

2 +w7x13 +w8x2

3

+w9x32 +w10x1x2 +w11x1x3

+w12x2x3

Skewness Standard

Deviation

Kurtosis

Rollup

Stats

Data Model

Data

SeriesPDF

Fig. 1: Data Change Modeling uses descriptive statistics as shown above that characterize the PDF or histogram of the input data to

build a model that is sensitive to small changes in input feature distribution.

Fig. 2 : This SPIE paper transformed into a fractal curve (left), and the histogram showing its distribution.

0 Correlation 1

0N

umbe

r of

Spe

ctra

37

Classes O/B

Class B/A/F

Class A/F

Class B/A

Class F/G/KClass M

Class G/K/M

Fig. 3 (Left) and Fig. 4 (Right): (Left) Correlation is not an acceptable classifier. 140 out of 161 objects score correlations above 90;

(Right), Graphics based search engine applied to Jacoby features.

0

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

3510 wavelength (angstroms) 7430 0 1

0 13510 wavelength (angstroms) 7430

O5V

B0V

B6V

A1V

A5V

F0V

F5V

G0V

G6V

K0V

K5V

M0V

M5V

HistogramsJacoby Spectra

O5V

B0V

B6V

A1V

A5V

F0V

F5V

HistogramsJacoby Spectra

G0V

G6V

K0V

K5V

M0V

M5V

Fig 5: Jacoby spectra examples. Full Jacoby spectra ranges from 3510 Å to 7427.2 Å. The mathematical description of these histograms using 6 descriptive statistics are proposed as index tags.

DEFCUR A-Z:pi=4@*atn(1@):open "input" for input as #1:input #1, stdev:input #1, skew:input #1, kurt:input #1, m6

input #1, m8:close #1:stdev=(stdev-0.1187)/0.0001:skew=(skew-0.3675)/0.002:kurt=(kurt+0.767)/0.0037

m6=(m6+7.844)/0.0285:m8=(m8+76.5682)/0.2019:l1o1=-0.5787-0.5977*skew+3.7225*kurt-3.0537*m8+0.1586*skew*skew

l1o1=l1o1+0.3774*kurt*kurt+1.2794*m8*m8+0.2712*skew*kurt+0.5787*skew*m8-1.9491*kurt*m8-0.503*skew*kurt*m8

l1o1=l1o1+0.0438*skew*skew*skew-1.1006*kurt*kurt*kurt+1.6117*m8*m8*m8-0.5895*kurt*skew*skew+0.1889*skew*kurt*kurt

l1o1=l1o1+0.5122*skew*m8*m8+0.738*m8*skew*skew+2.9299*m8*kurt*kurt-3.6302*kurt*m8*m8

l1o2=-0.821+0.3054*m6+1.1897*stdev+0.6089*kurt+5.2783*m6*m6+0.1364*stdev*stdev+3.2261*kurt*kurt+0.0459*m6*stdev

l1o2=l1o2-8.5618*m6*kurt+0.0052*stdev*kurt+0.7006*m6*stdev*kurt+1.5339*m6*m6*m6-0.112*stdev*stdev*stdev

l1o2=l1o2-1.6305*kurt*kurt*kurt-0.3927*stdev*m6*m6-0.7456*m6*stdev*stdev+4.9791*m6*kurt*kurt-5.0348*kurt*m6*m6

l1o2=l1o2+0.6639*kurt*stdev*stdev-0.4826*stdev*kurt*kurt

l2o1=0.1901+0.491*l1o1-0.8597*skew+0.9397*l1o2-0.059*l1o1*l1o1-0.2106*skew*skew-0.0241*l1o2*l1o2+0.1474*l1o1*skew

l2o1=l2o1+0.0242*l1o1*l1o2+0.1736*skew*l1o2-0.6347*l1o1*skew*l1o2+0.0232*l1o1*l1o1*l1o1+0.021*skew*skew*skew

l2o1=l2o1-0.1672*l1o2*l1o2*l1o2+0.1348*skew*l1o1*l1o1+0.4039*l1o1*skew*skew+0.3466*l1o1*l1o2*l1o2

l2o1=l2o1-0.254*l1o2*l1o1*l1o1-0.4502*l1o2*skew*skew+0.7386*skew*l1o2*l1o2

l2o3=0.2277+0.1695*l1o2+0.1386*m6+0.6821*l1o1-0.3724*l1o2*l1o2+0.1918*m6*m6-0.1865*l1o1*l1o1-0.0393*l1o2*m6

l2o3=l2o3+0.3757*l1o2*l1o1+0.121*m6*l1o1-0.717*l1o2*m6*l1o1+0.1848*l1o2*l1o2*l1o2-0.0936*m6*m6*m6

l2o3=l2o3-0.0106*l1o1*l1o1*l1o1+0.3308*m6*l1o2*l1o2-0.2212*l1o2*m6*m6+0.0678*l1o2*l1o1*l1o1

l2o3=l2o3-0.1594*l1o1*l1o2*l1o2+0.5126*l1o1*m6*m6+0.0819*m6*l1o1*l1o1

l3o3=-0.2074+0.8909*l2o1+0.3944*m8+0.2778*l2o3+0.2988*l2o1*l2o1+0.0589*m8*m8+0.2505*l2o3*l2o3-0.2058*l2o1*m8

l3o3=l3o3-0.4905*l2o1*l2o3+0.102*m8*l2o3+0.2339*l2o1*m8*l2o3+0.3629*l2o1*l2o1*l2o1-0.0622*m8*m8*m8

l3o3=l3o3-0.7296*l2o3*l2o3*l2o3+0.2209*m8*l2o1*l2o1-0.2667*l2o1*m8*m8+2.1015*l2o1*l2o3*l2o3

l3o3=l3o3-1.7526*l2o3*l2o1*l2o1+0.3038*l2o3*m8*m8-0.5106*m8*l2o3*l2o3

l4o2=0.0496+0.715*l3o3+0.353*m6+0.025*skew+0.0059*l3o3*l3o3-0.0501*m6*m6-0.0724*skew*skew+0.0167*l3o3*m6

l4o2=l4o2-0.16*l3o3*skew+0.1811*m6*skew+0.1065*l3o3*m6*skew+0.0429*l3o3*l3o3*l3o3-0.0413*m6*m6*m6

l4o2=l4o2+0.0148*skew*skew*skew-0.0067*m6*l3o3*l3o3-0.0393*l3o3*m6*m6+0.1431*l3o3*skew*skew

l4o2=l4o2-0.0834*skew*l3o3*l3o3+0.0357*skew*m6*m6-0.1426*m6*skew*skew:l4o2 = l4o2* (0.0004) + (0.5)

If l4o2>.4 and l4o2<.6 then iflag=0 else iflag=1

open "out1" for output as #1:print #1, l4o2,iflag:close #1:end Fig. 6: Change Detection Data Model for example spectra from Jacoby database. QBASIC source code is autonomously generated.

DECLARE SUB writefunc (a!(), b!(), cc!, m!, ycept!, iseg!, n!)

DEFSNG A-Z: ON ERROR RESUME NEXT: SCREEN 9: CLEAR : CLS

INPUT "File => ", zz$: INPUT "Number of points => ", n

INPUT "Filter Coeff (0.001) => ", cc: IF cc = 0! THEN cc = .001

REDIM xdata(n), ydata(n): OPEN zz$ FOR INPUT AS #1

FOR i = 1 TO n: INPUT #1, ydata(i): xdata(i) = (i - 1) / (n - 1): NEXT i

CLOSE #1: ymin = ydata(1): ymax = ydata(1): xmin = xdata(1): xmax = xdata(1)

FOR i = 2 TO n

IF ydata(i) < ymin THEN ymin = ydata(i)

IF ydata(i) > ymax THEN ymax = ydata(i)

IF xdata(i) < xmin THEN xmin = xdata(i)

IF xdata(i) > xmax THEN xmax = xdata(i)

NEXT i

FOR i = 1 TO n: ydata(i) = (ydata(i) - ymin) / (ymax - ymin)

xdata(i) = (xdata(i) - xmin) / (xmax - xmin): NEXT i

iseg = 1: REDIM mslope(n - 1), dslope(n - 1), a(n - 1), b(n - 1)

FOR i = 1 TO n - 1: xden = xdata(i + 1) - xdata(i)

IF xden <> 0 THEN mslope(i) = (ydata(i + 1) - ydata(i)) / (xdata(i + 1) - xdata(i))

NEXT i

FOR i = iseg + 1 TO n - 1: dslope(i) = mslope(i) - mslope(i - 1): a(i) = dslope(i) * cc: b(i) = xdata(i): NEXT i

m = mslope(iseg): ycept = ydata(iseg) - m * xdata(iseg)

CLS : WINDOW (0, 1)-(1, 0): xstep = 1 / n

FOR i = 1 TO n - 1: LINE (xdata(i), ydata(i))-(xdata(i + 1), ydata(i + 1)), 4: NEXT i

REDIM ytmp(n): ijk = 0: FOR k = 1 TO n: i = xdata(k): y = m * i + ycept

FOR j = iseg + 1 TO n - 1: y = y + a(j) * LOG(1 + 10 ^ ((i - b(j)) / cc)) / LOG(10): NEXT j: ytmp(k) = y: NEXT k

OPEN "datamdl.out" FOR OUTPUT AS #1: FOR i = 1 TO n: PRINT #1, ymin + ytmp(i) * (ymax - ymin): NEXT i: CLOSE #1

k = 0: FOR i = 1 TO n - 1: LINE (xdata(i), ytmp(i))-(xdata(i + 1), ytmp(i + 1)), 15: IF a(i) <> 0! THEN k = k + 1

NEXT i: REDIM aa(k), bb(k): k = 0: FOR i = 1 TO n - 1

IF a(i) <> 0! THEN k = k + 1: aa(k) = a(i): bb(k) = b(i)

NEXT i

OPEN "datamdl.bas" FOR OUTPUT AS #10

PRINT #10, "defsng a-z : on error resume next :cls: c="; cc

PRINT #10, "n="; k

PRINT #10, "redim a(n-1),b(n-1)"

PRINT #10, "for i=1 to n-1 : read a(i) : read b(i) : next i"

PRINT #10, "open " + CHR$(34) + "datamdl.out" + CHR$(34) + " for output as #1"

PRINT #10, "for x1=0 to"; n - 1

PRINT #10, "x=x1/"; (n - 1)

PRINT #10, "y = "; m; " * x + "; ycept

PRINT #10, "for j=1 to n-1 : y = y + a(j) * LOG(1 + 10 ^ ((x - b(j)) / c)) / LOG(10) : NEXT j"

PRINT #10, "y="; ymin; "+y*("; ymax; "-"; ymin; "):print #1, y : next x1 : close:end"

k1 = 1: FOR j = 1 TO k STEP 4

PRINT #10, "Data "; : PRINT #10, aa(j); ","; bb(j); ","; : k1 = k1 + 1

IF k1 > k THEN EXIT FOR

PRINT #10, aa(j + 1); ","; bb(j + 1); ","; : k1 = k1 + 1


PRINT #10, aa(j + 2); ","; bb(j + 2); ","; : k1 = k1 + 1


PRINT #10, aa(j + 3); ","; bb(j + 3): k1 = k1 + 1


NEXT j: CLOSE #10: ERASE aa, bb: LOCATE 23, 1: INPUT "", za$: END Fig. 7: Source code for generating Turlington polynomial models.

DEFCUR A-Z:pi=4@*atn(1@):open "input" for input as #1:input #1, peakloc

input #1, oskew:input #1, okurt:close #1

peakloc=(peakloc-8.8947)/4.3213:oskew=(oskew-1.1492)/0.0438:okurt=(okurt-0.2802)/0.1467

l1o2=-0.3336+0.0971*sin(6.2832*oskew)+1.7721*cos(6.2832*oskew)-1.1296*sin(6.2832*peakloc)

l1o2=l1o2+0.8456*cos(6.2832*peakloc)-0.3148*sin(6.2832*okurt)+0.4756*cos(6.2832*okurt)

l2o2=0.3536+1.1824*l1o2-0.3348*peakloc+0.6246*okurt-0.0529*l1o2*peakloc+0.0322*l1o2*okurt

l2o2=l2o2-0.1666*peakloc*okurt+0.4881*l1o2*peakloc*okurt

l1o1=0.143-1.7749*oskew-0.2264*peakloc+1.6572*okurt+3.3617*oskew*peakloc-0.0453*oskew*okurt

l1o1=l1o1-3.806*peakloc*okurt+0.0729*oskew*peakloc*okurt

l2o3=-0.1688+0.879*l1o2-0.3262*l1o1+0.2256*l1o2*l1o2+0.6561*l1o1*l1o1-0.9226*l1o2*l1o1

l2o3=l2o3-0.0518*l1o2*l1o2*l1o2+0.4181*l1o1*l1o1*l1o1+0.8435*l1o1*l1o2*l1o2-0.9513*l1o2*l1o1*l1o1

l3o2=-0.1369+0.17*l2o2+0.0712*l2o3-0.0834*l2o2*l2o2-0.1485*l2o3*l2o3+0.2223*l2o2*l2o3

l3o2=l3o2-0.7185*l2o2*l2o2*l2o2+2.5702*l2o3*l2o3*l2o3+4.437*l2o3*l2o2*l2o2-6.1732*l2o2*l2o3*l2o3

l3o2 = l3o2*0.0004+0.5001

IF l3o2 > .4988 AND l3o2 < .5012 THEN iflag = 0 ELSE iflag = 1

open "out" for output as #1:print #1, l3o2,iflag:close #1:end

Fig. 8: Change Detection Data Model for separating “O” class from other temperature class spectra. When output for iflag is zero, temperature class is “O”. QBASIC source code is autonomously generated.

Binary Stars&

Exo-Planets

Binary Stars&

Exo-Planets

Exo-Planet

Binary Star

Binary Star

Fig. 9: Example binary star and exo-planet light curves (left); zoom in to 3 specific curves on right to show overlap.

DEFCUR A-Z:pi=4@*atn(1@):open "input" for input as #1:input #1, stdev:input #1, m8:close #1

stdev = (stdev-0.0119) / 0.0071 : m8 = (m8-397208.9192) / 2.5846

l1o2=-0.1275+1.3376*stdev+0.2687*m8

l1o1=0.2485-1.0431*stdev-0.0192*m8-0.7598*stdev*stdev+0.2475*m8*m8+0.2762*stdev*m8

l1o1=l1o1+0.9751*stdev*stdev*stdev-0.0004*m8*m8*m8-0.0943*m8*stdev*stdev+0.2795*stdev*m8*m8

l1o3=0.4628-0.8694*stdev-0.9955*stdev*stdev+1.0526*stdev*stdev*stdev)

l2o3=0.1471-0.2874*l1o2+0.4846*l1o1+0.3234*l1o3-0.1091*l1o2*l1o2+0.0417*l1o1*l1o1

l2o3=l2o3+0.2806*l1o3*l1o3+0.5575*l1o2*l1o1+0.3354*l1o2*l1o3-0.9087*l1o1*l1o3

l2o3=l2o3-0.0048*l1o2*l1o1*l1o3+0.1274*l1o2*l1o2*l1o2-0.0065*l1o1*l1o1*l1o1

l2o3=l2o3+0.0768*l1o3*l1o3*l1o3-0.6829*l1o1*l1o2*l1o2+0.6652*l1o2*l1o1*l1o1

l2o3=l2o3+0.0161*l1o2*l1o3*l1o3+0.095*l1o3*l1o2*l1o2-0.374*l1o3*l1o1*l1o1+0.1275*l1o1*l1o3*l1o3

l2o2=0.3835+0.0005*l1o3-0.926*stdev-0.422*l1o1+0.0013*l1o3*l1o3-0.5794*stdev*stdev+0.912*stdev*l1o1*l1o1

l2o2=l2o2-0.2772*l1o3*stdev+0.2485*l1o3*l1o1+0.6057*stdev*l1o1+0.0109*l1o3*stdev*l1o1 -0.53*l1o1*l1o1

l2o2=l2o2-0.1332*l1o3*l1o3*l1o3+0.7475*stdev*stdev*stdev+0.2173*l1o1*l1o1*l1o1-0.1737*stdev*l1o3*l1o3

l2o2=l2o2-0.687*l1o3*stdev*stdev-1.2963*l1o3*l1o1*l1o1+1.0067*l1o1*l1o3*l1o3+0.4285*l1o1*stdev*stdev

l3o3=-0.0042-0.2612*l2o3-0.1423*stdev+1.1172*l2o2+0.0017*l2o3*l2o3+0.0039*stdev*stdev-0.1494*l2o2*l2o2

l3o3=l3o3-0.9722*l2o3*stdev-0.0942*l2o3*l2o2+1.4563*stdev*l2o2-0.0026*l2o3*stdev*l2o2+0.077*l2o3*l2o3*l2o3

l3o3=l3o3+0.1056*stdev*stdev*stdev-0.2506*l2o2*l2o2*l2o2-0.2379*stdev*l2o3*l2o3+0.4022*l2o3*stdev*stdev

l3o3=l3o3+0.349*l2o3*l2o2*l2o2-0.2614*l2o2*l2o3*l2o3-0.6417*l2o2*stdev*stdev+0.556*stdev*l2o2*l2o2

l2o1=-0.8407-2.3665*l1o2+4.7385*stdev+0.5401*l1o1+0.0312*l1o2*l1o2+4.9348*stdev*stdev-0.0819*l1o1*l1o1

l2o1=l2o1-2.0565*l1o2*stdev-0.3378*l1o2*l1o1-0.4258*stdev*l1o1+0.7526*l1o2*stdev*l1o1+2.0705*l1o2*l1o2*l1o2

l2o1=l2o1-15.1765*stdev*stdev*stdev-0.0116*l1o1*l1o1*l1o1-10.5109*stdev*l1o2*l1o2+19.9498*l1o2*stdev*stdev

l2o1=l2o1+0.483*l1o2*l1o1*l1o1-1.0323*l1o1*l1o2*l1o2+2.6964*l1o1*stdev*stdev-0.9501*stdev*l1o1*l1o1

l3o2=0.0355-0.1555*l2o3-0.203*stdev+0.2399*l2o1+0.0036*l2o3*l2o3+0.1149*stdev*stdev-0.4096*l2o1*l2o1

l3o2=l3o2+0.0332*l2o3*stdev+0.1212*l2o3*l2o1+0.6159*stdev*l2o1-0.0709*l2o3*stdev*l2o1+0.9553*l2o3*l2o3*l2o3

l3o2=l3o2+0.0599*stdev*stdev*stdev-0.0111*l2o1*l2o1*l2o1-0.5923*stdev*l2o3*l2o3-0.1128*l2o3*stdev*stdev


l4o2=0.0026+0.3854*l3o3-0.0285*stdev+0.407*l3o2+0.0002*l3o3*l3o3+0.0057*stdev*stdev+0.0013*l3o2*l3o2

l4o2=l4o2+0.1058*l3o3*stdev-0.0742*l3o3*l3o2+0.1359*stdev*l3o2-0.1609*l3o3*stdev*l3o2+0.004*l3o3*l3o3*l3o3

l4o2=l4o2-0.0007*stdev*stdev*stdev-0.0513*l3o2*l3o2*l3o2+0.3105*stdev*l3o3*l3o3-0.3124*l3o3*stdev*stdev


l4o2=l4o2*0.0002+0.5

IF l4o2 > .4988 AND l4o2 < .5012 THEN iflag = 0 ELSE iflag = 1

open "out" for output as #1 : print #1, l4o2, iflag : close #1 : end

Fig. 10: Change Detection Data Model for distinguishing binary stars from exo-planets. An output of iflag=0 denotes an exo-planet. QBASIC source code is autonomously generated.

Y(x)=255(6.4651- 4.2064COS(6πx)+.9284SIN(6πx)-3.4306COS(18πx)-.0182SIN(18πx)+2.9895COS(12πx)+.9458SIN(12πx)+2.2042COS(36πx)-.1SIN(36πx)

-1.6436COS(10πx)-1.3934SIN(10πx)-2.1054COS(6πx)+.4545SIN(6πx)- 1.7485COS(2πx)-.851SIN(2πx)+1.7537COS(8πx)-.6431SIN(8πx)+1.5867COS(24πx)-.9171SIN(24πx)-1.7154COS(18πx)-.0328SIN(18πx)+1.4905COS(12πx)+.4867SIN(12πx)-1.3774COS(30πx)-.3051SIN(30πx)+1.1039COS(36πx)-.0196SIN(36πx)-1.0041COS(42πx)+.3935SIN(42πx)-.8165COS(10πx)-.703SIN(10πx)-1.0538COS(6πx)+.2224SIN(6πx)+.9674COS(22πx)-.4429SIN(22πx)+1.0181COS(72πx)-.1377SIN(72πx)-.3388COS(20πx)+.9632SIN(20πx)+.4204COS(4πx)+.9096SIN(4πx))/64

Fig. 11: Examples of image Data Modeling on M51. (Top Left) original; (Top Center) image converted to a Hilbert sequenced time series; (Top Right) resultant eigenfunction Data Model of M51; and (Bottom) automatically generated eigenfunction Data Model

equation using 20 sine and cosine terms.

rem initialize

redim xdata(n),ydata(n),extreme(n),totaldat(n),r(n),c(n),newdata(n),newdata1(n),r1(n),c1(n)

rem open x and y data array files

call open_data_files(xdata,n)

call open_data_files(ydata,n)

rem sort x into ascending order and sort y with x to retain x associations

call sort2(n,xdata,ydata)

rem put ydata into newdata

for i=1 to n

newdata(i)=ydata(i)

next i

rem specify objective correlation value

cobj=0.99

rem calculate Data Modeling eigenfunction model of y(x)

rem final result held in totaldat; initialize k1 to hold number of terms

k1=0

10 continue

k1=k1+1

rem generate Fourier transform of data and corresponding power spectra

rem use linear regression to fit a line through the dB power spectra

rem identify maxima in power spectra that occur above the linear fit

rem pull out maxima locations in array extreme

rem find maximum value in array extreme, and use location to

rem extract real and imaginary Fourier terms held in r (real) and c(imaginary)

rem for use in eigenfunction model

call calc_extremes(ydata,n,extreme,r,c)

rem find maximum value location ; only look at first half of data since symmetric

rem ignore zeroth order location

n1=int(n/2)

dmax = extreme(1)

dloc = 1

FOR i = 2 TO n1 - 2

IF extreme(i) > dmax THEN

dmax = extreme(i)

dloc = i

END IF

NEXT i

rem construct sine and cosine representation of dloc frequency at sampling equal to original

rem r1 and c1 hold eigenfunction coefficients, d the frequency term

r1(k1)=r2(dloc)

c1(k1)=c2(dloc)

d(k1)=dloc

FOR i = 1 TO n

newdata1(i) = newdata1(i) + CCUR(r2(dloc)) * COS(dloc * 2# * pi * (i / N))

newdata1(i) = newdata1(i) + CCUR(c2(dloc)) * SIN(dloc * 2# * pi * (i / N))

newdata1(i) = newdata1(i) / SQR(N)

NEXT i

rem calculate residual between original and summation of all terms generated so far

FOR i = 1 TO n

newdata(i) = newdata(i) - newdata1(i)

NEXT i

rem add current term to previous terms

FOR i = 1 TO n

totaldat(i) = totaldat(i) + newdata1(i)

NEXT i

rem calculate correlation between original and current model (totaldat)

CALL correl(totaldat,ydata,N,ycorrel)

rem test to see if correlation criteria met

rem test to see if maxterms criteria met

IF ycorrel < cobj AND k1 < .125 * N THEN

GOTO 99

END IF

rem output final model

CALL output_eigen(r1,c1,d,k1,N)

END Fig. 12: Pseudo-code for Data Modeling eigenfunction model construction.

N=total number of cases

Call Generate_Waveform_Stats(ydata,n,ystats)

Call Generate_Rollup_Stats(ystats,n,rstats)

For i=1 to n

rem 10% Sub-sample nominal cases

Next i

M=number of sub-sampled points

For i=1 to m

rem Construct target output value of ½

rem Allow first 10% of points to be 0.499

rem Allow last 10% of points to be 0.501

Next i

10 continue

rem ______________________________________________________________

rem Data Relations Modeling

rem Construct Data Model using M Stats as input and the target output

rem Use multivariable linear regression to fit polynomial building blocks

rem using at most 3 input variables to map into the target output

rem Use correlation to rank inputs and reduce number if desired

Call correlate_inputs

Call prune_inputs

for i=1 to ninputs

for j=1 to ninputs

for k=1 to ninputs

rem select input #1 from stack

rem select input #2 from stack and different than input #1

rem select input #3 from stack and different than input #1 or #2

for L=1 to number_object_types

call mult_var_regression

call calc_error

next L

next k

next j

next i

rem select D=3 best solutions from above and put on the input stack as functionals

rem repeat until error criteria or max number of layers is achieved

rem score best current case to error criteria

if current_error<=minimum_error then

rem finished

else

goto 10

endif

call output_data_model

rem ______________________________________________________________

For i=1 to n

rem Interrogate Data Relations Model

Call Interrogate_data_model(features,Dmval)

If Dmval>.4988 and Dmval<.5012 then

rem nominal

iflag=0

else

rem tip-off

iflag=1

endif

Next i

end

n=total num ber of cases

Call G enerate_W aveform _Stats(ydata,n,ystats)

Call G enerate_Rollup_Stats(ystats,n,rstats)

For i=1 to n

rem Sub-sam ple of all cases called Group 1

Next i

m =num ber of subsam pled cases

rem use previous pseudo-code for generating a Data Change M odel

call generate_data_change_m odel

fori=1 to n

call Interrogate_data_m odel(features,Dm val)

call flag_tipoffs

rem Save tip-off data as new Group 2

next i

for i=1 to n

rem subsam ple another m points from new pool

rem place with 1st 10% subsam pled from original G roup 1

Next i

m =2*m


for i=1 to n


call flag_tipoffs

rem Save as Group 3

next i

rem extract all k tipoff cases from G roup 3 and

rem random ly assign to 0/1 c lass

for i=1 to k

if rnd>.5 then

out=1

else

out=0

endif

next i

10 continue

call generate_data_relations_m odel

rem score cases

for i=1 to k


if Dm val< .25 then

trainval=0

elseif Dm val>.75 then

trainval=1

else

if trainval=0 then

trainval=1

else

trainval=0

endif

endif

next i

rem check for lim it cycle or 4 layer Data M odel learning all cases

if lim it_cycle or all_learned then

rem sharp im ages are the sm aller class

rem if equal num ber in each class, select class w ith largest RSS

else

goto 10

endif

rem last m odel generated can be used in lieu of training a final m odel


for i=1 to n


call flag_tipoffs

next i

call write_output

end Fig. 13: Pseudo-code for supervised constructing Data Change Models (left) and unsupervised Data Modeling approach for selecting

images for stacking (right).

BackgroundReject

SharpData Model

[O(3n)]

StackData

ModelingEnhance

Image

Camera

Time Fig. 14: Block diagram of unsupervised Data Modeling approach for selecting images for stacking.

Image

Rzzzzzzzzzf =),,,,,,,,( 534335343332312313

Data Model of Equivalent Kernel

)]3([ nO

*

* f is

Image Pixel Locations Correspond To Center Row andColumn of Equivalent Kernel

5554535251

4544434241

3534333231

2424232221

1514131211

zzzzz

zzzzz

zzzzz

zzzzz

zzzzz

Center Image Pixel LocationsFrom Final Image

(After Applying Equivalent Kernel)

5554535251

4544434241

3534333231

2424232221

1514131211

zzzzz

zzzzz

zzzzz

zzzzz

zzzzz

Fig. 15: Image pixel locations corresponding to the center row and column of the equivalent kernel are transformed into the center

image pixel locations after applying the equivalent kernel. This yields the Data Model of the equivalent kernel.

Registax Sharp Selected and Wavelet Processed

Score = 0.1886

Data ModelingScore = 0.1917

Raw Frame #1


Score = 0.1886


(Note: JPEG artifacts removed,Noise suppressed)

Raw Frame #1(Note: JPEG artifacts present)

Fig. 16: Data Modeling for image stacking on Mars (top), and zoom in to Mars detail (bottom), Left image is raw image, center sharp

selected and wavelet processed using Registax, and the right image is using Data Modeling approach. A total of 18 images were stacked for both the Registax and Data Modeling final images.


Score = 0.2026


(Note: Noise suppressed)

Raw Frame #1


Score = 0.2026


Raw Frame #1

Fig. 17: Data Modeling for image stacking on lunar surface feature (top), and zoom in to lunar surface feature (bottom). Left image is raw image, center sharp selected and wavelet processed using Registax, and the right image is using Data Modeling approach. Note

the absence of noise and the presence of truly resolved detail in the Data Modeling image.

Fig. 18: Tags for images in Figures 16 and 17.

CLS

INPUT "ascii file => ", zz$

INPUT "tag output file => ", tag$

OPEN zz$ FOR INPUT AS #1

OPEN tag$ FOR OUTPUT AS #3

fsum = 0: mean = 0: mean1 = 0: stdev = 0: skew = 0

kurt = 0: m6 = 0: m8 = 0: m = 0: min = 1: max = 0

DO UNTIL EOF(1)

LINE INPUT #1, b$

n = LEN(b$)

FOR i = 1 TO n

z = ASC(UCASE$(MID$(b$, i, 1)))

IF z = 32 THEN

'skip procesing of spaces

GOTO 999

ELSEIF z >= 65 AND z <= 90 THEN

'A-Z

z = 10 + (z - 65)

ELSE

'numbers

z = VAL(MID$(b$, i, 1))

END IF

FOR j = 5 TO 0 STEP -1

IF z >= 2 ^ j THEN

z = z - 2 ^ j

f = 1

ELSE

f = 0

END IF

f = 2 * f - 1

fsum = fsum + (f - mean1)

m = m + 1

mean1 = (mean1 * (m - 1) + f) / m

mean = (mean * (m - 1) + fsum) / m

stdev = SQR((((stdev ^ 2) * (m - 1) / m)) + (((mean - fsum) ^ 2) / m))

IF stdev <> 0 THEN

skewnew = ((fsum - mean) / stdev) ^ 3

skew = (skew * (m - 1) + skewnew) / m

kurtnew = ((fsum - mean) / stdev) ^ 4

kurt = (((kurt + 3) * (m - 1) + kurtnew) / m) - 3

m6new = ((fsum - mean) / stdev) ^ 6

m6 = (((m6 + 15) * (m - 1) + m6new) / m) - 15

m8new = ((fsum - mean) / stdev) ^ 8

m8 = (((m8 + 105) * (m - 1) + m8new) / m) - 105

END IF

NEXT j

999 :

NEXT i

LOOP

PRINT #3, mean1, mean, stdev, skew, kurt, m6, m8

CLOSE : END Fig. 19: Source code for generating the tag estimates in Figure 18 and the tag for this paper included in the abstract.

1110010111001101101011010011001

12 −= ii wz

∑=

=i

j

ji zy0

-11

Fig. 20: Data Modeling for text mining. Resulting 1/f data set (right) is now amenable for processing as described in Figure 1.

TABLES

Table 1: Complete Table of Jacoby spectra tag values.

Table 2: Input parameters for generating binary star light curves, along with derived features and Data Modeling results. Luminosity

ratio and radii ratio are constant near the top of the chart for training data on a single object type and varying orbital inclinations. Important data is found in columns 4-8, which describe the histogram of each light curve. Columns 5-7 are the inputs derived for the

Data Model. Rollups showing overlap are at the bottom of each column.

Table 3: Input parameters for exo-planets, along with features derived and Data Modeling results. Luminosity ratio and orbital

inclination are constant values for the first 50 cases while planet radius varies between 0.1 and 0.5 of star radius. The second 50 cases contain equal variation in planet radius and orbital inclination variation between 80 and 90 degrees. Columns 4-8 describe the

histogram of each light curve. Rollups of the columns are provided, which show the existence of total overlap. 7 cases in the last 50 were classified as being like the training data (Accept), while the remaining 43 were labeled as different (Reject).

VIRTUAL INSTRUMENT PROTOTYPING WITH DATA MODELING

H. M. Jaenisch and J. W. Handley Sparta, Inc.

Huntsville, AL

J. C. Pooley III and S. R. Murray Amtec, Inc.

Huntsville, AL

ABSTRACT

Virtual instruments and sensors are mathematical embodiments of physically unrealizable devices. The mathematical theory for prototyping virtual sensors using Data Modeling is defined. The theory of change

detection using Data Modeling is presented. Several examples of prototype virtual sensors are given including: physical parameter sensor using trajectory, THz anthrax detector, gear tooth health sensor, and

fault detector.

INTRODUCTION

A virtual instrument is comprised of a multitude of real and/or virtual sensors. A virtual sensor is a physically unrealizable sensor that exists only as a mathematical construct. It uses as input measurements from other physical devices. This mathematical function fuses measurements into a

derived measurement. The fusion process includes: data reduction, normalization, filtering, transformation, and processing; all of which are encompassed by a mathematical function termed a Data Fusion Model, or abbreviated as Data Model. In this work, we explore the use of Data Models as virtual

sensor prototypes. Such prototypes become physical virtual sensors when they are embedded in firmware in conjunction with physical sensors as a hardware device.

THEORY

For example, consider the measurement of a novel physical parameter termed the Dynamic

Coefficient of Thermal Expansion. The Dynamic Coefficient of Thermal Expansion differs from the classic coefficient of thermal expansion by including the elongation effects due to the coupling of vibration and thermal effects in the material. Clearly, this Dynamic Coefficient of Thermal Expansion does not

currently exist and cannot be measured with existing sensors. It can, however, be inferred from existing sensors and suitable mathematical processing, or the construction of a suitable virtual sensor. This would be achieved as follows:

In the laboratory, strain gauges governed by

GF

R∆=ε (1)

placed in the axial and longitudinal directions measure size change in each axis. For strain gauges, ε is

strain, ∆R change in resistance, while GF is strain sensitivity or gauge factor. Accelerometers are

governed by

m

kxa = (2)

derived from Hooke’s Law and Newton’s 2nd

law of motion. These measure vibration frequency. In (2), k is the spring constant, x displacement, and m mass. Thermocouples governed by

Approved for public release; distribution is unlimited.

a

a

S

VT =∆ (3)

measure temperature change. In (3), the Seebeck coefficient is Sa, the voltage across the thermocouple

Va, and the temperature change between the sides of the thermocouple ∆T. From these measurements,

the change of material size due to changes temperature (∆T) are obtained from

TLL ∆=∆ α (4)

where L is length and α is the coefficient of thermal expansion. The total change including vibration is defined as

ωβ∆=∆ LL (5)

where ∆ω is change in frequency derived from accelerometer measurements and β is proportional to the amplitude of the forcing function. The total expansion of the material would be the sum of effects in (4)

and (5), which yields the Dynamic Coefficient of Thermal Expansion (γ) given by

γω

LL

T

L =∂∂+

∂∂

(6)

Because (6) is a partial differential equation based on physical properties in the form of different

material properties (T and ω), the Dynamic Coefficient of Thermal Expansion cannot be directly measured. Therefore, strain gauges, thermocouples, and accelerometers are input variables to a Data

Model. This Data Model fuses these inputs into meta-variables of the form

),,( Tafuh ε= (7)

These meta-variables form the nested polynomial for the Dynamic Coefficient of Thermal

Expansion (γ).

∑∑= =

=N

h

k

j

j

hhj us1 0

,γ (8)

STOCHASTIC (fBm) DIFFERENTIAL EQUATION MODELING

Invariably, sensor measurements involve sample and process noise. This leads to the modeling of sensor measurements using the one-dimensional stochastic differential equation

1, 2

)()()( tnxgxfdt

dx += (9)

which assumes the self-similar or fractal fine structure of a fBm process is modeled by an additive noise

term g(x)n(t). A solution to (9) would be

[ ] .)()()()( ∫ += dttnxgxftx (10)

Lacking a repeatable numerical solution, the classic approach is modeling the fBm ensemble average behavior of (10) using the generalized Fokker-Planck equation of the form

))((2

1))((

2

2

2

Pxgx

Pxfxt

P

∂∂+

∂∂−=

∂∂

(11)

for P(t,x) with P(0,x)=d(x-z). The Fokker-Planck equation is a model of dynamic continuous probability density functions (PDF) for averages of solutions to stochastic differential equations in (9) arising from Brownian motion increments. The Fokker-Planck equation only characterizes an ensemble of

realizations; it does not model specific fBm trajectories.

Jaenisch derived a specific formulation for a particular fBm trajectory realization x(t)

{ } [ ]∫ +≅≅ dttnxgxftftx f )()()()()( D (12)

This uses self-similarity inherent in the quasi -periodic function f(x) to define fractal fine structure (micro-trend). This takes the form of an interpolating operator Df applied to a continuous function f(t)

(macro-trend)3, 4

.

This approach shows that the process is modeled by a first order differential equation of the form

)(tfdt

dx = (13)

where t is the independent variable and f(t) is the fBm process. Differential equation based processes are comprised of continuous differentiable functions. F(t) is cast as a Turlington polynomial T(x) of the form

( )

∑

+++=

−m

d

xx

mm

m

DBxAxT1

10 101log)( (14)

Turlington polynomials are constructed in a piecewise orthogonal term fashion, yielding a continuous

function. Handley previously obtained the general form of the derivative2 from (14). From this, the first

derivative is

( )

( ) .

101

10

1

∑ −

−

+

+=m

d

xx

d

xx

m

m

m

m

m

m

d

DB

dx

dT (15)

Data Relations Modeling discovers the functional relationship between the first derivative and its forcing functions

5.

DATA RELATIONS MODELING

Data Relations Modeling6 derives a mathematical expression from measured data. This

inverse modeling and parameter estimation yields functional equations and numeric coefficients. These equations are the functional form of the prior distribution required for Bayesian methods.

Discovering the form of the prior model requires approximating the Kolmogorov-Gabor polynomial

L++++= ∑∑∑∑∑∑i j k

kjiijki j

jiiji


using Ivakhnenko Group Method of Data Handling (GMDH) discrete nested functions of the form

)))))))))((((((((,()( 21 tbxtbxbxtftx nK= (17)

which is a form of orthogonal nested polynomial network. This nested function is comprised of low order basis functions (meta-variables) defined by truncated 3

rd order Taylor series approximations to the

Logistic Function. In (17), n represents the number of layers in the final Data Model and x(bi(t)) the inputs mapped from the previous layer. Each layer forms a polynomial model that is a fusion of up to 3 variables into a meta-variable.

In order to perform Data Relations Modeling for a stochastic differential equation, one selects the

form of the forcing functions driving T’(x). These forcing functions are inputs to the modeling process, and

T’(x) is the output. If specific prior forms of the forcing functions are known, they can be specified. If the priors are unknown, eigenfunctions derived from dominant positive frequency components are used as the model of the fBm trajectory forcing functions.

A useful model forcing function can be obtained from the correlation function and the

characteristic spectral function for the sampled process.

Our eigenfunction extraction method uses a variation of Wiener filtering to identify individual

peaks in a dB power spectrum. First, least squares regression fits a straight line to the entire dB power

spectrum, resulting in a slope and a y intercept. The y intercept is the noise floor across the spectrum. Peaks are identified by slope changes using the 1

st derivative of the dB power spectrum. The

associated Fourier coefficients (cosine (frequency) and sine (phase)) for each peak are used as input into

the Data Modeling process equation. Data Models scale as a nested O[3

n] process, resulting in nonlinear differential equations that

may have hundreds of terms associated with it. This results in the autonomously generated Data Driven Differential Equation

2, 5

)))))))))((((((((,(21

tbxtbxbxtfdt

dxn

K= (18)

Data Modeling generates the Data Driven Differential Equation shown in (18) that can be solved using numerical integration. These differential equations of motion are used to obtain state

transition matrices for Kalman filtering of nonlinear processes and motions. When such differential equations are coupled with the fractal operator Df , fBm dynamic processes are modeled as

{ }.)()( tftx fD≅ (19)

where f(t) is the Runge-Kutta solution to Equation (18) above2.

DATA SEQUENCE MODELING Data sets (time series, image, or other vectors of information) can be reconstructed from a sparse

sub-sampling1 , 6

. Sparse sub-sampling is achieved by simple naive decimation. The sparsely sub-sampled points are a specimen or Data Model of the original data set. Because they are demonstrated to be the support vector

[ ] 01),( =−+ bwxy iiiα (20)

of the set, the sample size need never exceed 10% of the original number of points, akin to reconstructing

a musical score by saving only every 10th note. Data Sequence Modeling is based on a unique fractal pseudo-derivative operator previously defined

2 as

{ } ).()())(()(

110111110111

−+−−−+−−+− −−+−+

−−

−−

≡ nnninniNnnnn

if yyyyyyyxN

yyyy

N

yyyD (21)

This fractal operator evolved from extensive work with the Iterated Function System (IFS) contractive mapping

+

=

n

n

nn

n

nf

e

y

x

dc

a

y

xw

0 (22)

Unlike IFS, Df generates fractal fine structure in a single pass without resorting to random number

generators or Markov processes relying strictly on the Collage and Ruelle-Takens Theorem.

CHANGE DETECTION

Change detection is achieved by creating a Data Model hypersensitive to small changes in

sample distribution. Changes in distribution are often too small to resolve with classical methods. To exploit small changes, the multiplicative high-order terms and cross terms yielded by the Data Model

are required. This is akin to producing a multivariate statistical process control snapshot for very non-Gaussian and nonlinear processes.

Analogous to Shewhart’s X-Bar chart7, a model of the complex multi-variate process that is in

statistical control is derived by observing the time history of the governing distributions. The highly volatile sensitivity of the Data Model equation signals a tip-off when the process dynamics drift or

change8, 9

. A Data Model is built by training on nominal cases against a categorical variable value.

Overfitting makes the model sensitive to feature values outside training boundaries. Change detection models detect combinations of incoming feature values that within valid ranges but simply not in combinations encountered in the nominal training set.

RESULTS AND DISCUSSION

PHYSICAL PARAMETER VIRTUAL SENSOR

Can the mass of an object be directly measured from position? Using classical sensors this

seems untenable. However, a virtual sensor can yield the mass from x and y positions alone. Virtual sensors for cross sectional area and drag coefficient are also developed from the x and y FPA position data

2, 5.

To illustrate, Fig. 1 depicts a measured maneuvering object trajectory. The physics of the target is unknown. The trajectory is biased and distorted by the sensor measurement process and therefore

unknown. A Kalman filter cannot be applied because this object is maneuvering and its equations of motion are unknown. No a priori model exists for the sensor measurement bias and noise. This yields a problem of blind system identification and parameter estimation.

To estimate the equations of motions for this object, we seek a second order differential equation

),,,,,,,(112

2

mnuucc

dt

dyyf

dt

ydKK= (23)

where f is a continuous function. Also, dt

dy and 2

2

dt

yd are first and second derivatives of y with respect to

time, c1 to cn are constants, and u1 to um are forcing functions. U1 through um are autonomously

selected from a priori domain physics related derivative forcing functions:

2r

GMaGravity =

m

pAvCa D

Drag

25.0= ∑ ∑

∞

=

∞

=+=

0 0

)sin()cos(n n

nni nxbnxau (24)

The control parameters are estimated using a genetic algorithm following Koza.

Optimization of parameters uses Levenberg-Marquardt nonlinear least squares fitting10

. The genetic algorithm is a directed random search parameter estimator defined as follows:

1. Select a gene pool size of n

n, where n is the number of coefficients in the forcing function.

2. Populate pool with random values (apply prior probabilities if known).

3. Rank according to selected cost function (determine points falling outside ¼ standard deviation boundary of objective function, and sum their Euclidean distances).

4. Select frequency of occurrence of mutation and its form (negation or random perturbation).

5. Select number of offspring to propagate forward as ½ nn.

6. Populate offspring pool with top ½ of current pool and spliced offspring (splicing selects a random location in the coefficient array and splices the left side of one parent and the right side of another

into a new offspring). 7. Pool is re-ranked as in 3). 8. Stopping criteria: Points within ¼ standard deviation or user specified maximum number of

iterations. Best candidate solutions pass through Levenberg-Marquardt optimization.

For the trajectory in Fig. 1 (upper left), the genetic algorithm estimated:

2037.0 mA = (Equivalent body area)

"5.8217.0 == mD (Diameter equivalent body) 69.0=DC

lbskgM 7318.33 == (25)

The differential equation modeling steps are:

1. Identifying target physics model

a. Position and velocity derivative b. Physics based forcing function derivatives c. Thrust derivative

d. Lumped parameter stability derivatives 2. Identifying sensor bias model

a. Position bias

b. Orientation error c. Measurement error d. Boresight error

Details are found in previous work5. The final results are in Fig. 1. The mode of the error lies in the 8th bin

of the histogram, corresponding to a most frequently occurring error bound of negative one (-1.0) to ½

(0.5) meters. This matches the original Cramer-Rao bound ambiguity. This results in the final second order differential equation form

++++= )20cos,18cos,14cos,8cos,5cos,2(cos)()()(2

2

fCfgfTfdt

ydD

+)36cos,31cos,25cos,23cos,17cos,8cos,5cos,3(cosf

+)36cos,31cos,25cos,23cos,17cos,8cos,5(cosf )20cos,17cos,5cos,2(cosf (26)

where T is thrust, g is gravity, CD is drag, and each cosine term is a dominant positive frequency

component. The number for each cosine is the number of cycles.

Residual

Error

Original and

Model Overlay

0.0

-6151.40.0 1200.0

5.16

-13.18

Thrust

0.0 1200.0

Gravity

Drag

Lumped Stability Derivatives

0.0 1200.0

0.0 1200.0

Boresight

Measurement

Orientation

Meters

Meters

-1.0 m ½ m

Fig. 1. Measured trajectory data compared with model overlay (top left), physics based and lumped stability derivative (fin number, aspect, and orientation) forcing functions (top right), user input thrust acceleration profile (bottom left), and orientation, measurement, and boresight error derivatives (bottom right).

Bt SIMULANT VIRTUAL THz SENSOR

Fig. 2 shows 5 different anthrax Bt simulant concentrations measured with a THz sensor (1.6, 1.8, 2.7, 3.8, and 4.4%). For this study, no field data, simulation, or code existed, only a single viewgraph with crude representation of measurements. This viewgraph had to be scanned in from a paper copy, and

each of the data curves shown extracted from the viewgraph and into a data file. This was done by plotting the bitmap image from the scan to the screen, isolating each of the data curves separately, and reading pixel values relative to the screen coordinates to determine the values of the curves. Once this

was done, the curves were normalized to the values given on the absorption axis of the original chart, and interpolated up to the correct number of points using Data Sequence Modeling.

Because the presence and locations of peaks in each Bt spectrum varied as a function of concentration (particularly above 9 THz), neither simple linear nor higher order Lagrangian interpolation was applicable. Both Data Relations Modeling and Data Sequence Modeling were required to construct

this vi rtual sensor prototype.

Creating this virtual prototype required the following process.

1. Generate prototype spectrum by averaging the 5 known spectra. 2. Sub-sample prototype 20% (52 points).

3. Convert specimen points into an equation by fitting a Turlington polynomial. 4. Sub-sample resulting polynomial and each of the original spectra 8:1 (12.5%). 5. Calculate difference between the polynomial and each spectrum.

6. Concentration, frequency, and prototype are inputs. Differences are the training output. 7. Generate equation (Fig. 3) that generates THz absorption spectrum. 8. Apply Df to (7) providing fine structure.

5.0 Frequency 10.0

1.3

A

bsorp

tion

5.5

1.61.71.8

2.25

2.7

3.25

4.4

4.1

3.8

Fig. 2. Truth Bt anthrax simulant THz spectra (1.6, 1.8, 2.7, 3.8, and 4.4% concentrations) and Data Model Virtual Sensor Prototype spectra predictions (1.7, 2.25, 3.25, and 4.1% concentrations).

The predicted spectra are shown in Fig. 2, and the Data Model virtual sensor prototype is shown in the Appendix. This virtual sensor prototype requires as input the desired concentration, and writes out into a file named Output the absorption spectrum between 5 and 9.8 THz for the input concentration. The

source code is written in QBASIC, and can easily be converted into any other language of choice. A virtual sensor prototype that estimates the Bt concentration from measured absorption

spectra is also given in the Appendix. The Bt spectrum virtual sensor prototype is run in a loop between

minimum and maximum frequency, and its absorption output (γ) and associated frequency (ν) are used as

input into the Bt concentration virtual sensor prototype. The average concentration (ζ) is then determined

by averaging all of the concentration outputs (n) using the following functional form

∑=

=n

i

fn 1

),(1 νγζ (27)

The Bt concentration model in conjunction with the Bt spectrum model are an end-to-end high

fidelity simulation. Once each spectrum has been generated, Data Modeling can be applied to the data to provide a

transfer function that transforms the Bt simulant data into the live anthrax absorption spectrum expected in field tests. This defines a virtual sensor that responds to a stimulus available in controlled environment settings, and transforms them into the response expected in-situ.

