Probability Density ProfilesProbability Density Profiles

A New Perspective on

Subject Behavior Profiles

by

Martin Colwell

SAR Technology Inc. © 2007

  Illustrate the Implementation of PDEN GIS

Probability Density ProfilesProbability Density Profiles A New Perspective on Subject Behavior Profiles

Review traditional Probability Profile Plots

Demonstrate their typical use in SAR planning 

Determine their strengths & weaknesses

Consider a new method for determining Subject Probability Profiles

Describe the theory of PDEN ProfilesDescribe the theory of PDEN Profiles

Create real-data PDEN Profiles

Describe the mathematical basis for PDEN Profiles

Describe the benefits of PDEN Profiles

Demonstrate the generation of PDEN Profile Maps

Traditional Probability Profile Plots

Mean (Average): The middle value of all the distances found from the PLS (or IPP).

Utilize the standard mathematical terms of Median and Mean (Average) distances the subjects were found from the Point Last Seen (or IPP).

Median: The distance at which there are an even number of cases on either side.

Traditional Probability Profile PlotsThe original distribution of the distances the subjects were found from the PLS.

Median Mean

Median: The distance at which there are an even number of cases on either side.

Mean (Average): The middle value of all the distances found from the PLS.

Traditional Probability Profile PlotsProbability Band (Annulus): Defines a distance either side of the median in which a percentage of the subjects were found (eg 25%, 50%, 75% 100%)

Median

Traditional Probability Profile Plots

Probability Zones: Define a circular zone, from the PLS, in which a percentage of the subjects were found (eg 25%, 50%, 75% 100% POA zones)

25% POA Zone50% POA Zone

Traditional Probability Profile Plots

75% POA Zone

Probability Bands from the MedianProbability Zone from the PLS

Typically display subject location POA’s within specific zones or bands

Traditional Probability Profile Plots

Disadvantages:

Do not take into account either the direction of the subject or the increasing size of the area with distance from thePoint Last Seen.

Distribution Plots

Advantages:

Illustrate the linear frequency of actual distances the subjects were found from the Point Last Seen.

Weakness of Traditional POA Probability Profile Plots

• Probability bands do not take into account the smaller search area inside, compared to outside, the mean.

Median and Mean (average) do not describe the actual distribution of the subjects.

There may be no subjects found at the median!

Do not portray uniformly dispersed or clustered locations either side of the mean.

Do not indicate the probable direction of travel.

Probability bands do not describe whether subjects were found inside or outside of the mean.

-         Displayed as Probability Density Maps in true GIS Applications

Probability Density ProfilesProbability Density Profilesof Missing Persons

- Based on the original subject Distance & Direction Information

-  Uses an Optimized Bandwidth Procedure to extract useful information

-         Creates Weibull Type 2 (Raleigh) Distribution Profiles 

Utilizes Polar (Directional) Coordinate Data

-         Creates a matrix of Cartesian (XYZ) Grid Data   (With thousands of XYZ data points) 

-         Displayed as Two Dimensional & Three DimensionalProbability Density Maps

-         Generates ESRI GIS Shape and TIFF Files

-         Can be Geo-Referenced to GeoTiff Files

Probability Density ProfilesProbability Density Profiles – – Uniform Distribution

Probability Density ProfilesProbability Density Profiles – – Single Cluster Distribution

Probability Density ProfilesProbability Density Profiles – – Two Cluster Distribution

Probability Density ProfileProbability Density Profile – – Children 1-6 years oldProbability Density vs Radius from PLS

0% 40% 50% 60% 100%90%80%70%20%

10% 30%

0

10

20

30

40

50

60

70

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50Distance from PLS, km

Prob

abili

ty D

ensi

ty %

POA

/Sq.

Km

Probability Density

% Probability of Area

Children 1 to 6 years old

0.1km bandwidth

Probability of Area

Radial profile from the PLS

Conclusions: 1. A major PDEN peak occurs at 0.55km Radius, approx 20% POA.2. All the high PDEN region is between the PLS and the median.3. The PDEN at the median (0.95 km, 50%POA) is significantly lower than closer to the PLS.4. Searching from PLS out towards the median is more productive, per unit area, than beyond the median.5. Search radially outwards from the PLS.6.Search outwards beyond the median only after searching from the PLS towards the median.

Median0.95 km

0.55 km

PLS

Probability Density ProfileProbability Density Profile – – Children 1-6 years old

Narrow Narrow

Optimal Optimal

Broad

Broad

Probability Density ProfilesProbability Density Profiles – – Resemble Weibull DistributionsNaturally occurring distributions, with mathematically described scale & shape

Shape=2

Probability Density ProfilesProbability Density Profiles

0

0.1

0.2

0.3

0.4

0.5

0.6

0 5 10 15 20 25 30 35

Time

Do

llar

s (i

n $

M)

Funding Profile Over TimeFunding Profile Over Time

Probability density:

v(t) = 2adte-at2

- The Weibull distribution models the value’s buildup, peak and taper.