For cloud dispersion mapping, only a limited number of sensors may be available for spatial deployment. For example, in the case of measuring anthrax Bt simulant concentrations with a terahertz (THz) sensor, a very coarse grid of sensors is deployed as shown in Fig. 3. This limited number of

sensors sub-samples the measurement space. Using Data Modeling, these sensors yield spatial interpolation and extrapolation. A proof of concept for such virtual 3-D spatial sensors was previously demonstrated

12.

Bio Agent Gas Cloud Meteorological cloud propagation model

Agent concentration vs time – t*+ ∆t Prevailing Wind

Detonation at time t*

Standoff Detector

Point Detector Grid (Hollow = Virtual, Filled = Real)

Fig. 3. Solid point detectors represent those in the field for the test, while hollow squares represent virtual sensors whose measurements are generated with Data Modeling.

GEAR MESH FREQUENCY VIRTUAL SENSOR

The Gear Mesh Frequency Virtual Sensor prototype uses as input the measured accelerometer data, which is transformed into the instantaneous frequency corrected acceleration from which the gear mesh frequency is extracted as output.

This sensor prototype uses corrected shaft velocity variation ))(( tθ& during multiple shaft speed

periods and acquisition snapshot time intervals is Rotordynamic Kinetics Variation Correction (RKVC)

13, 14, 15, 16. RKVC does not require a Tach Pulse, but does require expected nominal shaft speeds

and allowable velocity variations from nominal speed.

The input acceleration signal x(t) may contain several spectral components at different carrier

frequencies fc. This x(t) signal is band pass filtered about a carrier fc(k)

)()()( fXfHfX k→→ (28)

where H(f) = 1 if fc - ∆f ≤ f ≤ fc + ∆f and H(f) = 0 if fc - ∆f < f < fc + ∆f. This modified Xk(f) is then Hilbert

transformed (H.T.) into

)(~..)( fTHf kk χχ →→ (29)

which yields the analytical signal after inverse Fourier transforming

)(~)()(ˆ txtxtx kkk += (30)

from which the Instantaneous Phase Φk (t) for the k

th synchronous waveform of carrier fc(k)

=

)(

)(~)(

tx

txArcTant

k

k

kφ (31)

Then the primary signal x(t) can be instantaneously phase corrected for velocity variations within periods.

)(

)()(

t

tt

ωφ=∆ (32)

This is the instantaneous temporal correction for each time sample. This regenerated time sampling history is interpolated to recreate a new time signal with constant angular velocity. From this

new time signal with constant angular velocity, the gear mesh frequency is determined, and the new time signal used to indirectly measure gear tooth health status.

VIRTUAL FAULT DETECTOR

Since determination of current helicopter gearbox conditions as nominal or anomalous is

physically unrealizable directly from accelerometer measurements alone , a virtual sensor is required. The Virtual Fault Detector uses dominant eigenfunctions of the accelerometer measurements. These eigenfunctions are calculated in the same manner described for the Physical Parameter Virtual

Sensor. The vibration data passes through a change detector. If tip-offs occur, the data is flagged for verification. Prototyping occurs in three parts:

1. Construction of a Data Model for the nominal data. 2. Construction of a Data Model for the anomalous data. 3. Construction of a combined Data Model for the difference between the second and first models.

This third model is the actual model used for anomalous behavior. A 100 kHz data stream was sampled for .05 seconds, yielding 4096 points. This data was sub-

sampled by 10% to 410 points. Eigenfunction extraction yielded the inputs for constructing the nominal and anomalous Data Models. Df reconstructed the equation output at the same sampling rate as the

original (4096 points). Comparison is made between the Data Model and interpolation spectrum. These power spectra

are obtained from linear interpolation of the 410 sub-sampled points and the Data Model. Fig. 4 shows the integrated error between these spectra. Both linear interpolation (top) and Data Modeling (bottom) captured low frequency content.

The Nyquist location in the power spectrum corresponds to ½ the number of sub-sampled points,

or in this case 205 terms into the power spectrum (½ of the 410 sub-sampled points). However, the Data

Model was able to more accurately capture the higher frequency information (shown by the higher value for the linear interpolation curve over the Data Model on the right of the graph in Fig. 4) than the classical linear interpolation approach.

0 F r e q u e n c y T e r m 5 1 2

0

I n t e g r a t e d E r r o r

5 3 0 1 L i n e a r I n t e r p o l a t i o n

D a t a M o d e l

N y q u i s t

Fig. 4. Integrated error between power spectrum for nominal Data Model and original (bottom) and linear interpolation and original (top). Error is less for the Data Model than for linear interpolation at the higher frequencies, and nearly equal up to the point predicted by Nyquist as the maximum frequency captured with the sub-sample rate.

Once the nominal and anomalous equation Data Models are built, a third and final combined

equation Data Model is built using the difference between the nominal and anomalous equation Data Models. The difference data is generated by subtracting the nominal equation output from the anomalous equation output. A combined equation Data Model is then generated in the same manner as for the

nominal and anomalous data using Data Relations Modeling with eigenanalysis.

The power spectrum for the nominal and anomalous Data Modeling outputs are shown in Fig. 5,

depicting the first 512 points (first ¼) in the power spectrum for each. These power spectrum curves actually fall on top of one another, but have been shifted in mean so that the structure of each can be seen. The final model is a signal amplifier that minimizes entropy from insignificant sidebands and

folding frequencies and redistributes the energy into the dominant peaks.

N o m i n a l A n o m a l o u s

Orig inal

Data

M o d e l

Fig. 5. Power spectra for nominal (left), anomalous (middle), and histogram of anomalous time series (right). Main gear mesh frequency (980 Hz) indicated by arrow.

The gear mesh frequency (980 Hz) is the first peak from the left (indicated by an arrow in Fig. 5). The side lobes around 980 Hz are used to detect anomalies.

The combined Data Model equation used to generate the spectra values is given in the Appendix.

This Data Model replaces the one-million point measurement set with the short source code found in the Appendix. This Data Model uses as input a flag of fault or no-fault conditions (specified by a 1 or 0), resulting in

+= )199cos,131cos,110cos,95cos,79cos,66cos,40(cos)( fgi

λ

)201cos,199cos,133cos,131cos,112cos,110cos,95cos,79cos,66cos,40(cosfw ⋅ (33)

A change detector is given in the Appendix. 4096 points from the nominal Data Model were divided into 64 point windows. These windows were characterized using mean, standard deviation, skewness, kurtosis, 6

th and 8

th moment, entropy fractal dimensions

14. This change detector

performed as shown in Table 1.

Correct Classification False Alarm BDR

Nominal 64 / 64 (100%) 0 / 64 (0%) 95.5%

Anomalous 61 / 64 (95%) 3 / 64 (5%) 95.5% Table 1. Virtual fault detector results.

Concatenating nominal and anomalous windows resulted in the Bayesian Detection Rate (BDR)

)1( pfrp

rpBDR

−+= (34)

where r is the true positive rate (64 out of 128 = ½), p is the probability of fault (number of faults / total number of measurements) (64 out of 128 = ½), and f is the false positive rate (3 out of 128).

SUMMARY AND CONCLUSIONS The creation of a virtual instrument, and therefore virtual sensors, is made possible through the

use of Data Modeling. The mathematical theory behind Data Modeling has been explained, and several practical examples of prototype virtual sensors were implemented. These examples include virtual sensors for physical parameter measurement from trajectory position, absorption spectra from view

graphs, indirect measurement of gear tooth health through measurement of gear mesh frequency, and fault detection without a priori knowledge of fault conditions during training.

ACKNOWLEDGEMENTS

The authors would like to thank Dr. Bill Pickel and Dr. Mehmet Erengil of the University of Texas at Austin Institute for Advanced Technology (IAT) for providing the measured trajectory used in constructing the physical parameter virtual sensor. The authors would also like to thank John Deacon of

Sparta, Inc; Tim Aden and Scott McPheeters of US Army PEO-SHORAD for their support of the authors’ work; Dr. Andrew Greene, Nils Höltge, Dr. Marcel Thuerk, and Sion Balass of Matrix-ADS Limited for supporting the genetic algorithm method of selecting parameters; and Licht Strahl Engineering INC and

Dr. Marvin Carroll of Tec-Masters, Inc. for use of the Majqanda tool suite in the analysis in this paper.

REFERENCES

1. Jaenisch, H. and Handley, J., Data Modeling of 1/f noise sets, Proceedings of SPIE Fluctuations

and Noise 2003, Santa Fe, NM, (June 2003).

2. Jaenisch, H.M., Handley, J.W., and Faucheux, J.P., Data Driven Differential Equation Modeling of fBm processes, Proceedings of SPIE, (August 2003).

3. Handley, J.W., Jaenisch, H.M., Bjork, C.A., Richardson, L.T., and Carruth, R.T., Chaos and Fractal

Algorithms Applied to Signal Processing and Analysis, Simulation, 60:4, 261-279 (1993). 4. Handley, J.W. On The Existence Of A Transition Point In Sampled Data Using Fractal Methods.

Master's Thesis, Ann Arbor, MI: UMI, 1995.

5. Jaenisch, H.M., and Handley, J.W., Automatic Differential Equation Data Modeling for UAV Situational Awareness, Society for Computer Simulation, Huntsville Simulation Conference 2003, (October 2003).

6. Jaenisch, H.M. and Handley, J.W., Data Modeling for Radar Applications, Proceedings of the IEEE Radar Conference 2003, (May 2003).

7. Grant, E. Statistical Quality Control. New York: McGraw-Hill, 1952.

8. Jaenisch, H.M., Handley, J.W., and Bonham, L.A., Enabling Calibration on Demand for Situational Awareness, Army Aviation Association of America (AAAA), Tennessee Valley Chapter, (February 2003).

9. Jaenisch, H.M., Handley, J.W., Faucheux, J.P., and Harris, B., Data Modeling of Network Dynamics, Proceedings of SPIE, (August 2003).

10. Koza, J.R., Genetic Programming: On the Programming of Computers by Means of Natural

Selection, Cambridge, MA: MIT Press, 1992. 11. Jaenisch, H.M., Handley, J.W., White, K.H., Watson Jr., J.W., Case, C.T, and Songy, C.G., Virtual

prototyping with Data Modeling, Proceedings of SPIE, Seattle, WA, (July 2002).

12. Jaenisch, H.M., Handley, J.W., Roberts, S.K., Case, C.T., and Songy, C.G. Mapping of underwater topology using indirect sensing and Data Modeling, Proceedings of SPIE, Seattle, WA, (July 2002).

13. Pooley, J.C., Murray, S.R., Jaenisch, H.M., and Handley, J.W., Fault Detection via Complex Hybrid Signature Analysis, JANNAF 39th Combustion, 27th Airbreathing Propulsion, 21st Propulsion Systems Hazards, and 3rd Modeling and Simulation Subcommittees Joint Meeting, Colorado Springs,

CO, (December 2003). 14. Jaenisch, H.M., Handley, J.W., Pooley, J.C., and Murray, S.R. Data Modeling For Complex

Epicyclical Gearbox Fault Detection, JANNAF 39th Combustion, 27th Airbreathing Propulsion, 21st

Propulsion Systems Hazards, and 3rd Modeling and Simulation Subcommittees Joint Meeting, Colorado Springs, CO, (December 2003).

15. Dempsey, P.J., Handschuh, R.F., and Afjeh, A.A., Spiral Bevel Gear Damage Detection Using

Detection Fusion Analysis, 5th

International Conference on Information Fusion, Annapolis, MD, (July 2002).

16. McFadden, P.D., A Technique For Calculating The Time Domain Averages of the Vibration of

the Individual Planet Gears and the Sun Gear in an Epicyclic Gearbox, Journal of Sound and Vibration, 1991, 144(1), 163-172.

APPENDIX Bt spectrum virtual sensor prototype (QBASIC source code is in two columns. Place second column listing immediately after first column to run.)

DEFCUR A-Z:DECLARE FUNCTION yt @(x@,conc@):DECLARE FUNCTION proto@(x@)INPUT "Concentration (1.6% - 4.4%) => ", concOPEN "output" FOR OUTPUT AS #1ntotal = 256: nsub = 32: ymin = 1.3@: ymax = 5.73@

10 : n = INT(x * ( nsub - 1) / (ntotal - 1)) + 1istart = CINT((( ntotal - 1) * (n - 1)) / (nsub - 1))ifinish = CINT((( ntotal - 1) * n) / ( nsub - 1))iend = CINT(((ntotal - 1) * (n + 1)) / ( nsub - 1))

IF iend > 255 THEN iend = iend - 255xpt = (x - istart ) / (ifinish - istart )ix = ( ntotal - 1) * xpt

s = (( yt( iend , conc) - ymin ) / ( ymax - ymin ))s = s - (( yt(istart, conc) - ymin ) / ( ymax - ymin ))Rem Fractal operator (Data Sequence Model)Rem applied to yt (Data Relations Model)

dfde = ((( yt(ifinish , conc ) - yt(istart, conc)) / ( ntotal - 1))) * ixdfde = dfde- ((s*( yt( ntotal -1, conc )-yt(0, conc ))/(ntotal -1)))*ixdfde = dfde + s * yt (ix, conc ) + yt (istart, conc)dfde = dfde - yt (0, conc ) * s

PRINT #1, dfde: IF x < 255 THEN : x = x + 1: GOTO 10CLOSE : ENDFUNCTION proto (x)Rem Turlington polynomial

y = -.0783 * x + 2.6417y = y + - .0009 * LOG(1 + 10 ^ ((x - 5.1724) / .01)) / LOG(10)y = y + .0022 * LOG(1 + 10 ^ ((x - 5.3448) / .01)) / LOG(10)

y = y + .0066 * LOG(1 + 10 ^ ((x - 5.5172) / .01)) / LOG(10)y = y + - .0006 * LOG(1 + 10 ^ ((x - 5.6897) / .01)) / LOG(10)y = y + - .0042 * LOG(1 + 10 ^ ((x - 5.8621) / .01)) / LOG(10)y = y + .0072 * LOG(1 + 10 ^ ((x - 6.0345) / .01)) / LOG(10)

y = y + - .0075 * LOG(1 + 10 ^ ((x - 6.2069) / .01)) / LOG(10)y = y + .0127 * LOG(1 + 10 ^ ((x - 6.3793) / .01)) / LOG(10)y = y + - .0114 * LOG(1 + 10 ^ ((x - 6.5517) / .01)) / LOG(10)y = y + .0104 * LOG(1 + 10 ^ ((x - 6.7241) / .01)) / LOG(10)

y = y + - .0007 * LOG(1 + 10 ^ ((x - 6.8966) / .01)) / LOG(10)y = y + - .0431 * LOG(1 + 10 ^ ((x - 7.069) / .01)) / LOG(10)y = y + .009 * LOG(1 + 10 ^ ((x - 7.2414) / .01)) / LOG(10)y = y + .0215 * LOG(1 + 10 ^ ((x - 7.4138) / .01)) / LOG(10)

y = y + .013 * LOG(1 + 10 ^ ((x - 7.5862) / .01)) / LOG(10)y = y + .0028 * LOG(1 + 10 ^ ((x - 7.7586) / .01)) / LOG(10)y = y + .0024 * LOG(1 + 10 ^ ((x - 7.931) / .01)) / LOG(10)

y = y + - .0232 * LOG(1 + 10 ^ ((x - 8.1034) / .01)) / LOG(10)y = y + - .0114 * LOG(1 + 10 ^ ((x - 8.2759) / .01)) / LOG(10)y = y + .0006 * LOG(1 + 10 ^ ((x - 8.4483) / .01)) / LOG(10)y = y + .0114 * LOG(1 + 10 ^ ((x - 8.6207) / .01)) / LOG(10)

y = y + .0102 * LOG(1 + 10 ^ ((x - 8.7931) / .01)) / LOG(10)y = y + - .0037 * LOG(1 + 10 ^ ((x - 8.9655) / .01)) / LOG(10)y = y + .0166 * LOG(1 + 10 ^ ((x - 9.1379) / .01)) / LOG(10)y = y + - .0297 * LOG(1 + 10 ^ ((x - 9.3103) / .01)) / LOG(10)

y = y + .0266 * LOG(1 + 10 ^ ((x - 9.4828) / .01)) / LOG(10)y = y + - .0237 * LOG(1 + 10 ^ ((x - 9.6552) / .01)) / LOG(10)y = y + .0349 * LOG(1 + 10 ^ ((x - 9.8276) / .01)) / LOG(10)proto = y

END FUNCTIONFUNCTION yt (x, conc )ON LOCAL ERROR RESUME NEXT

Rem Kolmogorov- Gabor polynomial (Data Relations Model)x1 = xf = 5@ + 4.8@ * x1 / 255p1 = proto(f)

psub = (p1 - 2.8587) / .3133fsub = (f - 7.5) / 1.2623csub = (conc - 2.86) / .992l1o1 = .1757

l1o1 = l1o1 + (1.0155) * ( csub )l1o1 = l1o1 + (.1212) * (fsub )l1o1 = l1o1 + (.0355) * (psub )l1o1 = l1o1 + ( -.1403) * ( csub * csub )

l1o1 = l1o1 + ( -.002) * (fsub * fsub)l1o1 = l1o1 + (.0058) * (psub * psub)l1o1 = l1o1 + (.0652) * (csub * fsub)l1o1 = l1o1 + (.0926) * (csub * psub)

l1o1 = l1o1 + ( -.0108) * ( fsub * psub)l1o1 = l1o1 + ( -.0223) * ( csub * fsub * psub )l1o1 = l1o1 + ( -.0086) * ( csub * csub * csub )

l1o1 = l1o1 + ( -.002) * (fsub * fsub * fsub )l1o1 = l1o1 + ( -.0149) * ( psub * psub * psub )l1o1 = l1o1 + ( -.096) * (fsub * csub * csub )l1o1 = l1o1 + (.0266) * (csub * fsub * fsub )

l1o1 = l1o1 + (.004) * ( csub * psub * psub)l1o1 = l1o1 + ( -.0108) * ( psub * csub * csub )l1o1 = l1o1 + ( -.0109) * ( psub * fsub * fsub )l1o1 = l1o1 + (.0177) * (fsub * psub * psub)

l2o2 = .0615l2o2 = l2o2 + (.0341) * (l1o1)l2o2 = l2o2 + (.6378) * (csub )l2o2 = l2o2 + ( -.1514) * ( psub )

l2o2 = l2o2 + ( -.3858) * (l1o1 * l1o1)l2o2 = l2o2 + ( -1.2049) * ( csub * csub )l2o2 = l2o2 + ( -.0637) * ( psub * psub)

l2o2 = l2o2 + (1.5586) * (l1o1 * csub )l2o2 = l2o2 + (.9776) * (l1o1 * psub)l2o2 = l2o2 + ( -1.0147) * ( csub * psub )l2o2 = l2o2 + (.7615) * (l1o1 * csub * psub)

l2o2 = l2o2 + ( -.7653) * (l1o1 * l1o1 * l1o1)l2o2 = l2o2 + ( -.012) * (csub * csub * csub )l2o2 = l2o2 + (.0165) * (psub * psub * psub)l2o2 = l2o2 + (2.2658) * ( csub * l1o1 * l1o1)

l2o2 = l2o2 + ( -1.2412) * (l1o1 * csub * csub)l2o2 = l2o2 + (.118) * (l1o1 * psub * psub)l2o2 = l2o2 + ( -.0762) * ( psub * l1o1 * l1o1)l2o2 = l2o2 + ( -.6542) * ( psub * csub * csub )

l2o2 = l2o2 + ( -.2157) * ( csub * psub * psub )l1o2 = .1727l1o2 = l1o2 + (1.0447) * ( csub )

l1o2 = l1o2 + (.101) * ( psub)l1o2 = l1o2 + ( -.1425) * ( csub * csub )l1o2 = l1o2 + ( -.0026) * ( psub * psub)l1o2 = l1o2 + (.1389) * (csub * psub)

l1o2 = l1o2 + ( -.0068) * ( csub * csub * csub)l1o2 = l1o2 + ( -.0028) * ( psub * psub * psub) l1o2 = l1o2 + ( -.0792) * ( psub * csub * csub)l1o2 = l1o2 + ( -.0094) * ( csub * psub * psub)

l2o3 = .0072l2o3 = l2o3 + (.0073) * (l1o1)l2o3 = l2o3 + (.0215) * ( fsub )l2o3 = l2o3 + (.9155) * (l1o2)

l2o3 = l2o3 + ( -.021) * (l1o1 * l1o1)l2o3 = l2o3 + ( -.032) * ( fsub * fsub)l2o3 = l2o3 + ( -.3618) * (l1o2 * l1o2)

l2o3 = l2o3 + ( -.1584) * (l1o1 * fsub )l2o3 = l2o3 + (.3667) * (l1o1 * l1o2)l2o3 = l2o3 + (.122) * (fsub * l1o2)l2o3 = l2o3 + ( -.4431) * (l1o1 * fsub * l1o2)

l2o3 = l2o3 + ( -.0294) * (l1o1 * l1o1 * l1o1)l2o3 = l2o3 + ( -.0279) * ( fsub * fsub * fsub)l2o3 = l2o3 + (3.7686) * (l1o2 * l1o2 * l1o2)l2o3 = l2o3 + (.9564) * ( fsub * l1o1 * l1o1)

l2o3 = l2o3 + (.1896) * (l1o1 * fsub * fsub )l2o3 = l2o3 + ( -8.7844) * (l1o1 * l1o2 * l1o2)l2o3 = l2o3 + (5.1023) * (l1o2 * l1o1 * l1o1)l2o3 = l2o3 + ( -.1916) * (l1o2 * fsub * fsub)

l2o3 = l2o3 + ( -.4921) * ( fsub * l1o2 * l1o2)l3o2 = - .0065l3o2 = l3o2 + (.0354) * (l2o2)

l3o2 = l3o2 + (.1506) * ( csub )l3o2 = l3o2 + (.7698) * (l2o3)l3o2 = l3o2 + (.6768) * (l2o2 * l2o2)l3o2 = l3o2 + (.5409) * ( csub * csub)

l3o2 = l3o2 + (.3753) * (l2o3 * l2o3)l3o2 = l3o2 + ( -1.145) * (l2o2 * csub )l3o2 = l3o2 + ( -.5801) * (l2o2 * l2o3)l3o2 = l3o2 + (.1062) * ( csub * l2o3)

l3o2 = l3o2 + ( -.5078) * (l2o2 * csub * l2o3)l3o2 = l3o2 + (.9953) * (l2o2 * l2o2 * l2o2)l3o2 = l3o2 + ( -1.6416) * (csub * csub * csub )l3o2 = l3o2 + ( -1.3315) * (l2o3 * l2o3 * l2o3)

l3o2 = l3o2 + ( -8.1034) * (csub * l2o2 * l2o2)l3o2 = l3o2 + (9.13) * (l2o2 * csub * csub )l3o2 = l3o2 + ( -1.0112) * (l2o2 * l2o3 * l2o3)

l3o2 = l3o2 + (2.7962) * (l2o3 * l2o2 * l2o2)l3o2 = l3o2 + ( -4.3226) * (l2o3 * csub * csub )l3o2 = l3o2 + (4.0238) * ( csub * l2o3 * l2o3)l2o1 = .1936

l2o1 = l2o1 + (.0136) * (l1o1)l2o1 = l2o1 + (.0761) * ( fsub )l2o1 = l2o1 + (.9395) * ( csub )l2o1 = l2o1 + ( -3.3321) * (l1o1 * l1o1)

l2o1 = l2o1 + ( -.0653) * ( fsub * fsub )l2o1 = l2o1 + ( -3.5339) * (csub * csub )l2o1 = l2o1 + (.4446) * (l1o1 * fsub)l2o1 = l2o1 + (6.7663) * (l1o1 * csub )

l2o1 = l2o1 + ( -.3496) * ( fsub * csub )l2o1 = l2o1 + ( -1.1369) * (l1o1 * fsub * csub )l2o1 = l2o1 + ( -1.0585) * (l1o1 * l1o1 * l1o1)

l2o1 = l2o1 + ( -.0324) * ( fsub * fsub * fsub)l2o1 = l2o1 + (1.0497) * ( csub * csub * csub)l2o1 = l2o1 + (1.009) * ( fsub * l1o1 * l1o1)l2o1 = l2o1 + (.2401) * (l1o1 * fsub * fsub )

l2o1 = l2o1 + ( -3.1781) * (l1o1 * csub * csub )l2o1 = l2o1 + (3.2531) * ( csub * l1o1 * l1o1)l2o1 = l2o1 + ( -.2992) * ( csub * fsub * fsub)l2o1 = l2o1 + (.1068) * ( fsub * csub * csub )

l3o3 = - .0189l3o3 = l3o3 + (.1171) * (l2o2)l3o3 = l3o3 + ( -.18) * (l2o1)l3o3 = l3o3 + (1.0049) * (l2o3)

l3o3 = l3o3 + (.0969) * (l2o2 * l2o2)l3o3 = l3o3 + (.4771) * (l2o1 * l2o1)l3o3 = l3o3 + ( -.8041) * (l2o3 * l2o3)l3o3 = l3o3 + ( -1.3671) * (l2o2 * l2o1)

l3o3 = l3o3 + (1.0996) * (l2o2 * l2o3)l3o3 = l3o3 + (.5197) * (l2o1 * l2o3)l3o3 = l3o3 + (.2083) * (l2o2 * l2o1 * l2o3)

l3o3 = l3o3 + ( -.8395) * (l2o2 * l2o2 * l2o2)l3o3 = l3o3 + ( -1.9646) * (l2o1 * l2o1 * l2o1)l3o3 = l3o3 + (1.7981) * (l2o3 * l2o3 * l2o3)l3o3 = l3o3 + (2.2974) * (l2o1 * l2o2 * l2o2)

l3o3 = l3o3 + ( -.4148) * (l2o2 * l2o1 * l2o1)l3o3 = l3o3 + (1.0008) * (l2o2 * l2o3 * l2o3)l3o3 = l3o3 + ( -1.339) * (l2o3 * l2o2 * l2o2)l3o3 = l3o3 + (5.2977) * (l2o3 * l2o1 * l2o1)

l3o3 = l3o3 + ( -6) * (l2o1 * l2o3 * l2o3)l4o2 = .0312l4o2 = l4o2 + ( -.0024) * (l3o2)l4o2 = l4o2 + (.175) * (csub )

l4o2 = l4o2 + (.8012) * (l3o3)l4o2 = l4o2 + ( -.0577) * (l3o2 * l3o2)l4o2 = l4o2 + ( -.1658) * ( csub * csub )

l4o2 = l4o2 + ( -.1476) * (l3o3 * l3o3)l4o2 = l4o2 + (.0647) * (l3o2 * csub)l4o2 = l4o2 + (.0622) * (l3o2 * l3o3)l4o2 = l4o2 + (.2193) * ( csub * l3o3)

l4o2 = l4o2 + ( -.2148) * (l3o2 * csub * l3o3)l4o2 = l4o2 + ( -.4841) * (l3o2 * l3o2 * l3o2)l4o2 = l4o2 + ( -.4275) * ( csub * csub * csub)l4o2 = l4o2 + (.4404) * (l3o3 * l3o3 * l3o3)

l4o2 = l4o2 + ( -1.3707) * (csub * l3o2 * l3o2)l4o2 = l4o2 + (1.3713) * (l3o2 * csub * csub)l4o2 = l4o2 + ( -2.5034) * (l3o2 * l3o3 * l3o3)l4o2 = l4o2 + (2.8303) * (l3o3 * l3o2 * l3o2)

l4o2 = l4o2 + ( -.2438) * (l3o3 * csub * csub)l4o2 = l4o2 + (.627) * (csub * l3o3 * l3o3)l4o2 = l4o2 * (.9814) + (.0007)

IF l4o2 < ( -1.8572) - (1.1209) THEN l4o2 = (- 1.8572) - (1.1209)IF l4o2 > (2.2382) + (1.1209) THEN l4o2 = (2.2382) + (1.1209)yt = l4o2 + p1END FUNCTION

Bt concentration virtual sensor prototype. (QBASIC source code is in two columns. Place second column listing immediately after first column to run.)

Rem Kolmogorov- Gabor polynomial (Data Relations Model)input "Absorption = ", absorpinput "Frequency = ", freqabsorp = ( absorp - 2.8609) / .917freq = (freq- 7.5164) / 1.2582l1o3 = .0874l1o3 = l1o3 + (1.2226) * ( absorp)l1o3 = l1o3 + ( -.0199) * ( absorp * absorp )l1o3 = l1o3 + ( -.0923) * ( absorp * absorp * absorp )l1o1 = - .1685l1o1 = l1o1 + (1.2886) * ( absorp)l1o1 = l1o1 + ( -.117) * (freq)l1o1 = l1o1 + (.0405) * ( absorp * absorp )l1o1 = l1o1 + (.162) * (freq* freq)l1o1 = l1o1 + ( -.1277) * ( absorp * freq)l1o1 = l1o1 + ( -.1125) * ( absorp * absorp * absorp )l1o1 = l1o1 + ( -.0905) * (freq* freq* freq)l1o1 = l1o1 + (.0417) * (freq* absorp * absorp )l1o1 = l1o1 + (.0456) * ( absorp * freq* freq)l1o2 = 0l1o2 = l1o2 + (.9782) * ( absorp )l1o2 = l1o2 + ( -.257) * (freq)l2o1 = - .0678l2o1 = l2o1 + ( -.4069) * (l1o3)l2o1 = l2o1 + (1.216) * (l1o1)l2o1 = l2o1 + ( -.1502) * (l1o2)l2o1 = l2o1 + (.7836) * (l1o3 * l1o3)l2o1 = l2o1 + (.8533) * (l1o1 * l1o1)l2o1 = l2o1 + (.0785) * (l1o2 * l1o2)l2o1 = l2o1 + ( -1.3953) * (l1o3 * l1o1)l2o1 = l2o1 + (.0431) * (l1o3 * l1o2)l2o1 = l2o1 + ( -.3564) * (l1o1 * l1o2)l2o1 = l2o1 + ( -1.2731) * (l1o3 * l1o1 * l1o2)l2o1 = l2o1 + (1.0529) * (l1o3 * l1o3 * l1o3)l2o1 = l2o1 + ( -.9622) * (l1o1 * l1o1 * l1o1)l2o1 = l2o1 + (.4182) * (l1o2 * l1o2 * l1o2)l2o1 = l2o1 + ( -3.9566) * (l1o1 * l1o3 * l1o3)l2o1 = l2o1 + (3.8149) * (l1o3 * l1o1 * l1o1)l2o1 = l2o1 + ( -1.2622) * (l1o3 * l1o2 * l1o2)l2o1 = l2o1 + (2.0645) * (l1o2 * l1o3 * l1o3)l2o1 = l2o1 + (.0742) * (l1o2 * l1o1 * l1o1)l2o1 = l2o1 + (.1696) * (l1o1 * l1o2 * l1o2)l2o3 = - .0818l2o3 = l2o3 + (.0759) * (l1o3)l2o3 = l2o3 + ( -1.0568) * ( absorp )l2o3 = l2o3 + (1.2708) * (l1o1)l2o3 = l2o3 + (.2173) * (l1o3 * l1o3)l2o3 = l2o3 + ( -.7217) * ( absorp * absorp )l2o3 = l2o3 + (.4713) * (l1o1 * l1o1)l2o3 = l2o3 + (.9198) * (l1o3 * absorp )l2o3 = l2o3 + ( -2.5729) * (l1o3 * l1o1)l2o3 = l2o3 + (1.6621) * ( absorp * l1o1)l2o3 = l2o3 + ( -.1178) * (l1o3 * absorp * l1o1)l2o3 = l2o3 + (4.7812) * (l1o3 * l1o3 * l1o3)l2o3 = l2o3 + ( -1.0852) * ( absorp * absorp * absorp )l2o3 = l2o3 + ( -.8142) * (l1o1 * l1o1 * l1o1)l2o3 = l2o3 + ( -5.9031) * ( absorp * l1o3 * l1o3)l2o3 = l2o3 + (3.913) * (l1o3 * absorp * absorp )l2o3 = l2o3 + (5.4668) * (l1o3 * l1o1 * l1o1)l2o3 = l2o3 + ( -4.7801) * (l1o1 * l1o3 * l1o3)l2o3 = l2o3 + (1.6853) * (l1o1 * absorp * absorp )l2o3 = l2o3 + ( -2.8258) * ( absorp * l1o1 * l1o1)l2o2 = - .166l2o2 = l2o2 + (.1439) * (l1o3)l2o2 = l2o2 + (.1699) * ( absorp )l2o2 = l2o2 + (.5341) * (l1o2)l2o2 = l2o2 + ( -3.3055) * (l1o3 * l1o3)l2o2 = l2o2 + (.2443) * ( absorp * absorp )l2o2 = l2o2 + (2.6863) * (l1o2 * l1o2)l2o2 = l2o2 + (5.2213) * (l1o3 * absorp )l2o2 = l2o2 + (.6905) * (l1o3 * l1o2)l2o2 = l2o2 + ( -5.5128) * ( absorp * l1o2)l2o2 = l2o2 + ( -.324) * (l1o3 * absorp * l1o2)

l2o2 = l2o2 + (2.9655) * (l1o3 * l1o3 * l1o3)l2o2 = l2o2 + ( -2.9985) * ( absorp * absorp * absorp )l2o2 = l2o2 + (4.7594) * (l1o2 * l1o2 * l1o2)l2o2 = l2o2 + ( -6.6428) * ( absorp * l1o3 * l1o3)l2o2 = l2o2 + (3.9217) * (l1o3 * absorp * absorp )l2o2 = l2o2 + ( -2.3861) * (l1o3 * l1o2 * l1o2)l2o2 = l2o2 + (2.2219) * (l1o2 * l1o3 * l1o3)l2o2 = l2o2 + (8.6865) * (l1o2 * absorp * absorp )l2o2 = l2o2 + ( -10.1277) * ( absorp * l1o2 * l1o2)l3o1 = - .026l3o1 = l3o1 + ( -.5655) * (l2o1)l3o1 = l3o1 + (1.8251) * (l2o3)l3o1 = l3o1 + ( -.1683) * (l2o2)l3o1 = l3o1 + ( -.2062) * (l2o1 * l2o1)l3o1 = l3o1 + (2.3963) * (l2o3 * l2o3)l3o1 = l3o1 + (.6281) * (l2o2 * l2o2)l3o1 = l3o1 + ( -1.9053) * (l2o1 * l2o3)l3o1 = l3o1 + (1.9318) * (l2o1 * l2o2)l3o1 = l3o1 + ( -2.833) * (l2o3 * l2o2)l3o1 = l3o1 + (1.1231) * (l2o1 * l2o3 * l2o2)l3o1 = l3o1 + (5.0889) * (l2o1 * l2o1 * l2o1)l3o1 = l3o1 + ( -6.9325) * (l2o3 * l2o3 * l2o3)l3o1 = l3o1 + (3.2972) * (l2o2 * l2o2 * l2o2)l3o1 = l3o1 + ( -9.7348) * (l2o3 * l2o1 * l2o1)l3o1 = l3o1 + (9.4629) * (l2o1 * l2o3 * l2o3)l3o1 = l3o1 + (5.598) * (l2o1 * l2o2 * l2o2)l3o1 = l3o1 + ( -5.8589) * (l2o2 * l2o1 * l2o1)l3o1 = l3o1 + (12.2058) * (l2o2 * l2o3 * l2o3)l3o1 = l3o1 + ( -14.3) * (l2o3 * l2o2 * l2o2)l3o3 = - .0486l3o3 = l3o3 + ( -.1775) * (l2o2)l3o3 = l3o3 + (2.4852) * (l2o3)l3o3 = l3o3 + ( -1.1885) * (l2o1)l3o3 = l3o3 + (1.6957) * (l2o2 * l2o2)l3o3 = l3o3 + (2.0175) * (l2o3 * l2o3)l3o3 = l3o3 + (2.0096) * (l2o1 * l2o1)l3o3 = l3o3 + ( -1.2979) * (l2o2 * l2o3)l3o3 = l3o3 + ( -1.7907) * (l2o2 * l2o1)l3o3 = l3o3 + ( -2.615) * (l2o3 * l2o1)l3o3 = l3o3 + (.0304) * (l2o2 * l2o3 * l2o1)l3o3 = l3o3 + (.8335) * (l2o2 * l2o2 * l2o2)l3o3 = l3o3 + ( -3.6036) * (l2o3 * l2o3 * l2o3)l3o3 = l3o3 + (3.1642) * (l2o1 * l2o1 * l2o1)l3o3 = l3o3 + ( -6.2871) * (l2o3 * l2o2 * l2o2)l3o3 = l3o3 + (4.7083) * (l2o2 * l2o3 * l2o3)l3o3 = l3o3 + ( -3.1113) * (l2o2 * l2o1 * l2o1)l3o3 = l3o3 + (4.2263) * (l2o1 * l2o2 * l2o2)l3o3 = l3o3 + (6.8997) * (l2o1 * l2o3 * l2o3)l3o3 = l3o3 + ( -6.9167) * (l2o3 * l2o1 * l2o1)l4o3 = .0318l4o3 = l4o3 + ( -.0515) * (l3o1)l4o3 = l4o3 + (.0789) * (absorp )l4o3 = l4o3 + (.9848) * (l3o3)l4o3 = l4o3 + (.1984) * (l3o1 * l3o1)l4o3 = l4o3 + (.0332) * (absorp * absorp )l4o3 = l4o3 + ( -4.6211) * (l3o3 * l3o3)l4o3 = l4o3 + ( -4.345) * (l3o1 * absorp )l4o3 = l4o3 + (4.515) * (l3o1 * l3o3)l4o3 = l4o3 + (4.2166) * ( absorp * l3o3)l4o3 = l4o3 + ( -.0451) * (l3o1 * absorp * l3o3)l4o3 = l4o3 + (.1034) * (l3o1 * l3o1 * l3o1)l4o3 = l4o3 + ( -.0114) * ( absorp * absorp * absorp )l4o3 = l4o3 + (4.8489) * (l3o3 * l3o3 * l3o3)l4o3 = l4o3 + (1.441) * (absorp * l3o1 * l3o1)l4o3 = l4o3 + ( -.0934) * (l3o1 * absorp * absorp )l4o3 = l4o3 + ( -7.0619) * (l3o1 * l3o3 * l3o3)l4o3 = l4o3 + (2.0594) * (l3o3 * l3o1 * l3o1)l4o3 = l4o3 + (.0613) * (l3o3 * absorp * absorp )l4o3 = l4o3 + ( -1.3036) * (spec * l3o3 * l3o3)conc = l4o3 * (.8247) + (2.8608)IF conc < (1.6) - (1.0025) THEN conc = (1.6) - (1.0025)IF conc > (4.4) + (1.0025) THEN conc = (4.4) + (1.0025)print "Concentration = ", conc : end

DEFCUR A-Z: DECLARE FUNCTION yt @ (x@)20:input "No Fault (0) or Fault (1) => ",fOPEN "output" FOR OUTPUT AS #1ntotal = 4096: nsub = 410: ymin = -17@: ymax = 15@10 : n = 1 + INT(CINT(x - 1) / 10)istart = CINT((( ntotal - 1) * (n - 1)) / ( nsub - 1))ifinish = CINT((( ntotal - 1) * n) / ( nsub - 1))xpt = (x - istart ) / ( ifinish - istart )ix = ( ntotal - 1) * xpts = ( yt (n + 1,f) - ymin ) / ( ymax - ymin )s = s - (yt (n - 1,f) - ymin ) / (ymax - ymin)Rem Fractal operator (Data Sequence Model)Rem applied to yt (Data Relations Model)dfde = ((( yt(n,f) -yt(n -1,f))/( ntotal - 1)))*ixdfde =dfde - ((s*( yt(nsub -1,f) -yt (0,f))/( ntotal - 1)))*ixdfde = dfde + s*yt(xpt *(nsub - 1),f)+( yt(n-1,f)*( nsub -1)/( ntotal- 1))dfde = dfde - yt (0,f) * s * ( nsub - 1) / ( ntotal - 1)PRINT #1, dfde : IF x < 4095 THEN : x = x + 1: GOTO 10CLOSE : ENDFUNCTION yt (x,f)pi = 4@ * ATN(1@)npts = 410Rem Eigenfunctions for nominal termcos40 = COS((2 * pi * 40 * x / (npts - 1)) + .1895)cos66 = COS((2 * pi * 66 * x / (npts - 1)) - .5521)cos79 = COS((2 * pi * 79 * x / (npts - 1)) - 1.5047)cos95 = COS((2 * pi * 95 * x / (npts - 1)) + 1.5572)cos110 = COS((2 * pi * 110 * x / ( npts - 1)) - .2232)cos131 = COS((2 * pi * 131 * x / ( npts - 1)) - .6192)cos199 = COS((2 * pi * 199 * x / ( npts - 1)) + .4987)cos40 = (cos40 - .0024) / .6375cos66 = (cos66 - .0021) / .6371cos79 = (cos79 - .0002) / .6352cos95 = (cos95 - 0) / .6351cos110 = (cos110 - .0024) / .6374cos131 = (cos131 - .002) / .6371cos199 = (cos199 - .0021) / .6372Rem Kolmogorov- Gabor polynomial (Data Relations Model)Rem for nominal terml1o1 = 0l1o1 = l1o1 + ( -.3234) * (cos66)l1o1 = l1o1 + ( -.2608) * (cos95)l1o1 = l1o1 + ( -.1904) * (cos131)l1o1 = l1o1 + ( -.1473) * (cos199)l1o1 = l1o1 + (.1321) * (cos110)l1o1 = l1o1 + ( -.1214) * (cos79)l1o1 = l1o1 + (.0828) * (cos40)l1o1 = l1o1 * (8.4211) + ( -.4932)l1o2=0if f=1 then

Rem Eigenfunctions for anomalous termcos40 = cos ((2*pi*40*x/( npts -1)) - 0.8528)cos66 = cos ((2*pi*66*x/( npts -1))+0.0104)cos79 = cos ((2*pi*79*x/( npts -1))+0.2700)cos95 = cos ((2*pi*95*x/( npts -1)) - 0.1601)cos110 = cos((2*pi*110*x/( npts- 1)) - 0.4579)cos112 = cos((2*pi*112*x/( npts- 1))+0.6647)cos131 = cos((2*pi*131*x/( npts- 1)) - 0.4524)cos133 = cos((2*pi*133*x/( npts- 1))+0.0544)cos199 = cos((2*pi*199*x/( npts- 1))+1.0276)cos201 = cos((2*pi*201*x/( npts- 1))+0.6440)cos40 = (cos40 - 0.0016) / 0.6367cos66 = (cos66 - 0.0024) / 0.6375cos79 = (cos79 - 0.0024) / 0.6374cos95 = (cos95 - 0.0024) / 0.6375cos110 = (cos110 - 0.0022) / 0.6372cos112 = (cos112 - 0.0019) / 0.637cos131 = (cos131 - 0.0022) / 0.6373cos133 = (cos133 - 0.0024) / 0.6375cos199 = (cos199 - 0.0013) / 0.6363cos201 = (cos201 - 0.002) / 0.637Rem Kolmogorov - Gabor polynomial (Data Relations Model)Rem for anomalous terml1o2=0l1o2=l1o2+ ( - 0.7048) * (cos95)l1o2=l1o2+ (0.3304) * (cos66)l1o2=l1o2+ (0.2267) * (cos131)l1o2=l1o2+ (0.1533) * (cos199)l1o2=l1o2+ ( - 0.1535) * (cos110)l1o2=l1o2+ (0.1167) * (cos112)l1o2=l1o2+ (0.1135) * (cos79)l1o2=l1o2+ ( - 0.0753) * (cos133)l1o2=l1o2+ ( - 0.031) * (cos40)l1o2=l1o2+ (0.0009) * (cos201)l1o2=l1o2* (7.0817) + ( - 3.412)

endifyt = l1o1+l1o2END FUNCTION

Combined Data Model for nominal and anomalous helicopter gearbox data including frequency components used. (QBASIC source code is in two columns. To run, place second column listing immediately after first column.)

Change detector. (QBASIC source code is in three columns. To run, place second column listing immediately after first column, and the third after the second.)

DEFCUR A-Zpi=4@* atn (1@)open "input" for input as #1input #1, meaninput #1, stdevinput #1, skewinput #1, kurtinput #1, m6input #1, Jaeninput #1, Handclose #1mean = (mean --0.4454) / 0.5152stdev = ( stdev- 4.0706) / 0.8783skew = (skew -0.0162) / 0.3504kurt = ( kurt --0.8099) / 0.3423m6 = (m6 --8.1288) / 2.5084Jaen = ( Jaen -0.5198) / 0.0396Hand = (Hand -1.2527) / 0.0192l1o2= -0.5913l1o2=l1o2+ ( -0.1476) * (skew)l1o2=l1o2+ ( -0.9041) * (mean)l1o2=l1o2+ ( -0.9286) * (kurt )l1o2=l1o2+ (0.4752) * (skew * skew)l1o2=l1o2+ (0.2562) * (mean * mean)l1o2=l1o2+ ( -0.2235) * (kurt * kurt )l1o2=l1o2+ (0.5887) * (skew * mean)l1o2=l1o2+ ( -0.4035) * (skew * kurt )l1o2=l1o2+ ( -1.0192) * (mean * kurt )l1o2=l1o2+ (0.1964) * (skew * mean * kurt)l1o2=l1o2+ ( -0.1016) * (skew * skew * skew)l1o2=l1o2+ (0.0464) * (mean * mean * mean)l1o2=l1o2+ (0.1473) * ( kurt * kurt * kurt)l1o2=l1o2+ ( -0.2244) * (mean * skew * skew)l1o2=l1o2+ ( -0.2274) * (skew * mean * mean)l1o2=l1o2+ (0.3348) * (skew * kurt * kurt)l1o2=l1o2+ ( -0.0717) * (kurt * skew * skew)l1o2=l1o2+ (0.2728) * ( kurt * mean * mean)l1o2=l1o2+ (0.516) * (mean * kurt * kurt )l1o3=0.0066l1o3=l1o3+ (1.5425) * ( Jaen)l1o3=l1o3+ ( -0.4563) * (mean)l1o3=l1o3+ (0.44) * (kurt )l1o3=l1o3+ (0.3005) * ( Jaen * Jaen)l1o3=l1o3+ ( -0.05) * (mean * mean)l1o3=l1o3+ ( -0.7195) * (kurt * kurt )l1o3=l1o3+ ( -0.2532) * (Jaen * mean)l1o3=l1o3+ ( -0.1315) * (Jaen * kurt )l1o3=l1o3+ ( -1.4195) * (mean * kurt )l1o3=l1o3+ (0.4939) * ( Jaen * mean * kurt)l1o3=l1o3+ ( -1.2495) * (Jaen * Jaen * Jaen)l1o3=l1o3+ ( -0.1227) * (mean * mean * mean)l1o3=l1o3+ (0.3834) * ( kurt * kurt * kurt)l1o3=l1o3+ ( -0.0849) * (mean * Jaen * Jaen)l1o3=l1o3+ ( -0.5829) * (Jaen * mean * mean)l1o3=l1o3+ ( -0.7079) * (Jaen * kurt * kurt)l1o3=l1o3+ ( -2.3192) * (kurt * Jaen * Jaen)l1o3=l1o3+ ( -0.5023) * (kurt * mean * mean)l1o3=l1o3+ (0.857) * (mean * kurt * kurt )l2o3=0.0816l2o3=l2o3+ (1.0612) * (l1o2)l2o3=l2o3+ (0.1459) * (m6)

l2o3=l2o3+ ( -0.1725) * (l1o3)l2o3=l2o3+ (0.1167) * (l1o2*l1o2) l2o3=l2o3+ (0.2472) * (m6*m6)l2o3=l2o3+ ( -0.0337) * (l1o3*l1o3)l2o3=l2o3+ (0.1863) * (l1o2*m6)l2o3=l2o3+ ( -0.1812) * (l1o2*l1o3)l2o3=l2o3+ ( -0.1442) * (m6*l1o3)l2o3=l2o3+ (0.0003) * (l1o2*m6*l1o3)l2o3=l2o3+ ( -0.0953) * (l1o2*l1o2*l1o2)l2o3=l2o3+ ( -0.0601) * (m6*m6*m6)l2o3=l2o3+ (0.1436) * (l1o3*l1o3*l1o3)l2o3=l2o3+ (0.0718) * (m6*l1o2*l1o2)l2o3=l2o3+ ( -0.086) * (l1o2*m6*m6)l2o3=l2o3+ ( -0.3954) * (l1o2*l1o3*l1o3)l2o3=l2o3+ (0.4013) * (l1o3*l1o2*l1o2)l2o3=l2o3+ (0.0081) * (l1o3*m6*m6)l2o3=l2o3+ ( -0.0409) * (m6*l1o3*l1o3)l2o2=0.1892l2o2=l2o2+ (0.8725) * (l1o2)l2o2=l2o2+ ( -0.4387) * (skew)l2o2=l2o2+ (0.5315) * (l1o3)l2o2=l2o2+ (0.1053) * (l1o2*l1o2)l2o2=l2o2+ ( -0.0652) * (skew*skew)l2o2=l2o2+ ( -0.0461) * (l1o3*l1o3)l2o2=l2o2+ ( -0.4263) * (l1o2*skew)l2o2=l2o2+ ( -0.1575) * (l1o2*l1o3)l2o2=l2o2+ (0.1623) * (skew*l1o3)l2o2=l2o2+ ( -0.0269) * (l1o2*skew*l1o3)l2o2=l2o2+ ( -0.1766) * (l1o2*l1o2*l1o2)l2o2=l2o2+ (0.1399) * (skew*skew*skew)l2o2=l2o2+ ( -0.032) * (l1o3*l1o3*l1o3)l2o2=l2o2+ ( -0.0525) * (skew*l1o2*l1o2)l2o2=l2o2+ (0.0611) * (l1o2*skew*skew)l2o2=l2o2+ ( -0.0615) * (l1o2*l1o3*l1o3)l2o2=l2o2+ (0.2965) * (l1o3*l1o2*l1o2)l2o2=l2o2+ ( -0.1964) * (l1o3*skew*skew)l2o2=l2o2+ ( -0.0044) * (skew*l1o3*l1o3)l3o2=0.0876l3o2=l3o2+ (0.3014) * (l2o3)l3o2=l3o2+ (0.0996) * (Hand)l3o2=l3o2+ (0.9825) * (l2o2)l3o2=l3o2+ (0.0043) * (l2o3*l2o3)l3o2=l3o2+ ( -0.0983) * (Hand*Hand)l3o2=l3o2+ (0.2049) * (l2o2*l2o2)l3o2=l3o2+ (0.019) * (l2o3*Hand)l3o2=l3o2+ ( -0.2056) * (l2o3*l2o2)l3o2=l3o2+ (0.1945) * (Hand*l2o2)l3o2=l3o2+ ( -0.0195) * (l2o3*Hand*l2o2)l3o2=l3o2+ ( -0.0011) * (l2o3*l2o3*l2o3)l3o2=l3o2+ (0.0194) * (Hand*Hand*Hand)l3o2=l3o2+ (0.001) * (l2o2*l2o2*l2o2)l3o2=l3o2+ (0.1624) * (Hand*l2o3*l2o3)l3o2=l3o2+ ( -0.1872) * (l2o3*Hand*Hand)l3o2=l3o2+ ( -0.0355) * (l2o3*l2o2*l2o2)l3o2=l3o2+ (0.024) * (l2o2*l2o3*l2o3)l3o2=l3o2+ ( -0.0368) * (l2o2*Hand*Hand)l3o2=l3o2+ ( -0.1612) * (Hand*l2o2*l2o2)l3o1= -0.0593l3o1=l3o1+ (0.2493) * (l2o3)l3o1=l3o1+ ( -0.2785) * ( stdev)l3o1=l3o1+ (0.8792) * (l2o2)

l3o1=l3o1+ (0.0068) * (l2o3*l2o3)l3o1=l3o1+ (0.0318) * ( stdev *stdev)l3o1=l3o1+ (0.0869) * (l2o2*l2o2)l3o1=l3o1+ (0.1524) * (l2o3* stdev )l3o1=l3o1+ ( -0.0899) * (l2o3*l2o2)l3o1=l3o1+ ( -0.1232) * (stdev *l2o2)l3o1=l3o1+ (0.0041) * (l2o3* stdev *l2o2)l3o1=l3o1+ (0.0004) * (l2o3*l2o3*l2o3)l3o1=l3o1+ (0.0455) * ( stdev *stdev* stdev)l3o1=l3o1+ (0.0042) * (l2o2*l2o2*l2o2)l3o1=l3o1+ ( -0.2995) * (stdev *l2o3*l2o3)l3o1=l3o1+ ( -0.0799) * (l2o3* stdev* stdev)l3o1=l3o1+ (0.0093) * (l2o3*l2o2*l2o2)l3o1=l3o1+ ( -0.0269) * (l2o2*l2o3*l2o3)l3o1=l3o1+ ( -0.0337) * (l2o2* stdev* stdev)l3o1=l3o1+ (0.2907) * ( stdev *l2o2*l2o2)l4o3= -0.077l4o3=l4o3+ (1.1419) * (l3o2)l4o3=l4o3+ ( -0.0212) * (Jaen )l4o3=l4o3+ ( -0.5045) * (l3o1)l4o3=l4o3+ ( -0.0003) * (l3o2*l3o2)l4o3=l4o3+ (0.0168) * ( Jaen*Jaen)l4o3=l4o3+ ( -0.205) * (l3o1*l3o1)l4o3=l4o3+ ( -0.0134) * (l3o2* Jaen )l4o3=l4o3+ (0.1819) * (l3o2*l3o1)l4o3=l4o3+ ( -0.1237) * (Jaen *l3o1)l4o3=l4o3+ ( -0.0025) * (l3o2* Jaen*l3o1)l4o3=l4o3+ ( -0.0018) * (l3o2*l3o2*l3o2)l4o3=l4o3+ (0.0072) * ( Jaen*Jaen*Jaen)l4o3=l4o3+ (0.0166) * (l3o1*l3o1*l3o1)l4o3=l4o3+ ( -0.7481) * (Jaen *l3o2*l3o2)l4o3=l4o3+ ( -0.1203) * (l3o2* Jaen *Jaen )l4o3=l4o3+ (0.2037) * (l3o2*l3o1*l3o1)l4o3=l4o3+ ( -0.2098) * (l3o1*l3o2*l3o2)l4o3=l4o3+ (0.0858) * (l3o1* Jaen*Jaen)l4o3=l4o3+ (0.7326) * ( Jaen*l3o1*l3o1)l5o2=0.0092l5o2=l5o2+ (1.0851) * (l4o3)l5o2=l5o2+ ( -0.013) * (mean)l5o2=l5o2+ (0.106) * (Hand)l5o2=l5o2+ (0.0326) * (l4o3*l4o3)l5o2=l5o2+ ( -0.078) * (mean*mean)l5o2=l5o2+ ( -0.0273) * (Hand*Hand)l5o2=l5o2+ (0.109) * (l4o3*mean)l5o2=l5o2+ ( -0.2199) * (l4o3*Hand)l5o2=l5o2+ ( -0.4196) * (mean*Hand)l5o2=l5o2+ (0.2675) * (l4o3*mean*Hand)l5o2=l5o2+ (0) * (l4o3*l4o3*l4o3)l5o2=l5o2+ ( -0.1212) * (mean*mean*mean)l5o2=l5o2+ ( -0.0074) * (Hand*Hand*Hand)l5o2=l5o2+ ( -0.012) * (mean*l4o3*l4o3)l5o2=l5o2+ ( -0.035) * (l4o3*mean*mean)l5o2=l5o2+ ( -0.1349) * (l4o3*Hand*Hand)l5o2=l5o2+ (0.0378) * (Hand*l4o3*l4o3)l5o2=l5o2+ ( -0.0361) * (Hand*mean*mean)l5o2=l5o2+ (0.2311) * (mean*Hand*Hand)l5o2 = l5o2* (0.0002) + (0.5)open "out1" for output as #1print #1, l5o2close #1end

Data Modeling of deep sky images

James Handley*a

, Holger Jaenischa,b

, Albert Lima, Graeme White

a,

Alex Honsa, Miroslav Filipovic

c,a, Matthew Edwards

b

aJames Cook University, Centre for Astronomy, Townsville QLD 4811, Australia

bAlabama Agricultural and Mechanical University, Department of Physics, Huntsville, AL 35811

cUniversity of Western Sydney, Locked Bag 1797 PENRITH SOUTH DC NSW 1797, Australia

ABSTRACT

We present a method for simulating CCD focal plane array (FPA) images of extended deep sky objects using Data

Modeling. Data Modeling is a process of deriving functional equations from measured data. These tools are used to

model FPA fixed pattern noise, shot noise, non-uniformity, and the extended objects themselves. The mathematical

model of the extended object is useful for correlation analysis and other image understanding algorithms used in Virtual

Observatory Data Mining. We apply these tools to the objects in the Messier list and build a classifier that achieves

100% correct classification.

Keywords: Data Modeling, Virtual Observatory, astronomy, deep-sky, image modeling, noise modeling, Messier

classification, functional modeling, component modeling

1. INTRODUCTION

Data Modeling1,2

is the process of deriving a mathematical expression from measured data.

Two approaches for modeling focal plane array (FPA) images are component based modeling and functional modeling.

In component based modeling, the deep sky object and each noise effect are modeled independently. Arrays for

each effect are stacked into a final image. Component based modeling uses simple equations to model deep sky objects

and noise effects, or it can use statistical models. In contrast, functional modeling of deep sky objects is done strictly

using univariate or multivariate equations. These equations are continuous and their nth order derivatives are available

for analysis. Functional modeling also provides a common basis for robust image analysis.

2. COMPONENT BASED DATA MODELING

Component based Data Modeling of deep sky objects and noise effects (thermal, shot, fixed pattern, FPA nonuniformity,

and hot/dead pixels) are generated independently. Each independent model is stacked to build a Component Based Image

Data Model.

The Moffat function3 βγα

ρ

−

+

−+−= 1

)()(),( 00 yyxx

yxf (1)

is used for modeling point-spread functions. The control parameters α, β, γ, and ρ modify the size, shape, and extent of

the 2-D distribution. Deep sky objects are modeled using a series of Moffat functions and properly selecting coefficients.

This Data Model of M51 used 2 Moffat functions with coefficients obtained using a genetic algorithm (GA)4 yielding

*[email protected]; phone 1 256 337 3769; James Cook University; Centre for Astronomy; Townsville, QLD

4811; AUSTRALIA.


6.04.14.1

5.055.155.1

16.0

))342.0)(45(()28(5.11

6.0

))707.0)(20(()35(0.3),(

−−

+

−−+−+

+

−+−=

yxyxyxf (2)

Next, noise effects on the FPA and the surrounding field stars are modeled using different types of random number

distributions. Use of random numbers yield statistical rather than functional models. To eliminate random number

generators, we propose using sin xe (Sxe function) as a pseudo random number approximating function

5

−

−=

∑∑MinMax

Minx

Djif x

e

2

)(sin

),( (3)

−

−=

∑MinMax

Minx

Cjif x

e

2

)(sin

),( (4)

2

1sin),(

+=

exAjif (5)

(Nonuniformity) (Thermal) (Fixed Pattern)

2

2

1sinsize

xFN

e += (6) 0),( =jif 2

2

1sinsize

xEN

e += (7) 255),( =jif

(Dead pixels) (Hot pixels)

σπ )2

1sin2sin()

2

1sinlog(2),(

++−=

ee xxBjif size

xN

e

2

1sin += (Shot Noise and Field Stars) (8)

These functions use only an x-index value, but map into 2-D (i,j) using Hilbert sequencing. The number of Moffat

functions needed in the final model and their coefficients is determined using a genetic algorithm (GA). Component

models yield short numerical equations but of limited fidelity and not in real-time.

Figure 1 depicts a Component Data Model of M51. Comparison shows good representation of the salient image features.

Because M51 exhibits two bright cores, the genetic algorithm chose two Moffat functions to represent it. Comparison of

the original image on the bottom left with the final model shows similarity in placement of the main cores, similar

magnitude and extent of the cores, a similar number of field stars, and even hints of spiral structure.

3. FUNCTIONAL DATA MODELING

3.1 Univariate model – Turlington polynomial

The Turlington polynomial yields one equation with continuous derivative per data set of the form

( )

∑−

=

−

−

+−+−+=

1

2

001.101111 101log)(001.)()(

n

j

xx

jj

j

mmxxmyxT

jj

jj

jxx

yym

−

−=

+

+

1

1 (9)

where x and y are the original (x, y) data points and n the number of points from the original used for building the

Turlington polynomial. The variable n can be either all of the points or a sub-sampled set of the original.

The Turlington polynomial is built in a piecewise linear fashion one point at a time as data arrives. This lends itself to

real-time construction. The logarithm term makes Turlington polynomials both orthogonal and differentiable. One

drawback to the Turlington polynomial is in its use of n terms, equal to the number of points in the data set being

modeled. This yields fast and streaming real-time derivatives of terms, but an exceptionally large final model6,7

.

3.2 Univariate model - Eigenfunction

Eigenfunctions1 approximate T(x) in compact form. T(x) models require coefficients be stored describing each data

segment. The size is dependent on the data-sampling rate. Using eigenfunctions and the method of residuals, T(x) is

approximated by

∑=

+≅n

j

jjjj xiBxAxT1

sincos)( ωω (10)

In Equation (10) Aj is the amplitude of the jth eigenfunction term, Bj is the phase of the jth eigenfunction term, and ωj is

2π times the frequency f defined by the jth derived dominant Fourier frequency term. Dominant eigenfunction terms are

added one at a time until correlation between T(x) and the integral function in (9) converges. Pseudo code used for

creating eigenfunction models is shown in Figure 2.

These smaller models are orthogonal, differentiable, and Lebesgue integrable. However, they require multiple

Fourier Transforms, making them memory and computer intensive.

3.3 Multivariate functional Data Modeling

A functional is a function whose variables are themselves functions. We approximate the Kolmogorov-Gabor

polynomial by using Ivakhnenko’s Group Method of Data Handling8 (GMDH) using nested functionals of the form


LxxxyO 3),,,( 21 =K (11)

This structured approach forms intermediate meta-variables from combinations of three inputs combined into a single

new fused output. This fused meta-variable becomes a new input variable available at the next layer. Since the algorithm

only uses inputs necessary to achieve convergence, pruning of inputs is automatic and requires no external intervention,

enabling unsupervised learning under proper circumstances.

This Functional Data Model is derived in near real-time like T(x) and yields a final model substantially shorter like the

eigenfunction model. If derived from derivative data, it will yield a differential equation model. The algorithm for this

process is listed in Reference 1.

3.4 2-D image to 1-D conversion

Enabling functional modeling of images requires transformation from 2-D into 1-D without 2-D decorrelation. Several

methods exist, including raster scanning and fixed pattern readout such as zigzag sequencing. However, only Hilbert

sequencing preserves 2-D correlations at dyadic sample sizes. Hilbert sequencing is illustrated graphically in Figure 3

and is given by

)()()()( 43211 nnnnn PwPwPwPwH ∪∪∪=+ )()()()( 43211 nnnnn PwPwPwHwP ∪∪∪=+ (12)

subject to Lindenmayer’s L-system grammars represented and defined by

°=−→−+→+

→−++−→+−−+→

90δ

FF

FLRFRLFR

FRLFLRFL

),,(),,(

),,(),,(

),sin,cos(),,(

δααδαα

αααα

+→⇒−−→⇒+

++→⇒

yxyx

yxyx

lylxyxF (13)

Psuedo code for generating the Hilbert sequence is readily available3.

4. APPLICATIONS

4.1 Training cases

We chose a bitmap database of 110 Messier objects for use in classifier construction. Classifier flowcharts in Figures 4

and 5 will be described in Section 4.4 and 4.5. In our database, M102 (repeat of M101) was removed and replaced with

NGC58669. Each bitmap is a 24-bit color image with varying sizes. Bitmaps were resized to 64x64, generating a

thumbnail of each image. Thumbnails were then converted to gray scale and Hilbert sequenced into 1-D.

4.2 Turlington Data Models

Turlington polynomials given in Equation (9) were used to construct Data Models of the entire image. Because

Turlington polynomials are continuous functions, they can be interpolated to any resolution. Figure 6 compares the

original and Turlington Data Models for M13 (cluster), M20 (nebula), and M101 (galaxy). T(x) models suppress noise

by increasing the fitting parameter currently set to 0.001 in Equation (9).

4.3 Eigenfunction Data Models

Eigenfunction models of Messier objects consisting of 5, 10, and 20 terms were built. Using more terms than 5 gave

marginal improvement in correlation, yet doubled and quadrupled the model size. Therefore, we limited our models to 5

terms.

Figures 7-9 contain a library of eigenfunction Data Models, one for each of the 110 Messier objects. Each object is

plotted next to 3 data graphs. The middle graph is the Hilbert sequenced original waveform, the bottom graph the

eigenfunction model, and the top graph is a special histogram derived from the eigenfunction model that will be

discussed in more detail in Section 4.4.

Data Models of images with the object centrally located and somewhat symmetrical (except for open clusters)

contained three dominant peaks. The width of these peaks was generally wider for extended objects. Open

clusters did not show these peaks. Rather, the waveforms displayed 1/f structure5 (Figures 7-9).

4.4 Change detection Data Model

The top graphs in Figures 7-9 are created by generating a double histogram with mode subtraction. First, the

histogram of the data is calculated with number of bins equal to number of points. This histogram is normalized 0-255

and a new histogram (same number of bins) calculated. The mode of this histogram is removed, leaving the modified

2nd

order histograms shown. Characterization of these histograms using descriptive statistics provides features for a

hybrid change detector. Reference 1 describes change detection theory and descriptive features in detail. Our statistical

feature classifier correctly identified 107 out of 110 objects. Figure 4 is a flowchart of the change detector, and the

details are as follows:

Our first change detector (8 layer) ([O(3)8]polynomial) was constructed to identify clusters as nominal, and other

objects (nebula and galaxy) as off-nominal. When this classifier was tested against cluster data, it achieved 100% correct

classification. However, when galaxies and nebula were presented, 19 were mislabeled as clusters. Finally, a classifier

was constructed resolving differences between clusters and mislabeled objects (galaxies and nebulae).

The resolver is a 10 layer ([O(3)10]polynomial) Data Model. The classifier only post processes the data sets labeled as

“clusters”. Our resolver correctly identified all nebulae, all clusters except M35, and all galaxies except M108.

With clusters removed, the remaining objects are passed through a second change detector ([O(3)8]polynomial) that

correctly identified all presented galaxies and all nebulae except M78 (which was mislabeled galaxy).

Additional unusual and potentially difficult test cases were selected to explore performance envelopes of the classifier.

We chose NGC869 (double cluster), the Large Magellanic Cloud (LMC), Small Magellanic Cloud (SMC), Comet

Hale-Bopp, and Comet Neat, shown in Figure 10. These cases were presented to the change detector for assignment:

Object Cluster CD Resolver Galaxy CD Interpretation

Double Cluster 0.5002 -5.2 x108 N/A Unlike any Messier list object

LMC 0.5 0.9558 N/A Cluster

SMC 0.5517 N/A 3.1 x107 Nebula

LMC+SMC 3.1 x109 N/A 3.4 x10

4 Nebula

Hale-Bopp 4.7 x1010 N/A 0.5 Galaxy

Neat 9.9 x109 N/A 4.0 x10

7 Nebula

4.5 Stellar object classification

5-term eigenfunction models are constructed for each Messier object10,11, yielding 16 coefficients shown as a cluster plot

in Figure 11 and given in tabular form in Figure 12. These features are used to build a Data Model classifier. First, we

reduced the dimension of the output to two classes forming a cascade. The first classifier identifies clusters from the

initial pool of objects (clusters, nebula, and galaxies). Once identified, a second classifier distinguishes between nebula

and galaxies. Figure 5 is a flowchart of this classifier.

Using this approach, a 10 layer Data Model ([O(3)10]polynomial) was constructed that correctly distinguished all 57

clusters from galaxies and nebula. Next, a 2 layer Data Model ([O(3)2]polynomial) was constructed that correctly

classified the remaining 40 galaxies and 13 nebulae into their proper classes. The total classifier uses both Data Models;

the first one identifies clusters, while the second determines if the object is a galaxy or nebula.

We obtained 100% correct classification for the Messier list of objects. Also, the 2 combined Data Models needed

only 13 of the 16 available features from Figure 12. The three features not used were B(1), w(4)/2π, and B(5). Our

classifier also detects novelties or changes. If the equation yielded a value outside of -0.0862< x < 1.1313 bounds, the

object is flagged unique; exhibiting characteristics not observed in the Messier training set.

Figure 11 shows a graph using the best two discriminating features (A0 and A1) from Figure 12. These features were

automatically selected by calculating the separability between the 3 classes of objects in all 16 dimensions of the feature

space, and plotting the features that maximized the minimum cluster separability using a K-factor defined as

1

2

1

2

2

2

12

NN

Kσσ

µµ

+

−= (14)

where µi is the mean of each group, Ni is the number of points in each group, and σi is the standard deviation of each

group. Unusual cases were presented to the classifier to score. Our classifier assigned them as follows:

Object DM Value Interpretation

NGC 869 (Double Cluster) 1.5 x1014 Unlike any Messier list object

Large Magellanic Cloud (LMC) 0.2843 Nebula

Small Magellanic Cloud (SMC) 1.0173 Galaxy

Combined LMC and SMC 3.5 x109 Unlike any Messier list object

Comet Hale-Bopp 0.9991 Cluster

Comet Neat 1.0269 Cluster

These cases are unusual because they do not resemble any Messier object. The Double Cluster and combined LMC

and SMC were flagged as novelties. We found our Data Model classifier can determine when new class definitions are

required without supervision.

5. SUMMARY

In conclusion, two approaches for modeling deep sky images was successfully demonstrated. Component based

modeling allows very simple equations to be built. Functional modeling was demonstrated using two different

techniques. Turlington polynomials were demonstrated for real-time applications, and eigenfunctions for short models.

Very good exception handling of novel examples is exhibited using Change Detectors. Functional Data Modeling

resulted in a classifier that correctly identified all 110 of the Messier objects and performed reasonably well classifying

unusual objects.

ACKNOWLEDGMENTS

The authors would like to thank Scott McPheeters, Tim Aden, and John Deacon for their continued support during the

course of this work.

REFERENCES

1. Jaenisch, H., Handley J., Lim A., M.D. Filipovic, White G., Hons A. , Crothers S., Deragopian G., Schneider M.,

Edwards M., “Data Modeling for Virtual Observatory data mining”, Proceedings of SPIE Vol. 5493 (2004).

2. Lim, A., Jaenisch, H., Handley, J., Berrevoets, C., White, G., Deragopian, G., Payne, J., Schneider, M., “Image

Resolution and Performance Analysis of Webcams for Ground Based Astronomy”, Proceedings of SPIE Vol. 5489

(2004).

3. Jaenisch, H.M., Handley, J.W., Scoggins, J., Carroll, M.P., “ISIS: An IR Seeker Model Incorporating Fractal

Concepts”, Proceedings of SPIE, Vol 2225 (1994).



5. Jaenisch, H. and Handley, J., “Data Modeling of 1/f noise sets”, Proceedings of SPIE Vol. 5114 (2003).



7. Jaenisch, H.M., Handley, J.W. , Faucheux, J.P., “Data Driven Differential Equation Modeling of fBm processes”,

Proceedings of SPIE Vol. 5204(2003).

8. Madala, H.R., Ivakhnenko, A.G., Inductive Learning Algorithms for Complex Systems Modeling, Boca Raton, FL:

CRC Press, 1994.

9. “Messier Objects”, http://www.3towers.com/messier.htm, June 1, 2004.

10. Jaenisch, H.M., Filipovic, M.D., “Classification of Jacoby Stellar Spectra Using Data Modeling”, Proceedings of

SPIE Vol.4816 (2002).

11. Jaenisch, H.M., Collins, W.J., Handley, J.W., Hons, A., Filipovic, M.D., Case, C.T., Songy, C.G., “Real-time visual

astronomy using image intensifiers and Data Modeling”, Proceedings of SPIE Vol. 4796 (2002).

FIGURES

Thermal Noise Shot Noise & Hot/Dead PixelsFixed Pattern NoiseNonuniformity

M51 Data Model M51 with Noise EffectsM51 Fig. 1: Component based Image Data Model for M51, including surrounding star field and FPA noise effects.

http://www.3towers.com/messier.htm

rem initialize

redim xdata(n),ydata(n),extreme(n),totaldat(n),r(n),c(n),newdata(n),newdata1(n),r1(n),c1(n)

rem open x and y data array files

call open_data_files(xdata,n)

call open_data_files(ydata,n)

rem sort x into ascending order and sort y with x to retain x associations

call sort2(n,xdata,ydata)

rem put ydata into newdata

for i=1 to n

newdata(i)=ydata(i)

next i

rem specify objective correlation value

cobj=0.99

rem calculate Data Modeling eigenfunction model of y(x)

rem final result held in totaldat; initialize k1 to hold number of terms

k1=0

10 continue

k1=k1+1

rem generate Fourier transform of data and corresponding power spectra

rem use linear regression to fit a line through the dB power spectra

rem identify maxima in power spectra that occur above the linear fit

rem pull out maxima locations in array extreme

rem find maximum value in array extreme, and use location to

rem extract real and imaginary Fourier terms held in r (real) and c(imaginary)

rem for use in eigenfunction model

call calc_extremes(ydata,n,extreme,r,c)

rem find maximum value location ; only look at first half of data since symmetric

rem ignore zeroth order location

n1=int(n/2)

dmax = extreme(1)

dloc = 1

FOR i = 2 TO n1 - 2

IF extreme(i) > dmax THEN

dmax = extreme(i)

dloc = i

END IF

NEXT i

rem construct sine and cosine representation of dloc frequency at sampling equal to original

rem r1 and c1 hold eigenfunction coefficients, d the frequency term

r1(k1)=r2(dloc)

c1(k1)=c2(dloc)

d(k1)=dloc

FOR i = 1 TO n

newdata1(i) = newdata1(i) + CCUR(r2(dloc)) * COS(dloc * 2# * pi * (i / N))

newdata1(i) = newdata1(i) + CCUR(c2(dloc)) * SIN(dloc * 2# * pi * (i / N))

newdata1(i) = newdata1(i) / SQR(N)

NEXT i

rem calculate residual between original and summation of all terms generated so far

FOR i = 1 TO n

newdata(i) = newdata(i) - newdata1(i)

NEXT i

rem add current term to previous terms

FOR i = 1 TO n

totaldat(i) = totaldat(i) + newdata1(i)

NEXT i

rem calculate correlation between original and current model (totaldat)

CALL correl(totaldat,ydata,N,ycorrel)

rem test to see if correlation criteria met

rem test to see if maxterms criteria met

IF ycorrel < cobj AND k1 < .125 * N THEN

GOTO 99

END IF

rem output final model

CALL output_eigen(r1,c1,d,k1,N)

END

Fig. 2: Pseudo-code for Data Modeling eigenfunction model construction.

Fig. 3: Hilbert sequence scanning maintains dyadic neighbor correlation.

Cluster

Change Detector

Galaxy

Change Detector

Resolver

ClassifierCluster?

No

Yes

Galaxy?

No

Yes

START

Cluster?

No

YesCluster

Change Detector

Galaxy

Change Detector

Resolver

ClassifierCluster?

No

Yes

Galaxy?

No

Yes

START

Cluster?

No

Yes

Cluster

Nebula

Galaxy

Cluster

Nebula

Galaxy

Fig. 4: Flowchart for Data Modeling Change Detector.

Galaxy

ClassifierCluster

Classifier

START

Cluster?

Yes

No

Galaxy?

Yes

NoGalaxy

ClassifierCluster

Classifier

START

Cluster?

Yes

No

Galaxy?

Yes

No

Nebula

Cluster Galaxy

Nebula

Cluster Galaxy

Data Model

Original Data Model (256 x 256)

Data Model

Original Data Model (256 x 256)

Data Model

Original

Data Model

(64 x 64)

Data Model (256 x 256)

Original

(64 x 64)

Data Model

(64 x 64)

Original

(64 x 64)

Data Model

(64 x 64)

Original

(64 x 64)

Fig. 5: Flowchart for Data Modeling Classifier.

Fig. 6: Comparison of Turlington Data Model output and original for M13 (top), M20 (middle), and M101 (bottom).

M1 M2 M3 M4

M5 M6 M7 M8

M9 M10 M11 M12

M13 M14 M15 M16

M17 M18 M19 M20

M21 M22 M23 M24

M25 M26 M27 M28

M29 M30 M31 M32

M33 M34 M35 M36

M37 M38 M39 M40

M41 M42 M43 M44

M45 M46 M47 M48

M1 M2 M3 M4M1 M2 M3 M4

M5 M6 M7 M8M5 M6 M7 M8

M9 M10 M11 M12M9 M10 M11 M12

M13 M14 M15 M16M13 M14 M15 M16

M17 M18 M19 M20M17 M18 M19 M20

M21 M22 M23 M24M21 M22 M23 M24

M25 M26 M27 M28M25 M26 M27 M28

M29 M30 M31 M32M29 M30 M31 M32

M33 M34 M35 M36M33 M34 M35 M36

M37 M38 M39 M40M37 M38 M39 M40

M41 M42 M43 M44M41 M42 M43 M44

M45 M46 M47 M48M45 M46 M47 M48 Fig. 7: Data Models for Messier objects M1-M48.

M49 M50 M51 M52

M53 M54 M55 M56

M57 M58 M59 M60

M61 M62 M63 M64

M65 M66 M67 M68

M69 M70 M71 M72

M73 M74 M75 M76

M77 M78 M79 M80

M81 M82 M83 M84

M85 M86 M87 M88

M89 M90 M91 M92

M93 M94 M95 M96

M49 M50 M51 M52M49 M50 M51 M52

M53 M54 M55 M56M53 M54 M55 M56

M57 M58 M59 M60M57 M58 M59 M60

M61 M62 M63 M64M61 M62 M63 M64

M65 M66 M67 M68M65 M66 M67 M68

M69 M70 M71 M72M69 M70 M71 M72

M73 M74 M75 M76M73 M74 M75 M76

M77 M78 M79 M80M77 M78 M79 M80

M81 M82 M83 M84M81 M82 M83 M84

M85 M86 M87 M88M85 M86 M87 M88

M89 M90 M91 M92M89 M90 M91 M92

M93 M94 M95 M96M93 M94 M95 M96

Fig. 8: Data Models for Messier objects M49-M96.

M97 M98 M99 M100

M101 M102 M103 M104

M105 M106 M107 M108

M109 M110 M7 M8

M97 M98 M99 M100M97 M98 M99 M100

M101 M102 M103 M104M101 M102 M103 M104

M105 M106 M107 M108M105 M106 M107 M108

M109 M110 M7 M8M109 M110 M7 M8

Fig. 9: Data Models for Messier objects M97-M110.

Fig. 10: Comparison of test objects: M51 (left), NGC869 (left center), LMC and SMC (right center), Comet Hale-Bopp (right top), and

Comet Neat (right bottom).

N e b u l a

G a l a x y

C l u s t e r

Fig. 11: Cluster plot showing distribution of Data Modeling features. Features plotted are A(0) on the x-axis and A(1) on the y-axis.

Bottom plot shows Messier number designation, and top plot coded by object classification.

Fig. 12: Table of coefficients for the 5-term eigenfunction based Data Model for each Messier object.

Image resolution and performance analysis of webcams

for ground based astronomy

Albert Lim*a, Holger

Jaenisch

a,b, James Handley

a, Miroslav Filipovic

c,a, Graeme White

a, Alex Hons

a,

Cor Berrevoets, Gary Deragopiana, Jeffrey Payne

a, Mark Schneider

a, Matthew Edwards

b

a James Cook University, Centre for Astronomy, Townsville QLD 4811, Australia

bAlabama Agricultural and Mechanical University, Department of Physics, Huntsville, AL 35811 cUniversity of Western Sydney, Locked Bag 1797 PENRITH SOUTH DC NSW 1797, Australia

ABSTRACT

We present here a novel concept for achieving real-time super-resolution ground-based imagery for small aperture

telescopes. We explore the combination of existing stacking and registration software in conjunction with real-time

equation based Data Models. Our research indicates that for anisoplanatic imagery, a real-time video/software

enhanced analog to conventional speckle imaging is possible. This paper highlights the technique and theory for

creating such a system.

Keywords: Video/Software adaptive optics, image stacking, speckle imaging, webcam, Data Model, Registax, image

enhancement

1. INTRODUCTION

The use of broadband imaging as an alternative to speckle imaging for small telescope applications has been previously

demonstrated. Images were dewarped and sharp images selected for registration and stacking. Averaging of the corrected

sequence removes motion blur, improves the signal to noise (SNR), and causes the remaining short-exposure blur to

become position stable. For narrow field of view (FOV) observations, the point-spread function (PSF) is considered

constant across the image (isoplanatic); for wider FOV’s the atmosphere introduces a spatially dependent PSF, termed

anisoplanatic distortion. Figure 1 shows anisoplanatic distortion on Mars and lunar crater Gassendi1,2,3.

Lambert demonstrated theoretical and practical examples of achieving super-resolved images by exploiting the

anisoplanatic distortion to enable observation of spatial frequency content which would normally be lost beyond the

telescope aperture4. Super-resolution is only possible under anisoplanatic conditions. Most proposals to exploit

super-resolution are based on active control and alteration of the optical system to capture the high frequency content5.

We propose not moving or altering the optical system, and simply waiting until an extra sharp cell with its extra high

spatial frequency content is imaged into the telescope. Data Modeling Change Detectors are used to indicate when

these short exposure sharp and distortion free broadband images occur. Once detected, these extra sharp cells are

filtered, registered, stacked, and enhanced to maximize the effects of the high spatial frequency content of the final

image. By displaying the current enhanced image on a high-resolution monitor and leaving it intact (persistent)

throughout the next improved resolution update, continuous video enhanced imagery occurs even if updates are

intermittent4,6,7,8.

We demonstrate how Data Modeling can enable real-time adaptive identification of sharp frames from live video

streams, which are then processed using algorithms currently available in the form of freeware such as Registax9. To

achieve this, we use video collected at real-time rates (1-15Hz) using webcams or video sampled imagery. The real-time

Data Modeling algorithm enables the user to either select a few examples of sharp images to calibrate the system and get

the process up and going quickly, or it may be left entirely unsupervised allowing the Data Model to learn how to discern

sharp distortion free images form blurry ones10.

*[email protected]; phone +65 65674163; James Cook University, Centre for Astronomy, Townsville, QLD 4811,

AUSTRALIA


Once constructed, the Data Modeling Change Detector is then used to indicate images that should be filtered, registered,

stacked, and enhanced in real-time for display at the telescope on a monitor or at the eyepiece in the form of a miniature

high-resolution television display akin to the Collins Electro-Optics I3 Image Intensifier eyepiece11,12,13.

2. EQUIPMENT

2.1 Telescope

In the course of this work, Mars images were collected on the 25th August 2003 using the Singapore Science Centre

Observatory telescope, a 40 cm (16”), F13, Cassegrain telescope with a focal length of 5200 mm (205”). A Celestron

Ultima 2x Barlow lens was used with the above telescope for negative projection to the below webcam achieving a

combined F number of F26.5. Local observers rated the seeing on this night as 7 out of 10 per improvised scale

described in page 2 of “Photographic Atlas of the Moon”14.

2.2 Webcam

We used the Philips ToUcam Pro with a ¼” CCD with 1 lux sensitivity capable of 60 frames per second video capture at

640 x 480 resolution. It is equipped with a F/2.0 lens of 6mm focal length, integrated microphone and USB interface.

For collecting the Mars AVI video, the lens on this webcam was removed and replaced with a 1.25” standard eyepiece

sleeve and used with the above telescopic setup. Sound was disabled and the download frame rate was set at 15 frames

per second with exposure times of 1/33 second per frame.

2.3 Image registration/stacking software

Software was required to register and stack the sharp images from the webcam. Programs available included several

freeware programs: AstroStack, AstroVideo, IRIS, and Registax. For this application, Registax was chosen for

registration and stacking.

2.4 Image processing software

Once the sharp images were stacked, enhancement was done to the stacked image using Registax (wavelet based) and

MAJQANDA (Data Modeling). These methods are described later in this work.

3. SHARP IMAGE DETECTION AND SELECTION

3.1 Absolute chi-by-eye

Traditionally, sharp images are detected and selected using I×χ , defined as “absolute chi-by-eye”. This approach

provides the highest quality and sensitivity in image selection, but is also the least efficient and slowest because it

depends on human interaction and comparison between multiple images in a stream of frames that easily exceed

hundreds to thousands of image frames with varying blurriness.

3.2 Registax quality factor

Alternatively, a step towards automation is provided by software such as Registax. It provides a figure of merit termed

“Quality factor” to assist in identification of sharp frames.

The Quality factor is calculated from information in the spatial frequency graphs (Figure 1) and the filter thresholds are

set interactively by the user. A sum of the values below the red line is calculated and the fraction that is located right of

the green Quality-filter line is the value. This is based on the fact that sharper/clearer images will show relatively higher

amounts of fine (high spatial frequency) details than more blurred images. The position of the filter is crucial to

proper image quality ranking. If the filter is set near zero, all the images appear to have the same quality

(approximately 1).

The different filter settings have marked effects on the quality estimate. Figure 1 (top row) shows power spectra for

quality filter settings of 1, 5, and 15 pixels on the same image. When the filter is set at 1, most of the values are right of

the filter. In the bottom row of Figure 1, graphs are given that show quality difference (top red line) for all the images

(ordered by quality).

Also shown in the bottom row graphs of Figure 1 are curves using pixel neighborhoods of various sizes for comparing

the differences between each image and the reference image (blue line, bottom of the two graphs). This method, though

powerful, must be used with care because improved quality factors can be dominated by spatial noise. Both spatial

noise and sharp detail will yield greater RSS differences with the reference image. This means relying on this quality

factor for sharp image selection remains interactive and non-robust.

We also used simple JPEG compression file size as sharp image indication with the smaller file size indicating a

sharp image. The lunar image in focus has regions of essentially constant value giving higher JPEG compression. Mars

at best focus has regions of graded shading. When graduated regions are blurred or noisy, the image has constant regions

yielding higher compression for blurry Mars images.

3.3 Data Modeling sharp image detection

We assert as a fundamental premise that nominal occurs more frequently than off-nominal. If the reverse is true,

the definition swaps and the premise still hold.

We propose a method for identifying sharp frames by creating an equation based Data Model that is tuned to examples of

sharp frames and then used as a real-time bulk filter. We later explore the generalization of this approach to the

unsupervised case. In order to understand how Data Modeling can achieve this, we will examine Data Modeling theory.

Data Modeling15

is the process of deriving a mathematical expression from measured data. We approximate the

Kolmogorov-Gabor polynomial of the form16

LK ++++= ∑∑∑∑∑∑i j k

kjiijk

i j

jiij

i


by using Ivakhnenko’s Group Method of Data Handling (GMDH) using nested functionals of the form16


LxxxyO 3),,,( 21 =K (2)

This structured approach forms intermediate meta-variables from combinations of three inputs combined into a single

new fused output. This fused meta-variable becomes a new input variable available at the next layer. Since the algorithm

only uses the inputs necessary to achieve convergence, pruning of inputs occurs automatically and requires no external

intervention.

The O [3n] functional (function whose variables themselves are functions) is now used as a snapshot model of a group

of sharp images, or conversely a group of blurry images, depending on the application. This enables the functional

to be used as a fast real-time bulk filter for identifying sharp images bases on change detection18,19.

3.3.1 Change detector construction

Each Data Model is generated using only a portion (50% or less) of the available nominal data, with the

remainder held back for testing the Data Model after construction. This supports the ergodic Data Modeling re-

sampling theory that only a portion of the available data is required to produce adequate change models20.

For analyzing images, samples of pixel neighborhoods (image sub-sections) are extracted and characterized by five (5)

descriptive statistics given by

∑=

− −=N

j

jNxx

1

2

11 )(σ (3) ∑

=

−=

N

j

j

N

xxSkew

1

3

1

σ (4)

31

4

1 −

−= ∑

=

N

j

j

N

xxKurt

σ (5) 15

1

6

16 −

−= ∑

=

N

j

j

N

xxM

σ (6)

1051

8

18 −

−= ∑

=

N

j

j

N

xxM

σ (7)

Once the features in Equations (3) - (7) are calculated for each nominal data set, a Data Change Model is generated using

a target regression value of ½ as the nominal output. Once the equation model is derived, boundaries are placed on either

side of the target regression value that characterizes the fluctuation of the functional about ½. Typically, these boundaries

are set at 0.499 and 0.501 with a standard deviation of .0012. Figure 2 illustrates construction of a change detector21,22.

In unsupervised mode, no a priori knowledge of images being sharp or non-sharp is required. It is only assumed

that the majority of the images are background (non-sharp). Pseudo-code for constructing a Data Change Model

unsupervised is found in Figure 3. For the images analyzed for this paper, the final Data Change Model was determined

by placing a tight boundary of 0.0012 on either side of the final sharp image class designation. All images that flagged

inside of the two boundaries are considered candidate sharp images, while all others are considered non-sharp images

and are discarded from further consideration23,24,25.

4. IMAGE REGISTRATION AND STACKING

4.1 Registax for image registration

A good reference image is selected and an alignment area, depicted by a cursor box in Figure 4 (left), is framed over a

feature of interest such as a planetary disk or a lunar crater. The alignment area boxed will now serve as a basis for the

alignment procedure. During initial alignment, the program will align the images with the reference area using a Fourier-

based cross-correlation with normalization. To achieve this, both the reference-area and the image-area are transformed

into real and imaginary parts using Fast Fourier Transform (FFT). The cross-correlation products (real and imaginary)

are calculated from26

),(),(),(),(),( yxSEQyxREFyxSEQyxREFyxShift IMAGIMAGREALREALREAL += (8)

),(),(),(),(),( yxSEQyxREFyxSEQyxREFyxShift IMAGREALREALIMAGIMAG += (9)

This is then processed using an inverse FFT to create a new “image”. The brightest spot in this image is located at the

expected shift position. The initial FFT also estimates image quality, which is directly derived from the FFT-transform of

the image being processed. The FFT-transform is first converted into a “real” image by calculating

),(),(),(),(),( yxSEQyxSEQyxSEQyxSEQyxIMAGE IMAGIMAGREALREAL += (10)

A radial average value called a power spectrum is then calculated for all radii from the center of this “image” outwards.

Figure 4 (left) shows the power spectrum and a graph displaying registration properties. Basically, after running the FFT

on an image, the program sums for every possible radius (from the center) the intensities in the FFT-ed image per radius.

This sum per radius in integer steps is then divided by the number of pixels that contributed to the intensity value (the

smaller the radii, the fewer the contributing pixels). An array with radii starting from 1 onwards is thus generated. A

bandwidth filter is then used to calculate the sum for the radii in the band and presented as a proportion based on the sum

in the band divided by the sum for all radii. In this way, the sum of the values in the section of the power-spectrum

defined by user is thus compared with the total sum of the power-spectrum. This process is depicted in Figure 1.

Next, the images are sorted according to quality levels. Only those images are processed which exceed a certain user-

defined quality threshold in comparison to the image with the highest measured quality. The next stage is

optimizing the initial alignment. During optimization, every image is first block-matched with the reference image to

find a pixel-shift that result in the smallest sum of squared differences between the images.

Every image is normalized before these squared differences are computed to prevent the changes in overall brightness

affecting the outcome. The block-match search is done in a user-specified search window, and the program keeps track

of the improvement for every image in alignment (lower sum of squared differences, SSD). Using a second user defined

percentage threshold, optimization terminates when subsequent optimizations do not yield SSD improvement. Once

completed, the registered images are ready to be stacked to minimize noise and eliminate motion by spatial correlation

4.2 Image stacking increases SNR

Stacking enough correctly selected sharp (low noise) images results in significant increase in the signal-to-noise ratio.

Technically, the signal-to-noise ratio (SNR) is what makes the difference between a good and a poor image. Adding 2

images will combine their signals S1 and S2, however, their noise also quadruple as27

21 SSS += (11)

2

2

2

1

2 NNN += (12)

Their signal to noise ratio is defined as

( )2

12

2

2

1

21

NN

SSSNR

+

+= (13)

From (13), we derive,

2

1

2

1

2

2

1

2

2

1

2

1

11

+

+

+

=

N

N

SNR

N

N

SNRSNR

(14)

Assuming both images have similar SNR and noise, adding 2 images increases the signal-to-noise by a factor of square

root 2 resulting in 1.414 times the initial SNR of each image.

During stacking, we can assume multiple exposures taken under similar conditions will result in adding images with

similar SNR and noise. (i.e. S1 = S2 = S3 = …… and N1 = N2 = N3 = …… )

Extending from the above, Equation (14) may be expressed as

...

...1...1

2

1

2

1

2

2

1

2

2

1

2

1 +

+

+

+

+

+

=

N

N

SNR

N

N

SNRSNR

(15)

2

1

)111(

)()()( 321

K

K

+++

++=

SNRSNRSNRSNR (16)

For n images therefore, the final signal-to-noise ratio is function of square root of n, i.e.

1)(*2

1

SNRnSNR = (17)

Thus, it can be seen that signal to noise varies to the square root of n, while stacking 2 images will increase SNR by

1.414 times, stacking 9 images will increase the SNR by 3 times and stacking 625 images would result in an increase

SNR factor of 25 times.

During stacking, all the pixel-intensities of the images considered of good quality are stacked as seen in Figure 4

(center). Stacking and all other subsequent processing is done in floating point matrices to maintain information for post

image processing. RGB layers may be stacked separately or used to create luminance out of RGB-layers which are

maintained for post processing in either color option.

5. IMAGE ENHANCEMENT

5.1 Registax wavelet based image enhancement

After stacking, post-processing is done using wavelets. The wavelet-processing incorporated by Registax uses a simple

filter to separate the stacked image into 6 spatial layers. The process is similar to successive unsharp masking runs.

Initially, the stack-image gets convoluted with the wavelet-filter. The difference between the stack and this convolved

image is calculated and moved into the first wavelet-layer. The second layer is constructed using the blurred image

which is convolved once more with the wavelet-filter, and the difference between these is stored in layer 2.

This process continues until the initial intensities of the stack are divided across 6 layers and a final, blurred, base layer.

The advantage with wavelet-techniques are orthogonality and invertibility. This insures no information is lost during

processing. Adding the layers together without alteration retrieves the original stacked image. The 6 layers represent 6

spatial resolution layers, the highest (layer 1) will have the finest details and the lowest (layer 6) the coarsest details.

Layer 1 will often have most noise present, and removing this layer reduces noise. Individual layers may be post

processed to increase the contrast for details present in that layer. This powerful image enhancement method requires

some skill and interactivity to use, and is difficult to make robust for automatic processing.

5.2 Data Modeling image enhancement

The authors have determined that the Van Cittert deconvolution nonlinear image processing method can be

approximated through the application of a series of high pass and low pass kernel filters to an image. Additionally, using

knowledge of the number of times the filters are applied enables the individual high pass and low pass filters to be

combined into an equivalent kernel. An equivalent kernel is one that when applied to the image gives the same value as

a series of individual kernels applied to the image sequentially.

To illustrate, consider an image processing sequence consisting of applying a high pass filter, followed by a low pass

filter, followed finally by a Gaussian filter to an image. Examples of these filters are

111

1151

111

−−−−−−−−

111

131

111

121

242

121

High Low Gaussian (18)

Convolving each of these three kernels together yields a larger kernel that is called the equivalent kernel.

111

1151

111

−−−−−−−−

⊗111

131

111

⊗121

242

121

=

4810841

4210161024

8109415294108

10161522521521610

8109415294108

4210161024

14810841

−−−−−−−−−−−−−−−−−−−−−−

1

−−−−−−

(19)

When the equivalent kernel is applied to the image, the result is the same as convolving the image with each of the

three smaller kernels separately. Mathematically, the order the kernels are applied in does not matter. If a high pass

filter of size 3x3 is applied to an image alternatively with a low pass filter that is also of size 3x3, the resultant for one

application of these 2 filters is the same as having used a single equivalent kernel of 5x5. Taking this one step further, a

series of 5 sets of 3x3 high pass and 3x3 low pass filters could be applied to the image so that

== )(5),,,,( HPLPHPLPHPLPHPLPHPLPHPLP 21 x 21 Equivalent Kernel (20)

For this example, an equivalent kernel of size 21x21 would yield the same answer in a single pass. To keep from having

to spatially convolve such a large kernel or to avoid using a Fast Fourier Transform (FFT) for convolution, the authors

have derived a Data Model of the equivalent kernel.

We found the equivalent kernel process is approximated with Data Models. The Data Model uses either

equivalent features from local pixel neighborhoods operated on by the equivalent kernel, or a subset of the pixel

neighborhood values directly as input. The subset of the pixel neighborhood chosen corresponds to the center row and

center column of the kernel as shown in Figure 5.