Probability Density ProfilesProbability Density Profiles - The Weibull distribution describes the curves buildup, peak and taper.

Probability Density = 2adle-al2

L = Distanced = Scale factor

a = Shape parameter

The Weibull distribution curve can have it’s shape described in mathematical terms.

A general PDEN profile, using the distribution formula, could then be applied to other subject types

Predicted Probability Density Profile

0

10

20

30

40

50

60

70

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50Distance from PLS, km

Prob

abili

ty D

ensi

ty %

POA

/Sq.

Km

Probability Density

Predicted PDEN

0.1km bandwidthPDEN Profile Prediction: - Given the shape (a), scale factor (d) and median distance (l) for the known PDEN profile Children 1 to 6, Average Age, 3.5 years old.

- And given the median distance (l) for Children 2 to 8, average 5.5, years old. -We can draw a predicted PDEN profile for Children of Average Age 5.5 years old

Median0.95 km

0.55 km

PLS Children 1 to 6 years old (Average 3.5 years old)

Predicted PDEN Profile: Children 2 to 8 years old (Average Age 5.5 years old)

Median1.25 km

50% POA

Probability Density = 2adle-al2

Probability Density ProfilesProbability Density Profiles - PDEN Profile Prediction

Probability Density ProfileProbability Density Profile – – Children 1-6 years old

Demonstrating 360o Radial probability density profile from the PLS

Median 0.95km

Probability Density ProfileProbability Density Profile – – Children 1-6 years old

Demonstrating Directional probability density profile from the PLS

Probability Density ProfileProbability Density Profile – – Children 1-6 years oldDemonstrating Directional probability density profile from the PLS

ExpectedDirectionof Travel

75%POA

75%POA

50%POA

50%POA

25%POA

25%POA

Probability Density ProfilesProbability Density Profiles – – Children 1-6 years old

Conversion of Radial PDEN Data to X,Y,Z Grid for GIS Display

GIS Surface plot

Probability Density ProfilesProbability Density Profiles – – Children 1-6 years old

PDEN Data in an X,Y,Z Grid format for GIS Display

360 radial data points converts to 6,708 X,Y,Z GIS data points

Probability Density ProfilesProbability Density Profiles – – Children 1-6 years old

PDEN Data in X,Y,Z Grid format for GIS Display

Probability Density Profile displayed as a 3D GIS surface plot

PLS

Probability Density ProfilesProbability Density Profiles – – Children 1-6 years oldProbability Density Profile displayed as a 2D GIS contour map

PLSExpectedDirectionof Travel

km

km

With geo-referencing,PDEN contour profiles can be displayed as map layers over a GIS base map.

High PDEN region

Low PDENregion

2D PDEN Contour map, showing the highest PDEN region in dark red, lowest PDEN region in pale blue.

The distance from the central PLS is displayed on X & Y scalebars.

Probability Density ProfilesProbability Density Profiles – – Children 1-6 years old

2D Probability Density Profile - GIS Contour Map Layer

Associated geo-referencing data displays the PDEN contour map in it’s true geographical location and to the correct map scale.

High PDEN region

Low PDENregion

ExpectedDirectionof Travel

PLS

2 km

2 km

The 2D PDEN Contour map is displayed as a geo-referenced map layer above a street map layer, defining the high PDEN search region on the street map.

Probability Density ProfilesProbability Density Profiles – – Children 1-6 years old

2D Probability Density Profiles – As GIS Map Layer

The 2D PDEN Contour and Surface Maps can be overlayed on GIS map layers for defining the high probability density areas.

2D and 3D Probability Density Profile Maps can be created in a varietyof formats to easily visualize the high probability density areas.

Probability Density ProfilesProbability Density Profiles

Hikers –North America

50%POA

25%POA

Median2.4km

75%POA

Median2.4km

Preliminary Conclusions

Children, Hikers and Hunters PDEN profiles have been examined so far.

PDEN Map Profiles provide more useful radial and directional planning data than traditional subject POA profiles.

Need to re-process the original subject distribution datasets using the Optimized Bandwidth Procedure.

‘Template’ profiles could be created for other subject types with limited datasets.

Create a data library of 2D & 3D Subject PDEN Profile maps.

Distribute PDEN Profile Maps in various formats – GIS Shape files, Geotiffs, JPG’s, DXF files etc.

Weibull distributions appear to occur for many subject types.

Probability Density ProfilesProbability Density ProfilesTraditional POA Subject Profile

PDEN Map Subject Profile

Median2.4km

25%POA

50%POA

75%POA

Interim Recommendations ForSubject Profile Planning

Based on a preliminary examinationof the PDEN Profile Maps, comparedto traditional Subject POA Profiles.

Search radially outwards from

the PLS towards the median.Almost all of the high PDEN region is between the PLS and well before the median.

The median region may have a fairly low PDEN.

Search outwards beyond the median only after searching from the PLS towards the median.

Remember that 50% of subjects are still found beyond the median!