6. RESULTS

Figures 6 and 7 display results of image selection, registration, stacking and processing using Data Modeling and

Registax for Mars and the moon. These results can be summarized as:

Mars Moon

Total pool 66 225

Data Model flagged to stack 18 38

Registax images used 18 38

Final Registax Image Quality Factor = 0.1886 0.2026

Final Data Model Image Quality Factor = 0.1917 0.2053

The Mars results showed striking improvement by Registax over the original, but an additional marginal improvement

by the Data Model. For the moon, although interactive Registax clearly improved on the raw, stacked image Data

Modeling showed marked improvement over it. Also, in this case the Data Modeling was performed unsupervised.

Finally, legacy images constructed with stacking are shown in Figure 8.

7. FUTURE WORK

Our present setup uses webcams at the telescope to capture AVI sequences (mostly blurred frames with sporadic sharper

ones) for storage in a computer for post processing.. Ideally, the Data Modeling approach for selecting only the sharpest

images and enhancing them even further can be performed inline and in real-time as images are acquired at the telescope

thereby facilitating the use of the Data Model itself as a real-time video/software enhanced analog to speckle

imaging. This is illustrated in Figure 9.

Achieving real time sharp image selection also saves storage space (each 2 minute AVI sequence on 640 x 480

resolution is over 1GB in size) and valuable time. We need to also apply Data Modeling to the automatic image

registration problem. The advent of powerful computers, high speed USBs and related technology will enable webcams

using software to perform real-time registration28.

8. SUMMARY

We demonstrated webcam based image deblurring and enhancement using existing software and algorithms for small

aperture ground based telescopes. We also demonstrated the concept of auto selection of sharp frames, constructed a

Data Change Detection Model using local pixel statistics as features to find other sharp images in real-time, and

modeled a Van Cittert deconvolution using an equivalent kernel from a series of high and low pass filters29.

ACKNOWLEDGEMENTS

The authors wish to thank committee members of The Astronomical Society of Singapore (TASOS) and Mr. Tan Wei

Leong for their support and the images; and Scott McPheeters, Tim Aden, and John Deacon for their continued support

during the course of this work.

REFERENCES

1. Charnotskii, M.I., “Imaging in turbulence beyond diffraction limits”, Proceedings of SPIE Vol. 2534 (1995)

2. Charnotskii, M.I., et. al., “Observation of superresolution in nonisoplanatic imaging through turbulence”, J. Opt.

Soc. Am. A., Vol. 7, No. 8, August 1990, pp. 1345-50.

3. Lubosz, W., “Optical systems with resolving powers exceeding the classical limit. II”, J. Opt. Soc. Am. A., Vol. 57,

1967, pp. 932-41.

4. Roddier, F. Adaptive Optics in Astronomy. Cambridge, UK: Cambridge University Press, 1999.

5. Hubin, N. & L. Noethe 1993, “What is Adaptive Optics?”, Science, 262 : 1345 – 1484.

6. Jaenisch, H.M., Handley, J.W., Scoggins, J., Carroll, M.P., "Atmospheric Optical Turbulence Model (ATOM) Based

On Fractal Theory, " Proceedings of SPIE, Los Angeles, CA January 24, 1994.

7. Ellerbrook, B.L. “First Order Performance Evaluation of Adaptive-Optics Systems for Atmospheric-Turbulence

Compensation in Extended-Field-of-View Telescopes.” Journal of the Optical Society of America A 11:783-805

(1994)

8. Fraser, D., Lambert, A., Jahromi, M.R.S., Clyde, D., Donaldson, N., “Can broad-band image restoration rival

speckle restoration?”, Proceedings of SPIE Vol. 4792 (2002).

9. Berrevoets C., “Processing Webcam Images with Registax ”, Sky and Telescope, Apr 2004, p 130 – 135.

10. Lambert, A., Fraser, D., Jahromi, M.R.S., Hunt, B.R., “Super-resolution in image restoration of wide area images

viewed through atmospheric turbulence”, Proceedings of SPIE Vol. 4792 (2002).

11. Shemer, A., et al., “Superresolving optical system with time multiplexing and computer decoding”, Applied Optics,

Vol. 38 No. 35, 10 December 1999, pp. 7245-51.

12. Jaenisch, H.M., Collins, W.J., Handley, J.W., Case, C.T., Songy, C.G., “Real-time visual astronomy using image

intensifiers and Data Modeling”, Proceedings of SPIE, Vol. 4796 (2002).

13. Collins, W., “Collins Electro-Optics”, http://www.ceoptics.com/, May 24, 2004.

14. Lim, A., Chong, S.M., Ang, P.S., “Photographic Atlas of the Moon”, Cambridge University Press, 2002.



16. Westwick, D.T. Kearney, R.E., Identification of Nonlinear Physiological Systems, Piscataway, NJ: IEEE Press,

2003.

17. Madala, H.R., Ivakhnenko, A.G., Inductive Learning Algorithms for Complex Systems Modeling, Boca Raton, FL:

CRC Press, 1994.



19. Handley, J., Jaenisch, H., Lim, A., White, G., Pennypacker, C., Edwards, M., “Data Modeling of Deep Sky Images”,

Proceedings of SPIE Vol. 5497 (2004).

20. Jaenisch, H.M., Handley, J.W., Pooley J.C., Murray S.R., “Data Modeling for Fault Detection”, Society for

Machinery Failure Prevention Technology (MFPT), (April 2003), published in Proceedings of MFPT (2004).

21. Pooley, J.C. and Murray, S.R., “Rotordynamic Fault Diagnostics Using Phase Coherent Filtering”, Proceedings of

the Society for Machinery Failure Prevention Technology (MFPT), (April 2003).

22. Pooley, J.C., Murray, S.R., Jaenisch, H.M., and Handley, J.W., “Fault Detection via Complex Hybrid Signature

Analysis”, JANNAF 39th Combustion, 27th Airbreathing Propulsion, 21st Propulsion Systems Hazards, and 3rd

Modeling and Simulation Subcommittees Joint Meeting, Colorado Springs, CO, (December 2003).

23. Jaenisch, H., Handley J., Lim A., M.D. Filipovic, White G., Hons A. , Crothers S., Deragopian G., Schneider M.,

Edwards M., “Data Modeling for virtual observatory data mining”, Proceedings of SPIE Vol. 5493 (2004).

24. Jaenisch, H.M., Handley, J.W., Bonham, L.A., “Enabling Calibration on Demand for Situational Awareness”, Army

Aviation Association of America (AAAA), Tennessee Valley Chapter, (February 2003).

25. Jaenisch, H.M., Handley, J.W., Faucheux, J.P., Harris, B., “Data Modeling of Network Dynamics”, Proceedings of

SPIE, Vol. 5200 (2003).

26. Berrevoets, C. “Registax Website”, http: http://aberrator.astronomy.net/registax/, May 24, 2004.

27. Newberry M.V. “The Signal to Noise Connection – Part II”, CCD Astronomy, Fall 1994, p 13 – 14 (1994)

28. Lim, A., “Astro Scientific Centre Pte. Ltd.”, http://www.astro.com.sg/, May 24, 2004.

29. James Cook University, Centre for Astronomy, http://www.jcu.edu.au/school/mathphys/astronomy/, May 24, 2004.

http://www.ceoptics.com/

http://aberrator.astronomy.net/registax/

http://www.astro.com.sg/

http://www.jcu.edu.au/school/mathphys/astronomy/

FIGURES

Fig. 1: Examples of individual sharp (left), intermediate (center) and blur frames (right) as scored by quality factor. Graphs on bottom

show ranking of approximately 50 images by quality (top curve), and difference between reference image and individual frames

(bottom curve).

Tip-Off

Tip-Off

Nominal

Tip-Off

Tip-Off

Nominal

Equation

Output

σ

T = w0+w1x1+w2x2+w3x3 +w4x12

+w5x22 +w6x3

2 +w7x13 +w8x2

3

+w9x32 +w10x1x2 +w11x1x3

+w12x2x3

Skewness Standard

Deviation

Kurtosis

Rollup

Stats

Data Model

Data

SeriesPDF

Fig. 2: Data Change Modeling uses descriptive statistics as shown above that characterize the PDF or histogram of the input data to

build a model that is sensitive to small changes in input feature distribution.

n=total num ber of cases

Call G enerate_W aveform _Stats(ydata,n,ystats)

Call G enerate_R ollup_Stats(ystats,n,rstats)

For i=1 to n

rem Sub-sam ple of all cases called G roup 1

Next i

m =num ber of subsam pled cases

rem use prev ious pseudo-code for generating a D ata Change M odel


fori=1 to n

call In terrogate_data_m odel(features,D m val)

call flag_tipoffs

rem Save tip-off data as new G roup 2

next i

for i=1 to n

rem subsam ple another m points from new pool

rem place w ith 1st 10% subsam pled from original G roup 1

Next i

m =2*m


for i=1 to n

call In terrogate_data_m odel(features,D m val)

call flag_tipoffs

rem Save as G roup 3

next i

rem extract a ll k tipoff cases from G roup 3 and

rem random ly assign to 0/1 c lass

for i=1 to k

if rnd>.5 then

out=1

else

out=0

endif

endif

nex t i

10 continue

call generate_data_re la tions_m odel

rem score cases

for i=1 to k

ca ll In terrogate_data_m odel(features,D m val)

if D m val< .25 then

tra inval=0

elseif D m val> .75 then

tra inval=1

else

if tra inva l=0 then

tra inva l=1

else

tra inva l=0

endif

endif

nex t i

rem check for lim it cyc le or 4 layer D ata M odel learn ing a ll cases

if lim it_cycle or all_ learned then

rem sharp im ages are the sm aller c lass

rem if equal num ber in each c lass, se lect c lass w ith largest R SS

else

goto 10

endif

rem last m odel generated can be used in lieu of tra in ing a fina l m odel

ca ll generate_data_change_m odel

for i=1 to n

ca ll In terrogate_data_m odel(features,D m val)

ca ll flag_tipoffs

nex t i

ca ll w rite_output

end Fig. 3: Pseudo-code for unsupervised Data Change Detector generation.

Fig. 4: (Left) Alignment in progress at 12%. The power spectrum and Registration Properties are shown; (Center) Stacking. Users can

manually deselect blur frames that were ranked incorrectly by Registax; (Right) Registax’s 6 layer wavelet processing works well with

most planetary astronomical images.

Image

Rzzzzzzzzzf =),,,,,,,,( 534335343332312313

Data Model of Equivalent Kernel

)]3([ nO

*

* f is

Image Pixel Locations

Correspond To Center Row and

Column of Equivalent Kernel

5554535251

4544434241

3534333231

2424232221

1514131211

zzzzz

zzzzz

zzzzz

zzzzz

zzzzz

Center Image Pixel Locations

From Final Image

(After Applying Equivalent Kernel)

5554535251

4544434241

3534333231

2424232221

1514131211

zzzzz

zzzzz

zzzzz

zzzzz

zzzzz

Fig. 5: Data Modeling captures center row and center column of equivalent kernel and builds an equation model that captures the

image enhancement algorithm.

Registax Sharp Selected and Wavelet

Processed

Score = 0.1886

Data Modeling

Score = 0.1917

Raw Frame #1


Processed

Score = 0.1886

Data Modeling

Score = 0.1917

(Note: JPEG artifacts removed,

Noise suppressed)

Raw Frame #1

(Note: JPEG artifacts present)

Fig. 6: Data Modeling for image stacking on Mars (top) and zoom into Mars (bottom).


Processed

Score = 0.2026

Data Modeling

Score = 0.2053

(Note: Noise suppressed)

Raw Frame #1


Processed

Score = 0.2026

Data Modeling

Score = 0.2053

Raw Frame #1

Fig. 7: Data Modeling for image stacking on lunar surface feature (top), and zoom in to lunar surface feature (bottom).

Jupiter Saturn Alpine Valley

Fig. 8: Examples of high-resolution images produced by processing stacked images from AVIs using Registax.

BackgroundReject

SharpData Model

[O(3n)]

StackData

Modeling

EnhanceImage

Camera

Time

Fig. 9: Block diagram of unsupervised Data Modeling approach for selecting images for stacking.

Classification of Jacoby stellar spectra data using Data Modeling

Holger M. Jaenisch*a and Miroslav D. Filipovic**aaUniversity of Western Sydney

Keywords: Data Modeling, Neural network, Stellar Spectra Classification, Astronomy, Decision Architecture

ABSTRACT

This paper presents the results of creating a Data Model by using neural equation networks of high order polynomials toachieve 100% correct classification of the Jacoby stellar spectra. The Jacoby set is a challenging group of 161 spectraspanning the full range of temperature, sub-temperature and luminosity groupings of standard star types. To achieve fulllearning, the development of a cascaded decision architecture linking an extensive network of polynomial decisionequations was required. The two dominant features were extracted, and complex decision maps generated. Also, thesensitivity of the equation architecture to misclassification due to measurement noise was analyzed.

1. BACKGROUND

1.1. History of Stellar SpectraIn the early 1800’s, the German physicist Joseph von Fraunhofer first observed the patterns of sun spectrum absorptionlines. In the late part of the 19th century, astronomers were able for the first time to routinely examine the spectra of starsin large numbers. Astronomers like Angelo Secchi and E.C. Pickering noted that the stellar spectra could be divided intogroups by the general appearance of their spectra. They proposed classification schemes where stars were groupedtogether by the prominence of certain spectral lines. In Secchi’s scheme, stars with strong hydrogen lines were calledtype I, stars with strong lines from metallic ions like iron and calcium were called type II, stars with wide bands ofabsorption that got darker toward the blue were called type III, and so on. Building upon this early work, astronomers atthe Harvard Observatory refined the spectral types and renamed them with letters, A, B, C, and so forth. The observatoryalso embarked on a massive project to classify spectra, carried out by astronomers Williamina Fleming, Annie JumpCannon, and Antonia Maury. The results of their work was published between 1918 and 1924 as the Henry DraperCatalog (he was the benefactor of this work), and provided classifications of 225,300 stars representing only a tinyfraction of the stars in the sky. In the course of this Harvard study, some of the old spectral types were consolidatedtogether, and the types were rearranged to reflect a steady change in the strengths of representative spectral lines. Thisrearrangement of order caused the order of the spectral classes to become O, B, A, F, G, K, and M. These letterdesignations have no physical meaning but have remained to this day. Each spectral class is divided into tenths (sub-temperature class), so that a B0 star follows an O9, and an A0, a B9. In this scheme the sun is designated a type G2.

These early studies generally lumped stars together into classes based upon the appearance of the spectra. Those thatlooked the same got lumped together. However, in the 1930’s and 1940’s, a better understanding of the physics behindthe spectra was developed. It was then understood that the predominant factor driving a star’s spectra was its surfacetemperature. It was found that O class stars are the hottest with surface temperatures on the order of 40,000 ˚K, while Mclass stars the coolest at 3000 ˚K. The decimal divisions of sub-temperature class follow the same pattern, with a B0 starhotter than a B9.

*[email protected]; phone 1 256 337 3768; fax 1 256 830 0287; Sparta, Inc.; 4901 Corporate Dr. Ste 102;Huntsville, Alabama, USA 35805;**[email protected]; Intl. phone 61 2 47360135; local phone (02) 47360135;Intl fax 61 2 47360129;http://www.uws.edu.au/astronomy/index.html; Astronomy, School of Engineering and IndustrialDesign, University of Western Sydney, Locked Bag 1797, PENRITH SOUTH DC, NSW 1797, AUSTRALIA.

In the 1940’s and 1950’s, W. W. Morgan and P.C. Keenan at Yerkes Observatory introduced the MK system. The MKsystem is a refinement of the earlier Harvard work that takes into account that stars at the same temperature can havedifferent sizes. For example, a star one hundred times larger than our sun but with the same surface temperature, willshow subtle spectrum differences and will have a much higher luminosity. The MK system adds a Roman numeral to theend of the spectral type to indicate the luminosity class: I a supergiant, III a giant star, and V a main sequence star. Oursun as a typical main-sequence star would be designated a G2V1.

1.2. Explanation of Jacoby SpectraThe spectral type of a star is fundamental to astronomy, so much so that an astronomer beginning the study of any starwill first try to find out its spectral type. The spectral type of a star allows the astronomer to know not only thetemperature of the star, but also its luminosity (expressed often as the absolute magnitude of the star) and its color. Theseproperties help in determining the distance, mass, and many other physical quantities associated with the star, itssurrounding environment, and its past history. If the star has not already been catalogued or if there is doubt about thelisted classification, then the classification must be done by taking a spectrum of a star with a spectrograph andcomparing it with an Atlas of well-studied spectra of bright stars. Fig. 1 shows 13 standard digital spectra from theprincipal MK spectral types measured with a spectrograph. The range of wavelength (the x axis) is 3900 Å to 4500 Åand the intensity (the y axis) of each spectrum is normalized, which means that it has been multiplied by a constant sothat the spectrum fits into the picture, with a value of 1.0 for the maximum intensity, and 0 for no light at all.

3900 wavelength (angstroms) 4500


0

1

0

1

0

1

0

10

1

0

1

0

1

0

1

0

10

1

0

1

0

1

0

1


O5V

B0V

B6V

A1V

A5V

F0V

F5V

G0V

G6V

K0V

K5V

M0V

M5V



0

1

0

1

0

1

0

10

1

0

1

0

1

0

1

0

10

1

0

1

0

1

0

1


O5V

B0V

B6V

A1V

A5V

F0V

F5V

G0V

G6V

K0V

K5V

M0V

M5V

Fig 1: Examples of Jacoby spectra. Shown here are 13 examples of Jacoby spectra extractedfrom the original between 3900 Å and 4500 Å and measured with a spectrograph. Normalizedintensity image graphs taken as photographs of the same portion of the Jacoby spectra areshown on the lower right of each of the object data graphs. This portion of the Jacoby spectrais free of measurement error and is the same portion used by CLEA. The full Jacoby spectraranges from 3510 Å to 7427.2 Å.

The classification of stellar enables the reduction of a large sample of diverse stars to a manageable number of naturalgroups with similar characteristics. Since the group members presumably have similar physical characteristics, they canbe studied as groups, not as isolated stars. By the same token, unusual stars may readily be identified because of theirobvious differences from the natural groups. These peculiar stars can then be subjected to intensive study in order toattempt to understand the reason for their unusual nature. These exceptions to the rule often help in understanding broadfeatures of the natural groups and provide evolutionary links between the groups.

The Jacoby stellar catalog is the cleanest and most comprehensive data collected in the visible wavelength band to date.This catalog provides a comprehensive catalog for modeling and predicting deep sky object spectra. The data for thiscatalog was taken over 26 different nights between December 1980 and December 1981 with the Intensified ReticonScanner (IRS) on the No. 1 90 cm telescope located at the Kitt Peak National Observatory in Tucson, Arizona. The IRSis a dual beam spectrophotometer that measures both the sky and the star simultaneously. Data for this catalog was takenin the beam switch mode, where the object is alternatively observed through one aperture and then the other. These twoapertures were 13 inches in diameter and were separated physically by 61 inches. Jacoby, et.al.2 documented in theiroriginal work that this instrument was under development until February 1981, and that great care was taken to insurethat no fixed pattern noise effects from the instrument were introduced into the measurements.

The spectra for each of the 161 objects published in the Jacoby catalog were measured in three (3) overlappingsegments. These segments consisted of: 1) blue, grating No. 35 (600 1 mm-1) in second order with copper sulfate andSchott WG3 blocking filters to cover 3430 or 3500 Å to 4870 or 4950 Å (dependent on exact instrument setup for thecase being observed); 2) green, grating No. 56 (600 1 mm-1) in second order with either a Schott GG455 or Corning 3-75blocking filter to cover from 4760 to 6220 Å; and 3) red, grating No. 36 (1200 1 mm-1) in first order with either a SchottGG15 or WG2 blocking filter to cover from 6000 to 7450 Å. The profile of the instrument can be represented by aGaussian function, and the full width of the half maximum is approximately 4.5 Å (which varied across the spectrum andwith seeing variations, but not more than 10%), and the data was corrected for any noted measurement artifacts.

1.3. Wavelength FeaturesIn order to perform classification, a set of numbers that describe or characterize each of the spectra above must bechosen and developed. These descriptive numbers are called features3. In the case of this work, actual values sampledfrom the spectra themselves were used as the characterizing features. These features were not chosen in a random or adhoc manner from the data, they were chosen to be the different line strengths located between 3900 Å and 4500 Å thatscientists and astronomers have in the past 100 years determined as being descriptors of specific star types. Thesefeatures are listed in Fig 2.

3900 Start of CLEA file3934 Ca II (K line), visible A0-A9, strongest G8-K23965 He I, visible O6-B9, strongest B0-B23968 Ca II (H line), H line(blend) = K line at ~F0, strongest from G8-K23970 H I (H Epsilon), H Balmer Series, Max@A04026 He I, strongest B24031 Mn I, compare to 4130 Si II line for discrimination in F stars4046 Fe I, visible F0 and cooler, strongest K2-K5, discerns G stars4068 C III, strongest O6-O94073 O II, visible O9-B24089 Si IV, strongest O6-O94097 N III, visible B0-B24100 He II, strongest O4102 H I (H Delta), H Balmer Series, Max@A0

4121 He I, strongest B, maximum around B24131 Si II, visible B2-A54144 Fe I, Appear F0 and cooler, strongest K2-K54227 Ca I, first visible F0, Strengthens to cooler K and M stars4267 C II, visible B0-B8, strongest at B44300 CH & Metals (G Band), visible F5 in main sequence, G0 in supergiants.4317 O II, visible in O9-B24340 H I (H Gamma), H Balmer Series, Max@A04384 Fe I, strongest K2-K54388 He I, maximum around B24471 He I, maximum around B24472 He I, maximum around B24481 Mg II, evident in B2-A5, Strongest at A1.4500 End of CLEA file

Fig 2: The 28 wavelength features from the Jacoby spectra used in this work, and theirphysical significance in stellar spectra.

2. CLASSIFICATION METHODS

2.1. Classical MethodsHistorically, expert systems, Bayesian classifiers, clustering methods, and neural networks are methods that have beenavailable and used unsuccessfully to classify stellar spectra. In the case of expert systems, the if-then-else rules that mustbe developed are complex and difficult to test. For the type of decision architecture encompassed by this work, an expertsystem of if-then-else rules would need on the order of 28 factorial features times 7 temperature classes times 19 sub-temperature classes times 6 luminosity classes, or on the order of 2x1054 rules. Even if this expert system providedperfect class separability and classification, the if-then-else rule set would be totally unmanageable and could notadequately be tested for failure modes.

The benefit of the Bayesian classifier is that it does not have such a large and unwieldy number of rules or features tokeep track of and test. However, the Bayesian classifier only uses a statistical model to represent the classes using themean and covariance of the features. If a Bayesian classifier is used to model the decision architecture required for thisstellar classification problem, a great deal of overlap in the 28 dimensional feature space in mean and covariance wouldexist. This would result in a large amount of misclassification.

Clustering methods such as K-means are similar to the Bayesian classifier approach in that the 28 dimensional featurespace is large, but they provide a better representation of the spectra to be modeled. Instead of using pure mean andcovariance, the K-means algorithm uses geometrical distance from the mean of the cluster to determine classmembership. However, clustering methods require trying out combinatorial subsets of groupings (basically allcombinations) to determine the optimal grouping and class membership, which for this type of data set is computationalunfeasible.

Neural networks are universal function approximators that find coefficients for the fit, but the form of the equation thatthey use is unknown to the trainer. The Rumelhardt back propagation neural network for example uses the sigmoidalfunction to represent the transition from layer to layer. In Fig 3, the sigmoidal function would transform at the hiddenlayer the values from the input layer into those propagated to the output layer. Once the output layer is determined, it iscompared with the truth (differences are generated) and the differences are propagated back to the input layer where thecoefficients (called weights) are adjusted to minimize this difference. Neural networks do not allow the user or trainer toextract the equation that the neural network has determined, and the contribution of any one feature to the final outcomeis unknown. In addition, neural networks require an a priori unknown amount of training time and have several inputparameters (such as convergence and learning rate) that must be tweaked by the user, thereby making them non-robustand ad hoc. These methods have been tried on stellar spectra classification in the past and have demonstrated onlymarginal improvement over the best Bayes classifiers and fail on subclass decisions. These networks can also quicklybecome complex.

InputLayer

HiddenLayer

OutputLayer

x1

x2

x3

x4

x5

x6

x28

“O”

“B”

“A”

“F”

“G”

“M”

“K”

N1

N2

N3

N4

N5

N6

N28

...

...

...

InputLayer

HiddenLayer

OutputLayer

x1

x2

x3

x4

x5

x6

x28

“O”

“B”

“A”

“F”

“G”

“M”

“K”

N1

N2

N3

N4

N5

N6

N28

...

...

...

Fig 3: Traditional neural network architecture for determining temperature class of stellarobjects from the 28 wavelength input features of interest.

2.2. Data Modeling Using Neural Equation Stellar Classification ArchitectureThe Neural Equation Stellar Classification architecture is an example of a neural equation based method that uses DataModeling to generalize multivariate regression and neural networks by deriving high order polynomial equations ofdatabase relations using simple low order building blocks. This method is robust and efficient, and is based in part on theGroup Method of Data Handling (GMDH) polynomial method developed by Ivakhnenko4,5. Jaenisch6 has taken thisprocess a step further by developing an information processing architecture for generating equations representing theknowledge contained in hybrid neural networks and expert systems. This architecture is defined by the following salientfeatures: 1) Feature selection process is fully autonomous; 2) Generates neural polynomial decision equation; 3) Inputvariables may call external simulations and executables for their value(s); 4) Output variables may call externalsimulations and executables with value(s); 5) Output may become input to new neural equation; 6) Expert systems maybe linked with neural equation as either input or output; and 7) Neural equation modeling is totally open-ended andscaleable. This capability allows multi-variable problems to be represented in a different form from standard linear andnon-linear programming formats and provides a new format that is amenable to neural architecture solution methods.Any solution that can be found by a neural architecture can be recast in Ivakhnenko polynomial form and can then beused to generate an objective function directly. This new simple algebraic model can then be optimized using othermethods in Data Modeling developed by Jaenisch to provide a fast and efficient model. Fig. 4 shows an example usingthird order building blocks (labeled as triples) to build a higher order (in this case 27th order) polynomial model.

x1

x3

x5

x3

x5

x7

x1

x5

x7

N

N

N

N

N

N

N

N

N

Triple

Triple

Triple Triple U “K”or “M”

N = NormalizerU = Unitizer

Equation 1

Equation 2

Equation 3 Equation 4x1

x3

x5

x3

x5

x7

x1

x5

x7

N

N

N

N

N

N

N

N

N

Triple

Triple

Triple Triple U “K”or “M”

x1

x3

x5

x3

x5

x7

x1

x5

x7

x1

x3

x5

x3

x5

x7

x1

x5

x7

N

N

N

N

N

N

N

N

N

N

N

N

N

N

N

N

N

N

Triple

Triple

Triple

Triple

TripleTriple TripleTriple UU “K”or “M”

“K”or “M”

N = NormalizerU = Unitizer

Equation 1

Equation 2

Equation 3 Equation 4

Fig 4: Neural equation stellar classification architecture example. Here, four differentequations (represented as Triple for their type) are used in a decision architecture to determinetemperature class K from temperature class M. This architecture can be embedded in a higherorder architecture that determines temperature classes K and M from other classes such as G.

In this type of equation based neural architecture, the tree structure can continue to expand if the complexity of thedecision architecture becomes more complex. This allows for entire branches to be added to the tree if necessary withoutretraining the entire tree. Also, each individual equation in the architecture can be represented by an entire network ofpolynomial based equation models, thereby allowing for the nesting of decision architectures within other decisionarchitectures. This method is fully adaptive, and is allowed to choose which features are needed, and only uses thosefeatures that are necessary to accurately make the correct decision.

3. RESULTS

3.1. Decision ArchitecturesUsing the equation based neural architecture approach, independent decision architectures were constructed for each ofthe classes to be determined: temperature, sub-temperature, and luminosity. Once these architectures were generated,they were taken and fused together into one overall network that calls each architecture and yields the final full stellarclassification. This is shown graphically in Fig. 5, where the example case shown yields G for the temperature class, 2for the sub-temperature class, and V for the luminosity class. It had been hoped that a fractional subset of the featuresprovided to the algorithm would be chosen as necessary to adequately model the classes, and this was the case for thetemperature class, where only 23 of the 28 features were needed. This, however, was not the case with the harder sub-temperature and luminosity classes, where the algorithm independently used all 28 features for its decision architecture.

(O,B,A,F,G,K,M)3965,3968,4500,4026,41214144,4340,3900,3934,39704046,4089,4097,4100,41024131,4227,4300,4317,4384

4388,4472,4481

Order=81

Temperature Class

(0,0.5,1.0,…,9.5)

4500,3970,3934,3968,41444046,4068,4073,4100,42674388,4481,4031,4300,44723900,3965,4089,4097,41314227,4340,4384,4026,4102

4121,4317,4471

Order=729Sub-Temperature

Class

(I,II,III,IV,V,Ve)Order=6561

Luminosity Class

Stellar Class(e.g. G2V)

4089,3968,3970,4026,40974317,4384,3934,4388,44814068,4102,4267,4500,40734340,4472,3900,3965,40464131,4144,4227,4300,4031

4121,4471,4100

Wavelengths(Angstroms)**

(O,B,A,F,G,K,M)3965,3968,4500,4026,41214144,4340,3900,3934,39704046,4089,4097,4100,41024131,4227,4300,4317,4384

4388,4472,4481

Order=81

Temperature Class

(0,0.5,1.0,…,9.5)

4500,3970,3934,3968,41444046,4068,4073,4100,42674388,4481,4031,4300,44723900,3965,4089,4097,41314227,4340,4384,4026,4102

4121,4317,4471

Order=729Sub-Temperature

Class

(I,II,III,IV,V,Ve)Order=6561

Luminosity Class

Stellar Class(e.g. G2V)

4089,3968,3970,4026,40974317,4384,3934,4388,44814068,4102,4267,4500,40734340,4472,3900,3965,40464131,4144,4227,4300,4031

4121,4471,4100

Wavelengths(Angstroms)**

Fig 5: Overall decision architecture. Wavelength features used for each class are shown on theleft listed in order of overall contribution along with the maximum polynomial order for thedecision architecture in the center and the possible outcomes on the right. The decision ofeach of these three architectures is fused together to give the final stellar class, which in thisexample case is G2V.

If we zoom into each of the ovals shown in Fig. 5, we find the decision architecture for each class decision. Thearchitecture for determining the temperature class is shown below in Fig. 6. In Fig. 6, each oval is represented by anequation architecture with the order of the equation required to make each decision shown. The decision architecture isread from left to right. The process of using this architecture to determine the temperature class would consist ofextracting the necessary features for each equation (shown to the left of each ellipse), feeding them into the equationmodel as inputs, and extracting the output. The output would then be compared to the decision boundary shown on theright of the oval. For example, for Equation 1, wavelength features 3968, 4144, 4500, 3965, 4102, 4131, 4472, 4121, and4100 (all in Å) would be input, and the output compared to the decision boundary (in this case, 0.5). If the output valueis less than 0.5, then the temperature class is O; else another equation must be run through for further determination(labeled Remaining in Fig. 6). This cascade approach guarantees convergence and learning of all examples uses in thelearning process, and also provides an excellent decision architecture roadmap for determining the class of an unknownstellar object.

This same process is repeated for both the sub-temperature and luminosity classes. In the case of the sub-temperatureclass determination, a larger (in terms of the number of ovals) architecture was required because there were morepossible outcomes. The results of this architecture construction are shown in Fig. 7, Fig. 8, and Fig. 9. These figures areconnected together so that when the architecture flows off the bottom of Fig. 7, it begins again at the top of Fig. 8, andFig. 8 flows right into Fig. 9 the same way. The luminosity class decision architecture is shown in Fig. 10 and is read thesame as Fig. 6. It should be noted that cases in which the order was high (81 or greater) were considered cases that weredifficult to learn. These were cases where the equations had to learn very subtle differences; ones that the othertechniques described earlier in this work would have failed on.

It should also be noted that in the luminosity class, one oval required an equation of order 6561. This case is one thatindicates that possibly more classes need to be created in the stellar catalog to accurately distinguish between stellarluminosities. In addition, 4 cases in the sub-temperature class determination and 2 cases in the luminosity classdetermination had to be siphoned, meaning that learned examples were removed and a separate equation generated fordetermining the rest of the class from the remaining classes. This also represents cases that were difficult to learn and arecases where the number of truth classes should be increased and the spectra reassigned.

<.5 then “B”

Else“R emaining”

Order=3

Equation 2393439684317

<.5 then “O”

Else“ Remaining”

Order=81

3968

Equation 1

41444500396541024131447241214100

<.5 then “A”


Order=9

Equation 3

40894500397040264340

<.4 then “F”

Else“Re maining”

Order=9

Equation 4

43004121438843844340

<.5 then “G”


Order=9

Equation 5

40463965448140973900 <.5 then “K”

Order=3

Equation 6

402639654227

Else “M”

It is difficult to distinguish Class “O”from the other classes because an 81 st

order polynomial was needed.

<.5 then “B”


Order=3

Equation 2393439684317

<.5 then “O”


Order=81

3968

Equation 1

41444500396541024131447241214100

<.5 then “A”


Order=9

Equation 3

40894500397040264340

<.4 then “F”


Order=9

Equation 4

43004121438843844340

<.5 then “G”


Order=9

Equation 5

40463965448140973900 <.5 then “K”

Order=3

Equation 6

402639654227

Else “M”

It is difficult to distinguish Class “O”from the other classes because an 81 st

order polynomial was needed.

Fig 6: Temperature class decision architecture. Decision architecture is read from left to right.It should be noted that the determination of Class O from the other classes modeled withEquation 1 was a difficult case because an 81st order polynomial was needed.

<.55 then “0”


Order=729Equation 1B

4388,4481,39683970,4317,45004267,4073,4384

<.50 then “0”


Order=243Equation 1A

3934,44724089,40314384,39704481,44714046,3968

<.5 then “0.5”


Order=9

Equation 2

4481,4102408941213900

<.5 then “1.0”


Order=9

Equation 3A

4026,4472

<.5 then “1.0”


Order=9

Equation 3B

4102,39344046,4068

408942674500 <.5 then “1.5”

Order=3

Equation 4

413143844388


<.5 then “2.0”

Order=81Equation 5A

4144,4340,4317,41024471,4384,39704089,4073,4481 Else Next

Page

4068,4144,41314340,4227

4121410240974068

It is difficult to distinguish Class “0” fromthe other classes because of the high orderof both of the polynomials (1A and 1B)needed.

Class had to be separated (siphoned) tolearn. This indicates a need for anothersub-classification.

Difficult caseindicated by high

order

<.55 then “0”



4388,4481,39683970,4317,45004267,4073,4384

<.50 then “0”



3934,44724089,40314384,39704481,44714046,3968

<.5 then “0.5”


Order=9

Equation 2

4481,4102408941213900

<.5 then “1.0”


Order=9

Equation 3A

4026,4472

<.5 then “1.0”


Order=9

Equation 3B

4102,39344046,4068

408942674500 <.5 then “1.5”

Order=3

Equation 4

413143844388


<.5 then “2.0”

Order=81Equation 5A

4144,4340,4317,41024471,4384,39704089,4073,4481 Else Next

Page

4068,4144,41314340,4227

4121410240974068

It is difficult to distinguish Class “0” fromthe other classes because of the high orderof both of the polynomials (1A and 1B)needed.

Class had to be separated (siphoned) tolearn. This indicates a need for anothersub-classification.


order

Fig 7: Sub-temperature class decision architecture A. Decision architecture is read from left toright. When leaving Equation 5A without a sub-temperature class decision, proceed toarchitecture B in Fig 8. It should be noted that Equations 1A, 1B, and 5A which determinesub-temperature classes 0 and 2 were difficult cases.

<.5 then “2.5”


Order=9

Equation 6396841214267

<.5 then “2.0”


Order=9

Equation 5B

3934,40683900409740314102

<.4 then “3.0”


Order=27

Equation 7A

4089,40974388,39004500,3968

4100<.5 then “3.0”


Order=9

Equation 7B

4026,4073

<.5 then “4.0”


Order=243Equation 9

3965,4300,43174384,3968,39344089,4340,40264472,4097,3970

4046,3900 <.5 then “4.5”

Order=3

Equation 10

396539684227


<.5 then “5.0”

Order=9

Equation 11

4068,4144,43174097,3934,4300

4267,4089

41314227

4097413140894046

Previous Page

Else NextPage


order

<.5 then “2.5”


Order=9

Equation 6396841214267

<.5 then “2.0”


Order=9

Equation 5B

3934,40683900409740314102

<.4 then “3.0”


Order=27

Equation 7A

4089,40974388,39004500,3968

4100<.5 then “3.0”


Order=9

Equation 7B

4026,4073

<.5 then “4.0”


Order=243Equation 9

3965,4300,43174384,3968,39344089,4340,40264472,4097,3970

4046,3900 <.5 then “4.5”

Order=3

Equation 10

396539684227


<.5 then “5.0”

Order=9

Equation 11

4068,4144,43174097,3934,4300

4267,4089

41314227

4097413140894046

Previous Page

Else NextPage


order

Fig 8: Sub-temperature class decision architecture B. Decision architecture is read from left toright. When leaving Equation 11 without a sub-temperature class decision, proceed toarchitecture C in Fig 9. It should be noted that Equation 9 determining sub-temperature class4 was a difficult case.

<.5 then “6.0”


Order=9

Equation 134384,4073

40263970

<.5 then “5.5”


Order=3

Equation 12

403140263970

<.5 then “6.5”


Order=3

Equation 14

447245003970

<.5 then “7.0”


Order=9

Equation 15

4073,4471

<.5 then “7.5”


Order=3

Equation 16

422740263968

<.5 then “8.0”

Order=3

Equation 17

408940973965


<.5 then “8.5”

Order=3

Equation 18

39683970

39684481

4300,4102396843404317

Previous Page

<.5 then “9.0”

Order=3

Equation 19

42674500

Else “9.5”Else

“ Remaining”

No Difficult Cases!

<.5 then “6.0”


Order=9

Equation 134384,4073

40263970

<.5 then “5.5”


Order=3

Equation 12

403140263970

<.5 then “6.5”


Order=3

Equation 14

447245003970

<.5 then “7.0”


Order=9

Equation 15

4073,4471

<.5 then “7.5”


Order=3

Equation 16

422740263968

<.5 then “8.0”

Order=3

Equation 17

408940973965


<.5 then “8.5”

Order=3

Equation 18

39683970

39684481

4300,4102396843404317

Previous Page

<.5 then “9.0”

Order=3

Equation 19

42674500

Else “9.5”Else

“ Remaining”

No Difficult Cases!

Fig 9: Sub-temperature class decision architecture C. Decision architecture is read from left toright. It should be noted that there were no difficult cases here.

<.60 then “I”



4073,39704481,43844500,3934

<.38 then “I”


Order=81Equation 1A

4267,44723934,45004144,43404068,39684102

<.5 then “II”


Order=27

Equation 2

4046,45004031,44713968,41444472,4300 <.3 then “III”



4500,4100,4267

<.5 then “III”


Order=81Equation 3B

4481,40684388,39704384,45004340,42273900,4097 <.5 then “IV”

Order=27

Equation 4

4046,4300,39344097,4144,41003970,4031


<.5 then “V”

Order=3

Equation 5

396841443900

Else “Ve”

4472,41214031,4317

3965,4089,39684388,4131,40264068,4144,42273934,4481

Very difficult classdecision!

Possibly new Sub-class needed

Possibly new Sub-classes needed!

<.60 then “I”



4073,39704481,43844500,3934

<.38 then “I”


Order=81Equation 1A

4267,44723934,45004144,43404068,39684102

<.5 then “II”


Order=27

Equation 2

4046,45004031,44713968,41444472,4300 <.3 then “III”



4500,4100,4267

<.5 then “III”


Order=81Equation 3B

4481,40684388,39704384,45004340,42273900,4097 <.5 then “IV”

Order=27

Equation 4

4046,4300,39344097,4144,41003970,4031


<.5 then “V”

Order=3

Equation 5

396841443900

Else “Ve”

4472,41214031,4317

3965,4089,39684388,4131,40264068,4144,42273934,4481

Very difficult classdecision!

Possibly new Sub-class needed

Possibly new Sub-classes needed!

Fig 10: Luminosity class decision architecture. Decision architecture is read from left to right.It should be noted that Equations 1A and 1B for class I, and Equations 3A and 3B for class IIIwere very difficult cases and denote the possibility that new sub-classes are needed.

3.2. Decision Map and Source CodeIn addition to constructing the decision architectures above, Data Modeling developed by Jaenisch also provides amethod for characterizing the interaction in the decision space between the dominant features. For example, usingwavelength 3968 Å and the wavelength feature that is the highest available contributor to the overall class determination(highest bar other than 3968 Å in the temperature, sub-temperature, and luminosity class bar charts in Fig. 11), decisionmaps such as the ones shown in Fig. 12 below can be constructed. These maps hold all other input features constantwhile varying the two dominant features. Fig. 12 shows three different slices through the decision space. To constructthe maps of Fig. 12, all input features except 3968 Å and 3965 Å were set to their minimum values (left map), middlerange values (middle map), and maximum values (right map). Features 3968 Å and 3965 Å were allowed to vary acrosstheir entire valid training ranges for each of the three maps. These maps display the complicated boundaries that existbetween decisions in a graphical manner in the temperature class, with each class displayed as a different color andlabeled in white. Each map represents a single slice through n-dimensional decision space that is fractal in nature. This isin stark contrast with the classical methods described earlier in this work that use simple mean and covariance models ofcircles and ellipses to represent the decision boundaries and overlap regions. Because of the fractal nature of the decisionspace, it is impossible to generate if-then rules that adequately characterize the complicated decision process and provide100% classification. Equivalent maps can also be constructed for the sub-temperature and luminosity classes, as well asfor the overall decision.

In the Jacobi data sets, only 161 stellar spectra were measured and repetition in overall stellar classes exists. In all, thereare more than 1000 possible classifications of stellar objects available as outcomes to the equation model, and theequation model was generated using all available data representing only about 10% of the possible classificationoutcomes. This equation model network was built with only a subset of the possible outcomes, and is allowed togeneralize to determine if a new object is of a class that as yet has not been measured.

Luminosity Class

0

20

40

60

80

100

120

1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 16 17 1 8 19 20 21 22 23 24 25 2 6 27 28

3900

3934

3965

3968

3970

4026

4031

4046

4068

4073

4089

4097

4100

4102

4121

4131

4144

4227

4267

4300

4317

4340

4384

4388

4471

4472

4481

4500

Luminosity Class

0

20

40

60

80

100

120

1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 16 17 1 8 19 20 21 22 23 24 25 2 6 27 28

3900

3934

3965

3968

3970

4026

4031

4046

4068

4073

4089

4097

4100

4102

4121

4131

4144

4227

4267

4300

4317

4340

4384

4388

4471

4472

4481

4500

3900

3934

3965

3968

3970

4026

4031

4046

4068

4073

4089

4097

4100

4102

4121

4131

4144

4227

4267

4300

4317

4340

4384

4388

4471

4472

4481

4500

Total Usage

0

50

100

150

200

250

300

350

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

3900

3934

3965

3968

3970

4026

4031

4046

4068

4073

4089

4097

4100

4102

4121

4131

4144

4227

4267

4300

4317

4340

4384

4388

4471

4472

4481

4500

Total Usage

0

50

100

150

200

250

300

350

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

3900

3934

3965

3968

3970

4026

4031

4046

4068

4073

4089

4097

4100

4102

4121

4131

4144

4227

4267

4300

4317

4340

4384

4388

4471

4472

4481

4500

3900

3934

3965

3968

3970

4026

4031

4046

4068

4073

4089

4097

4100

4102

4121

4131

4144

4227

4267

4300

4317

4340

4384

4388

4471

4472

4481

4500

Sub-Temperature Class

0

20

40

60

80

1 00

1 20

1 40

1 60

1 80

2 00

1 2 3 4 5 6 7 8 9 10 11 12 1 3 14 15 16 17 18 1 9 20 21 22 23 24 2 5 26 27 28

3900

3934

3965

3968

3970

4026

4031

4046

4068

4073

4089

4097

4100

4102

4121

4131

4144

4227

4267

4300

4317

4340

4384

4388

4471

4472

4481

4500

Sub-Temperature Class

0

20

40

60

80

1 00

1 20

1 40

1 60

1 80

2 00

1 2 3 4 5 6 7 8 9 10 11 12 1 3 14 15 16 17 18 1 9 20 21 22 23 24 2 5 26 27 28

3900

3934

3965

3968

3970

4026

4031

4046

4068

4073

4089

4097

4100

4102

4121

4131

4144

4227

4267

4300

4317

4340

4384

4388

4471

4472

4481

4500

3900

3934

3965

3968

3970

4026

4031

4046

4068

4073

4089

4097

4100

4102

4121

4131

4144

4227

4267

4300

4317

4340

4384

4388

4471

4472

4481

4500

Temperature Class

0

5

10

15

20

25

30

35

40

45

50

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

3900

3934

3965

3968

3970

4026

4046

4089

4097

4100

4102

4121

4131

4144

4227

4300

4317

4340

4384

4388

4472

4481

4500

Temperature Class

0

5

10

15

20

25

30

35

40

45

50

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

3900

3934

3965

3968

3970

4026

4046

4089

4097

4100

4102

4121

4131

4144

4227

4300

4317

4340

4384

4388

4472

4481

4500

3900

3934

3965

3968

3970

4026

4046

4089

4097

4100

4102

4121

4131

4144

4227

4300

4317

4340

4384

4388

4472

4481

4500

Fig 11: Summary of feature usage by class decision and total. Features that are used moreoften are more significant contributors to each decision. Wavelength 3968 Å was the mostsignificant contributor.

Fig 12: Decision maps for the temperature class. The graph on the left was constructed byholding the values of all wavelength features except 3968 Å (x-axis) and 3965 Å (y-axis) attheir minimum values and allowing the two features of interest to vary. The center graph usesthe mid range values of each of these features and the right graph the maximum values ofthese features. In each graph, 3968 Å (ranges from 0.0502 to 0.8651) and 3965 Å (rangesfrom 0.0567 to 0.9086) are varied from the min to the max of their ranges in 100 equal steps.It should be noted that each map represents a single slice through a n-dimensional decisionspace that is fractal in nature.

To determine the sensitivity of the equation architecture to sensor measurement errors, noised signatures were generated.These noised signatures were created by adding up to one standard deviation (one sigma) of random error to the originalJacobi spectra. Wavelength features were then extracted, and these features were constrained to be within the trainingboundaries of the Data Modeling equations. The Data Modeling classifications were then sorted into ascended orderindependently for temperature, sub-temperature, and luminosity classes. These results are shown in Fig. 13. Fig 13

shows each outcome as a separate row, with the first row across the top of each map representing the true Jacobi spectraand 100% correct classification. Individual stars can be traced down the map to determine how classification proceededas more noise was added to the measured spectra. It should be noted that for temperature class, as more noise was added,class O looked more like B, and B more like O, and K and M also both tended to look like O. For sub-temperatureclasses, all classes tended to look like 0 and 0.5 as more noise was added. And for luminosity, almost all stars lookedlike either I or III with increased noise. Also included below in Fig. 14 is an example of an equation generated with themethod developed by Jaenisch, in this case the equation that determines the O temperature class from all othertemperature classes.

Fig 13: Misclassification sensitivity to increasing noise. In the temperature class, class Otended to look like B, B like O, and both K and M like O with increased noise. For sub-temperature, almost all stars looked like 0 or 0.5, and for luminosity, most stars converged toeither I or III with increasing noise.

l1o3 = 2.1688 - 0.3521*t4500 + 2.038*t3965 -1.743*t4144l1o3 = l1o3 - 0.8056*t4500*t4500 + 0.1852*t3965*t3965l1o3 = l1o3 + 0.3188*t4144*t4144 + 0.323*t4500*t3965l1o3 = l1o3 + 0.6292*t4500*t4144 - 1.316*t3965*t4144l1o3 = l1o3 + 0.4574*t4500*t3965*t4144 - 0.1163*t4500*t4500*t4500l1o3 = l1o3 + 0.0236*t3965*t3965*t3965 + 0.0526*t4144*t4144*t4144l1o3 = l1o3 - 1.2608*t3965*t4500*t4500 + 0.5746*t4500*t3965*t3965l1o3 = l1o3 - 0.1599*t4500*t4144*t4144 + 0.4705*t4144*t4500*t4500l1o3 = l1o3 - 0.0637*t4144*t3965*t3965 - 0.0888*t3965*t4144*t4144l1o2 = 2.1356 + 1.7757*t3968 - 0.5989*t4472 - 1.3982*t4144l1o2 = l1o2 - 0.0689*t3968*t3968 - 0.8957*t4472*t4472l1o2 = l1o2 + 0.0356*t4144*t4144 - 0.1635*t3968*t4472l1o2 = l1o2 - 1.4428*t3968*t4144 + 1.172*t4472*t4144l1o2 = l1o2 + 1.1471*t3968*t4472*t4144 - 0.1052*t3968*t3968*t3968l1o2 = l1o2 + 0.1781*t4472*t4472*t4472 + 0.0473*t4144*t4144*t4144l1o2 = l1o2 + 0.2106*t4472*t3968*t3968 - 1.016*t3968*t4472*t4472l1o2 = l1o2 - 0.1649*t3968*t4144*t4144 - 0.2573*t4144*t3968*t3968l1o2 = l1o2 + 0.1521*t4144*t4472*t4472 - 0.1863*t4472*t4144*t4144l2o3 = 0.6373 + 0.2525*l1o3 - 0.1045*t4100 + 0.0243*l1o2l2o3 = l2o3 - 0.7416*l1o3*l1o3 + 0.0463*t4100*t4100l2o3 = l2o3 - 0.2711*l1o2*l1o2 - 0.0633*l1o3*t4100l2o3 = l2o3 + 0.5585*l1o3*l1o2 + 0.1682*t4100*l1o2l2o3 = l2o3 - 2.006*l1o3*t4100*l1o2 + 0.085*l1o3*l1o3*l1o3l2o3 = l2o3 + 0.0023*t4100*t4100*t4100 + 0.3119*l1o2*l1o2*l1o2l2o3 = l2o3 + 1.0911*t4100*l1o3*l1o3 + 0.0022*l1o3*t4100*t4100l2o3 = l2o3 - 0.4909*l1o3*l1o2*l1o2 + 0.0063*l1o2*l1o3*l1o3l2o3 = l2o3 - 0.145*l1o2*t4100*t4100 + 0.7645*t4100*l1o2*l1o2l3o2 = 0.1866 + 1.1442*l2o3 + 0.1711*t3968 - 0.2543*t4121l3o2 = l3o2 - 0.6571*l2o3*l2o3 - 0.0781*t3968*t3968l3o2 = l3o2 + 0.3587*t4121*t4121 - 0.571*l2o3*t3968l3o2 = l3o2 - 0.0396*l2o3*t4121 - 0.896*t3968*t4121l3o2 = l3o2 + 1.7283*l2o3*t3968*t4121 - 0.0585*l2o3*l2o3*l2o3l3o2 = l3o2 + 0.0728*t3968*t3968*t3968 - 0.0041*t4121*t4121*t4121l3o2 = l3o2 + 0.0751*t3968*l2o3*l2o3 + 0.0006*l2o3*t3968*t3968l3o2 = l3o2 - 0.6821*l2o3*t4121*t4121 + 0.7264*t4121*l2o3*l2o3l3o2 = l3o2 + 0.1007*t4121*t3968*t3968 - 0.0401*t3968*t4121*t4121l2o2 = 0.6221 + 0.7539*l1o3 - 0.0836*t4131 - 0.4128*l1o2l2o2 = l2o2 - 0.5396*l1o3*l1o3 - 0.1235*t4131*t4131l2o2 = l2o2 + 0.0406*l1o2*l1o2 - 0.7962*l1o3*t4131

l2o2 = l2o2 + 0.0073*l1o3*l1o2 + 0.986*t4131*l1o2l2o2 = l2o2 - 1.9057*l1o3*t4131*l1o2 - 0.2229*l1o3*l1o3*l1o3l2o2 = l2o2 - 0.0107*t4131*t4131*t4131 + 0.3671*l1o2*l1o2*l1o2l2o2 = l2o2 + 1.4539*t4131*l1o3*l1o3 + 0.0094*l1o3*t4131*t4131l2o2 = l2o2 - 0.9617*l1o3*l1o2*l1o2 + 0.7312*l1o2*l1o3*l1o3l2o2 = l2o2 + 0.1344*l1o2*t4131*t4131 + 0.3263*t4131*l1o2*l1o2l2o1 = 0.4775 + 0.3633*l1o3 + 0.0247*t4102 + 0.1438*l1o2l2o1 = l2o1 - 0.456*l1o3*l1o3 + 0.0627*t4102*t4102l2o1 = l2o1 - 0.4008*l1o2*l1o2 - 0.1214*l1o3*t4102l2o1 = l2o1 + 0.4992*l1o3*l1o2 + 0.0815*t4102*l1o2l2o1 = l2o1 - 0.9426*l1o3*t4102*l1o2 + 0.085*l1o3*l1o3*l1o3l2o1 = l2o1 + 0.0094*t4102*t4102*t4102 + 0.0038*l1o2*l1o2*l1o2l2o1 = l2o1 + 0.4241*t4102*l1o3*l1o3 - 0.0709*l1o3*t4102*t4102l2o1 = l2o1 + 0.0891*l1o3*l1o2*l1o2 - 0.2419*l1o2*l1o3*l1o3l2o1 = l2o1 - 0.05*l1o2*t4102*t4102 + 0.3658*t4102*l1o2*l1o2l3o3 = 0.6049 + 0.8283*l2o3 + 0.5852*l2o2 - 1.0118*l2o1l3o3 = l3o3 - 0.0688*l2o3*l2o3 - 0.239*l2o2*l2o2l3o3 = l3o3 - 0.3041*l2o1*l2o1 - 1.9498*l2o3*l2o2l3o3 = l3o3 + 0.7738*l2o3*l2o1 + 1.0837*l2o2*l2o1l3o3 = l3o3 – 0.0248*l2o3*l2o2*l2o1 - 0.4811*l2o3*l2o3*l2o3l3o3 = l3o3 - 1.0575*l2o2*l2o2*l2o2 + 1.4796*l2o1*l2o1*l2o1l3o3 = l3o3 + 1.8239*l2o2*l2o3*l2o3 - 2.1622*l2o3*l2o2*l2o2l3o3 = l3o3 - 0.5003*l2o3*l2o1*l2o1 + 0.0719*l2o1*l2o3*l2o3l3o3 = l3o3 + 4.5777*l2o1*l2o2*l2o2 - 3.8468*l2o2*l2o1*l2o1l4o3 = 0.4485 + 0.0845*l3o2 - 0.196*t4500 + 0.562*l3o3l4o3 = l4o3 + 0.0271*l3o2*l3o2 - 0.1464*t4500*t4500l4o3 = l4o3 - 1.0001*l3o3*l3o3 - 0.9481*l3o2*t4500l4o3 = l4o3 + 0.2703*l3o2*l3o3 + 1.3307*t4500*l3o3l4o3 = l4o3 + 0.0357*l3o2*t4500*l3o3 + 0.3315*l3o2*l3o2*l3o2l4o3 = l4o3 + 0.0062*t4500*t4500*t4500 + 0.2246*l3o3*l3o3*l3o3l4o3 = l4o3 + 0.9*t4500*l3o2*l3o2 - 0.1734)*l3o2*t4500*t4500l4o3 = l4o3 + 0.3387*l3o2*l3o3*l3o3 - 0.9927*l3o3*l3o2*l3o2l4o3 = l4o3 + 0.4172*l3o3*t4500*t4500 - 1.0498*t4500*l3o3*l3o3Class = l4o3*0.1665 + 0.8056if Class < 0.1 then Class = 0.1if Class > 0.9 then Class = 0.9

l1o3 = 2.1688 - 0.3521*t4500 + 2.038*t3965 -1.743*t4144l1o3 = l1o3 - 0.8056*t4500*t4500 + 0.1852*t3965*t3965l1o3 = l1o3 + 0.3188*t4144*t4144 + 0.323*t4500*t3965l1o3 = l1o3 + 0.6292*t4500*t4144 - 1.316*t3965*t4144l1o3 = l1o3 + 0.4574*t4500*t3965*t4144 - 0.1163*t4500*t4500*t4500l1o3 = l1o3 + 0.0236*t3965*t3965*t3965 + 0.0526*t4144*t4144*t4144l1o3 = l1o3 - 1.2608*t3965*t4500*t4500 + 0.5746*t4500*t3965*t3965l1o3 = l1o3 - 0.1599*t4500*t4144*t4144 + 0.4705*t4144*t4500*t4500l1o3 = l1o3 - 0.0637*t4144*t3965*t3965 - 0.0888*t3965*t4144*t4144l1o2 = 2.1356 + 1.7757*t3968 - 0.5989*t4472 - 1.3982*t4144l1o2 = l1o2 - 0.0689*t3968*t3968 - 0.8957*t4472*t4472l1o2 = l1o2 + 0.0356*t4144*t4144 - 0.1635*t3968*t4472l1o2 = l1o2 - 1.4428*t3968*t4144 + 1.172*t4472*t4144l1o2 = l1o2 + 1.1471*t3968*t4472*t4144 - 0.1052*t3968*t3968*t3968l1o2 = l1o2 + 0.1781*t4472*t4472*t4472 + 0.0473*t4144*t4144*t4144l1o2 = l1o2 + 0.2106*t4472*t3968*t3968 - 1.016*t3968*t4472*t4472l1o2 = l1o2 - 0.1649*t3968*t4144*t4144 - 0.2573*t4144*t3968*t3968l1o2 = l1o2 + 0.1521*t4144*t4472*t4472 - 0.1863*t4472*t4144*t4144l2o3 = 0.6373 + 0.2525*l1o3 - 0.1045*t4100 + 0.0243*l1o2l2o3 = l2o3 - 0.7416*l1o3*l1o3 + 0.0463*t4100*t4100l2o3 = l2o3 - 0.2711*l1o2*l1o2 - 0.0633*l1o3*t4100l2o3 = l2o3 + 0.5585*l1o3*l1o2 + 0.1682*t4100*l1o2l2o3 = l2o3 - 2.006*l1o3*t4100*l1o2 + 0.085*l1o3*l1o3*l1o3l2o3 = l2o3 + 0.0023*t4100*t4100*t4100 + 0.3119*l1o2*l1o2*l1o2l2o3 = l2o3 + 1.0911*t4100*l1o3*l1o3 + 0.0022*l1o3*t4100*t4100l2o3 = l2o3 - 0.4909*l1o3*l1o2*l1o2 + 0.0063*l1o2*l1o3*l1o3l2o3 = l2o3 - 0.145*l1o2*t4100*t4100 + 0.7645*t4100*l1o2*l1o2l3o2 = 0.1866 + 1.1442*l2o3 + 0.1711*t3968 - 0.2543*t4121l3o2 = l3o2 - 0.6571*l2o3*l2o3 - 0.0781*t3968*t3968l3o2 = l3o2 + 0.3587*t4121*t4121 - 0.571*l2o3*t3968l3o2 = l3o2 - 0.0396*l2o3*t4121 - 0.896*t3968*t4121l3o2 = l3o2 + 1.7283*l2o3*t3968*t4121 - 0.0585*l2o3*l2o3*l2o3l3o2 = l3o2 + 0.0728*t3968*t3968*t3968 - 0.0041*t4121*t4121*t4121l3o2 = l3o2 + 0.0751*t3968*l2o3*l2o3 + 0.0006*l2o3*t3968*t3968l3o2 = l3o2 - 0.6821*l2o3*t4121*t4121 + 0.7264*t4121*l2o3*l2o3l3o2 = l3o2 + 0.1007*t4121*t3968*t3968 - 0.0401*t3968*t4121*t4121l2o2 = 0.6221 + 0.7539*l1o3 - 0.0836*t4131 - 0.4128*l1o2l2o2 = l2o2 - 0.5396*l1o3*l1o3 - 0.1235*t4131*t4131l2o2 = l2o2 + 0.0406*l1o2*l1o2 - 0.7962*l1o3*t4131

l2o2 = l2o2 + 0.0073*l1o3*l1o2 + 0.986*t4131*l1o2l2o2 = l2o2 - 1.9057*l1o3*t4131*l1o2 - 0.2229*l1o3*l1o3*l1o3l2o2 = l2o2 - 0.0107*t4131*t4131*t4131 + 0.3671*l1o2*l1o2*l1o2l2o2 = l2o2 + 1.4539*t4131*l1o3*l1o3 + 0.0094*l1o3*t4131*t4131l2o2 = l2o2 - 0.9617*l1o3*l1o2*l1o2 + 0.7312*l1o2*l1o3*l1o3l2o2 = l2o2 + 0.1344*l1o2*t4131*t4131 + 0.3263*t4131*l1o2*l1o2l2o1 = 0.4775 + 0.3633*l1o3 + 0.0247*t4102 + 0.1438*l1o2l2o1 = l2o1 - 0.456*l1o3*l1o3 + 0.0627*t4102*t4102l2o1 = l2o1 - 0.4008*l1o2*l1o2 - 0.1214*l1o3*t4102l2o1 = l2o1 + 0.4992*l1o3*l1o2 + 0.0815*t4102*l1o2l2o1 = l2o1 - 0.9426*l1o3*t4102*l1o2 + 0.085*l1o3*l1o3*l1o3l2o1 = l2o1 + 0.0094*t4102*t4102*t4102 + 0.0038*l1o2*l1o2*l1o2l2o1 = l2o1 + 0.4241*t4102*l1o3*l1o3 - 0.0709*l1o3*t4102*t4102l2o1 = l2o1 + 0.0891*l1o3*l1o2*l1o2 - 0.2419*l1o2*l1o3*l1o3l2o1 = l2o1 - 0.05*l1o2*t4102*t4102 + 0.3658*t4102*l1o2*l1o2l3o3 = 0.6049 + 0.8283*l2o3 + 0.5852*l2o2 - 1.0118*l2o1l3o3 = l3o3 - 0.0688*l2o3*l2o3 - 0.239*l2o2*l2o2l3o3 = l3o3 - 0.3041*l2o1*l2o1 - 1.9498*l2o3*l2o2l3o3 = l3o3 + 0.7738*l2o3*l2o1 + 1.0837*l2o2*l2o1l3o3 = l3o3 – 0.0248*l2o3*l2o2*l2o1 - 0.4811*l2o3*l2o3*l2o3l3o3 = l3o3 - 1.0575*l2o2*l2o2*l2o2 + 1.4796*l2o1*l2o1*l2o1l3o3 = l3o3 + 1.8239*l2o2*l2o3*l2o3 - 2.1622*l2o3*l2o2*l2o2l3o3 = l3o3 - 0.5003*l2o3*l2o1*l2o1 + 0.0719*l2o1*l2o3*l2o3l3o3 = l3o3 + 4.5777*l2o1*l2o2*l2o2 - 3.8468*l2o2*l2o1*l2o1l4o3 = 0.4485 + 0.0845*l3o2 - 0.196*t4500 + 0.562*l3o3l4o3 = l4o3 + 0.0271*l3o2*l3o2 - 0.1464*t4500*t4500l4o3 = l4o3 - 1.0001*l3o3*l3o3 - 0.9481*l3o2*t4500l4o3 = l4o3 + 0.2703*l3o2*l3o3 + 1.3307*t4500*l3o3l4o3 = l4o3 + 0.0357*l3o2*t4500*l3o3 + 0.3315*l3o2*l3o2*l3o2l4o3 = l4o3 + 0.0062*t4500*t4500*t4500 + 0.2246*l3o3*l3o3*l3o3l4o3 = l4o3 + 0.9*t4500*l3o2*l3o2 - 0.1734)*l3o2*t4500*t4500l4o3 = l4o3 + 0.3387*l3o2*l3o3*l3o3 - 0.9927*l3o3*l3o2*l3o2l4o3 = l4o3 + 0.4172*l3o3*t4500*t4500 - 1.0498*t4500*l3o3*l3o3Class = l4o3*0.1665 + 0.8056if Class < 0.1 then Class = 0.1if Class > 0.9 then Class = 0.9

Fig 14: Example source code from Data Modeling to determine O temperature class fromother temperature classes.

4. FUTURE WORK

Although only the full clean portion of the Jacoby spectra was analyzed in this work, this technique should also beamenable to processing across the full spectra and provide 100% classification using only a subset of the wavelengthfeatures available to the process. The authors looked at using the traditional UBV regions for any performanceimprovement, but there was no discernable improvement using these. There may be wavelength regions in these spectrathat lie outside of the traditional UBV regions that would provide performance enhancement, and these should be lookedat further. The complexity of the classifier could be reduced if the number of wavelengths required for classificationwere reduced. This may be possible by using wavelength feature ratios that combine multiple wavelengths togetherinstead of using the raw spectra numbers directly.

5. CONCLUSIONS

Until now, no classification method applied to stellar spectra has given 100% classification in a closed form solution.Data Modeling has given both a closed form solution for stellar classification (albeit a high order solution in some cases)and has given 100% correct classification on the temperature class, sub-temperature class, and luminosity for the 161Jacoby spectra. This method independently found that all 28 features that were available were needed for classification,and that wavelength 3968 Å (Ca II (H line), H line (blend)) was the most significant wavelength contributor toclassification. In the temperature class, class O was the hardest to distinguish from the other classes and required an 81st

order polynomial to do so. In the sub-temperature class, 0 was the hardest to classify and required a 729th orderpolynomial, and in the luminosity class, III was the hardest and required a 6561st order polynomial. In addition,siphoning was required on 4 classes in the sub-temperature cases and on 2 classes in the luminosity, which indicates thatadditional stellar sub-classes are needed in stellar catalogs. This is readily apparent in the luminosity class, where both a6561st polynomial and siphoning were required to determine class III from IV, V, and Ve. Classes II, IV, and Ve in theluminosity class had fewer samples than the other luminosity classes in the Jacobi spectra, which unfairly constrains thefeature dynamic ranges for those classes. More stellar spectra are needed for these classes. The classical UBVwavebands were tried by the authors and did not yield a less complicated solution than the one presented here. And useof this Data Model equation to determine star classification from discrete wavelength measurements along with theobservation of its visual magnitude makes available to the astronomer a direct estimate of the distance to the star.

ACKNOWLEDGMENTS

This stellar spectra classification project was originally begun to partially fulfill the requirements for the completion of aMasters of Science in Astronomy at the University of Western Sydney. The authors would like to thank the following fortheir contributions to this effort: Professor A. Hons (email:[email protected]) and Professor P. Jones(email:mailto:[email protected]), University of Western Sydney (UWS), G. Snyder, Contemporary LaboratoryExperiments in Astronomy (CLEA) (website:http://www.gettysburg.edu/academics/physics/clea/CLEAhome.html)(email:[email protected]), and C. Case, C. Songy, and J. Handley, Sparta, Inc. for their support and encouragement.

REFERENCES

1) Project CLEA Home Page,Contemporary Laboratory Experiences in Astronomy, University of Gettysburg,Gettysburg, PA, http://www.gettysburg.edu/academics/physics/clea/CLEAhome.html, 25 October 20012) Jacoby, G.H., Hunter, D.A. and C. A. Christian. “A Library of Stellar Spectra”,Astrophysical Journal SupplementSeries, 56:257-281, October 1984.3) Fukunaga, K. “Introduction to Statistical Pattern Recognition”, 2nd ed., Boston, MA: Academic Press, c1990.4) Ivakhnenko, G.,GMDH: Group Method of Data Handling for complex systems modeling and forecasting, CyberneticCenter, Ukraina, http://www.inf.kiev.ua/GMDH-home, December 1998.5) Madala, H.R. and A.G. Ivakhnenko.Inductive Learning Algorithms for Complex Systems Modeling, Boca Raton, FL:CRC Press, c1994.6) Jaenisch, H.M. “Neural Equation Stellar Classification”, University of Western Sydney, Master’s Thesis, Sydney,Australia, October 2001.

Enabling Unattended Data Logging and Publication by Data Model

Change Detection and Environmental Awareness

Holger M. Jaenisch*

James Cook University, Townsville QLD 4811, Australia

ABSTRACT

This paper presents a novel self-initializing algorithm using Change Detection to achieve self-awareness of unusual

conditions without a prior modeling assumptions. Deviations from baseline nominal conditions yield a tip-off and the

variation off baseline indicates a novelty to be logged for publication. Incremental processing of the data log enables

common transients to be ignored and viewed as nominal. In this framework, only second pass novelties invoke enough

interest for publication. The mathematical methods for enabling this exploit both classical control theory transfer

functions to model the environment and O(3n) Volterra series type polynomials as an innovative change detection

method without explicit modeling.

Keywords: Formal V&V Analysis, Data Modeling, Agent, UAV, UMV, Deep Space Probes, Transfer Function Modeling,

Automatic Proofing, Automatic Differential Equation Derivation, Ezekiel

1. INTRODUCTION

This is a new and novel self-initializing algorithm1 based on classical methods (both control and Fourier theory) applied

in a completely unique way. It can be deployed to watch any type of scene or environment and will determine what is

normal or out of the ordinary simply from monitoring small regions of the entire scene. Whenever something new or

unusual occurs, it is able to record the scene and concurrently update its internal model of what it has seen and

experienced. The algorithm continues to collect samples of novel scenes representing unusual changes as they are

encountered. Once filled to capacity, it automatically offloads what is learned and maintains a memory trace for

continuity. This paper presents the details of this algorithm, including its application to novel imagery from Mars,

Mercury, Asteroid Ida, Moon Titan, and Earth.

2. EZEKIEL CONCEPT

Environmental scene awareness is enabled with a novel Artificially Intelligent Algorithm called Ezekiel1. Notionally,

various environmental math models are stored in slide carousel fashion that are retrievable by indexing on similarity of

present conditions to ones encountered in the past. The carousel of environment models coupled with carousels of

encountered variations is notionally illustrated in Figure 1.

Fig. 1. Illustrations of the Ezekiel concept.

Ezekiel captures linear models of sensor input and corresponding interpreted output as a classical control-theory transfer

function (TF) model. Using the notional concept of a carousel slide projector, each new linear model created by Ezekiel

is stored as a “slide” in the carousel. As the agent scans its surroundings, it compares mathematically the new sensor

input to the linear model TF slides it has in memory (slide carousel indexed on numerical similarity).



If the new sensor input does not numerically resemble any of the existing TF slides, the two closest matches are used to

derive a virtual slide for comparison. If this fails to yield a match, the new data is added as a new slide and the collection

re-indexed based on mathematical similarity. However, if similarity was discovered immediately upon inspection, then

the second carousel of known variations is activated and equivalence checked. If a match occurs the sample is ignored. If

no match occurs, the variation carousels associated with the two closest TF matches are equivalence checked; if still no

match; the new data is added as a new known-variant and placed in the carousel. The variation carousel is then updated

and re-indexed.

In addition to the baseline transfer function model, each TF slide has another entire carousel of variations associated with

it (wheel-within-wheel). In this fashion, Ezekiel continues to scan its environment building a representational model and

belief system of its surroundings. When its slide capacity is reached, it off loads the data for telemetry and saves a small

percentage of the slide data to provide continuity with previous experience. Ezekiel is now able to store new novelties as

they occur. This empowers Ezekiel to construct linear system TF models in a fully adaptive piece wise fashion, enabling

highly nonlinear regions to be automatically captured with higher density linear models and linear regions to be

represented adaptively with sparse models.

3. THEORY

Data Modeling1-2 is a unique term coined to describe the process of deriving mathematical equations as models from

simulated or measured data. What follows is a specific embodiment that achieves Data Modeling using classical control

and Fourier theory. The final differential equations generated by the process are generically termed Data Models.

Classic transfer functions3-4 are defined using the raw input and output data from a plant (in this case scene of interest).

The z transform of the input and output data is defined as

n

n zazazazX −−− ++++= K2

2

1

11)( (1) n

n zbzbzbbzY −−− ++++= K2

2

1

10)(

In this form, the classic transfer function is formed by

n

n

n

n

zazaza

zbzbzbb

zX

zYzH −−−

−−−

++++++++

==K

K2

2

1

1

2

2

1

10

1)(

)()( (2)

where X(z) defines the characteristic polynomial of the system, and whose roots (poles) define the sensitivity of the

system. This represents the TF for a single input variable with multiple realizations resulting in a single output variable

of multiple associated variations. For multiple input/output variable cases, the transfer function is

[ ]

=

)()()(

)()()(

)()()(

)(

21

22221

11211

zHzHzH

zHzHzH

zHzHzH

zH

KKKK

K

K

L

MOLM

L

L

(3)

where each resultant output is a linear combination of all inputs. The transfer function from input j to output i is Hij(z).

This matrix captures the influence of each individual input element to all outputs.

This process can be greatly simplified by adopting a unique and novel interpretation or perspective of definitions. By

taking a vector of multiple simultaneous inputs and an equal length vector of multiple simultaneous outputs (MIMO) and

transposing them (transposing column vectors to rows), the result is simplified to an equivalent single input/output

system (SISO) where the multiple realizations of the single input value are actually the single simultaneous input values.

This variable transfiguration (transpose followed by new interpretation of resulting vector) is represented by

−

=

=

0.42

63.19

2.42

)(

,

,2

,1

MM

iN

i

i

x

x

x

zX [ ] [ ]0.4263.192.42)( )(,1)1(,1,1 −== ++ℑ LL Niii xxxzX (4)

N inputs, single realization Single virtual input, N realizations

Once the input vector is transfigured, the process in (4) is repeated for the output vector. Now a virtual single input

single output transfer function is defined following the form in (2) above.

A MIMO system has now been transfigured into a virtual SISO form that is readily converted into differential equation

form as follows

])[(])1[()][(])[(])2[(])1[(][ 1021 TnixbTixbiTxbTniyaTiyaTiyaiTy nn −++−++−++−+−= LL (5)

This form represents a simple differential equation model derived from a single input/output exemplar, where the powers

of the z transform terms are converted into lags of the historical data (both input and previous output).

If the number of I/O terms is large, this form becomes unwieldy because of the need to keep track of n inputs and n-1

previous outputs. And if this differential equation model is stored and used as a transfer function, the required precision

to make the calculations rises rapidly with as the order of the denominator increases, which is directly dependent on the

n-1 previous outputs here.

Both of these problems are mitigated by recasting the TF and ultimately the differential equation form into Fourier

frequency space. This is easily accomplished using the classic z transform defined for z = eiω , which is the same as the

discrete Fourier transform (DFT) defined by

∑=k

j kn

ji

nC )

2exp(

1 π (6)

Therefore, the TF itself can be easily represented in frequency space using the fast Fourier transform (FFT) to determine

the Fourier series of the input and output sequences. Classically, these two Fourier series would be combined into a TF

given by

∑=

+=n

j

jjjj xBxAxy1

sincos)( ωω )(

)(

InputFFT

OutputFFTTF = (7)

The Fourier series is a lengthy representation that includes many alias terms due to frequency folding. A simplified form

identifies the dominant frequency components or eigenmodes, labeling them as lambda terms after zeroing all other

frequency locations.


BAfxyN

ij

j

j

N

ij

jk ==

+=≅ ∑ )sin()cos(),,,()(22

21

ππ λλλλλ K

xBxA jjjjj ωωλ sincos += (8)

The Ezekiel Transfer Function (ETF) is now formed from these lambda terms

=))((

))(()(

HeptorFFT

sPixelValueFFTETF

λλλ (9)

where an individual λETF is easily converted into differential form as

( )

( ) 0

101

101

2001.

001.

1

1

1

1

12

12 =

+

−−

−−−

−−−

− ∑−

=−

−

−

−

+

+

nslides

iff

ff

ii

ii

ii

ii

i

i

ff

yy

ff

yy

ff

yy

df

dA (10)

where y is the integral of the real part of the ETF given by ΣA. Likewise, the expression in (10) must also be repeated for

the imaginary part of the ETF.

Since each ETF is a linear model of a vastly nonlinear scene, a series of ETFs are saved in piecewise linear fashion to

accommodate such behavior. For this reason, the subscript i is added to (9)

i

iHeptorFFT

sPixelValueFFTETF

=))((

))(()(

λλλ (11)

The collection of ETF’s comprise a carousel, with each ETF occupying a slot. The matrices of remaining values after

eigenfunction modeling results in the following (i x k) matrices, where i is the number of ETFs and k is the number of

non-zero rows remaining in the U (real) or V (imaginary) matrix. A notional U matrix of real values is given by

=

0000

000

00

000

000

4,5

4,43,4

5,33,32,3

3,21,2

3,12,1

λλλ

λλλλλλλ

U (12)

where the real portion of the eigenmodels of the ith ETF is located in the ith column of the U matrix. A second matrix V

like this one is also generated for the imaginary terms.

Next a differential equation model is derived for each λ order across the λETFs by transposing the A matrix and B

matrix and repeating the differential equation construction process in (10) above. It should be noted that the λETFs are in

an order of ranked similarity which will be explained in the next section. This results in an analogous derivation of the

Volterra type functional differential equation (a differential equation whose variables are themselves either differential or

integral equations). In this case, we call this unique form the Ezekiel Integro-Differential equation given as

0)()()(11 0

12

=ℵℵ−ℵ

−

∫∫ ∫ℵ

−

ℵ

−

−nn

k

d ETFdfdf fdf

d

f

λλλ (13)

where is the similarity FOM defined as the unique characterization of each ETF and described in detail in the next

section. The Appendix contains a numerical implementation of evaluating (13).

ℵ

4. EZEKIEL IMPLEMENTATION

The Ezekiel transfer function is defined using as Output pixel values and as Input a vector of 7 statistics (coined Heptor)

describing the pixel neighborhood. These 7 statistics are the standard deviation (m2), skewness (m3), kurtosis (m4), m6,

m8, DH (novel fractal statistic comparing shortest path to actual path), and DJ (novel fractal statistic comparing range of

data to its standard deviation). This process is depicted in Figure 2.

Fig. 2. Heptor Statistics Calculation. A 7 x 7 neighborhood is pulled out of the overall image (top) and is shown on the bottom left.

This neighborhood is enlarged in the second bottom fame, and then shown as the actual image values in the third bottom frame.

Statistics are calculated on these 49 numbers, resulting in the Heptor stats shown on the bottom right.

Since each ETF is a linear model of a vastly nonlinear scene, a series of ETFs are saved in piecewise linear fashion to

accommodate such behavior. The collection of ETF’s comprise a carousel, with each ETF occupying a slot.

Because each ETF is only valid over a short range of values, it is necessary to identify for a new input which ETF is the

closest matching to the current input. Since the current input will always fall in between two (2) existing ETFs in the

carousel, a new virtual ETF is formed by weighted (proximity to new input) averaging of the eigenmodes in the

neighboring ETFs. This is done with (13).

ETF identification uses a novel type of Formal V&V (symbolic V&V) which treats the existing ETFs as “gold standard”

software models and new incoming data as test software. A mathematical proof of congruence between the two is

accomplished using a novel similarity FOM5. This is calculated using each input component of the ETF, enabling ETF

sorting by structural similarity.

))(2cos()( BABASignFOMSimilarity oo π==ℵ (14)

The similarity FOM (ℵ ) characterizes the input Heptor portion of each ETF. Conversely, the variation carousel is

populated by raw output pixel values stored in the order the neighborhoods occur. A comparison is now done between

the pixel neighborhood under consideration and each variant by calculating Equivalence (correlation of the magnitude of

the eigenmodes), which is defined for k variants by

( )( )( ) ( )∑∑

∑

==

=

−−

−−=

N

jk

kj

N

j

j

N

jk

kjj

k

VariantVariantCurrentCurrent

VariantVariantCurrentCurrent

eEquivalenc

1

2

,

1

2

1

,

100 (15)

Current and Variant are the magnitudes of the TF (PSD) models for the simulation and measured data given by

22

jjj BAurrentC += (16)

and Current and Variant in (15) are the respective means of the result of (16). A threshold for determining a match to

an ETF is automatically determined using

∑−

=+ −=

1

1

12

1 NETFSlides

m

mm FOMSimilarityFOMSimilarityNETFSlides

SimThresh (17)

which is ½ the average difference between the similarity FOM for neighboring slides in the ETF. Therefore, this is a

dynamic and adaptive threshold that changes according to ETFs in memory. For the variants, the Equivalency threshold

is also dynamic and adaptive, but is calculated as the running mean of the all Equivalency tests performed in the current

analysis, using the Equivalency relationship in (15). The Equivalency threshold is given as

∑−

=

=1

1

1 sNVariation

m

meEquivalencsNVariation

hEquivThres (18)

Carousel update is based on these adaptive thresholds. The current exemplar is compared to all members of the carousel

to find the most similar. Next, if the similarity is within the threshold, the exemplar is compared to the variants of this

ETF for equivalence. If equivalent, it is discarded, otherwise it is added as a new variant.

However, if the exemplar falls outside the threshold for similarity to the closest matching ETF, the process is more

involved. A decision to add this exemplar as an ETF is made. First, the variants of 4 ETFs are tested. These are the most

similar ETF, the ETF one index ahead of it, the ETF one index behind it, and the weighted average ETF formed from the

two ETFs the exemplar falls between. If the exemplar proves equivalent, then discard; otherwise, add a new ETF and its

associated variant.

Given the current stored corpus of experiential knowledge in Ezekiel, a prediction is made mathematically of what other

neighborhood sites should exist and what they would look like described by their input Heptor. These sites are called

relics and are significant because, if found, they are mathematically predicted to exist based on prior experience.

Prediction is accomplished using the stored ETF models to reconstruct an ensemble of Heptors for the variants

associated and stored with each. This is accomplished by inverting the ETF model and multiplying the resultant by the

pixel neighborhood values of the variant, most of which were associated and stored with the ETF when the exemplar was

similar enough to the ETF. The process of generating these predicted Heptors is given by

))(1

( VariantFFTETF

InvFFTeptorPredictedH = (19)

These Heptors are assigned a categorical value of ½ meaning relic. A change detector polynomial is derived using these

exemplars which enables distant sampled neighborhoods to be quickly compared to the mathematical predictions.

Change detection is achieved by creating a multi-variable polynomial Data Model that exhibits hypersensitivity to small

changes of input values. Tuning makes the model sensitive to feature values and combinations not encountered in

derivation. Threshold detecting outliers is set using the polynomial output from training data. The upper and lower

boundaries are assigned to be one-half of a standard deviation above and below the mean value of the polynomial output

from the training data.

The simulation created for this paper uses two different novel algorithms called change detectors. The first is a change

detector that uses the Heptor for the current pixel neighborhood as input and tips-off (outside the nominal boundary

around ½) whenever a new novel pixel neighborhood is encountered (unlike anything experienced before). This change

detector serves as a first level bulk filter to determine if a new pixel neighborhood should be processed or discarded.

The second is a novel “relic” detector (a relic is a mathematically predicted environmental scene that is expected to exist

based on current experience) described above, for which the definitions of nominal and tip-off are reversed. It looks for

data similar to training data. For the relic detector, tip-off (relic detection) occurs in the tight boundary around ½.

Whenever values in this region are found, the existence of a relic is flagged. Since both change detectors are self-

initializing and adaptive while providing goals with checks and balances; a novel form of machine situational

awareness is achieved.

To give the Ezekiel agent a navigational aide, a sense of direction is provided. The next heading for Ezekiel is chosen by

taking an extended range sensor reading in each of the 4 cardinal directions from the current heading. In the simulation

for this paper, this was simplified to be the fixed global cardinal points and heading independent. Because of the coarse

resolution of this simulation, very little difference in method is resolved; however, in practice the difference is

appreciable because goal following is enhanced using the local orientation. These headings are processed through the

Guide TF. The output value from each of the 4 cardinal directions is a vector whose resultant is normalized to construct a

unit pointing vector. The length of the final vector defining the new location is based on the pixel neighborhood size and

the distance away from the current pixel neighborhood that the extended range sensor readings are taken.

Generally, computer processing and memory resources are constrained. This makes it necessary for the ETF and

associated variation carousel to limit the maximum number of slides. When the maximum number of slides is reached,

the carousels write out (data log and publish) while telemetering and re-initializing. If the ETF carousel is filled, the

associated variation carousel is also re-initialized. If the variation carousel is re-initialized, the ETF carousel is left intact.

During re-initialization, multiple ETF slides are averaged together into one while multiple variation slides are averaged

into one. These averaged slides are then maintained by the Ezekiel algorithm to maintain historical continuity.

The entire corpus of offloaded carousels can be exchanged between sensor platforms for initialization by capturing the

information content in differential equation form as shown in (6) through (8). This information can also be represented in

lookup table (LUT) format. The LUT allows for real-time execution of equations, even those that cannot run in real-time

by constructing a table of solutions to the carousel ODEs. Reference 5 contains a detailed description of constructing

LUTs for computer processing and memory limited applications.

5. ALGORITHM TESTING

5.1 Description

The ETF can now be used to model and reconstruct the data. This will demonstrate both exact reconstruction of training

points stored in the carousels, and that the ETF carousel results do not blow up when working between carousels. This is

demonstrated using different frequency sine curves for output and increasing index counter for input. By changing the

starting point on the index counter, different sine curves should be extracted that represent interpolated frequencies

between those used to construct the ETF slides. This is shown in the Appendix as a surface plot of similarity sorted sine

curves, the number of which is greater than the original number of ETF slides. This demonstrates the interpolation power

of the ETF carousel. Each sine curve is the output derived from multiplying the input by the interpolated ETF, and

performing an inverse FFT.

As a system level proof, the 128 point long 1/f data set in Figure 3 was created. Windows of data were selected from this

1/f data set for input, and corresponding data windows 1 point ahead in the x-axis were selected for output. The Ezekiel

algorithm was then applied to determine the number of slides and variations required to capture this data. To demonstrate

that the transfer functions and associated variations captured the data process, the data set was appended to itself and

processed three times. Algorithm learning can be seen by the graph in Figure 3, where the final third is all zeros,

indicating that no novelties were detected in the final copy of the data. The middle third of the data still contained

novelty indications by the change detector, indicating that it had not fully learned all of the data; however, it should be

noted that the ETF flagged each case in the middle third selected by the change detector as a false alarm (change detector

novelty found to be nominal according to ETF).

Fig. 3. Change detectors and TF carousel models converge when no new or novel information is presented.

The above shows that ETF and change detectors in the Ezekiel algorithm converged in terms of information content and

did not add any new novel information after carousels were populated (no 2 or 3 after the first 1/6 of the run) and final

change detectors constructed (no 1, 2, or 3 in final 1/3 of the run).

Now that the ETF has been shown to capture the information presented, and that the Ezekiel algorithm with offloading

converges, the overall process in the simulation of applying the algorithm to an image is described. First, initialization of

the simulation must occur. One method is to use off-loaded carousel information from a previous run. This is the method

used by the author for all Mars images after the first, in order to maintain continuity between platforms and runs. The

second method of initialization is to allow the simulation to process a fixed number of pixel neighborhoods without use

of either the guide or change detector and assuming that each is a novel neighborhood. This is done by placing the sensor

platform in the center of the image and allowing it to march outward in a spiral pattern, as seen in Figure 4 (top right).

Once initialized, each new pixel neighborhood selected from the image is processed through the following basic steps:

1. Calculate Heptor Stats of neighborhood

2. Run Heptor through Guide to flag relics

3. Run Heptor through Change Detector to flag novelty

4. Run Guide on extended sensor range pixel neighborhoods,

combine output into pointing vector.

5. If Not relic and Not novelty, Goto 7

6. If relic or If novelty, check carousels as described in

previous section to confirm novelty. If novelty confirmed,

add information to carousels.

7. Use pointing to move to next neighborhood and repeat at 1.

Start

End

(0,0)

NX

Y

Fig. 4. Novelty neighborhoods from simulation run of overall Mars image, with neighborhoods connected by lines using closest

distance (right, top). Final reduced neighborhood group and path to be handed over to next level mission (right, bottom).

Flag denoting novelty/relic results for all neighborhoods processed (bottom left). Here 0 is non-novelty, 1 a novelty flagged by the

change detector later found to be a false alarm, 2 novelties requiring addition of an ETF to the carousel, 3 novelties requiring only

addition of a variant to an existing carousel, and 4 relics.

As multiple pixel neighborhoods are processed through the above steps, a novelty flag for each neighborhood (Figure 4)

is generated. Here, a value of 0 is non-novelty, 1 is a preliminary novelty flagged by the change detector that is later

confirmed to be a false alarm, 2 is a novelty resulting in the addition of an ETF to the carousel, 3 is a novelty resulting in

the addition of a variant to an existing carousel, and 4 is a relic.

As neighborhoods are visited, only those that are novelties are of potential further interest for the mission. These novelty

neighborhoods are shown in Figure 4 on the left, with lines connecting neighborhood centers together by starting at the

first point (denoted by A) and finding the closest distance to current point. This process is repeated until all novelty

neighborhoods are mapped. Both false alarm and overlapping neighborhoods are removed.

Next, the group is ranked and reduced for handover to a sensor platform at the next level. This is performed by ranked

based on standard deviation within each class (novelty flag = 2, 3 and 4), with the highest standard deviation being

ranked the highest. Next, the top 10% of these regions are kept and placed together into a single group and the minimum

distance path that connects the centers of the regions together is calculated. This provides the final ranked route plan for

use at the next level, shown in Figure 4 on the right.

5.2 Global Layer To A

The algorithm process at the satellite level the global Mars flat map given in Table 1, resulting in the selection of the

local neighborhoods shown including the path. Table 1 displays the characterization of the selected local neighborhoods

in ranked order, including intended course of action by the next level of UMV. Table 1 shows neighborhood A.1 ranked

highest, with A.4 also noteworthy of further study at the UAV flyover level as a relic.

Bitmap Center

(x, y)

Simil

Rank

(10-4)

Std

Dev

Rank

Heptor Statistics

(StdDev, Skew, Kurt, M6, M8, DJ, DH)

A.1

(237,207)

(Relic) 1.9 38.08

20.942, 0.053, -1.321, -11.59, -97.396,

1.883, 1.373

Predicted scape. Two mounds center Q3.

Rich structure with volcanic crater Q4.

DEPLOY UAV flyover.

A.2

(259,284)

(Var) 87.8 26.00

8.042, 0.141, -0.844, -8.941, -86.235, 1.938,

1.276

Variant scape. Large plateau with mound

and impact crater Q2. Mountain range Q1,

shelf Q4.

A.3

(306,337)

(TF) 118.3 16.46

3.317, 0.332, -0.269, -4.651, -60.362, 1.624,

1.367

Unique scpe. Very rich fractured area. Many

mountains and flow structure. Mountain Q4,

flow channel Q3.

A.4

(83,295)

(Relic) 80.8 24.50

8.511, -0.364, -0.674, -7.7, -78.273, 1.793,

1.363

Predicted scape. Interesting high altitude

plateau Q1,Q4, possible large shelf and drop

off Q3, or mountain range shadow.

DEPLOY UAV flyover.

Table 1. Characterization of final selected neighborhoods in ranked order from overall Mars flat map.

5.3 Layer A.1 to B

The algorithm process at the UAV flyover level the Mars image given in Table 2 corresponding to A.1. This results in

the selection of the local neighborhoods shown in Table 2 including the path. Table 2 displays the characterization of the

selected local neighborhoods in ranked order, including intended course of action by the next level of UMV. Table 2

shows neighborhood B.1 ranked highest, with B.3 and B.5 also noteworthy of further study at the next level as a relic.

Each is flagged for a UAV recon mission, and B.1 will be looked at in further detail in this work.

Bitmap Center

(x, y)

Simil

Rank

(10-4)

Std

Dev

Rank

Heptor Statistics


B.1

(71,518)

(TF) 41.25 33.82

20.796, -0.149, -0.973, -8.788, -

79.743, 1.666, 1.365

Unique scape. Large fractured Mound amid

shadowed craters. Part of large flow field.

Channel on top center.

Initiate UAV Recon Mission.

B.3

(229,379)

(Relic) 31.0 13.60

4.495, 0.811, -0.4, -3.048, -43.476, 1.791,

1.338

Predicted scape. Foothills of mountains Q4.

Crater Q3. Initiate UAV Recon Mission

B.5

(569,1072)

(Relic) 75.08 28.82

9.438, 0.115, -0.946, -10.332, -94.062, 1.771,

1.332

Predicted scape. Fascinating flow channel

hills and structure across Q3. Initiate UAV

Recon Mission.

(0,0)

X

Y

N

Start

End

Start

End

(0,0)

NX

Y

Table 2. Characterization of final selected neighborhoods in ranked order for zoom in to A.1.

5.4 Layer B.1 to C

The algorithm process at the UAV recon level the next Mars image in Table 3 (corresponding to B.1 above), resulting in

the selection of the local neighborhoods shown including the path. Table 3 displays the characterization of the selected

local neighborhoods in ranked order, including intended course of action by the next level of UMV. Table 3 shows

neighborhood C.1 ranked highest, with no relic present.

Bitmap

(7x7

Enlarged)

Center

(x, y)

Simil

Rank

(10-4)

Std

Dev

Rank

Heptor Statistics


C.1

(173,7)

(Var) 1070 53.05

45.996, -0.128, -0.783, -8.878, -86.141, 1.238, 1.369

Variant scape. Possible hill with craters near foothill

Q2,Q3. DEPLOY ROVER. Analyze and sample

terrain mission initiated.

C.2

(140,14

0)

(TF)

2170 13.81

20.537,-0.233, 0.072, 0.681, 9.549, 1.32, 1.384

Unique scape. Possible rise and slope Q1, Q2.

Mountains near Q4.

C.3

(205,18

6)

(TF)

13.51 26.21

26.05, -0.036, 0.149, -0.937, -31.079, 1.316, 1.374

Unique scape. Plateau with mountain Q2. Slope Q3.

C.4

(271,21

4)

(TF)

110 35.88

31.352, -0.14, -0.705, -6.19, -56.771, 1.299, 1.372

Unique scape. Mountainous region. Large variation of

structure.

Table 3. Characterization of final selected neighborhoods in ranked order for zoom in to B.1.

5.5 Terrain 1 (Rocky Zoom Into C.1)

Processing at ROVER level the Mars image (corresponding to C.1 above). Table 4 displays selected neighborhoods in

ranked order, including intended course of action by the next level of UMV. Table 4 shows neighborhood D.1 ranked

highest, with no relic present.

(0,0)

X

Y

NStart

End

(0,0)

X

Y

N

Start

End

Bitmap

(7x7

Enlarged

)

Center

(x, y)

Simil

Rank

(10-4)

Std

Dev

Rank

Heptor Statistics


D.1

(216,227)

(TF) 18.98 8.23

9.321, 0.073, 0.431, 2.331, -4.151, 1.319,

1.357

Unique rocks. Analyze. COLLECT

SAMPLES from different scale zones.

D.2

(207,231)

(Var) 3630 16.90

14.54, -0.498, -0.031, -1.22, -26.075, 1.534,

1.388

Variant rocks. Photo.

D.3

(195,263)

(TF) 63.84 36.08

22.584, -0.071, -0.88, -8.862, -84.041,

1.674, 1.381

Unique rocks. Analyze.

D.4

(154,220)

(TF) 210 53.79

42.312, .00059, -1.458, -12.259, -99.769,

1.737, 1.369

Unique rocks. Analyze.

D.5

(133,193)

(Var) 320 27.77

26.576, -0.803, -0.116, -2.197, -

35.877,1.755, 1.372

Variant rocks. Photo.

Table 4. Characterization of final selected neighborhoods in ranked order.

ACKNOWLEDGEMENTS

The author thanks the following for support and encouragement during this Doctor of Science thesis and this paper: Dr.

White, Dr. Filipovic, Dr. Blank, Dr. Fennelly, Mr. Hons, and Mr. Handley of James Cook University Department of

Astronomy; and Mr. Hicklen of dtech Systems, Inc.

REFERENCES

1. Jaenisch, H.M., Algorithms for Autonomous Unmanned Deep Space Probes, D.Sc. Thesis, James Cook University.

(Submitted to University for Approval), 2005.

2. Jaenisch, H.M. and Handley, J.W., “Data Modeling for Radar Applications”, Proc.IEEE Radar Conf. 2003.


4. Barnett, S., Cameron, R.G., Introduction to Mathematical Control Theory, 2nd ed., Oxford: Clarendon Press, 1993.

5. Jaenisch, H.M., Handley, J.W., Hicklen, M.L., “Data Model Predictive Control as a New Mathematical Framework

for Simulation and VV&A”, SPIE Defense and Security Symposium, Orlando, FL, April 2006.

APPENDIX Carousel Demonstration Holger M. Jaenisch November 3, 2005

nfiducial 10:= Number of original slides npts 64:= Length of each TF Eigflag 1:= Eigenfunction Model On=1

newpts 37:= Total number of slides (after interpolation) i 0 nfiducial 1−..:= j 0 npts 1−..:= w1i

10i 1+

nfiducial

⋅:=

Aj i, sin w1

i

j 2⋅ π⋅

npts⋅

:= Bj j, j:= zA

i⟨ ⟩cfft A

i⟨ ⟩( ):= zAi⟨ ⟩

FULLEIGzAi⟨ ⟩

Eigflag,( ):= zBi⟨ ⟩

cfft Bi⟨ ⟩( ):=

zBi⟨ ⟩

FULLEIGzBi⟨ ⟩

Eigflag,( ):= dotpi

CALCSIM zBi⟨ ⟩

npts,( ):= dotpi 1, i:= dotp csort dotp 0,( ):= A1

i⟨ ⟩A

dotp i 1,⟨ ⟩

:=

B1i⟨ ⟩

Bdotp i 1,⟨ ⟩

:= A A1:= B B1:= C reverse AT( ):= zA1

i⟨ ⟩zA

dotp i 1,⟨ ⟩

:= zA zA1:= dotp dotp0⟨ ⟩:= zB

i⟨ ⟩cfft B1

i⟨ ⟩( ):=

tfj i,

zAj i,

zBj i, 1+

:= Etfi⟨ ⟩

FULLEIGtfi⟨ ⟩

Eigflag,( ):= k 0npts

21−..:= zTF

j i npts⋅+ tfj i,:= i 0 newpts 1−..:= w1

i1 i .25⋅+:=

Generate

new

locations

in between

fiducial

points

RCAR GENCAROUSELRe zTF( ) dotp, npts, nfiducial,( ):= ICAR GENCAROUSELIm zTF( ) dotp, npts, nfiducial,( ):=

NewBj i, if floor w1

i( ) ceil w1i( ) B

j floor w1i( ) 1−,, Bj floor w1i( ) 1−, ceil w1

i( ) w1i

−( )⋅ Bj ceil w1i( ) 1−, w1

ifloor w1

i( )−( )⋅+, :=

zdotpi

if floor w1i( ) ceil w1

i( ) dotpfloor w1i( ) 1−, dotp

floor w1i( ) 1− ceil w1i( ) w1

i−( )⋅ dotp

ceil w1i( ) 1− w1i

floor w1i( )−( )⋅+,

:=

RAj i, RUNTURL RCAR

j⟨ ⟩zdotp

i,

:= IA

j i, RUNTURL ICARj⟨ ⟩

zdotpi

,

:= TFC

j i, RAj i, 1− IA

j i,⋅+:= zNewBi⟨ ⟩

cfft NewBi⟨ ⟩( ):=

Selected

Virtual TFzNewAj i, TFC

j i, zNewBj i, 1+( )⋅:= NewA

i⟨ ⟩icfft zNewA

i⟨ ⟩( ):= NewOut1 reverse NewAT( ):= selsim 29:=

w1selsim

8.25= zdotpselsim

0.977= TF1j

TFCj selsim,:= D

j i, NewBmod j i+ npts,( )( ) selsim,:=

zDi⟨ ⟩

cfft Di⟨ ⟩( ):=

zEj i, TF1

jzD

j i, 1+( )⋅:= Ei⟨ ⟩

icfft zEi⟨ ⟩( ):= E Re E( ):= FnlTF

j i, TFCj i,:= FnlTF reverse FnlTF

T( ):=

0 2 40

2

4

D j 0,

Bj ceil w1

selsim( ) 1−,

Bj floor w1

selsim( ) 1−,

j

TURLCOEFx y, n,( ) A

y1

y0

−

x1

x0

−←

B A x0

⋅( )− y0

+←

Dm

ym 1+ y

m−

xm 1+ x

m−

ym

ym 1−−

xm

xm 1−−

−←

m 1 n 2−..∈for

Em

xm

←

m 0 n 1−..∈for

F0

A←

F1

B←

F2

n←

F2 m+ D

m←

m 1 n 2−..∈for

Fn m+ 1+ E

m←

m 0 n 1−..∈for

F

≡

0 20 40 601

0

1

Ej 0,

Aj ceil w1

selsim( ) 1−,

Aj floor w1

selsim( ) 1−,

j

CALCSIM zA n,( ) dotp

0

ceiln

2

1−

k

Re zAk( ) Im zA

k( )⋅( )∑=

←

dotp cos dotp 2⋅ π⋅( ) sign dotp( )⋅←

dotp

≡

RUNTURLDERF x, norder,( ) A F0

←

B F1

←

n F2

←

c 1←

Em

Fn m+ 1+←

m 0 n 1−..∈for

Dm

F2 m+←

Cm

if

x Em

−

c300>

x Em

−

300, c,

←

m 1 n 2−..∈for

dj k, DCOEF norder j, k,( )←

k 1 norder..∈for

j 1 norder..∈for

yd if norder 1 A, 0,( )←

yderiv

1

norder

h 1

n 2−

m

Dm

dnorder h,⋅ C

m( )1 norder−⋅

10

x Em−

Cm

1 10

x Em−

Cm+

h

⋅ 1−( )h 1+

⋅ ln 10( )norder 1−

⋅

∑=

∑=

yd+←

yderiv

≡

FULLEIGzX eigflag,( )

n length zX( )←

maxloc 1←

maxval zX1

←

maxval zXi

←

maxloc i←

zXi

maxval>if

i 2 n 1−..∈for

An 1− 0←

Amaxloc

zXmaxloc

←

An maxloc− A

maxloc

←

A0

zX0

←

eigflag 1if

A zX← otherwise

A

≡RUNTURL F x,( ) A F

0←

B F1

←

n F2

←

c 1010−

←

Em

Fn m+ 1+←

m 0 n 1−..∈for

Dm

F2 m+←

Cm

if

x Em

−

c300>

x Em

−

300, c,

←

m 1 n 2−..∈for

yturl A x⋅ B+

1

n 2−

m

Cm

Dm

⋅ log 1 10

x Em−

Cm+

⋅

∑=

+←

yturl

≡

DCOEF iorder j1, k1,( )

di1 i2, 0←

i2 0 iorder..∈for

i1 0 iorder..∈for

d1 1, 1←

break iorder 2<if

di1 i2, d

i1 i2, di1 1− i3, i3⋅+←

i3 i2 1− i2..∈for

i2 1 i1..∈for

i1 2 iorder..∈for

i 1 1..∈for

dj1 k1,

≡

INTEGDATA x y, n,( )

Am 0,

xm

xm 1−+

2←

m 1 n 1−..∈for

A0 0,

3 x0

⋅ x1

−

2←

An 0, A

n 1− 0, xn 1−+ x

n 2−−←

A0 1, 0←

Am 1+ 1,

0

m

k

yk

Ak 1+ 0, A

k 0,−( )⋅ ∑=

←

m 0 n 1−..∈for

A

≡

NewOut1

FnlTF

FnlTF

GENCAROUSELA dotp, noutputs, nrep,( ) mindotp min dotp( )←

maxdotp max dotp( )←

xi

dotpi

←

i 0 nrep 1−..∈for

yj

Ai j noutputs⋅+←

j 0 nrep 1−..∈for

x1 x←

y1 y←

nrep1 nrep←

x1j 1+ x

j←

y1j 1+ y

j←

j 0 nrep 1−..∈for

x10

1−←

y10

ynrep 1−←

nrep1 nrep1 1+←

mindotp 1−>if

x1nrep1

1←

y1nrep1

y0

←

nrep1 nrep1 1+←

maxdotp 1<if

Bi⟨ ⟩

TURLCOEFx1 y1, nrep1,( )←

i 0 noutputs 1−..∈for

B

≡

Data Driven Differential Equation Modeling of fBm processes

Holger M. Jaenisch*a

, James W. Handley a, and Jeffery P. Faucheux

a

aSparta Inc., 4901 Corporate Drive, Suite 102, Huntsville, AL 35805

ABSTRACT

This paper presents a unique method for modeling fractional Brownian motion type data sets with ordinary differential

equations (ODE) and a unique fractal operator. To achieve such modeling, a new method is introduced using

Turlington polynomials to obtain continuous and differentiable functions. These functions are then fractal interpolated to

yield fine structure. Spectral decomposition is used to obtain a differential equation model which is then fractal

interpolated to forecast a fBm trajectory. This paper presents an overview of the theory and our modeling approach along

with example results.

Keywords: Data Modeling, fractal, noise, 1/f, inverse fractal modeling, differential equation, fractal operator,

interpolation, extrapolation, pseudo-derivative

1. INTRODUCTION

Fractional Brownian motion (fBm) processes which exhibit 1/fβ power spectra characteristics are continuously being

discovered as governing dynamics in natural systems. Functional modeling and prediction of fBm sequences can

significantly advance technologies such as MEMS, quantum computing, optical bistability, and biological processes

which exhibit self-organization and complexity.

The goal of this work is to develop:

1. Interpolating process that generates repeatable fine structure using only knowledge of the local neighborhood.

2. Extrapolation process that generates repeatable fine structure using only knowledge of the local neighborhood.

3. Simple autonomous method for generating differential equation models from sampled1 fBm data.

Currently, fBm models assume the form of a one-dimensional stochastic differential equation

)()()( tnxgxfdt

dx+= (1)

which assumes the self-similar or fractal fine structure of a fBm process is modeled by an additive noise term g(x)n(t). A

solution to (1) would be

[ ] .)()()()( ∫ += dttnxgxftx (2)

Lacking a repeatable numerical solution, the classic approach is modeling the fBm ensemble average behavior of (2)

using the generalized Fokker-Planck equation2,3,4 of the form

))((2

1))(( 2

2

2

Pxgx

Pxfxt

P

∂∂

+∂∂

−=∂∂

(3)

for P(t,x) with P(0,x)=δ(x-z). The Fokker-Planck equation is a model of the dynamic continuous probability density

function (PDF) of the solution to the stochastic differential equation in (1) arising from Brownian motion increments.

*[email protected]; phone 1 256 337-3768; fax 1 256 830-0287; sparta.com


In (2), if n(t) is modeled by a pseudo random number generator, non-repeatable fBm trajectories result for each model

run. If the random number generator is replaced by a function of the form

)sin()( ettn = (4)

repeatable fBm trajectories are generated. If the fBm fine structure could be synthesized from the local temporal

dynamics or f(x) directly, the need for injecting the additive noise term n(t) externally would be eliminated.

Instead of modeling only the dynamic PDF, we derive a specific alternative equation formulation for a particular fBm

trajectory x(t). We propose

{ } [ ]∫ +≅≅ dttnxgxftftx f )()()()()( D (5)

which uses the self-similarity inherent in the almost periodic function f(x) to define the fractal fine structure (micro-

trend) in the form of an interpolating operator Df applied to a continuous function f(t) (macro-trend). Not only can this be

done to the trajectory or motion history directly by modeling f(t) with polynomials, but it can also be applied to the

solution of differential equation models.

2. INTERPOLATION

Interpolation is used to evaluate f(x) between fiducial points or knots using knowledge of the local temporal

neighborhood. Interpolation functions are smooth curves passing through fiducial points. Fine structure is added from a

separate source g(x) in a non-repeating fashion.

2.1 Deslauriers-Dubuc general interpolant

This approach achieves functional fractal fine structure modeling from sub-sampled fBm data or continuous models. The

dyadic extension of the fundamental interpolant is


where a, b, c, and d are scaling coefficients, subject to (7)5. In (7), [t] denotes the integer portion of the number t.

∑+

−=

−=3][

2][

)()()(t

tn

ntFnyty (7)

The fundamental interpolant generates a value at each interval midpoint (t/2) in the sampled data using a weighted sum

of the two points occurring before and after the midpoint. These points correspond to the 1 step and 3 step intervals

before and after the interval midpoint at the new dyadic interpolation resolution. Method limitations include being too

open-ended, summation of the scaling coefficients must equal one, and inability to determine scaling coefficient values

directly from the data.

Since the solution of the coefficients in (6) cannot be obtained analytically and remain heuristic adjustment values, we

chose to recast Deslauriers-Dubuc dyadic interpolation as Hidden Variable Iterated Function Systems (IFS). IFS recasts

the interpolation into geometric form using contractive affine transformations6,7 defined as

,0

+

=

n

n

nn

n

nf

e

y

x

dc

a

y

xw (8)

and Elton8 proved in (9) that this process will converge for all continuous functions in (8). Such convergence is

guaranteed by modeling the fBm macro-trend with continuous Turlington polynomials described later in this paper.

∑ ∫=

∞→=

+

n

kX

kn

xdxfxfn 0

)()()(1

1lim µ (9)

2.2 Iterated Function Systems (IFS)

IFS interpolation is a contractive affine transform subject to the constraints

=

−

−

1

1

0

0

n

n

ny

x

y

xw and for n=1,2,…, N. (10)

=

n

n

N

N

ny

x

y

xw

Therefore, the five real numbers a, c, d, e, and f that specify the transformation are defined as

,10 −=+ nnn xexa ,nnNn xexa =+ ,100 −=++ nnnn yfydxc .nnNnNn yfydxc =++ (11)

This is a system of four equations and five unknowns, resulting in one free parameter in each transformation. This free

parameter is chosen to be dn and is called the vertical scaling factor. This choice allows the other four parameters to be

written in terms of the data and the parameter dn as

,)(

)(

0

1

xx

xxa

N

nnn −

−= − ,

)(

)(

0

01

xx

xxxxe

N

nnNn −

−= −

,)(

)(

)(

)(

0

0

0

1

xx

yyd

xx

yyc

N

Nn

N

nnn −

−−

−−

= − .)(

)(

)(

)(

0

00

0

01

xx

yxyxd

xx

yxyxf

N

NNn

N

nnNn −

−−

−−

= − (12)

We determine dn using a pseudo-derivative, defined as

.)()( 1111 −+−+ −=+−+= nnnnnnn yyyyyyd (13)

IFS interpolation starts with an initial point (usually one of the points given in (10)) and applies one of the transforms

given in (8) and defined by the coefficients in (12) to this point to generate a new point that lies on the attractor for the

function. This new point is then used as a starting point and the process is repeated. This process iterates until the

function or attractor fills in to within a specified tolerance given as

.ˆ ε≤− xx (14)

This test must be performed either at each iteration or the process allowed to progress for a fixed large number of

iterations and then checked. The IFS coefficients in (12) determine the current x distance from the starting point of the

data set (as a fractional percentage of the length of the entire data set) and maps the new y value to an x location the same

fractional percentage into the transform interval. The coefficients in (12) determine how far the vertical scaling

coefficient raises the previous y value above or below the sloped line connecting the beginning y and ending y of the

transform interval.

IFS interpolation can be implemented in two ways: a deterministic algorithm or a random algorithm. The deterministic

algorithm applies the transform in (8) and (12) in a combinatorial fashion to fill in the attractor at increasing scale

resolutions. The random algorithm uses the same transforms and coefficients, but they are applied in a Markov process

fashion so that the attractor fills in at random x locations. Neither of these approaches yields an f(x) without substantial

O[103] iterations. The number of iterations was bounded to O[102] by the authors in previous work using a Ruelle-

Takens theorem base algorithm9.

2.3 Data Modeling

In previous work, the authors demonstrated the ability to sub-sample data and reconstruct it by defining the Data Model

as the sub-sampled points9,12. These equally spaced sub-sampled points are the support vectors and the number of

points N is identified using

=→∆+

N

NJ

R

Ji

Ji

1log

log

limmin)Re(0

1

σ 5.01

log

log

lim)Re(0

1 =

=→∆+

N

NJ

R

Ji

Ji

σ (15)

The number of points N to sub-sample in building the model is bounded by (15) where R is the range of the data and ı is

the standard deviation of the data. N can be solved for between these bounds in order to minimize the variance of the

residual between reconstructed and original. We have found excellent results by solving the second equation in (15) for

N, and have determined that a good rule of thumb to go by in the absence of this information is to sub-sample 10% of the

original data set.

2.4 Fractal operator

The use of fractal operators applied to images is explored by Davis10 and Russ19. Neither show how image operators

can be applied to f(x).

From (8), the information required for IFS to evaluate f(x) at a specific x location consists of the coefficients from (12)

that define the IFS transformation associated with the nth segment containing the point and the value of the previous

point (xi-1 ,yi-1). This information is used to transform the previous point into the current (xi ,yi) location. Because we

have defined dn by a pseudo-derivative, each of the coefficients in (12) is based on the sub-sampled data values

themselves along with the beginning and ending points. Using equally spaced sampled data and assuming that the

sampled data process starts at x0 = 0 and ends at xN = N reduces (12) and (13) to

,)( 1

N

xxa nn

n−−

= ,1−= nn xe ,))(()( 0111

N

yyyy

N

yyc Nnnnn

n

−−−

−= −+− (16)

).( 1101 −+− −−= nnnn yyyyf

Values for xn and xn-1 can be obtained from the sub-sampling rate, length of the original data set, and the current xi

value. If estimates of yn-1, yn, and yn+1 can be generated from a f(x), then the coefficients in (16) can be solved for directly

whenever the transform is used. Calculating these y values requires knowledge of xn-1, xn, and xn+1. Equation (16) is

further reduced assuming the Data Model exists on the [0,1] interval so that N = 1 and x1 = xN = 1.

Since the IFS process is based on affine transformations, the xi-1 value for the previous point that was transformed into

the current point xi can be determined geometrically. Using the same f(x) above, the value yi-1 at previous xi-1 is

determined and transformed into the needed (xi ,yi) pair. Recasting (8) using the coefficients from (16) results in our

fractal operator on yi defined as Df

{ } ).()())(()(

110111110111

−+−−−+−−+− −−+−+

−−

−−

≡ nnninniNnnnn

if yyyyyyyxN

yyyy

N

yyyD (17)

where N is the number of points in the original process Data Model and vanishes if N = 1, y0 the value of the Data Model

at x = 0, yN the value at x = N, i the index associated with the (x,y) pair that is transformed into the needed function pair,

and n the index of the straight line segment connecting the original Data Model points defining the interval containing

the required xi.

2.5 Turlington polynomials

We propose obtaining fiducial yi values from a polynomial representation of the fBm macro-trend. This equation is

evaluated at any yi with associated xi. Chebyshev polynomials are an orthogonal model. However, this also means extra

terms cannot be added to the model equation as new data points are sampled. Conversely, the Turlington polynomial11

defined as

( )

( )

( )

.101log

101

10log)(

1

10

1

10 ∑∑

++

+

++=−

−

−

md

xx

m

i

c

xx

c

xx

im

m

i

i

i

i

DCBxAxT (18)

where coefficients A and B are the slope and y intercept of the reference asymptote, ci and dm are the asymptote transition

region that captures major changes in the first derivative of the data, xi and xm are the intersection points between two

neighboring asymptotes, and Cm and Dm are the slopes of the asymptotes multiplied by the transition regions.

Determination of these coefficients are discussed in more detail in Reference 11.

Turlington polynomials are built in a piecewise fashion relative to an arbitrary reference asymptote. Terms can be

appended to the polynomial as new trajectory points occur. Turlington polynomials are constructed by assuming one of

the segments connecting neighboring points in the data set is the reference asymptote. The largest derivative location in

the data set is usually chosen for the reference asymptote; however, choice of the reference asymptote is arbitrary and

will be selected as the first segment in our application as shown below.

Although this function is built from the Data Model data points (support vectors) piecewise, the final equation

representation is both continuous and differentiable. Unlike spline based methods, one term is added to the equation for

each of the sub-sampled data points used in its construction, and because of the logarithmic nature of the equation, new

terms can be appended for new data points without changing the T(x) values at previously sampled x locations.

The analytical derivative of the Turlington polynomial is

( )

( )∑∑ −

−+

+

+

+

++=

m

d

xx

d

xx

m

mi

xx

xxx

i

m

m

m

m

i

ii

d

DCB

dx

dT

112

)1exp()1exp(

)1exp()1exp(

101

10

1010

10100

)1exp(

(19)

Real time data applications assume the first data segment collected is the reference asymptote causing i in (18) to be

zero. This results in the simplified form

( )

∑

+++=

−m

d

xx

mm

m

DBxAxT1

10 101log)( (20)

and ( )

( ) .

101

10

1

∑ −

−

+

+=m

d

xx

d

xx

m

m

m

m

m

m

d

DB

dx

dT (21)

By taking successive derivatives on (20) and collecting like terms, it can be shown that the nth order derivative of the

reduced form of the Turlington polynomial given in (20) is

( )

( )

( ).)10ln(

101

101

)(1

,

1

1

1

∑ ∑

+

−+= −

= −

−

+m

in

nn

ii

d

xx

i

d

xx

n

m

m

i

n

n

m

m

m

m

d

DnB

dx

Td ψ (22) ∑≠≠−=

−=i

njj

ij

jnin j

01

,1, ψψ

where B(1) = B and B(n) = 0 for all n > 1, and ψn,i defined for n > 1 with ψ1,1 = 1. This results in the identity

( )

( )

( )

( )

.101log)10ln(

101

101

)(1

10

1

,

1

1

1

∑∑ ∑∫∫

+++=

+

−+

−−

= −

−

+ md

xx

m

m

in

nn

ii

d

xx

i

d

xx

n

m

m

i

n

m

m

m

m

m

m

DBxAdxdxd

DnB L

321L ψ (23)

Source code listed in the Appendix contains a program for calculating both the Turlington polynomial for a given data

set and the nth order derivative using the formulation shown in (22). This source code takes as input a data file name, the

number of points in the file, and the order of the derivative (0 gives T(x) instead of a derivative). Output is a source

listing that can be interrogated at any given x value on the interval [0,1], with 0 representing the input data starting point

and 1 the ending point. A loop can be added to generate multiple data points on the [0,1] interval.

2.6 Fractal operator (Df) on Turlington polynomials

Once the beginning and ending x values for the nth IFS transform are determined, T(x) is used to determine y values for

each end point and yi-1 for the nth segment transformation. With this knowledge, it is only necessary to perform a single

IFS transform on (xi-1 ,yi-1) to find the desired value for (xi ,yi), which leads to the functional form of the fractal operator

on T(x) as

{ }.)()( xTxf fD= (24)

IFS

Original

T(x)

Diff Eq

0 255

Df{T(x)}

Fig. 1. Comparison of fractal operator derived function with original.

For the fractal operator method, the pseudo-derivative dn yields a lower bound value because it is derived from T(x)

rather than the raw data. Fig. 1 shows a comparison of this method with IFS and the original. The polynomial model

does not contain the range of fine structure that is captured in sub-sampling the raw data set. If the application requires,

the dn in (13) can be scaled up by σ(f(x))/σ(T(x)), the ratio of the standard deviation of the original data to the standard

deviation of T(x). These standard deviations can be calculated recursively on the data in real time as it occurs using (25)

where µ is the mean and σ is the standard deviation12.

1

1

1

1

+

+

++ +

=

i

i

i

iii

N

y

N

Nµµ ( ) ( )

.1

2

11

1

2

1

+

++

++

−+

=

i

ii

i

iii

N

y

N

N µσσ (25)

3. EXTRAPOLATION

Extrapolation is used to determine values at locations outside the neighborhood of all prior data points. This is akin to

evaluating a function beyond its original extent. This process can use knowledge of either the local or global

neighborhood. Extrapolation functions take the form of linear or smoothed trend estimates. Seldom is fine structure

preserved.

3.1 Linear prediction

Linear prediction is an autoregressive model that is linear in the sum of its terms. It performs extrapolation on data sets

that are equally spaced and stationary. Linear prediction is especially successful in extrapolating signals that are smooth

and oscillatory, though not necessarily periodic. Linear prediction uses N consecutive values in order to predict the N+1

value. Linear prediction is defined as

,1

∑=

− +=N

j

mjmjm xydy (26)

where each of the dj coefficients are the weighting factors in making the linear combinations of the N consecutive ym

values, and xm is the discrepancy between the predicted value and the actual. For linear prediction, the xm term is

assumed to be zero. The number of previous points to use in the calculation (N) is a user specified input. Measures must

also be taken with linear prediction to insure stability. The stability condition is given by the characteristic equation

01

=−∑=

−N

j

jN

j

N zdz 1≤z (27)

where all roots must be inside of the unit circle. When solving (26) above for the linear prediction coefficients, there is

no guarantee that the roots will fall within the unit circle13.

3.2 Novel fractal algorithm

Because IFS are contractive on the unit interval, they do not extrapolate. In order to use them for extrapolation while

maintaining contractivity, a new support vector outside of the original transform intervals is introduced. This is

accomplished using linear prediction or by appending the first point of the data set to the end of the data set according to

)( 11 iiNN xxxx −+= ++

iN yy =+1 Ni ,...,2,1= (28)

to form the extrapolation interval. New IFS transform coefficients using (12) are calculated, and the data sequence is

reconstructed to include the newly added support vector. Only the points generated in the first ½ of the extrapolation

interval are kept as extrapolated points. This process is repeated by increasing i in (28) and appending the next point in

the original data set to the end of the saved extrapolated points to create a new extrapolation interval. The reconstruction

process is continued until all of the original data points have been used, and can be repeated as many times as necessary

to generate the desired length forecast9.

3.3 Differential equation approach

Polynomial models of data are valid across the range of the original data set. Differential equation models do not suffer

this constraint because they are comprised of continuous functions that can model the dynamic process across a much

longer time projection than the original interval. If the underlying differential equation governing a fBm process can be

determined, the differential equation can then be used to determine the value of the function even for cases that fall

outside of the original boundaries.

Using Data Modeling, we have developed a simple method for determining a differential equation for a given fBm

process. We begin by assuming that the process can be modeled by a first order differential equation of the form

)(tfdt

dx= (29)

where t is the independent variable and f(t) is the fBm process. Since differential equation based processes are comprised

of continuous differentiable functions, we first build a T(x) of the form found in (20) of the function f(t). Once T(x) is

determined, its first derivative T’(x) can then be determined from (21).

Data Relations Modeling is then used to discover the functional relationship between the first derivative and its forcing

functions. Data Relations Modeling is the process of finding a mathematical expression that provides a good fit between

given finite sample values of the independent variables and the associated values of the dependent variables of the

process. The mathematical expression that fits the given sample of data is called a Data Model. This process involves

finding both the functional form of the Data Model and the numeric coefficients for the Data Model. The Data Relations

Modeling process of discovering automatic functions is an evolutionary and genetic directed approach where the set

elements consist entirely of simple polynomial building blocks and is described in detail in References 14-17.

The final form of the Data Relations Modeling equations and the polynomial order are derived from an approximation to

the Kolmogorov-Gabor polynomial

L++++= ∑∑∑∑∑∑i j k

kjiijk

i j

jiij

i


represented by

)))))))))((((((((,()( 21 tbxtbxbxtftx nK= [ ] ntxO 3)( = (31)

which is a form of an orthogonal nested polynomial network comprised of low order basis functions. In (31), n

represents the number of layers in the final Data Model and x(bi(t)) the inputs mapped from the previous layer. Data

Model polynomials on the order of ~ O[310] have been generated within minutes using only 3rd order polynomial basis

functions. Data Models adequately capture high degrees of process nonlinearities but still execute in real-time14,15,16,17.

In order to perform Data Relations Modeling, it is necessary to select the form of the forcing functions driving T’(x). If a

specific forms of the forcing functions are known, they can be specified as inputs into the Data Relations Modeling

process. If the form is unknown, Data Relations Modeling uses eigenfunctions or cosine and sine terms derived from the

dominant frequency components as the model of the fBm trajectory forcing functions.

To find the dominant frequency components, eigenanalysis is used to identify the eigenvalues of T’(x). These

eigenvalues correspond to the dominant peaks in the power spectra and are determined in ranked order of contribution.

However, the process of determining eigenvalues is complicated through the use of linear algebra methods to insure that

the matrix is Hermitian, normal, yields real roots, and in most cases is symmetric. If the matrix does not exhibit these

properties, it must be transformed. Since a great deal of care and preprocessing steps may potentially need to be taken in

applying this method, the authors chose to use the simple peak identification method listed below.

We propose a variation of Wiener filtering to identify individual peaks in the dB power spectra. This is achieved by

first performing regression to fit a straight line to the entire dB power spectra, resulting in a slope and a y intercept. The

value of the y intercept on the dB power spectra is the noise floor on the entire spectra and is illustrated in Fig. 2. Once

the noise floor is determined, dB power spectra values that fall below the noise floor can be zeroed and an inverse

Fourier transform performed. Maximum entropy methods can then be used to accurately determine the locations of

peaks in the power spectra and the corresponding number of cycles for each peak. This method was avoided in this work

because it can also lead to the enhancement of spurious peaks above the noise floor.

Peaks are identified by slope changes using the 1st derivative of the dB power spectra (dPSD). Zero values in dPSD

correspond to minima and maxima occurring one point earlier in the power spectra. For maxima, the derivative value

immediately before the zero crossing has a positive value and the derivative value after the crossing has a negative value.

Once flagged as a maxima, the value in the dB power spectra is then checked to insure that the maxima is above the

noise floor. This process is demonstrated in Fig. 2, where the graph on the left shows the linear fit and noise floor

superimposed onto the power spectra, and the graph on the right dPSD used in identifying maxima.

dB Power Spectrum dB Power Spectrum Derivative

Noise Floor

-58.6

0.0

-21.3

3

0.9

0 127 1 127

Linear FitZero Line

PSD peaks correspond

with negative slope at zero

crossing in PSD derivative

Fig. 2. Identification of maxima values above noise floor in power spectrum.

Cosine and sine terms as shown in the fractal operator and differential equation source code in the Appendix are

generated for each maxima. Care is taken to ensure that the number of cycles generated by each cosine and sine term

across the number of points used as input to Data Relations Modeling is the same as that represented by each dB power

spectra maxima. Simply adding these terms together and using a differential equation solver does not give the

solution to the differential equation. As shown in Section 4, nonlinear coupling exists between individual cosine

and sine terms, and the proper weighting coefficients for individual coupled terms must be generated14,15.

Since Data Relations Modeling scales as a nested O[3n] process, the resulting nonlinear differential equation may

have hundreds of terms associated with it. This results in the autonomously generated Data Driven Differential

Equation

)))))))))((((((((,( 21 tbxtbxbxtfdt

dxnK= (32)

Once Data Relations Modeling has found the Data Driven Differential Equation shown in (31), it can be solved using

numerical integration. Now, a solution of the differential equation can be found for independent variable values outside

of the original range. When this differential equation is coupled with the fractal operator Df, fBm dynamic processes are

modeled as

{ }.)()( tftx fD≅ (33)

Here f(t) is the Runge-Kutta solution to Equation (32) above. Although this method can be slow, the process of

integration can be sped up by saving intermediate terms. And if the application merits this approach, Turlington

polynomials for Df{T(x)} can be used directly bypassing the need for integration.

4. EXAMPLE

An example of this Data Modeling process was used to determine the differential equation for a specific fBm data set.

This model was extrapolated. Envision a continuous forcing function defined as

,)sin()( ∫= dxxxf e (34)

which is the integral of the Sxe function9,15. This function generates a fBm data set as shown in Fig. 3, where the data set

is 512 points in length, and the Data Differential Equation Modeling process was performed using the first 256 points of

the data set and leaving out the remaining 256 points for comparison with the extrapolation results18.

A variance based sub-sampling method was applied to the data set to determine the support vectors necessary for Data

Modeling. This optimal sampling method first sub-samples 10% of the data points evenly and uses straight-line

segments to interpolate between the points. The root sum squared (RSS) difference between the interpolated and

original curves is generated, and is declared to be the baseline. Next, the data set is sub-sampled and reconstructed

using the methods described above in Section 2.2, beginning at 5% and iteratively increasing the sub-sampling

percentage. When the RSS difference between the reconstructed and original is less than that calculated using the

straight-line segments above, the optimal sub-sampling rate is identified.

For the data set shown on the left of Fig. 3, the optimal sampling rate was found to be 5:1. Every 5th data point was sub-

sampled from the data set, and T(x) and T’(x) generated from Equations (20) and (21). This T’(x) represents the

differential equation, and is the output in the Data Relations Modeling process.

U s e d t o b u i l d

d i f f e r e n t i a l e q u a t i o n

H e l d b a c k t o t e s t

e x t r a p o l a t i o n

0 2 5 5 2 5 6 5 1 1

-11.4

8.0

Fig. 3. fBm data set generated using Sxe function.

Since no a priori knowledge exists for the form of the forcing functions, cosine and sine terms were generated for the

dominant frequency components. Fig. 2 shows the process of identifying nine peaks corresponding to 2, 6, 9, 12, 15, 18,

20, 23, and 26 cycles for this data set. A total of 18 separate inputs (9 cosine and 9 sine) were generated for use in Data

Relations Modeling, and each equation is built by multiplying the number of cycles above with 2π t. The equations for

these 18 terms are listed in the fractal operator and differential equation source code in the Appendix of this paper, and

correspond to

)),52cos(),46cos(),40cos(),36cos(),30cos(),24cos(),18cos(),12cos(),4(cos( tttttttttfdt

dx πππππππππ=

))52sin(),46sin(),40sin(),36sin(),30sin(),24sin(),18sin(),12sin(),4sin( ttttttttt πππππππππ

(35)

Using Data Relations Modeling, the functional relationship between these 18 inputs and the output was calculated.

Although it was possible that coupled terms in the final Data Model could reach orders as high as O[34] = 81, this case

yielded an acceptable solution using only the coupling of 3 forcing functions in any single term. A comparison of the

Data Model based first derivative and the Turlington derivative is shown in Fig. 4.

A fourth order Runge-Kutta differential equation solver was then applied to the Data Relations Model to generate the

macro-trend based on the differential equation. The fractal operator Df was then applied, resulting in the addition of the

micro-trend.

O r ig in a l

D a ta

M o d e l

R S S D if f = 3 .4 %

0 2 5 5-12

.6

8.0

Fig. 4. Turlington derivative and Data Model.

The Runge-Kutta process was allowed to continue to propagate for a full interval in the x-axis past the end of the data

used for generating the differential equation model. The results of this are shown in Fig. 5, including a comparison of

this extrapolation method and linear prediction. Source code to generate this function is given in the Appendix under the

fractal operator and differential equation heading. Removing the apostrophe in line 3 of the source generates the first ½

of the top curve (Df) in Fig. 5 along with the differential equation output, Df{T(x)} and T(x). Leaving the apostrophe in

place forces single point f(x) evaluation at the command line specified x value, and changing 255 to 511 in line 22 allows

the program to generate the curves on the full intervals shown in Fig. 5.

O rig in a l

D iffe re n tia l

E q u a tio n

L in e a r

P re d ic tio n

E x tra p o la te d

0 2 5 5 2 5 6 5 1 1

R S S D iff = 2 5 %

R S S D iff = 2 6 %

Fig. 5. Final results showing comparison of Data Driven Differential Equation Modeling and linear prediction.

Using the differential equation approach combined with the fractal operator Df generated a good estimate of the behavior

exhibited by the fBm process. This can be seen by comparing the graphs on the right of Fig. 5. The differential equation

method yielded a good filtered estimate of the fBm process, and can enable physics based modeling by the coupling of

physics based forcing functions to generate new fBm trajectories. On the other hand, the Turlington polynomial

approach yields real time solutions of a particular dynamic system configuration. The novel fractal operator predictably

generates model behavior fine structure in a functional form. All these tools are available for the investigator’s use and

discretion.

ACKNOWLEDGMENTS

The authors would like to thank Marvin Carroll, Tec-Masters, Level-13, Licht Strahl Engineering INC, and Kristi

Jaenisch and Technical Dive College for the use of the MAJQANDA algorithm suite during the course of this work.

REFERENCES

1. Handley, J. On The Existence of a Transition Point in Sampled Data using Fractal Methods. Ann Arbor, MI: UMI,

1995.

2. Zwillinger, D. Handbook of Differential Equations, 3rd edition. San Diego, CA: Academic Press, 1998.

3. Dubkov, A. B. Spagnolo, and N. Novgorod. “From theory of infinitely divisible distributions to derivation of

generalized master equation for Markov process”, Proceedings of SPIE: Noise in Complex Systems and Stochastic

Dynamics. June 2-4, 2003. Santa Fe, NM.

4. Soize, C. The Fokker-Planck Equation for Stochastic Dynamical Systems and Its Explicit Steady State Solutions.

Singapore: World Scientific, 1994.

5. Cherbit, G. Fractals: Non-integral Dimensions and Applications. New York: Wiley, 1991.

6. Massopust, P. Space Curves Generated By Iterated Function Systems. Ann Arbor, MI: UMI, 1986.

7. Scheinerman, E. Invitation to Dynamical Systems. Upper Saddle River, NJ: Prentice-Hall, 1996.

8. Elton, J. “An Ergodic Theorem for Iterated Maps,” Journal of Ergodic Theory and Dynamical Systems, 7:481-488

(1987).

9. Jaenisch, H. and J. Handley “Data Modeling of 1/f noise sets”. Proceedings of SPIE: Noise in Complex Systems and

Stochastic Dynamics. June 2-4, 2003. Santa Fe, NM. (CD version contains source code and executable)

10. Davis, G. “Implicit Image Models in Fractal Image Compression”, Invited paper, SPIE Conference on Wavelet

Applications in Signal and Image Processing IV, Denver, August 1996.

11. Turlington, T. Behavioral Modeling of Nonlinear RF and Microwave Devices. Boston, MA: Artech House, 2000.

12. Jaenisch, H. and J. Handley, “Data Modeling for Radar Applications,” Proceedings of the IEEE Radar Conference

2003, May 5-8, 2003. Huntsville, AL.

13. Press, W., S. Teukolsky, W. Vetterling, and B. Flannery. Numerical Recipes in FORTRAN, 2nd edition. Cambridge:

Cambridge University Press, 1992.

14. Jaenisch, H. and J. Handley “Data Modeling of network dynamics”. Proceedings of SPIE: Applications and Science

of Neural Networks, Fuzzy Systems, and Evolutionary Computation VI. August 5-6, 2003. San Diego, CA.

15. Jaenisch, H., J. Handley, K. White, J. Watson, C. Case, and C. Songy. “Virtual prototyping with Data Modeling”,

Proceedings of SPIE, July 7-11 2002. Seattle, WA.

16. Jaenisch, H., J. Handley, C. Case, C. Songy, “Graphics based intelligent search and abstracting using Data

Modeling”, Proceedings of SPIE, July 7-11 2002. Seattle, WA.

17. Jaenisch, H. “Fractal Interpolation for Patching Holes in Data Sets”, Proceedings of the Southeastern Simulation

Conference ’92, Pensacola, FL 1992.

18. Handley, J., H. Jaenisch, C. Bjork, L. Richardson, R. Carruth, “Chaos and Fractal Algorithms Applied to Signal

Processing and Analysis,” Simulation, 60:4, 261-279 (1993).

19. Russ, J. The Image Processing Handbook, Third Edition. Boca Raton, FL: CRC Press, 1999.

APPENDIX

Source listing for generating Turlington polynomial T(x) and n

th order derivative T

n(x) generation routine.

DEFCUR A-Z: INPUT "File to process => ", zz$: INPUT "Number of points => ", npts: INPUT "Derivative Order (0 for none) => ", n

CLS : OPEN zz$ FOR INPUT AS #1: OPEN "turling.bas" FOR OUTPUT AS #10

PRINT #10, "defcur a-z": PRINT #10, "cls": PRINT #10, "INPUT "; CHR$(34); "X (0-1) = "; CHR$(34); ", x"

INPUT #1, y0: INPUT #1, y1: y2 = y1: cc = .01: i = 2: m = (y1 - y0) * (npts - 1): ycept = y0

IF n > 0 THEN PRINT #10, "n="; n; ":redim d(n,n):d(1,1)=1": PRINT #10, "for i=2 to n": PRINT #10, " for k=1 to i"

IF n > 0 THEN PRINT #10, " for j=k-1 to k": PRINT #10, " d(i,k)=d(i,k)+d(i-1,j)*j"

IF n > 0 THEN PRINT #10, " next j": PRINT #10, " next k": PRINT #10, "next i"

IF n = 1 THEN PRINT #10, "y="; m ELSE PRINT #10, "y=0": PRINT #10, "y = "; m; " * x + "; ycept

IF n > 0 THEN PRINT #10, "FOR i = 1 TO n"

FOR n1 = 1 TO npts - 2

IF i > 2 THEN : y0 = y1: y1 = y2

INPUT #1, y2: i = i + 1: a = (npts - 1) * (y2 - 2 * y1 + y0) * cc: b = (i - 2) / (npts - 1)

IF n = 0 THEN PRINT #10, "y = y + "; a; " * LOG(1 + 10 ̂ ((x - "; b; ") / "; cc; ")) / LOG(10)"

IF n > 0 THEN PRINT #10, " y=y+d(n,i)*((-1)̂ (i+1))*"; a; "*(10̂ ((x-"; b; ")/"; cc; ")̂ i)*(log(10)̂ (n-1))/((("; cc; ")̂ n)*((1+10̂ ((x-"; b; ")/"; cc; "))̂ i))"

NEXT n1

IF n > 0 THEN PRINT #10, "NEXT i"

PRINT #10, "print "; CHR$(34); "F("; CHR$(34); "x"; CHR$(34); ") = "; CHR$(34); ",y": PRINT #10, "end": CLOSE : END

Source listing for applying fractal operator Df to T(x) and Data Driven Differential Equation Model. DEFCUR A-Z: DECLARE FUNCTION y1@ (x1@) : DECLARE FUNCTION yt@ (x@)

DECLARE SUB rk4 (y@, dydx@, x@, h@, yout@) : DECLARE SUB derivs (xh@, yt@, dym@)

CLS : x = VAL(COMMAND$): i = -1: OPEN "dqdmout" FOR OUTPUT AS #1: ' i = 0

ntotal = 256: nsub = 51: ymin = -5.375: ymax = 7.998

10 : IF x = ntotal THEN nsub = nsub + 1: ntotal = ntotal + ((ntotal - 1) / (nsub - 1))

n = CINT(x * (nsub - 2) / (ntotal - 1)) + 1

istart = CINT(((ntotal - 1) * (n - 1)) / (nsub - 1))

ifinish = CINT(((ntotal - 1) * n) / (nsub - 1))

iend = CINT(((ntotal - 1) * (n + 1)) / (nsub - 1))

xpt = (x - istart) / (ifinish - istart)

ix = (ntotal - 1) * xpt

dfde = ((y1(iend) - ymin) / (ymax - ymin)) - ((y1(istart) - ymin) / (ymax - ymin))

dfde = (((y1(ifinish) - y1(istart)) / (ntotal - 1)) - (dfde * (y1(ntotal - 1) - y1(0)) / (ntotal - 1))) * ix

dfde = dfde + (((y1(iend) - ymin) / (ymax - ymin)) - ((y1(istart) - ymin) / (ymax - ymin))) * y1(ix) + y1(istart)

dfde = dfde - y1(0) * (((y1(iend) - ymin) / (ymax - ymin)) - ((y1(istart) - ymin) / (ymax - ymin)))

x1 = (x * 1.1765 - CINT(6 * INT(x * 1.1765 / 6))) / (CINT(6 * (INT(x * 1.1765 / 6) + 1)) - CINT(6 * INT(x * 1.1765 / 6)))

dftx = ((yt((INT(x * 1.1765 / 6) + 2) * 2) - yt(2 * INT(x * 1.1765 / 6))) / 2) * yt(x1 * 100) * 2

dftx = dftx + yt(2 * INT(x * 1.1765 / 6)) - .4062 * (yt((INT(x * 1.1765 / 6) + 2) * 2) - yt(2 * INT(x * 1.1765 / 6)))

dftx = dftx + x1 * (yt((INT(x * 1.1765 / 6) + 1) * 2) - yt(2 * INT(x * 1.1765 / 6)))

dftx = dftx + x1 * (-(yt((INT(x * 1.1765 / 6) + 2) * 2) - yt(2 * INT(x * 1.1765 / 6))) * -.3029)

dftx = dftx * 13.373 + -5.375: tx = yt(x * 1.1765 / 3) * 13.373 + -5.375: de = y1(x)

PRINT #1, de, dfde, tx, dftx: IF i = 0 AND x < 255 THEN : x = x + 1: GOTO 10

CLOSE : END

SUB derivs (xh, yt, dym)

pi = 4@ * ATN(1@)

cos12 = (COS(24 * pi * xh) - .0033) / .6381: cos15 = (COS(30 * pi * xh) - .0033) / .6329: cos18 = (COS(36 * pi * xh) - .0033) / .6382

cos2 = (COS(4 * pi * xh) - .0033) / .6379: cos6 = (COS(12 * pi * xh) - .0033) / .6382: sin18 = SIN(36 * pi * xh) / .6337

cos20 = (COS(40 * pi * xh) - .0033) / .6392: cos23 = (COS(46 * pi * xh) - .0033) / .6378: cos26 = (COS(52 * pi * xh) - .0033) / .6379

cos9 = (COS(18 * pi * xh) - .0033) / .6377: sin12 = SIN(24 * pi * xh) / .6337: sin15 = SIN(30 * pi * xh) / .6293

sin2 = SIN(4 * pi * xh) / .6344: sin20 = SIN(40 * pi * xh) / .6322: sin23 = SIN(46 * pi * xh) / .6345: sin26 = SIN(52 * pi * xh) / .6344

sin6 = SIN(12 * pi * xh) / .6337: sin9 = SIN(18 * pi * xh) / .6343

l1o4 = -.0717 - .339 * sin2 - .5038 * cos6 - .2515 * cos9 + .1124 * sin2 * sin2 + .41 * cos6 * cos6 - .4611 * cos9 * cos9

l1o4 = l1o4 + .2167 * sin2 * cos6 - .0173 * sin2 * cos9 + .113 * cos6 * cos9 - .0825 * sin2 * cos6 * cos9

l1o4 = l1o4 + .4874 * sin2 * sin2 * sin2 + .3393 * cos6 * cos6 * cos6 + .1518 * cos9 * cos9 * cos9

l1o4 = l1o4 - .2052 * cos6 * sin2 * sin2 + .0462 * sin2 * cos6 * cos6 - .1794 * sin2 * cos9 * cos9

l1o4 = l1o4 + .1666 * cos9 * sin2 * sin2 - .0299 * cos9 * cos6 * cos6 - .1123 * cos6 * cos9 * cos9

l1o6 = .2828 - .6034 * sin9 - .7482 * sin6 + 1.2483 * cos2 + .198 * sin9 * sin9 - .3128 * sin6 * sin6

l1o6 = l1o6 - .1186 * cos2 * cos2 - .0887 * sin9 * sin6 + .3034 * sin9 * cos2 + .1787 * sin6 * cos2 - .2136 * sin9 * sin6 * cos2

l1o6 = l1o6 + .0484 * sin9 * sin9 * sin9 - .0084 * sin6 * sin6 * sin6 - .4628 * cos2 * cos2 * cos2 + .0119 * sin6 * sin9 * sin9

l1o6 = l1o6 - .0167 * sin9 * sin6 * sin6 + .1476 * sin9 * cos2 * cos2 - .1378 * cos2 * sin9 * sin9 - .087 * cos2 * sin6 * sin6

l1o6 = l1o6 + .3653 * sin6 * cos2 * cos2 - .0117 + .0189 * sin2 + .089 * sin9 + .023 * cos2

l1o5 = l1o5 - .0331 * sin2 * sin2 + .1918 * sin9 * sin9 - .1483 * cos2 * cos2 + .1793 * sin2 * sin9

l1o5 = l1o5 - .045 * sin2 * cos2 + .3006 * sin9 * cos2 - .221 * sin2 * sin9 * cos2 + .1851 * sin2 * sin2 * sin2

l1o5 = l1o5 + .0263 * sin9 * sin9 * sin9 - .0039 * cos2 * cos2 * cos2 - .2651 * sin9 * sin2 * sin2 + .1796 * sin2 * sin9 * sin9

l1o5 = l1o5 - .2937 * sin2 * cos2 * cos2 + .4368 * cos2 * sin2 * sin2 - .1382 * cos2 * sin9 * sin9 - .1277 * sin9 * cos2 * cos2

l1o1 = -.544 - .5585 * sin2 + .3854 * cos6 + .3821 * sin15 + .1043 * sin2 * sin2 + .3103 * cos6 * cos6

l1o1 = l1o1 + .0265 * sin15 * sin15 + .2171 * sin2 * cos6 - .0165 * sin2 * sin15 - .247 * cos6 * sin15

l1o1 = l1o1 + .1974 * sin2 * cos6 * sin15 + .4941 * sin2 * sin2 * sin2 - .2536 * cos6 * cos6 * cos6 + .024 * sin15 * sin15 * sin15

l1o1 = l1o1 - .2097 * cos6 * sin2 * sin2 + .0484 * sin2 * cos6 * cos6 - .0139 * sin2 * sin15 * sin15 - .2107 * sin15 * sin2 * sin2

l1o1 = l1o1 - .0523 * sin15 * cos6 * cos6 + .0563 * cos6 * sin15 * sin15

l1o2 = .1908 - .8303 * sin2 - .3383 * sin9 + 1.9176 * cos6 + .1123 * sin2 * sin2 - .6608 * sin9 * sin9

l1o2 = l1o2 + .3892 * cos6 * cos6 + .1806 * sin2 * sin9 + .2171 * sin2 * cos6 + .0592 * sin9 * cos6

l1o2 = l1o2 + .0139 * sin2 * sin9 * cos6 + .5024 * sin2 * sin2 * sin2 + .0399 * sin9 * sin9 * sin9 - 1.1168 * cos6 * cos6 * cos6

l1o2 = l1o2 + -.1353 * sin9 * sin2 * sin2 + .1881 * sin2 * sin9 * sin9 + .0512 * sin2 * cos6 * cos6 - .2117 * cos6 * sin2 * sin2

l1o2 = l1o2 + .1073 * cos6 * sin9 * sin9 + .0649 * sin9 * cos6 * cos6

l1o3 = .4092 * sin2 - .3528 * sin9 - .2975 * sin6 - .2772 * cos6 + .2512 * cos12 + .2035 * cos9 - .174 * sin20

l1o3 = l1o3 + -.1572 * cos18 + .1487 * cos15 - .1487 * cos23 + .1073 * sin15 + .114 * cos2 - .0981 * sin12

l1o3 = l1o3 + -.0834 * sin26 + .0472 * cos26 + .0421 * cos20 - .0275 * sin23 + .0245 * sin18

l2o3 = -.0003 + .3299 * l1o1 + .1325 * cos9 - .1543 * cos23 - .1321 * sin12 - .1084 * sin26 + .0923 * cos15

l2o3 = l2o3 + .1066 * sin15 + .0614 * cos26 - .0357 * sin23 - .1131 * sin20 + .0288 * sin18 + .1001 * cos18

l2o3 = l2o3 + -.093 * cos2 - .0263 * cos20 - .0748 * cos12 + .1838 * cos6 - .0479 * sin93 + .2072 * sin6

l2o3 = l2o3 + -.1522 * sin2 + .6088 * l1o6 + .0278 * l1o2 + .3372 * l1o5 - .2977 * l1o3 + .6479 * l1o4

dym = l2o3 * 3.7356 - .3038: END SUB

SUB rk4 (y, dydx, x, h, yout)

hh = h * .5: h6 = h / 6: xh = x + hh: yt1 = y + hh * dydx: CALL derivs(xh, yt1, dyt): yt1 = y + hh * dyt: CALL derivs(xh, yt1, dym)

yt1 = y + h * dym: dym = dyt + dym: CALL derivs(x + h, yt1, dyt): yout = y + h6 * (dydx + dyt + 2 * dym): END SUB

FUNCTION y1 (x1)

y = 0@: x = 0@: dydx = 1.1571: h = 1 / 255

FOR i = 1 TO x1: CALL rk4(y, dydx, x, h, yout): x = x + h: y = yout: CALL derivs(x, y, dydx): NEXT i: y1 = yout * 13.373: END FUNCTION

FUNCTION yt (x) : y0 = .0116 * x + .4062

y0 = y0 + -.0007 * LOG(1 + 10 ^ (x - 2)) + -.0529 * LOG(1 + 10 ^ (x - 4)) + .0778 * LOG(1 + 10 ^ (x - 6))

y0 = y0 + -.0127 * LOG(1 + 10 ^ (x - 8)) + .0057 * LOG(1 + 10 ^ (x - 10)) + -.0192 * LOG(1 + 10 ^ (x - 12))

y0 = y0 + .0243 * LOG(1 + 10 ^ (x - 14)) + -.0491 * LOG(1 + 10 ^ (x - 16)) + .0596 * LOG(1 + 10 ^ (x - 18))

y0 = y0 + -.0763 * LOG(1 + 10 ^ (x - 20)) + .0274 * LOG(1 + 10 ^ (x - 22)) + .0037 * LOG(1 + 10 ^ (x - 24))

y0 = y0 + -.0014 * LOG(1 + 10 ^ (x - 26)) + .0296 * LOG(1 + 10 ^ (x - 28)) + .0177 * LOG(1 + 10 ^ (x - 30))

y0 = y0 + -.0268 * LOG(1 + 10 ^ (x - 32)) + -.0704 * LOG(1 + 10 ^ (x - 34)) + .0091 * LOG(1 + 10 ^ (x - 36))

y0 = y0 + .0333 * LOG(1 + 10 ^ (x - 38)) + .05 * LOG(1 + 10 ^ (x - 40)) + -.0152 * LOG(1 + 10 ^ (x - 42))

y0 = y0 + -.0289 * LOG(1 + 10 ^ (x - 44)) + .013 * LOG(1 + 10 ^ (x - 46)) + .0116 * LOG(1 + 10 ^ (x - 48))

y0 = y0 + -.0414 * LOG(1 + 10 ^ (x - 50)) + .0419 * LOG(1 + 10 ^ (x - 52)) + -.002 * LOG(1 + 10 ^ (x - 54))

y0 = y0 + .0003 * LOG(1 + 10 ^ (x - 56)) + .0201 * LOG(1 + 10 ^ (x - 58)) + -.0488 * LOG(1 + 10 ^ (x - 60))

y0 = y0 + .0408 * LOG(1 + 10 ^ (x - 62)) + -.0013 * LOG(1 + 10 ^ (x - 64)) + -.016 * LOG(1 + 10 ^ (x - 66))

y0 = y0 + -.0098 * LOG(1 + 10 ^ (x - 68)) + -.0348 * LOG(1 + 10 ^ (x - 70)) + .0376 * LOG(1 + 10 ^ (x - 72))

y0 = y0 + .0334 * LOG(1 + 10 ^ (x - 74)) + -.0432 * LOG(1 + 10 ^ (x - 76)) + -.0107 * LOG(1 + 10 ^ (x - 78))

y0 = y0 + -.0038 * LOG(1 + 10 ^ (x - 80)) + -.0302 * LOG(1 + 10 ^ (x - 82)) + .0381 * LOG(1 + 10 ^ (x - 84))

y0 = y0 + -.002 * LOG(1 + 10 ^ (x - 86)) + .0344 * LOG(1 + 10 ^ (x - 88)) + -.0251 * LOG(1 + 10 ^ (x - 90))

y0 = y0 + .0087 * LOG(1 + 10 ^ (x - 92)) + .0093 * LOG(1 + 10 ^ (x - 94)) + -.0687 * LOG(1 + 10 ^ (x - 96))

y0 = y0 + .0727 * LOG(1 + 10 ^ (x - 98)) + -.0441 * LOG(1 + 10 ^ (x - 100))

yt = y0: END FUNCTION

Data Modeling augmentation of JPEG for real-time streaming video


and James W. Handleyb,c

adtech Systems Inc., P.O. Box 18924, Huntsville, AL 35804 bJames Cook University, Townsville QLD 4811, Australia

cSparta, Inc., 6000 Technology Dr., Bldg. 3, Huntsville, AL 35805

ABSTRACT

This paper explores sub-sampling in conjunction with JPEG compression algorithms. Rather than directly compressing

large high-resolution images, we propose decimation to thumbnails followed by compression. This enables Redundant

Array of Independent Disks (RAID) compression and facilitates real-time streaming video with small bandwidth

requirements. Image reconstruction occurs on demand at the receiver to any resolution required using Data Modeling

based fractal interpolation. The receive side first uncompresses JPEG and then fractal interpolates to any required

resolution. This device independent resolution capability is useful for real-time sharing of image data across virtual

networks where each node has a different innate resolution capability. The same image is constructed to whatever

limitations exist at each individual node, keeping image data device independent and image resolution scalable up or

down as hardware/bandwidth limitations and options evolve.

Keywords: Data Modeling, JPEG, JPEG2000, image compression, decimation, bandwidth, image resolution

1. CONCEPT

This paper examines sub-sampling used in conjunction with JPEG algorithms as shown in Figure 1. The approach starts

with an image and JPEG compression requiring a certain amount of time and memory. By first sub-sampling the image

down to a thumbnail, a smaller image to which to apply JPEG compression results. This thumbnail will compress much

faster and smaller than the original. The receive side then uncompresses the compressed thumbnail and fractal

interpolates to any resolution needed, including to super-resolution if desired. This device independent resolution

capability is useful for real-time sharing of image data across the virtual network because each node will have a different

innate resolution capability and limitations. This keeps image data device independent and image resolution scalable up

or down as hardware/bandwidth limitations and options evolve.

Original

ImageJPEG

Compressed

ThumbnailDecimated

Thumbnail

Decimated

ThumbnailSub-Sampling

Fig. 1. Data Modeling augmentation of JPEG.

The process of system identification followed by parameter estimation applied to solving the system inverse problem

yields an equation. This equation can be a differential equation, a particular solution in the form of an algebraic equation,

or a general solution in the form of a partial differential equation. In any case, the derived equation is what we refer to as

a Data Model. The Data Model is the smallest amount of information required to functionally model empirical dynamics.

This model can take the form of a group of real numbers, a single equation, or a network of equations. Data Models are

also variables that can be embedded in other functions as variables, therefore the term used is functional. Data Modeling

approximates differential equations and their forcing functions using algebraic basis functions in the specific form of the

Kolmogorov-Gabor (K-G) polynomial. The K-G polynomial is numerically derived using a hierarchy of third or lower

order polynomials as intermediate variables. Nesting these functional models yields the Data Relations Model1,2


LxxxyO 3),,,( 21 =K (1)



2. FRACTAL DECIMATION AND RECONSTRUCTION

Discrete cosine transform (DCT) and wavelet based compression algorithms adhering to the JPEG and JPEG2000

standards3 yield varying compression sizes on equal size images, because these compression techniques are based on

image complexity. In contrast, sub-sampling yields a fixed image thumbnail size that is known in advance and is not

dependent on image complexity. Rather, sub-sampling is dependent only on the sub-sample rate and the original size.

Using Data Modeling, the data is first converted from 2-D into a 1-D data sequence. Several methods exist, including

raster scanning and fixed pattern readout such as zigzag sequencing. However, only the Hilbert sequence space filling

fractal curve4 preserves 2-D correlations at dyadic sample sizes. Figure 2 demonstrates the Hilbert fractal curve. The

Data Model of the image is the Turlington polynomial5,6 of the resultant 1-D data sequence, which provides an equation

model of the thumbnail with continuous derivatives. The Turlington polynomial is of the form

( )

∑−

=

−

−⎥⎥⎦

⎤

⎢⎢⎣

⎡+−+−+=

1

2

001.101111 101log)(001.)()(

n

j

xx

jj

j

mmxxmyxT

jj

jj

jxx

yym

−

−=

+

+

1

1 (2)

Details on constructing these models are found in the References, and for brevity the Data Models are represented in this

paper by the actual sub-sampled points from the images.

Fig. 2. Example of Hilbert sequencing.

Once a dyadic block or image size is determined for the Hilbert sequence, the image is sampled independently for the 3

different color planes (red, green, and blue). This results in 3 separate data sequences whose values are between 0 and

255 (in the case of 24 bit images). Each data sequence can be processed separately and recombined into an image after

processing. The color gray is created when equal amounts of red, green, and blue are mixed together. To transform the

red, green, and blue data for each pixel into grayscale, the average of the red, green, and blue color planes is calculated.

3

bluegreenred

gray

yyyy

++= (3)

IFS fractal interpolation7,8,9 from the literature is recast into the functional form

),(),( iiiiii fydxcexayxF +++= (4)

where a new x and y value is derived on the left hand side of the equation from applying the ith transform to the previous

x and y locations in the iteration process shown on the right hand side of the equation. This process still requires iterating

through all of the previous steps to get to the proper (x, y) location pair and IFS transform. By assuming x is a

monotonically increasing index of uniform spacing, a unique fractal pseudo-derivative operator is defined as

{ } ).()())(()(

110111110111

−+−−−+−−+− −−+−+⎥⎦

⎤⎢⎣⎡ −−

−−

≡ nnninniNnnnn

if yyyyyyyxN

yyyy

N

yyyD (5)

where N is the number of points in the original process Data Model and vanishes if N = 1, y0 the value of the Data Model

at x = 0, yN the value at x = N, i the index associated with the (x,y) pair that is transformed into the needed function pair,

and n the index of the straight line segment connecting the original Data Model points defining the interval containing

the required xi. This operator is derived from the IFS equations in References 7-9, and Equation 5 is the Data Model of

the process10-13.

3. RESULTS

3.1 Image data sources

Two grayscale image classes stored in 24-bit color format were used for this study. Sizes in megabytes (MB) and in

pixels is given later in this work in Table 1. Because Hilbert sequencing requires images to be dyadic (power of two),

these images were padded up to the next highest power of two. One method of padding that would maintain the mean of

the image would be to actually pad the image its mean value. However, this does not preserve the other higher order

statistics and would provide sharp discontinuities all along the edges where the padding is added. These sharp

discontinuities would cause the fractal reconstruction algorithm to fail and were therefore avoided. Instead, the images

were padded with data from the image itself. Rows and columns taken from image and appended in reverse sequence to

top and right and the sub-sampling rates increased to insure equivalent thumbnails generated. An example of this dyadic

padding is shown in Figure 3.

Original

Padded

Fig. 3. Image 1 with no padding (left), and with padding on its top and right (right). Note symmetry along top and right edge.

3.2 Figures of merit for scoring results

To score image processing and reconstruction results, one or more figures of merit (FOM) must be selected. One simple

method is through use of standard statistics and higher order moments (M1 – M8 in Equations 6 - 11) given by

xxMN

j

jN== ∑

=1

11

01

=Mµ 21

=Mσ (6)

σ=−= ∑=

−

N

j

jNxxM

1

2

11

2 )( 12

=Mµ 32

=Mσ (7)

∑= ⎥

⎥⎦

⎤

⎢⎢⎣

⎡ −=

N

j

j

N

xxM

1

3

13 σ

03

=Mµ 153

=Mσ (8)

31

4

14 −

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −= ∑

=

N

j

j

N

xxM

σ 3

4=Mµ 96

4=Mσ (9)

151

6

16 −

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −= ∑

=

N

j

j

N

xxM

σ 15

6=Mµ 100

6=Mσ (10)

1051

8

18 −

⎪⎩

⎪⎨⎧

⎪⎭

⎪⎬⎫

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −= ∑

=

N

j

j

N

xxM

σ 105

8=Mµ 1407

8=Mσ (11)

where the µ term after each is the mean expected value of the raw moment calculation for a Gaussian distribution and the

σ term is the standard deviation of the moment for a Gaussian distribution, forming a confidence interval around the

values of the moment.

It was found that the mean squared error (MSE) and peak signal to noise ratio (PSNR) from the literature along with a

novel histogram correlation method worked well on the images described above, and therefore were applied to the Data

Model to compare each with the original. The mean squared error (MSE) and the peak signal to noise ratio (PSNR) are

given by

∑∑ −=i

ii yxMSE2)( ∑= )

255log(10

2

MSEPSNR (12)

Jaenisch created a novel FOM for use in numerical perceptual image scoring called Histogram correlation (Figure 4).

This method extracts a window from the two images being compared and generates a histogram for each. This histogram

is then converted into a time series of data and the correlation between the two calculated.

WindowExtracted

From SameLocation in

Each Image

Histogram

Generated

Histogram

Converted toTime Series

Correlation

Original

Reconstructed

WindowExtracted

From SameLocation in

Each Image

Histogram

Generated

Histogram

Converted toTime Series

Correlation

Original

Reconstructed

Fig. 4. Calculation of the Histogram correlation FOM.

Histogram correlation is given in equation form as

∑∑∑

∑−−

−−=

i

i

i

i

i

ii

yyxx

yyxx

r22 )()(

))(( (13)

For MSE, lower values represent a closer match to the original, while for PSNR, higher values represent a closer match.

For the correlation FOM, higher values represent closer matches to the original. For scoring images with Histogram

Correlation, MSE, or PSNR, windows ranging in size between 3 x 3 and 256 x 256 were used. 16 random placed

windows of each size were extracted from the images and the FOMs calculated inside of each window. For each window

size, the results from the 16 windows are summed together for each FOM and the resultants summed together for all

window sizes between 3 x 3 and 256 x 256. In the case of MSE, a lower value is better, indicating less error. For PSNR

and Histogram Correlation, higher values are better, indicating more correlation and higher SNR11.

3.3 Final images

The two image types were first decimated 10:1 and 100:1 and then reconstructed using Data Modeling based fractal

reconstruction. The original images along with 10:1 and 100:1 thumbnail sub-sampling and Data Modeling

reconstruction are shown in Figure 5. The Correlation FOM, PSNR, and MSE are given in Table 1 for images that were

decimated both at 10:1 and 100:1, JPEG compressed to the compression ratios varying between 50:1 and 704:1 as shown

in Table 1, and finally by reversing the process by uncompressing using JPEG and then applying Data Modeling fractal

reconstruction to inflate the uncompressed but decimated thumbnail back to its original size.

Fig. 5. Image type 1 and 2. 50:1 (top left), 89:1 (top right), original (middle), 459:1 (bottom left), and 704:1 (bottom right).

Image Name and

Sub-Sample Rate

Original

Size (KB)

Original

Size (Pixels)

Correlation

FOM

PSNR

(104)

MSE

(106)

JPEG Compressed

Size (KB)

Overall

Compression Ratio

Image 1 (10:1) 9700 1800 x 1800 2960 8.8 1.8 181 50:1

Image 2 (10:1) 9700 1800 x 1800 2208 12.1 0.3 109 89:1

Image 1 (100:1) 9700 1800 x 1800 2560 6.9 5.6 21 459:1

Image 2 (100:1) 9700 1800 x 1800 1440 10.7 0.3 13 704:1

Table 1. Statistics and FOM scores for images.

Table 1 summarizes FOM calculations for each reconstructed image compared with the original. As shown in Figure 5,

the Data Modeling approach provides good images at both 10:1 and 100:1 decimation and further JPEG compression.

Also shown in Table 1 is the resultant image size in kilobytes after compressing the thumbnail using JPEG set at a

quality factor of 90%. This quality factor was chosen to insure that no major information loss or artifacts were introduced

to the image by the JPEG algorithm. For each image, the overall compression ratio after using both fractal decimation

and JPEG compression were actually 5 to 9 times larger than the original thumbnail compression of 10:1 and 100:1.

From this, it is estimated that in order to obtain overall compression of 10:1, it is only necessary to sub-sample 2:1, and

to achieve overall compression of 100:1, it is only necessary to sub-sample 20:1. Fortran 95 source code for sub-

sampling, compressing, and reconstructing the images in Figure 5 is provided in the Appendix of this work.

Zooming into the resulting image type 1 after applying 100:1 decimation, JPEG compression, JPEG uncompression, and

Data Modeling fractal reconstruction is shown in Figure 6. Note that at this block size, only 4 pixels are saved as the

model of the block. Figure 6 shows that the fractal reconstructed method at fine resolution shows detail that is not

present in images that are inflated using methods such as linear interpolation. On the left in Figure 6 is a 16 x 16 pixel

neighborhood from the original image. In the middle of Figure 6 is the resultant of applying decimation, JPEG, and

linear interpolation, where only four values (model values) in the neighborhood were simply enlarged without adding

detail. On the right in Figure 6, the Data Modeling approach in the place of linear interpolation is shown. Our method

reconstructs detail from only the four pixel values used by linear interpolation.

Fig. 6. Zoom in to 16 x 16 pixel neighborhood on image type 1. Original (left), 100:1 decimation, JPEG compression/uncompression,

and linear interpolation to undo decimation showing lack of detail (center), and Data Modeling fractal reconstruction applied instead

of liner interpolation (right). Note that the Data Modeling result contains detail like that in the original.

3.4 Sub-sample rate and block size estimation Data Model

Final images shown in Figure 5 were sub-sampled and reconstructed to their original size using all available pixels in the

original padded image (block size of 2048x2048). For images where the dynamics of the local pixel neighborhoods vary,

better reconstruction results are obtained using image sub-blocks and performing reconstruction on each local

neighborhood independently. To explore the limiting performance of the algorithm, we varied the sub-sample rates from

1.5:1 to 16384:1 and the local pixel neighborhoods between 8x8 pixel blocks to 2048x2048. This is not mandatory in the

image decimation/compression/reconstruction process. The authors have found that a good rule of thumb given no

further information is to use 10:1 sub-sampling with block sizes of either 8x8 or 16x168. Figure 7 (top) shows the

Histogram Correlation FOM for image class 1 and 2 for varying. In the top 2 curves, increasing block sizes are shown

concatenated together for increasing sub-sample rates (which also increases from left to right on the top curves). Figure 7

(bottom) reverses the sequence by displaying groups of increasing sub-sample rates concatenated together for each

reconstruction block size (which also increases from left to right on the bottom curves).

0

500

1000

1500

2000

2500

3000

3500

4000

His

tog

ram

Co

rrela

tio

n F

OM

1.5 Sub-Sample Rate 16384

*Note that block size increments for each sub-sample rate

Image Class 1

0

500

1000

1500

2000

2500

3000

3500

4000

His

tog

ram

Co

rre

lati

on

FO

M

1.5 Sub-Sample Rate 16384

*Note that block size increments for each sub-sample rate

Image Class 2

0

500

1000

1500

2000

2500

3000

3500

4000

His

tog

ram

Co

rrela

tio

n F

OM

8 Block Size 2048

*Note that sub-sample rate increments for each block size

Image Class 1

0

500

1000

1500

2000

2500

3000

3500

4000

His

tog

ram

Co

rre

lati

on

FO

M

8 Block Size 2048

*Note that sub-sample rate increments for each block size

Image Class 2

Fig. 7. Varying sub-sample rates from 1.5:1 up to 16384:1 for image class 1 (top left) and image class 2 (top right). Varying block

sizes from 8x8 to 2048x2048 for image class 1 (bottom left) and image class 2 (bottom right).

The results curves shown in Figure 7 required many hours of CPU time to run the approximately 150 cases on each

image type (300 cases total), with each case requiring several hundred sub-cases to generate the windows to characterize

the figures of merit described in Section 3.2. Using the results of this analysis, a Data Model was constructed to

eliminate exhaustive characterization of new images that are of the same or similar class as those in Figure 5 for

determination of optimal block size and sub-sample rate. This Data Model was constructed using 1) desired Histogram

correlation FOM value for final result and 2) average standard deviation of 16 randomly selected 8x8 pixel block

neighborhoods in the original, and yielded as output an estimate of the optimal sub-sample rate and block size to use in

the Data Modeling fractal reconstruction process. This Data Model was created using the data curves in Figure 7 and

multi-variable linear regression using the nested Kolmogorov-Gabor polynomial form in Equation 11,2. Output results

from the Data Model compared with the training data is provided in Figure 8.

Original

DataModel

Su

b-S

am

ple

Ra

te

Blo

ck

Siz

e

Index Index Fig. 8. Predictive Data Model results compared to original for sub-sample rate (left) and block size (right).

Source code for this predictive Data Model is provided in Figure 9, and two lookup tables generated from these equation

models are provided in Figure 10. By finding the average standard deviation from 16 8x8 blocks from your image across

the bottom and the desired Histogram Correlation FOM along the side in the top Figure 10 lookup table, the optimal sub-

sample rate is read off directly. Using the same two inputs in the bottom Figure 10 lookup table, the optimal block size

used in the Data Modeling reconstruction process is read off directly.

OPEN "acorrel" FOR INPUT AS #1IN PUT #1, correl 'histogram correl fomCLOSE #1OPEN "astdev" FOR INPUT AS #1INPUT #1, stdev 'standard deviationCLOSE #1bcorrel = (correl - .4728) / .2133astdev = (stdev - 16.5976) / 8.0445GOSUB subsampasubsamp = l1o1 'log10 of subsamp rateombsuba = (asubsamp - 1.5932) / .8939mbstdev = (stdev - 16.3419) / 8.0329GOSUB blocksizablksiz = l4o1OPEN "output" FOR OUTPUT AS #1'subsamp rate and block sizePRINT #1, 10 ^ asubsamp, ablksizCLOSE #1END

subsamp:l1o1 = -.5407l1o1 = l1o1 + (-.4536) * (bcorrel)l1o1 = l1o1 + (.9381) * (astdev)l1o1 = l1o1 + (.3418) * (bcorrel * bcorrel)l1o1 = l1o1 + (.1327) * (astdev * astdev)l1o1 = l1o1 + (-.0858) * (bcorrel * astdev)l1o1 = l1o1 + (-.314) * (bcorrel * bcorrel * bcorrel)l1o1 = l1o1 + (-.2227) * (astdev * astdev * astdev)l1o1 = l1o1 + (-.1637) * (astdev * bcorrel * bcorrel)l1o1 = l1o1 + (-.0133) * (bcorrel * astdev * astdev)l1o1 = l1o1 * (.7468) + (1.3849)IF l1o1 < .1761 THEN l1o1 = .1761IF l1o1 > 4.2144 THEN l1o1 = 4.2144RETURN

blocksiz:l1o1 = -.2149l1o1 = l1o1 + (-.3765) * (mbstdev)l1o1 = l1o1 + (.5527) * (ombsuba)l1o1 = l1o1 + (-.2344) * (mbstdev * mbstdev)l1o1 = l1o1 + (.4856) * (ombsuba * ombsuba)l1o1 = l1o1 + (-.2067) * (mbstdev * ombsuba)l1o1 = l1o1 + (.6168) * (mbstdev * mbstdev * mbstdev)l1o1 = l1o1 + (-.154) * (ombsuba * ombsuba * ombsuba)l1o1 = l1o1 + (-.0497) * (ombsuba * mbstdev * mbstdev)l1o1 = l1o1 + (.0898) * (mbstdev * ombsuba * ombsuba)l1o2 = 0l1o2 = l1o2 + (.4639) * (mbstdev)l1o2 = l1o2 + (.2992) * (ombsuba)l2o3 = .7634l2o3 = l2o3 + (3.1154) * (l1o1)l2o3 = l2o3 + (5.9309) * (mbstdev)l2o3 = l2o3 + (-7.0514) * (l1o2)l2o3 = l2o3 + (-6.4967) * (l1o1 * l1o1)l2o3 = l2o3 + (-5.6109) * (mbstdev * mbstdev)l2o3 = l2o3 + (-26.2132) * (l1o2 * l1o2)

l2o3 = l2o3 + (1.8352) * (l1o1 * mbstdev)l2o3 = l2o3 + (10.0944) * (l1o1 * l1o2)l2o3 = l2o3 + (23.2086) * (mbstdev * l1o2)l2o3 = l2o3 + (-33.6865) * (l1o1 * mbstdev * l1o2)l2o3 = l2o3 + (-.3964) * (l1o1 * l1o1 * l1o1)l2o3 = l2o3 + (-6.604) * (mbstdev * mbstdev * mbstdev)l2o3 = l2o3 + (-14.0569) * (l1o2 * l1o2 * l1o2)l2o3 = l2o3 + (2.6917) * (mbstdev * l1o1 * l1o1)l2o3 = l2o3 + (5.7411) * (l1o1 * mbstdev * mbstdev)l2o3 = l2o3 + (31.1435) * (l1o1 * l1o2 * l1o2)l2o3 = l2o3 + (-1.3842) * (l1o2 * l1o1 * l1o1)l2o3 = l2o3 + (11.6813) * (l1o2 * mbstdev * mbstdev)l2o3 = l2o3 + (5) * (mbstdev * l1o2 * l1o2)l3o2 = -.2408l3o2 = l3o2 + (.58) * (l2o3)l3o2 = l3o2 + (.2969) * (ombsuba)l3o2 = l3o2 + (-.0829) * (mbstdev)l3o2 = l3o2 + (.219) * (l2o3 * l2o3)l3o2 = l3o2 + (-.2805) * (ombsuba * ombsuba)l3o2 = l3o2 + (.5857) * (mbstdev * mbstdev)l3o2 = l3o2 + (.3052) * (l2o3 * ombsuba)l3o2 = l3o2 + (-.1895) * (l2o3 * mbstdev)l3o2 = l3o2 + (-.1353) * (ombsuba * mbstdev)l3o2 = l3o2 + (.8291) * (l2o3 * ombsuba * mbstdev)l3o2 = l3o2 + (-.4954) * (l2o3 * l2o3 * l2o3)l3o2 = l3o2 + (.2144) * (ombsuba * ombsuba * ombsuba)l3o2 = l3o2 + (-.2555) * (mbstdev * mbstdev * mbstdev)l3o2 = l3o2 + (-.8619) * (ombsuba * l2o3 * l2o3)l3o2 = l3o2 + (.1994) * (l2o3 * ombsuba * ombsuba)l3o2 = l3o2 + (2.2746) * (l2o3 * mbstdev * mbstdev)l3o2 = l3o2 + (-.4046) * (mbstdev * l2o3 * l2o3)l3o2 = l3o2 + (-.2718) * (mbstdev * ombsuba * ombsuba)l3o2 = l3o2 + (-.9601) * (ombsuba * mbstdev * mbstdev)l4o1 = -.2943l4o1 = l4o1 + (2.4065) * (l3o2)l4o1 = l4o1 + (.3783) * (ombsuba)l4o1 = l4o1 + (-1.625) * (mbstdev)l4o1 = l4o1 + (1.5285) * (l3o2 * l3o2)l4o1 = l4o1 + (.1599) * (ombsuba * ombsuba)l4o1 = l4o1 + (.1683) * (mbstdev * mbstdev)l4o1 = l4o1 + (.0643) * (l3o2 * ombsuba)l4o1 = l4o1 + (-1.4337) * (l3o2 * mbstdev)l4o1 = l4o1 + (-.2452) * (ombsuba * mbstdev)l4o1 = l4o1 + (.7922) * (l3o2 * ombsuba * mbstdev)l4o1 = l4o1 + (-.5002) * (l3o2 * l3o2 * l3o2)l4o1 = l4o1 + (.1066) * (ombsuba * ombsuba * ombsuba)l4o1 = l4o1 + (.3336) * (mbstdev * mbstdev * mbstdev)l4o1 = l4o1 + (-.3695) * (ombsuba * l3o2 * l3o2)l4o1 = l4o1 + (-.3886) * (l3o2 * ombsuba * ombsuba)l4o1 = l4o1 + (.788) * (l3o2 * mbstdev * mbstdev)l4o1 = l4o1 + (-.4752) * (mbstdev * l3o2 * l3o2)l4o1 = l4o1 + (.2214) * (mbstdev * ombsuba * ombsuba)l4o1 = l4o1 + (-.8907) * (ombsuba * mbstdev * mbstdev)l4o1 = l4o1 * (761.0519) + (831.2941)IF l4o1 < 8 THEN l4o1 = 8IF l4o1 > 2048 THEN l4o1 = 2048RETURN

Fig. 9. QuickBASIC 4.5 source for determining optimal sub-sample rate and block size.

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 2 2

3 3 2 2 2 2 2 2 2 2 2 3 3 3 4 4 5 5 6 6 6 6 6 5 5

4 3 3 3 2 2 2 3 3 3 3 4 4 5 6 7 8 9 10 11 11 11 11 11 10

4 3 3 3 3 3 3 3 3 4 4 5 6 7 8 10 11 13 15 16 17 18 18 18 16

4 4 3 3 3 3 3 3 4 4 5 6 7 9 10 12 14 17 19 22 24 25 26 26 24

4 4 3 3 3 3 3 4 4 5 6 7 8 10 12 15 18 21 24 28 31 33 34 34 33

4 4 3 3 3 3 4 4 5 5 6 8 9 12 14 17 21 25 29 34 38 41 43 44 43

5 4 4 4 4 4 4 5 5 6 8 9 11 14 17 21 26 31 36 42 48 52 55 56 55

7 6 5 5 5 5 6 6 7 8 10 12 15 18 22 28 34 40 48 56 63 69 73 74 73

10 9 8 8 8 8 8 9 11 13 15 18 22 27 33 40 49 59 70 81 91 99 105 107 105

20 17 15 14 14 14 15 17 19 22 26 32 38 46 57 69 83 99 117 134 150 164 172 175 170

49 41 37 34 33 34 35 38 43 49 58 68 82 99 119 143 171 202 235 267 297 321 334 336 323

163 135 117 107 103 103 106 114 126 142 163 190 225 267 318 377 444 517 593 666 730 777 800 793 753

760 613 523 469 441 432 439 461 498 551 622 713 827 964 1126 1312 1518 1737 1956 2160 2326 2434 2464 2401 2243

5154 4062 3382 2962 2717 2599 2581 2648 2797 3026 3338 3741 4237 4831 5520 6293 7125 7978 8796 9505 10024 10272 10181 9719 8896

16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383

16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383

1.00

0.95

0.90

0.85

0.80

0.75

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0.65

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5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0 21.0 22.0 23.0 24.0 25.0 26.0 27.0 28.0 29.0

Standard Deviation (averaged from 16 8x8 blocks)

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 2 2

3 3 2 2 2 2 2 2 2 2 2 3 3 3 4 4 5 5 6 6 6 6 6 5 5

4 3 3 3 2 2 2 3 3 3 3 4 4 5 6 7 8 9 10 11 11 11 11 11 10

4 3 3 3 3 3 3 3 3 4 4 5 6 7 8 10 11 13 15 16 17 18 18 18 16

4 4 3 3 3 3 3 3 4 4 5 6 7 9 10 12 14 17 19 22 24 25 26 26 24

4 4 3 3 3 3 3 4 4 5 6 7 8 10 12 15 18 21 24 28 31 33 34 34 33

4 4 3 3 3 3 4 4 5 5 6 8 9 12 14 17 21 25 29 34 38 41 43 44 43

5 4 4 4 4 4 4 5 5 6 8 9 11 14 17 21 26 31 36 42 48 52 55 56 55

7 6 5 5 5 5 6 6 7 8 10 12 15 18 22 28 34 40 48 56 63 69 73 74 73

10 9 8 8 8 8 8 9 11 13 15 18 22 27 33 40 49 59 70 81 91 99 105 107 105

20 17 15 14 14 14 15 17 19 22 26 32 38 46 57 69 83 99 117 134 150 164 172 175 170

49 41 37 34 33 34 35 38 43 49 58 68 82 99 119 143 171 202 235 267 297 321 334 336 323

163 135 117 107 103 103 106 114 126 142 163 190 225 267 318 377 444 517 593 666 730 777 800 793 753

760 613 523 469 441 432 439 461 498 551 622 713 827 964 1126 1312 1518 1737 1956 2160 2326 2434 2464 2401 2243

5154 4062 3382 2962 2717 2599 2581 2648 2797 3026 3338 3741 4237 4831 5520 6293 7125 7978 8796 9505 10024 10272 10181 9719 8896

16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383

16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383 16383

1.00

0.95

0.90

0.85

0.80

0.75

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Sub-Sample Rate

5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0 21.0 22.0 23.0 24.0 25.0 26.0 27.0 28.0 29.0


2048 1032 428 125 8 8 8 8 8 2048 1391 2048 1693 74 356 238 324 833 1857 1974 1501 2048 8 8 8

2048 1032 428 125 8 8 8 8 8 2048 1391 2048 1693 74 356 238 324 833 1857 1974 1501 2048 8 8 8

2048 1032 428 125 8 8 8 8 8 2048 1391 2048 1693 74 356 238 324 833 1857 1974 1501 2048 8 8 8

8 1032 428 125 8 8 8 8 8 2048 1391 2048 1693 74 356 238 324 833 1857 1974 1501 2048 8 8 8

8 16 1370 121 8 8 8 8 8 2048 1391 2048 118 357 308 349 646 1078 1506 1775 1529 1779 8 8 8

8 2048 533 156 649 2048 8 2048 1841 2048 1757 38 55 18 108 352 488 437 414 413 506 741 2048 980 8

8 8 87 291 437 697 2048 172 1590 1397 232 48 101 349 448 60 34 173 8 391 143 860 1123 2048 8

8 8 68 428 331 209 152 572 1182 537 139 108 245 300 8 312 1361 923 8 577 8 2048 1942 1129 2048

8 8 75 468 242 326 295 686 775 315 126 140 226 70 8 1145 1977 723 8 679 8 2048 8 8 2048

8 8 82 485 189 356 382 622 540 227 120 140 150 8 215 1524 1814 138 233 395 549 1269 8 8 2048

8 8 86 480 154 320 388 485 359 171 110 117 65 8 414 1496 1232 8 818 8 1360 8 8 8 1136

8 8 43 388 197 211 283 297 213 129 91 71 8 8 470 1051 339 8 1258 8 2048 8 8 8 8

8 8 926 202 337 179 175 170 141 104 65 17 8 8 220 177 8 418 901 8 2048 558 763 8 125

8 8 2048 91 432 319 229 196 169 131 78 19 8 8 8 8 11 856 8 554 2048 2048 1721 8 8

8 8 1391 51 617 522 379 326 296 258 200 121 18 8 8 8 8 8 8 1795 2048 336 8 8 87

8 2048 885 1010 1032 444 331 345 362 370 365 332 253 118 8 8 8 8 1021 1784 1086 8 299 8 2048

2048 2048 2048 194 1179 350 2048 2048 2048 843 436 450 326 235 203 209 326 710 1296 1554 1234 1130 971 8 2048

2048 787 2048 990 8 8 8 8 8 8 8 2048 2048 2048 2048 2048 579 941 1653 1876 1745 1583 1438 2048 2048

2048 2048 1135 2048 8 8 8 8 8 8 8 2048 2048 2048 2048 1772 849 793 1564 2048 2048 2048 2048 2048 2048

8 8 2048 8 2048 2048 2048 2048 2048 2048 2048 2048 2048 246 8 8 8 2048 1290 8 1877 2048 2048 2048 2048

8 8 2048 8 2048 2048 2048 2048 2048 2048 2048 2048 2048 246 8 8 8 2048 1290 8 1877 2048 2048 2048 2048

1.00

0.95

0.90

0.85

0.80

0.75

0.70

0.65

0.60

0.55

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Block Size

5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0 21.0 22.0 23.0 24.0 25.0 26.0 27.0 28.0 29.0


2048 1032 428 125 8 8 8 8 8 2048 1391 2048 1693 74 356 238 324 833 1857 1974 1501 2048 8 8 8

2048 1032 428 125 8 8 8 8 8 2048 1391 2048 1693 74 356 238 324 833 1857 1974 1501 2048 8 8 8

2048 1032 428 125 8 8 8 8 8 2048 1391 2048 1693 74 356 238 324 833 1857 1974 1501 2048 8 8 8

8 1032 428 125 8 8 8 8 8 2048 1391 2048 1693 74 356 238 324 833 1857 1974 1501 2048 8 8 8

8 16 1370 121 8 8 8 8 8 2048 1391 2048 118 357 308 349 646 1078 1506 1775 1529 1779 8 8 8

8 2048 533 156 649 2048 8 2048 1841 2048 1757 38 55 18 108 352 488 437 414 413 506 741 2048 980 8

8 8 87 291 437 697 2048 172 1590 1397 232 48 101 349 448 60 34 173 8 391 143 860 1123 2048 8

8 8 68 428 331 209 152 572 1182 537 139 108 245 300 8 312 1361 923 8 577 8 2048 1942 1129 2048

8 8 75 468 242 326 295 686 775 315 126 140 226 70 8 1145 1977 723 8 679 8 2048 8 8 2048

8 8 82 485 189 356 382 622 540 227 120 140 150 8 215 1524 1814 138 233 395 549 1269 8 8 2048

8 8 86 480 154 320 388 485 359 171 110 117 65 8 414 1496 1232 8 818 8 1360 8 8 8 1136

8 8 43 388 197 211 283 297 213 129 91 71 8 8 470 1051 339 8 1258 8 2048 8 8 8 8

8 8 926 202 337 179 175 170 141 104 65 17 8 8 220 177 8 418 901 8 2048 558 763 8 125

8 8 2048 91 432 319 229 196 169 131 78 19 8 8 8 8 11 856 8 554 2048 2048 1721 8 8

8 8 1391 51 617 522 379 326 296 258 200 121 18 8 8 8 8 8 8 1795 2048 336 8 8 87

8 2048 885 1010 1032 444 331 345 362 370 365 332 253 118 8 8 8 8 1021 1784 1086 8 299 8 2048

2048 2048 2048 194 1179 350 2048 2048 2048 843 436 450 326 235 203 209 326 710 1296 1554 1234 1130 971 8 2048

2048 787 2048 990 8 8 8 8 8 8 8 2048 2048 2048 2048 2048 579 941 1653 1876 1745 1583 1438 2048 2048

2048 2048 1135 2048 8 8 8 8 8 8 8 2048 2048 2048 2048 1772 849 793 1564 2048 2048 2048 2048 2048 2048

8 8 2048 8 2048 2048 2048 2048 2048 2048 2048 2048 2048 246 8 8 8 2048 1290 8 1877 2048 2048 2048 2048

8 8 2048 8 2048 2048 2048 2048 2048 2048 2048 2048 2048 246 8 8 8 2048 1290 8 1877 2048 2048 2048 2048

1.00

0.95

0.90

0.85

0.80

0.75

0.70

0.65

0.60

0.55

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Block Size

5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0 21.0 22.0 23.0 24.0 25.0 26.0 27.0 28.0 29.0

Standard Deviation (averaged from 16 8x8 blocks) Fig. 10. Lookup table for determining optimal sub-sample rate (top) and block size (bottom).

3.5 Appendix sou

is Fortran 95. Input images are dyadic and in 4 column ASCII text. If input

ACKNOWLEDGEMENTS

The authors would like to thank the following individuals fo nce, direction, and support during the course of

REFER NCES

. Jaenisch, H.M., Handley, J.W., “Data Modeling for Radar Applications”, Proceedings of IEEE Radar Conference

2. urray, S., “Virtual Instrument Prototyping With Data Modeling”, JANNAF

3. , New York: Van Nostrand

4. oggins, C.J., Carroll, M.P., Handley, J.W., "Entropy Fractal Analysis of Medical Images Using

5. Edwards, M., “Data Modeling of deep sky

6. rds, M.,

7. mage Compression, Wellesley, MA: A.K. Peters, c1993.

1995.

pression”,

11. ., Deragopian, G., Payne, J.,

12. sing For Target

13. etection”, Society for

rce code description

The source code given in the Appendix

image is not dyadic, it must be padded as described in Section 3.1. Source code consists of 1) SUBSAMP, driver

program that decimates and calls the other programs; 2) BMP2TXT, creates a 4 column ASCII text file consisting of an

index, red color plane, green color plane, and blue color plane from a bitmap; 3) HILB, creates the dyadic Hilbert fractal

curve; 4) RECONST, inflates thumbnail back to original size; and 5) TXT2BMP, converts 4 column ASCII text file into

bitmap. Source code for BMP2TXT and TXT2BMP are not provided in this work, but can be obtained in EXE format

via the Internet. Also included is the source code for the JPEG compression algorithm applied to the decimated

thumbnails to obtain the final compression ratios.

r their guida

this work: Marvin Barnett, Computer Sciences Corporation; Richard Esslinger and Chester Rowe, Axiom, Inc.; Gary

Whitley, GME-C; Scott McPheeters and Tim Aden, MTHEL Project Office; and John Deacon, Sparta, Inc.

E

1

2003, Huntsville AL. May 18-19 2003.

Jaenisch, H., Handley, J., Pooley, J., M

39th Combustion, 27th Airbreathing Propulsion, 21st Propulsion Systems Hazards, and 3rd Modeling and

Simulation Subcommittees Joint Meeting, Colorado Springs, CO, December 1-5, 2003.

Pennebaker, W.B. and Mitchell, J.L., JPEG: Still Image Data Compression Standard

Reinhold, c1993.

Jaenisch, H.M., Sc

ROSETA, " Proceedings of SPIE, Los Angeles, CA, January 24, 1994.

Handley, J., Jaenisch, H., Lim, A., White, G., Hons, A., Filipovic, M.,

images”, SPIE Astronomical Telescopes and Instrumentation 2004, Glasgow, Scotland, UK, June 24, 2004.

Jaenisch, H., Handley, J., Lim, A., Filipovic, M., White, G., Hons, A., Deragopian, G., Schneider, M., Edwa

“Data Modeling for Virtual Observatory data mining”, SPIE Astronomical Telescopes and Instrumentation 2004,

Glasgow, Scotland, UK, June 24, 2004.

Barnsley M. F., and Hurd, L.P., Fractal I

8. Fisher, Y. ed. Fractal Image Compression: Theory and Application, New York: Springer-Verlag, c

9. Welstead, S. Fractal and Wavelet Image Compression Techniques, Bellingham, WA: SPIE Press, c1999.

10. Jaenisch, H.M., Taylor, S.C., Handley, J.W., Carroll, M.P., “Optimal Fractal Image and Data Com

Proceedings Of The Southeastern Simulation Conference ‘95, Orlando, FL, pp. 67-77.

Lim, A., Jaenisch, H., Handley, J., Filipovic, M., White, G., Hons, A., Berrevoets, C

Schneider, M., Edwards, M., “Image resolution and performance analysis of webcams for ground based astronomy”,

SPIE Astronomical Telescopes and Instrumentation 2004, Glasgow, Scotland, UK, June 24, 2004.

Jaenisch, H., Handley, J., Faucheux, J., “Data Modeling Enabled Real Time Image Proces

Discrimination”, Proceedings of SPIE, SPIE Defense and Security Symposium, March 2005.

Jaenisch, H.M., Handley, J.W., Pooley J.C., Murray S.R., “Data Modeling for Fault D

Machinery Failure Prevention Technology (MFPT), April, 2003.

APPENDIX

Driver Program (Image sub-sampling) (SUBSAMP) (Left) and Fractal Operator Algorithm (RECONST) (Right) program subsampimplicit noneinteger, allocatable :: rpt(:),gpt(:),bpt(:)integer, allocatable :: rpt1(:),gpt1(:),bpt1(:)integer*4 n,i,j,nth,smple1,ival,ijk,ijkoldinteger r,g,breal tmpcharacter*80 infilewrite (6,*)'Input bitmap name => 'read(5,*)infilewrite(6,*)'Sample rate (Every nth point, enter N) => 'read(5,*)nthwrite(6,*)write(6,*)'Converting bitmap to text'write(6,*)

c system call to create text file (4 cols, index, red, green, blue)call system("bmp2txt "//infile//" tmp1.txt")write(6,*)write(6,*)'Reading in data'write(6,*)open (1,file='tmp1.txt', status='unknown')

11 read(1,*,end=10)N,r,g,bgoto 11

10 rewind(1)n=n+1allocate (rpt(n),gpt(n),bpt(n),rpt1(n),gpt1(n),bpt1(n))do i=1,N

read(1,*)j,rpt(i),gpt(i),bpt(i)enddoclose(1)write(6,*)write(6,*)'Generate Hilbert transform'write(6,*)

c system call to run Hilbert programcall system("hilb")open(1,file='hilb.txt',status='unknown')write(6,*)write(6,*)'Apply Hilbert transform'write(6,*)do i=1,N

read(1,*)jrpt1(i)=rpt(j)gpt1(i)=gpt(j)bpt1(i)=bpt(j)

enddoclose(1)write(6,*)write(6,*)'Write out subsampled data'write(6,*)open(1,file='tmp1.txt',status='unknown')smple1=int(N/nth)smple1=int(sqrt(real(smple1)))smple1=smple1*smple1do i=1,smple1

tmp=real(N-1)/real(smple1-1)tmp=tmp*real(i-1)+1.0ival=nint(tmp)write(1,*)i-1,rpt1(ival),gpt1(ival),bpt1(ival)

enddoclose(1)write(6,*)write(6,*)'Reconstruct to original length'write(6,*)

c system call to run reconstruction programcall system ("reconst")write(6,*)write(6,*)'Read in reconstructed data'write(6,*)open(1,file='recon.txt',status='unknown')do i=1,N

read(1,*)j,rpt(i),gpt(i),bpt(i)enddoclose(1)open(1,file='hilb.txt',status='unknown')write(6,*)write(6,*)'Apply inverse Hilbert transform'write(6,*)do i=1,N

read(1,*)jrpt1(j)=rpt(i)gpt1(j)=gpt(i)bpt1(j)=bpt(i)

enddoclose(1)write(6,*)write(6,*)'Write out final result'write(6,*)open(1,file='tmp1.txt',status='unknown')do i=1,N

write(1,*)i-1,rpt1(i),gpt1(i),bpt1(i)enddoclose(1)

c cleanupcall system("txt2bmp tmp1.txt recon.bmp")call system("del tmp1.txt")call system("del hilb.txt")call system("del recon.txt")stopend

program reconstructimplicit noneinteger rpt,gpt,bptinteger*4 ilen,iwid,jlen,jwid,iflginteger*4 jsize,isize,i,j,ijk,ijk1integer rmin,rmax,gmin,gmax,bmin,bmaxcharacter*80 infileinteger fracoperinteger,allocatable::darray(:,:)write(6,*)'Input thumbnail file name => 'read(5,*)infilewrite(6,*)'Color (1) or Gray (2) => 'read(5,*)iflgif (iflg.ne.2)iflg=1write(6,*)'Thumbnail length => 'read(5,*)ilenwrite(6,*)'Thumbnail width => 'read(5,*)iwidwrite(6,*)'Final dyadic length => 'read(5,*)jlenwrite(6,*)'Final dyadic width => 'read(5,*)jwidopen (1,file=infile,status='unknown')jsize=ilen*iwidallocate (darray(3,jsize))ijk=0do i=1, ilen

do j=1, iwidijk=ijk+1read(1,*)ijk1,darray(1,ijk),darray(2,ijk),darray(3,ijk)

enddoenddoclose (1)isize=jlen*jwidopen (1,file='recon.txt',status='unknown')rmin=255rmax=0gmin=255gmax=0bmin=255bmax=0do i=1, ijk

if(darray(1,i).lt.rmin)rmin=darray(1,i)if(darray(1,i).gt.rmax)rmax=darray(1,i)if(darray(2,i).lt.gmin)gmin=darray(2,i)if(darray(2,i).gt.gmax)gmax=darray(2,i)if(darray(3,i).lt.bmin)bmin=darray(3,i)if(darray(3,i).gt.bmax)bmax=darray(3,i)

enddodo i=1, isize

rpt=fracoper(darray,1,i,jsize,isize,rmin,rmax)if (iflg.eq.2) then

write(1,*)i-1,rpt,rpt,rptelse

gpt=fracoper(darray,2,i,jsize,isize,gmin,gmax)bpt=fracoper(darray,3,i,jsize,isize,bmin,bmax)write(1,*)i-1,rpt,gpt,bpt

endifenddoclose (1)stopend

integer function fracoper(darray,k,i,nsub,ntotal,yn1,yx1)implicit noneinteger*4 i,k,n,istart,ifinish,iendreal xpt,ixreal y1,ys,yf,ye,y2,yn,yxinteger yn1,yx1integer*4 nsub, ntotalinteger darray(3,nsub)n = nint(real(i-1) * real(nsub - 2) / real(ntotal - 1)) + 1istart = nint((real(ntotal - 1) * real(n - 1)) / real(nsub - 1))ifinish = nint((real(ntotal - 1) * real(n)) / real(nsub - 1))iend = nint((real(ntotal - 1) * real(n + 1)) / real(nsub - 1))xpt = ((i-1) - istart) / (ifinish - istart)ix = (ntotal - 1) * xptys=real(darray(k,n))yf=real(darray(k,n+1))if (n+2.gt.nsub)then

ye=real(darray(k,1))else

ye=real(darray(k,n+2))endifyn=real(yn1)yx=real(yx1)y1=(ye-yn)/(yx-yn)y1=y1-((ys-yn)/(yx-yn))y1=-(y1*(real(darray(k,nsub))-real(darray(k,1)))/(ntotal-1))y1=y1+((yf-ys)/real(ntotal-1))y1=y1*ixix=int(xpt*nsub)y2=real(darray(k,ix))+(real(darray(k,ix+1))-real(darray(k,ix)))

+ *((xpt*nsub)-ix)y1=y1+(((ye-yn)/(yx-yn))-((ys-yn)/(yx-yn)))*y2+ysy1=y1-real(darray(k,1))*(((ye-yn)/(yx-yn))-((ys-yn)/(yx-yn)))if (y1.lt.yn)y1=ynif (y1.gt.yx)y1=yxfracoper=nint(y1)returnend

Modified IFS Random Iteration Algorithm (RECONST) (Left) and Hilbert Sequence Program (HILB) (Right) program reconstructimplicit noneinteger*4 ilen,iwid,jlen,jwid,iflginteger*4 jsize,isize,i,j,ijk,ijk1,num,nptsinteger rpt,gpt,bptcharacter*80 infilereal, allocatable:: z(:)real, allocatable:: rint(:)real, allocatable:: gint(:)real, allocatable:: bint(:)write(6,*)'Input thumbnail file name => 'read(5,*)infilewrite(6,*)'Color (1) or Gray (2) => 'read(5,*)iflgif (iflg.ne.2)iflg=1write(6,*)'Thumbnail length => 'read(5,*)ilenwrite(6,*)'Thumbnail width => 'read(5,*)iwidwrite(6,*)'Final dyadic length => 'read(5,*)jlenwrite(6,*)'Final dyadic width => 'read(5,*)jwidopen (1,file=infile,status='unknown')jsize=ilen*iwidisize=jlen*jwidnum=jsize-1npts=isizeallocate (z(0:num))allocate (rint(1:npts))allocate (gint(1:npts))allocate (bint(1:npts))do i=1,jsize

read(1,*)ijk1,rpt,gpt,bptrint(i)=real(rpt)gint(i)=real(gpt)bint(i)=real(bpt)

enddoclose (1)open (1,file='recon.txt',status='unknown')if (iflg.eq.2)then

do i = 0,numz(i)=rint(i+1)

enddocall fractalint(npts,z,num,rint)

elsedo i = 0,num

z(i)=rint(i+1)enddocall fractalint(npts,z,num,rint)do i = 0, num

z(i)=gint(i+1)enddocall fractalint(npts,z,num,gint)do i = 0, num

z(i)=bint(i+1)enddocall fractalint(npts,z,num,bint)

endifif (iflg.eq.2)then

do i=1,nptswrite(1,*)i-1,int(rint(i)),int(rint(i)),int(rint(i))

enddoelse

do i=1,nptswrite(1,*)i-1,int(rint(i)),int(gint(i)),int(bint(i))

enddoendifclose (1)stopend

subroutine FractalInt(points,z,num,yint)implicit noneinteger*4 i, k, num, points, ndonereal x(0:num),z(0:num),d(num)real yy(0:num), a(0:num), e(0:num), c(0:num)real yint(points)real tol,newx,newy,xx,yyy,rndum,breal h,dh,dd,correl,sdev,h0,perconf,dfhinteger nterm, nsetsdo i=1, points

yint(i)=-999.9999enddodo i=0,num

x(i)=real(i)/real(num)x(i)=x(i)*real(points-1)+1

enddoxx = x(0)yyy = z(0) do i=1,num-1

d(i)=(z(i+1)-z(i-1))/510.0enddod(num)=(z(0)-z(num-1))/510.0do i = 1, num

b = x(num)-x(0)a(i) = (x(i)-x(i-1)) / be(i) = (x(num) * x(i - 1) - x(0) * x(i)) / bc(i) = (z(i) - z(i - 1) - d(i) * (z(num) - z(0))) / byy(i) = (x(num) * z(i - 1) - x(0) * z(i) - d(i) *

+ (x(num) * z(0) - x(0) * z(num))) / benddo

202 do i=1,pointscall random_number(rndum)k=1+int(real(num)*rndum)if(k.gt.num)k=numif(k.lt.1)k=1newx = a(k) * xx + e(k)newy = c(k) * xx + d(k) * yyy + yy(k)yint(nint(newx)) = newyxx=newxyyy=newy

enddodo i = 1, points

If (yint(i).eq.-999.9999)thenGoTo 202

endifenddoreturnend

pro gram hilbimplicit noneinteger nptsinteger i,j,l1,arraysz,direc,parity,levelinteger ileft(5),iright(5)integer*4 incr, incr1common /hilby/ i,j,l1,incr,incr1,arraysz,direc,parity,level,

+ ileft,irightwrite(6,*)'Input number of points (dyadic) => 'read(5,*)nptsopen (1,file='hilb.txt',status='unknown')call hilbert (1,npts)close(1)stopend

RECURSIVE SUBROUTINE DOHILBERT (L, P)implicit noneinteger l,pinteger i,j,l1,arraysz,direc,parity,levelinteger ileft(5),iright(5)integer*4 incr, incr1common /hilby/ i,j,l1,incr,incr1,arraysz,direc,parity,level,

+ ileft,irightIF (l.ne.0) THEN

DIREC = ILEFT(P * DIREC + 3)CALL DOHILBERT(L - 1, -P)CALL FWDDIREC = IRIGHT(P * DIREC + 3)CALL DOHILBERT(L - 1, P)CALL FWDCALL DOHILBERT(L - 1, P)DIREC = IRIGHT(P * DIREC + 3)CALL FWDCALL DOHILBERT(L - 1, -P)DIREC = ILEFT(P * DIREC + 3)

END IFRETURNEND

INTEGER FUNCTION LINKINDEX (ROW, COLUMN)implicit noneinteger row,columninteger i,j,l1,arraysz,direc,parity,levelinteger ileft(5),iright(5)integer*4 incr, incr1common /hilby/ i,j,l1,incr,incr1,arraysz,direc,parity,level,

+ ileft,irightLINKINDEX = (COLUMN - 1) * arraysz + ROWRETURNEND

SUBROUTINE HSTEP (ROW, COLUMN, DIRECTION)implicit noneinteger row,column,directioninteger i,j,l1,arraysz,direc,parity,levelinteger ileft(5),iright(5)integer*4 incr, incr1common /hilby/ i,j,l1,incr,incr1,arraysz,direc,parity,level,

+ ileft,irightIF (ABS(DIRECTION).eq.1) THEN

ROW = ROW + DIRECTIONELSE

COLUMN = COLUMN + DIRECTION / 2END IFRETURNEND

SUBROUTINE hilbert (start, finish)implicit noneinteger start,finish,numinteger i1,h,k,linkindexinteger l,isuminteger i,j,l1,arraysz,direc,parity,levelinteger ileft(5),iright(5)integer*4 incr, incr1common /hilby/ i,j,l1,incr,incr1,arraysz,direc,parity,level,

+ ileft,irightnum = finish - start + 1ILEFT(1) = 1ILEFT(2) = -2ILEFT(3) = 0ILEFT(4) = 2ILEFT(5) = -1IRIGHT(1) = -1IRIGHT(2) = 2IRIGHT(3) = 0IRIGHT(4) = -2IRIGHT(5) = 1i = 0DO while (j.le.num)

i = i + 1j = (2 ** i) ** 2

enddol1 = i - 1INCR = 0INCR1 = 0ISUM = 0do I1 = 1, l1 - 1

ISUM = ISUM + (2 ** L) ** 2 - 1enddodo LEVEL = 1,l1

arraysz = 1do H = 1, LEVEL

arraysz = arraysz * 2enddoi = 1j = 1DIREC = 1PARITY = 1CALL DOHILBERT(LEVEL, PARITY)k = LINKINDEX(i, j)

enddowrite(1,*)arrayszRETURNEND

SUBROUTINE FWDimplicit noneinteger k,linkindex,isuminteger i,j,l1,arraysz,direc,parity,levelinteger ileft(5),iright(5)integer*4 incr, incr1common /hilby/ i,j,l1,incr,incr1,arraysz,direc,parity,level,

+ ileft,irightk = LINKINDEX(i, j)CALL HSTEP(i, j, DIREC)IF (l1.eq.LEVEL) THEN

INCR = INCR + 1IF (INCR.gt.ISUM) THEN

INCR1 = INCR1 + 1write(1,*)k

END IFEND IFRETURNEND

JPEG Compression Code (JPEG) blo ck data initjpegimplicit noneinteger isizeparameter (isize=8)integer zigzaguinteger zigzagvcommon /zigu/zigzagu(0:isize*isize-1)common /zigv/zigzagv(0:isize*isize-1)data zigzagu/0,0,1,2,1,0,0,1,2,3,4,3,2,1,0,0,1,2,3,4,5,6,5,

* 4,3,2,1,0,0,1,2,3,4,5,6,7,7,6,5,4,3,2,1,2,3,4,* 5,6,7,7,6,5,4,3,4,5,6,7,7,6,5,6,7,7/

data zigzagv/0,1,0,0,1,2,3,2,1,0,0,1,2,3,4,5,4,3,2,1,0,0,1,* 2,3,4,5,6,7,6,5,4,3,2,1,0,1,2,3,4,5,6,7,7,6,5,* 4,3,2,3,4,5,6,7,7,6,5,4,5,6,7,7,6,7/

endprogram jpegimplicit noneinteger isizeparameter (isize=8)integer jsizeparameter (jsize=2048)integer ksizeinteger pixelsdouble precision dctinteger zigzaguinteger zigzagvdouble precision reorderedinteger quanttableinteger rpix(jsize,jsize,3)integer final(jsize,jsize,3)character*80 pixfilecharacter*80 quantfilecharacter*80 outfileinteger lstmeaninteger i,j,k,l,m,ninteger previousdctermcommon /zigu/zigzagu(0:isize*isize-1)common /zigv/zigzagv(0:isize*isize-1)common /dctdat/dct(0:isize-1,0:isize-1),

* pixels(0:isize-1,0:isize-1)common /reorder/reordered(0:isize*isize-1)common /quant/quanttable(0:isize-1,0:isize-1)quantfile="qtable.dat"lstmean=128ksize=jsize/isizepixfile = "input"outfile = "output"call readblock(pixfile,rpix)call readquanttable(quantfile)do i=1,ksize

do j=1,ksizedo k=1,3

do l=0,isize-1do m=0,isize-1

pixels(l,m)=rpix(((i-1)*isize)+l+1,* ((j-1)*isize)+m+1,k)

enddoenddocall fdctcall quantizecall zigzagpreviousdcterm = lstmeancall unzigzagcall dequantizecall idctdo l=0,isize-1

do m=0,isize-1final(((i-1)*isize)+l+1,

* ((j-1)*isize)+m+1,k)=pixels(l,m)enddo

enddoenddo

enddoenddocall writeblock(outfile,final)stopendsubroutine fdctimplicit noneinteger x,y,u,vdouble precision sumdouble precision cdouble precision mpiinteger isizeparameter (isize=8)integer pixelsdouble precision dctcommon /dctdat/dct(0:isize-1,0:isize-1),

* pixels(0:isize-1,0:isize-1)mpi=2.0*asin(1.0)do u=0,isize-1

do v=0,isize-1sum = 0.0do x=0,isize-1

do y=0,isize-1sum=sum+(pixels(x,y)-128)*cos((2*x+1)*

* u*(mpi/16.))*cos((2*y+1)*v*(mpi/16.))enddo

enddodct(u,v)=0.25*C(u)*C(v)*sum

enddoenddoreturnendsubroutine idctimplicit noneinteger x,y,u,vdouble precision sumdouble precision mpidouble precision cinteger isizeparameter (isize=8)integer pixelsdouble precision dctcommon /dctdat/dct(0:isize-1,0:isize-1),

* pixels(0:isize-1,0:isize-1)mpi=2.0*asin(1.0)do x=0,isize-1

do y=0,isize-1sum = 0.0do u=0,isize-1

do v=0,isize-1sum=sum+C(u)*C(v)*dct(u,v)*cos((2*x+1)*

* u*(mpi/16.))*cos((2*y+1)*v*(mpi/16.))enddo

enddopixels(x,y)=128.5+0.25*sum

enddoenddoreturnend

double precision function c(n)implicit noneinteger nif (n.eq.0) then

c=1.0/sqrt(2.0)else

c=1.0endifreturnendsubroutine printdctimplicit noneinteger i,jinteger isizeparameter (isize=8)integer pixelsdouble precision dctcommon /dctdat/dct(0:isize-1,0:isize-1),

* pixels(0:isize-1,0:isize-1)do i=0,isize-1

write(6,*) i,(dct(i,j),j=0,isize-1)enddoreturnendsubroutine printpixelsimplicit noneinteger i,jinteger isizeparameter (isize=8)integer pixelsdouble precision dctcommon /dctdat/dct(0:isize-1,0:isize-1),

* pixels(0:isize-1,0:isize-1)do i=0,isize-1

write(6,*) i,(pixels(i,j),j=0,isize-1)enddo returnendsubroutine quantizeimplicit noneinteger i,jinteger isizeparameter (isize=8)integer pixelsdouble precision dctinteger quanttableinteger roundcommon /dctdat/dct(0:isize-1,0:isize-1),

* pixels(0:isize-1,0:isize-1)common /quant/quanttable(0:isize-1,0:isize-1)do i=0,isize-1

do j=0,isize-1dct(i,j)=round(dct(i,j)/quanttable(i,j))

enddoenddoreturnendsubroutine dequantizeimplicit noneinteger i,jinteger isizeparameter (isize=8)integer pixelsdouble precision dctinteger quanttablecommon /dctdat/dct(0:isize-1,0:isize-1),

* pixels(0:isize-1,0:isize-1)common /quant/quanttable(0:isize-1,0:isize-1)do i=0,isize-1

do j=0,isize-1dct(i,j)=dct(i,j)*quanttable(i,j)

enddoenddoreturnendinteger function round(x)implicit nonedouble precision xif (x.gt.0.0) then

round=int(x+0.5)else

round=int(x-0.5)endifreturnendsubroutine readblock(filename,rpix)implicit nonecharacter*80 filenameinteger i,jinteger xtempinteger jsizeparameter (jsize=128)integer rpix(jsize,jsize,3)open(1,file=filename,status='unknown')do i=1,jsize

read(1,*)(xtemp,rpix(i,j,1),rpix(i,j,2),+ rpix(i,j,3),j=1,jsize)

enddoclose(1)returnendsubroutine readquanttable(filename)implicit nonecharacter*80 filenameinteger i,jinteger isizeparameter (isize=8)real facparameter (fac=30)integer quanttablecommon /quant/quanttable(0:isize-1,0:isize-1)open(1,file=filename,status='unknown')do i=0,isize-1

read(1,*)(quanttable(i,j),j=0,isize-1)enddodo i=0,isize-1

do j=0,isize-1quanttable(i,j)=int(fac*real(quanttable(i,j)))

enddoenddoclose(1)returnend

integer function integercodesize(n)implicit noneinteger nif (n.lt.0) n=-nif (n.eq.1) then

integercodesize=1elseif(n.lt.4) then









integercodesize=10else

write(6,*)"Illegal coefficient value"stop

endifreturnendsubroutine zigzagimplicit noneinteger iinteger isizeparameter (isize=8)integer pixelsdouble precision dctinteger zigzaguinteger zigzagvdouble precision reorderedcommon /zigu/zigzagu(0:isize*isize-1)common /zigv/zigzagv(0:isize*isize-1)common /dctdat/dct(0:isize-1,0:isize-1),

* pixels(0:isize-1,0:isize-1)common /reorder/reordered(0:isize*isize-1)do i=0, isize*isize-1

reordered(i)=dct(zigzagu(i),zigzagv(i))enddoreturnendsubroutine unzigzagimplicit noneinteger iinteger isizeparameter (isize=8)integer pixelsdouble precision dctinteger zigzaguinteger zigzagvdouble precision reorderedcommon /zigu/zigzagu(0:isize*isize-1)common /zigv/zigzagv(0:isize*isize-1)common /dctdat/dct(0:isize-1,0:isize-1),

* pixels(0:isize-1,0:isize-1)common /reorder/reordered(0:isize*isize-1)do i=0, isize*isize-1

dct(zigzagu(i),zigzagv(i))=reordered(i)enddoreturnendsubroutine printrunlengths(previousdcterm)implicit noneinteger previousdcterminteger isizeparameter (isize=8)integer pixelsdouble precision dctdouble precision reorderedcommon /dctdat/dct(0:isize-1,0:isize-1),

* pixels(0:isize-1,0:isize-1)common /reorder/reordered(0:isize*isize-1)integer iinteger runlengthinteger dcdiffinteger integercodesizeopen(2,file="compress.out",status="unknown",access="append")runlength=0dcdiff=int(dct(0,0))-previousdctermwrite(2,*)integercodesize(dcdiff),dcdiffdo i=0,isize*isize-1

if (reordered(i).eq.0)thenrunlength=runlength+1

elsewrite(2,*)runlength,integercodesize(int(reordered(i))),

* int(reordered(i))runlength=0

endifenddoclose(2)returnendsubroutine writeblock(filename,final)implicit nonecharacter*80 filenameinteger i,j,kinteger jsizeparameter (jsize=128)integer final(jsize,jsize,3)open(1,file=filename,status='unknown')do i=1,jsize

do j=1,jsizedo k=1,3

if(final(i,j,k).lt.0)final(i,j,k)=0if(final(i,j,k).gt.255)final(i,j,k)=255

enddoenddo

enddok=0do i=1,jsize

do j=1,jsizewrite(1,10)k,final(i,j,1),final(i,j,2),final(i,j,3)k=k+1

enddoenddo

10 format(1x,i5,1x,i3,1x,i3,1x,i3)close(1)returnend

Shai-Hulud: The quest for worm sign

Holger M. Jaenisch*a, James W. Handleya, Jeffrey P. Faucheuxb and Ken Lamkinb aSparta, Inc., 6000 Technology Dr., Bldg. 3, Huntsville, AL 35805

bSparta, Inc., 7075 Samuel Morse Dr., Suite 200, Columbia, MD 21046

ABSTRACT

Successful worm detection at real-time OC-48 and OC-192 speed requires hardware to extract web based binary sequences at faster than these speeds, and software to process the incoming sequences to identify worms. Computer hardware advancement in the form of field programmable gate arrays (FPGAs) makes real-time extraction of these sequences possible. Lacking are mathematical algorithms for worm detection in the real time data sequence, and the ability to convert these algorithms into lookup tables (LUTs) that can be compiled into FPGAs. Data Modeling provides the theory and algorithms for an effective mathematical framework for real-time worm detection and conversion of algorithms into LUTs. Detection methods currently available such as pattern recognition algorithms are limited both by the amount of time to compare the current data sequence with a historical database of potential candidates, and by the inability to accurately classify information that was unseen in the training process. Data Modeling eliminates these limitations by training only on examples of nominal behavior. This results in a highly tuned and fast running equation model that is compiled in a FPGA as a LUT and used at real-time OC-48 and OC-192 speeds to detect worms and other anomalies. This paper provides an overview of our approach for generating these Data Change Models for detecting worms, and their subsequent conversion into LUTs. A proof of concept is given using binary data from a WEBDAV, SLAMMER packet, and RED PROBE attack, with BASIC source code for the detector and LUT provided. Keywords: Data Modeling, worm detection, change detection, OC-48, OC-192, fractal conversion, FPGA, WEBDAV,

SLAMMER, RED PROBE

1. INTRODUCTION Identification of web based binary sequences containing worms and viruses in order to prevent computer network attack and intrusion is of interest to the information assurance (IA) community. By flagging these events as they are presented to the network in real-time and not allowing them to infiltrate the network, or by removing them before they can modify or damage information on the network, savings both in terms of cost and man-hours is achieved. To date, there has been no ability to perform worm detection at real-time OC-48 and OC-192 speeds. Achieving this was limited both by the computer power required to extract the sequences and by the mathematical algorithms available to identify worms. Field programmable gate array (FPGA) based computer hardware technology has been developed that overcomes the hardware limitations. FPGAs are electronic chips that are programmed with a bit stream when they are powered up, and can be reprogrammed if necessary. Programs are compiled into the FPGA in the form of a lookup table (LUT). Algorithms converted into standard computer programs must then be converted into LUT format in order to run in a FPGA computer architecture. Data Modeling provides a method for single-pass identification of worm packets in binary data that can also be compiled into simple LUT format for use in FPGAs at OC-48 and OC-192 speeds1,2,3.

2. DATA MODELING If worm packets are detectable in real-time at OC-48 and OC-192 speeds, it is possible to intercept and quarantine them before they become resident inside or attack the computer system. In previous published work, Data Modeling was successfully applied to construct change detectors and classifiers for the 1999 KDD-Cup Challenge Data sponsored by DARPA4. This work successfully demonstrated Data Modeling’s ability to perform network intrusion detection using data derived from network logon sequences. For worm detection, this process is taken further by characterizing binary sequences of zeros and ones. This binary data is converted into a fractional Brownian motion (fBm) data sequence and characterized using parametric and non-parametric statistics. These statistics are then used to drive the Data Model change detector equation and subsequent LUT. * [email protected]; phone 1 256 337-3768; fax 1 256 890-2041; sparta.com

Data Models are variables that can be embedded in other functions as variables, therefore the term used is functional5,6. A hierarchy of functional models (equations whose variables are themselves equations) such as the Data Relations Model

))))))))),,(((((((((),,,( 2121 kjinL xxxybybybyfxxxy KK = [ ] nLxxxyO 3),,,( 21 =K (1)

can be built up to combine any group or subset of previously derived models and monitor waveforms for changes and alerts. By flagging tip-off conditions, rollup diagnostics and assessments become possible that reduce the information made available as waveform dynamics unfold for generating tip-off conditions. Data Relations Modeling finds a mathematical expression that provides a good fit between given finite sample values of the independent variables and the associated values of the dependent variables of the process. The mathematical expression that fits the given sample of data is called a Data Model. This process involves finding both the functional form of the Data Model and the numeric coefficients for the Data Model. The Data Relations Modeling process of discovering automatic functions is an evolutionary and genetic directed approach where the set elements consist entirely of simple polynomial building blocks7. We assert as a fundamental premise that nominal occurs more frequently than off-nominal. If the reverse is true, then the definition simply swaps, making it still hold8. Based on this assumption, we can proceed with an algorithm that will determine normal from abnormal without a priori knowledge or representation of either. Generation of this Data Change Model can be done in either supervised or unsupervised mode. Both supervised and unsupervised modes are covered in detail in Reference 8. Data Change Modeling4,8 is unique in that no a priori examples of what constitutes an anomalous condition need to be characterized or used for deriving the equation. Using only historical knowledge of nominal regions, a Change Detection equation is derived directly from the example feature set that is very highly tuned. The simple equation can take the form of a very large polynomial and is efficiently derived using Data Modeling algorithms. In practice, this O(3n) polynomial has been derived to be as high as 6,000th order and may be even higher order if the application requires it. This simply insures and builds confidence that any number of natural non-linearities or non-stationarities that exist even in the nominal region can be adequately described with a single, albeit high order, equation. Changes in complex dynamic processes such as manufacturing assembly lines can be monitored using X-Bar and R charts. This is known as Statistical Process Control, a field pioneered by Shewhart. The power of Shewhart’s statistical quality control charts is the ability to flag when a dynamic process is not in statistical control. The X-Bar chart upper control limit (UCL) and lower control limit (LCL) is defined by

.2

2

RAXLCL

RAXUCL

X

X

−=

+= (2)

and plots how far away from the average value (X ) of a process the current measurement falls. Values that repeatedly

fall outside of three standard deviations of the average (RA2 ) have assignable causes and are labeled as being out of statistical control (off-nominal). By creating a high order polynomial that learns the associations between inputs, the equation becomes sensitive to perturbations or changes in inputs due to combinations of these same moments not occurring during the nominal interval, just as Statistical Process Control is sensitive and flags whenever values repeatedly fall outside of the sigma boundary. As such, numerical instability causes the tightly tuned equation to “blow-up”, resulting in values in excess of ± 1x1010 rather than within the training boundaries of 0.5 ± .0012. These radical departures are treated as a flag in real-time that the statistics across the feature vector have changed from status quo. Figure 1 illustrates how binary data streams are converted into fractional Brownian motion (fBm), features are calculated, and then analyzed to detect if any change has occurred in system status from nominal.

Tip-Off

Tip-Off

Nominal

Tip-Off

Tip-Off

Nominal

Equation Output

σ

T = w0+w1x1+w2x2+w3x3 +w4x12

+w5x22 +w6x3

2 +w7x13 +w8x2

3

+w9x32 +w10x1x2 +w11x1x3

+w12x2x3

Skewness StandardDeviation


Data Model

ZScoresFractal

ConversionNetwork

FlowData

SeriesData

SeriesPDFPDF

Tip-Off

Tip-Off

Nominal

Tip-Off

Tip-Off

Nominal

Equation Output

σ

T = w0+w1x1+w2x2+w3x3 +w4x12

+w5x22 +w6x3

2 +w7x13 +w8x2

3

+w9x32 +w10x1x2 +w11x1x3

+w12x2x3

Skewness StandardDeviation


Data Model

ZScoresFractal

ConversionNetwork

FlowData

SeriesData

SeriesPDFPDF

Fig. 1. Change detection using Data Modeling.

Because the data is binary, it is first transformed using

12 −= ii wz (3)

where wi is the current binary digit and zi is the transformed value. This converts the data into –1 and +1 values. In the proof of concept contained in this paper, these numbers were converted to – ½ and + ½ by dividing the results of Equation 3 by 2. Once transformed, the data is then integrated using direct summation

)(1 zzyy iii −+= − (4)

resulting in a fractional Brownian motion (fBm) or 1/f type data set. In Equation 4, yi is the current integrated value, yi-1 the previous integrated value, zi is the current data sequence point, and z the mean of the data sequence points. This has been demonstrated by the authors in previous work. Each individual descriptive statistic characterizes the mth window of the incoming data sequence. Data Modeling has available over 40 different novel parametric and non-parametric statistics and autonomously decides which statistics are necessary. For this application, only 2 statistics (standard deviation and skewness) were required for input to the Data Change Model. These real-time features are given in Equations 6 and 7 by

1

1

11

+

+

+

+ +⎟⎟⎠

⎞⎜⎜⎝

⎛=

i

i

i

iii N

y

N

Nµµ (5)

( )( )

1

211

1

21

+

++

+

+

−+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

i

ii

i

iii N

y

N

N µσσ (6)

1

3

11

1+

++

+

⎟⎟⎠

⎞⎜⎜⎝

⎛ −+

=i

i

iiii

i N

yNSkew

Skewσ

µ

(7)

where i is the index corresponding to the previous time point, and i+1 the current time point. The real time calculation of the mean in Equation 5 is included for use in the real time estimates in Equations 6 and 7. Once constructed, the final Data Change Model is determined by placing a tight boundary of 0.0012 on either side of the final nominal class designation. All data sets inside of the two boundaries are considered candidate nominal, while all others are considered off-nominal and potential worms.

The equation Data Models are then converted into lookup table (LUT) format. One method for constructing LUTs for Data Relations Models is through the use of only the DOUBLE equation type given by

22111222211121221121 xxJxxxIxxxHxxxGxxFxxExxDxCxBxAy +++++++++= (8) This building block uses as input 2 channels of information (x1 and x2), which can be plotted on the x-axis and y-axis of the LUT separately. The value read from the LUT is the current estimate of the condition of the system variable (nominal or tip-off). If multiple DOUBLES are used in constructing the final Data Model, multiple LUTs can be used, each representing an individual DOUBLE. In this case, the value from one LUT is used as input into the next LUT. The proof of concept Data Model explained in the next section was constructed using a series of DOUBLE equations; however, the LUT was constructed as a final output LUT, meaning that the results of each intermediate DOUBLE equation are captured using a single LUT that maps the inputs directly into the output value. Figure 2 provides pseudo-code for generating this LUT.

ngrid = 25 : i = 0

REDIM smin(1), smax(1), sdev(1), dfeat(1), prob(ngrid,ngrid)

smin(0) = 0@ : smax(0) = 30@ : sdev(0) = 0@

smin(1) = -1.7964 : smax(1) = 1.2456 : sdev(1) = .5339

dfeat(0) = ((smax(0) + sdev(0)) - (smin(0) - sdev(0))) / (ngrid - 1)

dfeat(1) = ((smax(1) + sdev(1)) - (smin(1) - sdev(1))) / (ngrid - 1)

FOR featx = smin(0) - sdev(0) TO smax(0) + sdev(0) + .9 * dfeat(0) STEP dfeat(0)

i = i + 1 : j = 0

FOR featy = smin(1) - sdev(1) TO smax(1) + sdev(1) + .9 * dfeat(1) STEP dfeat(1)

j = j + 1

feat1 = (featx - 3.3274) / 1.1979 : feat2 = (featy - -.092) / .4131

‘Data Model equation here

call datamodel(feat1,feat2,outval)

prob(i, j) = outval

NEXT featy

NEXT featx

OPEN "cld4map.out" FOR OUTPUT AS #1

FOR i = ngrid TO 1 STEP -1

FOR j = 1 TO ngrid

PRINT #1, prob(i, j);

NEXT j

PRINT #1,

NEXT i

CLOSE : END

Fig. 2. Pseudo-code for generating an LUT.

3. PROOF OF CONCEPT The authors were supplied with four (4) data sets consisting of nominal and attack web server data in binary format. The first data set is a web server browsing session with no attacks (labeled nominal, or attack free). The second data set is a single slammer worm packet. The third data set is all the packets associated with a web server worm attack (both nominal and anomalous). The fourth data set is a second type of worm packet (code red probe) buried in among the nominal web server packets. Worm sign regions in data exist where the dynamics are different from the dynamics of nominal web server data. Statistics that capture these differences are used as input features to the Data Change Model. Worm sign regions will have one or more features that exist either outside of the nominal training boundaries, or in combinations that were not observed during training. This leads to the nominal Data Model flagging the regions as tip-off. First, the Data Change Model shown in Figure 3 was constructed using the nominal only data set. This Data Model uses the standard deviation and skewness calculated in Equations 6 and 7 and provided in a file named input. Written out by this program is a file named out1 that contains the actual Data Model equation value for nominal cases (ranging between 0.4988 and 0.5012), and 1 for potential worm detection cases. Only standard deviation (Equation 6) and skewness (Equation 7) were required by the Data Model to distinguish between the provided nominal and worm data, due in part to the larger standard deviations for the worm data than the nominal data, and for the existence of skewness for the worm data that were smaller than the currency precision used in the Data Model (resulting in being rounded to zero) and the

existence of worm packets in the fourth data set whose skewness was mathematically zero. However, as shown in the examples below, the standard deviation of these features is large, thereby causing overlap in the distributions of the features. To resolve this overlap, it is necessary to use the Data Model and subsequent lookup tables for real-time firmware implementation. Also, zero skewness also exists in the nominal data (shown in the Examples below), making it impossible to set a simple threshold to resolve nominal and worm.

DEFCUR A-Z

open "input" for input as #1

input #1, sdev

input #1, skew

close #1

sdev = (sdev-3.3274) / 1.1979

skew = (skew--0.092) / 0.4131

l1o1=0.0155

l1o1=l1o1+ (-0.0093) * (skew)

l1o1=l1o1+ (-0.0406) * (sdev)

l1o1=l1o1+ (0.0417) * (skew * skew)

l1o1=l1o1+ (-0.0472) * (sdev * sdev)

l1o1=l1o1+ (-0.1041) * (skew * sdev)

l1o1=l1o1+ (0) * (skew * skew * skew)

l1o1=l1o1+ (0.015) * (sdev * sdev * sdev)

l1o1=l1o1+ (0.0208) * (sdev* skew * skew)

l1o1=l1o1+ (0.0258)* (skew * sdev * sdev)

l2o1=-0.0608

l2o1=l2o1+ (1.7906) * (l1o1)

l2o1=l2o1+ (0.004) * (sdev)

l2o1=l2o1+ (1.3184) * (l1o1 * l1o1)

l2o1=l2o1+ (0.013) * (sdev * sdev)

l2o1=l2o1+ (0.0845) * (l1o1 * sdev)

l2o1=l2o1+ (-3.368)* (l1o1 * l1o1 * l1o1)

l2o1=l2o1+ (-0.004)* (sdev * sdev * sdev)

l2o1=l2o1+ (0.9435)* (sdev * l1o1 * l1o1)

l2o1=l2o1+ (-0.1378)*(l1o1 * sdev * sdev)

l2o2=-0.02

l2o2=l2o2+ (1.0804) * (l1o1)

l2o2=l2o2+ (0.0072) * (skew)

l2o2=l2o2+ (-0.0925) * (l1o1 * l1o1)

l2o2=l2o2+ (-0.0074) * (skew * skew)

l2o2=l2o2+ (0.0356) * (l1o1 * skew)

l2o2=l2o2+ (-4.5598)*(l1o1 * l1o1 * l1o1)


l2o2=l2o2+ (-0.4186)*(skew * l1o1 * l1o1)

l2o2=l2o2+ (0.1866)*(l1o1 * skew * skew)

l3o3=-0.0009

l3o3=l3o3+ (-0.1613) * (l2o1)

l3o3=l3o3+ (1.1353) * (l2o2)

l3o3=l3o3+ (1.6504) * (l2o1 * l2o1)

l3o3=l3o3+ (1.6929) * (l2o2 * l2o2)

l3o3=l3o3+ (-3.4651) * (l2o1 * l2o2)

l3o3=l3o3+ (17.8793)*(l2o1 * l2o1 * l2o1)

l3o3=l3o3+ (-2.9467)*(l2o2 * l2o2 * l2o2)

l3o3=l3o3+ (-36.9399)*(l2o2* l2o1 * l2o1)

l3o3=l3o3+ (20.7681)*(l2o1 * l2o2 * l2o2)

l3o2=0.0124

l3o2=l3o2+ (1.0601) * (l2o2)

l3o2=l3o2+ (0.0098) * (skew)

l3o2=l3o2+ (-0.7028) * (l2o2 * l2o2)

l3o2=l3o2+ (-0.0034) * (skew * skew)

l3o2=l3o2+ (0.0971) * (l2o2 * skew)

l3o2=l3o2+ (0.4137)* (l2o2 * l2o2 * l2o2)


l3o2=l3o2+ (-0.235)* (skew * l2o2 * l2o2)

l3o2=l3o2+ (-0.0059)*(l2o2 * skew * skew)

l4o1=-0.0177

l4o1=l4o1+ (-0.3696) * (l3o3)

l4o1=l4o1+ (1.4537) * (l3o2)

l4o1=l4o1+ (3.2139) * (l3o3 * l3o3)

l4o1=l4o1+ (4.2476) * (l3o2 * l3o2)

l4o1=l4o1+ (-7.0018) * (l3o3 * l3o2)

l4o1=l4o1+ (-0.1307)*(l3o3 * l3o3 * l3o3)

l4o1=l4o1+ (-7.5133)*(l3o2 * l3o2 * l3o2)

l4o1=l4o1+ (-5.6853)*(l3o2 * l3o3 * l3o3)

l4o1=l4o1+ (13.6263)*(l3o3 * l3o2 * l3o2)

l3o1=-0.0092

l3o1=l3o1+ (1.2443) * (l2o2)

l3o1=l3o1+ (-0.2622) * (l2o1)

l3o1=l3o1+ (1.623) * (l2o2 * l2o2)

l3o1=l3o1+ (1.4277) * (l2o1 * l2o1)

l3o1=l3o1+ (-3.1749) * (l2o2 * l2o1)

l3o1=l3o1+ (-3.0658)*(l2o2 * l2o2 * l2o2)

l3o1=l3o1+ (18.4779)*(l2o1 * l2o1 * l2o1)

l3o1=l3o1+ (21.2166)*(l2o1 * l2o2 * l2o2)

l3o1=l3o1+ (-37.9024)*(l2o2* l2o1 * l2o1)

l4o3=-0.0167

l4o3=l4o3+ (-0.2434) * (l3o1)

l4o3=l4o3+ (1.3298) * (l3o2)

l4o3=l4o3+ (2.6275) * (l3o1 * l3o1)

l4o3=l4o3+ (3.802) * (l3o2 * l3o2)

l4o3=l4o3+ (-5.9747) * (l3o1 * l3o2)

l4o3=l4o3+ (0.0312)* (l3o1 * l3o1 * l3o1)

l4o3=l4o3+ (-6.736)* (l3o2 * l3o2 * l3o2)

l4o3=l4o3+ (-5.3589)*(l3o2 * l3o1 * l3o1)

l4o3=l4o3+ (12.334)* (l3o1 * l3o2 * l3o2)

l5o1=-0.0265

l5o1=l5o1+ (0.2206) * (l4o1)

l5o1=l5o1+ (0.7282) * (l4o3)

l5o1=l5o1+ (0.0982) * (l4o1 * l4o1)

l5o1=l5o1+ (-2.7116) * (l4o3 * l4o3)

l5o1=l5o1+ (3.2379) * (l4o1 * l4o3)

l5o1=l5o1+ (-0.0158)*(l4o1 * l4o1 * l4o1)

l5o1=l5o1+ (3.2853)* (l4o3 * l4o3 * l4o3)

l5o1=l5o1+ (-9.306)* (l4o3 * l4o1 * l4o1)

l5o1=l5o1+ (6.1) * (l4o1 * l4o3 * l4o3)

l5o1= l5o1* (0.0002) + (0.5)

rem nominal l5o1 between .4988 and .5012

rem worm l5o1 set to 1

if l5o1 < .4988 then l5o1 = 1.0

if l5o1 > .5012 then l5o1 = 1.0

open "out1" for output as #1

print #1, l5o1

close #1

end

Fig. 3. QuickBASIC source code for the Worm Detector Data Model. For the nominal only data, each 256-point window flagged as being like the nominal training data. Next, statistics were generated for the unseen slammer packet data using 256-point windows. These statistics are then passed through the Data Model equation. Since this data file represents a single “worm” and no nominal packet data, all windows should flag as tip-off. Passing this data through the Data Model provided constant tip-off for all data windows as expected.

Third, statistics for an unseen worm attack (both worm packet and data packets returning the error codes generated by the attack) are presented to the Data Model for consideration. As was the case in processing the nominal data and the slammer packet data, areas that are nominal should pass through the Data Model while those that are flagged by the Data Model as being different than the training data should flag as off-nominal. Using the Data Model, data was flagged as different than the nominal data used during training, indicating the presence of a worm. Finally, a different type of worm based web attack (red probe) is presented to the Data Model. This data set consisted of a much larger region of nominal behavior before and after the worm, and a much shorter length worm than in the previous case. The Data Model flagged the new worm packet data as different than surrounding nominal packet data. Figure 4 shows the decision map generated from the Data Model and the location in the tip-off region of the worm packets for each of the data sets. This graphically captures the information contained in the LUT found in Figure 5.

0.0

Std

. Dev

.30

.0

0.4988

1.0

-2.3 Skewness 1.8

0.5012

Nominal

Tip -Off

WormPackets

Fig. 4. Decision map generated from the equation Data Model with the Red Probe worm packets superimposed.

The lookup table that can be embedded in FPGAs and used instead of the equation Data Model is given in Figure 5. The LUT gives the final Data Model nominal or tip-off decision. The size of the LUT here is 25x25; however, there is no limitation on size, nor is there a constraint that the LUT be square. In fact, the LUT can be whatever size is required for adequate characterization of the Data Model equation while keeping the LUT as small as possible. To fit the LUT on a single page and insuring legibility, the skewness axis on the LUT was split into two pieces, with the lower values presented in the top half and the higher skewness values in the lower half. Standard deviation and skewness are read directly off the bottom and left axes and the corresponding LUT value read from the intersection. Tip-off regions in the LUT are represented by ones (1) in the individual elements. There is no need for recording the value of the tip-off, and the use of ones helps minimize and fix the size of the LUT for presentation. As test cases, 8 raw binary sequences that can be passed through the LUT are provided in Table 1 (4 nominal cases followed by 4 worm cases). In reading the numbers from the LUT, if an input value fell between two columns on the LUT, the neighborhood surrounding where the value would occur on the table was read and the largest value away from ½ chosen to represent the calculation. The results of passing these 8 cases through the LUT are given in Table 2.

4. SUMMARY Data Modeling only requires examples of nominal behavior in order to train. Classical methods are limited both by the amount of time required to compare the current data sequence with a historical database of potential candidates and by the inability to accurately classify information that was unseen in the training process. Data Models do not suffer from these limitations; rather, they provide single pass solutions that accurately classify information unseen during training as being different. The results are highly tuned and fast running equation models that can be used at real-time OC-48 and OC-192 speeds to detect the presence of anomalies such as worms, which was successfully demonstrated in this work.

-2.3303

-2.1591

-1.9878

-1.8166

-1.6453

-1.4741

-1.3028

-1.1316

-0.9604

-0.7891

-0.6179

-0.4466

-0.2754

20.000021.250022.500023.7500

Skewness

25.000026.250027.500028.750030.0000

Sta

ndar

d D

evia

tion

1.25002.50003.75005.00006.25007.50008.7500

10.000011.250012.500013.750015.000016.250017.500018.7500

1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 0.5000 0.5000 0.50001 1 1 1 1 1 1 1 1 0.5001 0.5000 0.5000 0.50001 1 1 1 1 1 1 1 1 0.5001 0.5000 0.5000 0.50001 1 1 1 1 1 1 1 1 0.5001 0.5000 0.5000 0.50001 1 1 1 1 1 0.4997 0.5002 0.5001 0.5000 0.5000 0.5000 0.50001 1 1 0.5001 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.50001 1 1 1 1 1 0.5004 0.5001 0.5000 0.5000 0.5000 0.5000 0.50001 1 1 1 1 1 1 1 1 1 1 1 10.0000

-0.1042

0.0671

0.2383

0.4096

0.5808

0.7521

0.9233

1.0945

1.2658

1.4370

1.6083

1.7795

Skewness

1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1

0.5000 0.5000 0.5000 1 1 1 1 1 1 1 1 10.5000 0.5000 0.5000 0.5000 0.5000 0.5001 1 1 1 1 1 10.5000 0.5000 0.4999 0.4999 0.5000 0.5000 0.5001 0.5001 1 1 1 10.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5001 0.5001 0.5001 1 10.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5001 0.5001 0.5001 0.5001 0.5001 10.5000 0.5000 0.5000 0.5000 0.5001 0.5001 0.5001 0.5001 0.5001 0.5001 0.5000 10.5000 0.5000 0.5000 0.5000 0.5001 0.5001 0.5001 0.5000 0.5000 1 1 10.5000 0.4999 0.4999 0.5000 0.5001 0.5001 1 1 1 1 1 1

20.000021.250022.500023.750025.000026.250027.500028.750030.0000

Sta

ndar

d D

evia

tion

1.25002.50003.75005.00006.25007.50008.7500

10.000011.250012.500013.750015.000016.250017.500018.7500

0.0000

Fig. 5. LUT derived from equations and source code in Figures 2 and 3.

Case 1 0100011101000101010101000010000000101111001000000100100001010100010101000101000000101111001100010010111000110001000011010000101001001000011011110111001101110100001110100010000001110111011101110111011100101110011000110110110001101111011101010110010001110011

Case 2 0110100001101001011001010110110001100100001011100110001101101111011011010000110100001010010101010111001101100101011100100010110101000001011001110110010101101110011101000011101000100000010011010110111101111010011010010110110001101100011000010010111100110101

Case 3 1001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001

Case 4 0010000001110100011001010111100001110100001011110111100001101101011011000010110001100001011100000111000001101100011010010110001101100001011101000110100101101111011011100010111101111000011011010110110000101100011000010111000001110000011011000110100101100011

Case 5 1011000001000010101110000000000100000001000000010000000100110001110010011011000100011000010100001110001011111101001101010000000100000001000000010000010101010000100010011110010101010001011010000010111001100100011011000110110001101000011001010110110000110011

Case 6 1101100111111111100010110100010110110100100011010000110001000000100011010001010010001000110000011110001000000100000000011100001011000001111000100000100000101001110000101000110100000100100100000000000111011000100010010100010110110100011010100001000010001101

Case 7 0100100000000100010010000000010001001000000001000100100000000100010010000000010001001000000001000100100000000100010010001001000010010000100100001001000010010000100100001001000010010000100100001001000010010000100100001001000010010000100100001001000010010000

Case 8 0101100001011000010110000101100001011000010110000101100001011000010110000101100001011000010110000101100001011000010110000101100001011000010110000101100001011000010110000101100001011000010110000101100001011000010110000101100001011000010110000101100001011000

Table 1. 8 example cases.

Case Std. Dev. Skewness Result 1 5.1023 +/- 1.5098 0.7897 +/- 0.5339 Nominal 2 1.0863 +/- 1.5098 0.6205 +/- 0.5339 Nominal 3 0.3542 +/- 1.5098 0.0000 +/- 0.5339 Nominal 4 1.6501 +/- 1.5098 0.0992 +/- 0.5339 Nominal 5 10.5318 +/- 7.5622 0.4855 +/- 0.5270 Slammer Worm 6 12.2758 +/- 7.5622 -0.0819 +/- 0.5270 Slammer Worm 7 20.3279 +/- 7.5622 0.1443 +/- 0.5270 WebDav Worm 8 9.2613 +/- 7.5622 0.0000 +/- 0.5270 Red Probe Worm

Table 2. Results of passing statistics from 8 example cases through the LUT.

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SPIE, San Diego, CA, Aug 4, 2003. 5. Jaenisch, H.M., Handley, J.W., Faucheux, J.P. “Data Driven Differential Equation Modeling of fBm processes”,

Proceedings of SPIE, San Diego, CA, August 4, 2003. 6. Jaenisch, H., Handley, J., Pooley, J., Murray, S., “Virtual Instrument Prototyping With Data Modeling”, JANNAF

39th Combustion, 27th Airbreathing Propulsion, 21st Propulsion Systems Hazards, and 3rd Modeling and Simulation Subcommittees Joint Meeting, Colorado Springs, CO, (December 1-5, 2003).

7. Jaenisch, H., Handley, J., “Automatic Differential Equation Data Modeling for UAV Situational Awareness”, Society for Computer Simulation (Huntsville Simulation Conference 2003), Huntsville, AL, October 30-31, 2003.

8. Jaenisch, H., Handley, J., Lim, A., Filipovic, M., White, G., Hons, A., Deragopian, G., Schneider, M., Edwards, M., “Data Modeling for Virtual Observatory data mining”, SPIE Astronomical Telescopes and Instrumentation 2004, Glasgow, Scotland, UK, June 24, 2004.

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