Guofeng Cao

CyberInfrastructure and Geospatial Information Laboratory

Department of GeographyNational Center for Supercomputing Applications

(NCSA)University of Illinois at Urbana-Champaign

Geog 480: Principles of GISGeog 480: Principles of GIS

Spatial Reasoning and UncertaintySpatial Reasoning and Uncertainty

What you will learnWhat you will learn• What is spatial reasoning? • Why is spatial information imperfect?• What are the different types of imperfection in

spatial information? • How can we reason about spatial information

under uncertainty?• What qualitative and quantitative approaches to

uncertainty are there? • What sorts of applications exist for reasoning

under uncertainty?

Formal aspects of spatial reasoningFormal aspects of spatial reasoning

Spatial reasoningSpatial reasoning• Spatial reasoning has aspects that are:

o Cognitive o Computationalo Formal

• Formal aspects are derived from logic• Key logical distinction is between

o Syntax (see chapter 7)o Semantics (meaning)

• E.g., “Paris is in France”

Logic and deductionLogic and deduction

• Premiseso Facts: “Paris is the capital of France”o Rules: “All oak trees are broadleaved”

• Conclusions: deductive inferences• Soundness: All deductive inferences are true• Completeness: All true propositions may be

deduced

Paris is a city in France

All cities in France are European cities

Paris is a European city

x is a y

All y’s are z’s

x is a z

If it is snowing then John is skiing

It is snowing

John is skiing

All men are mortal

Socrates is a man

Socrates is mortal

Every day in the past the universe existed

The universe existed last Friday

Every day in the past the universe existed

The universe will exist next Friday

InferencesInferences

Spatial reasoning exampleSpatial reasoning exampleSuppose a knowledge base (KB) contains the following facts:1. Aland, Bland, Cland, and Dland are countries.2. Eye, Jay, Cay, and Ell are cities.3. Exe and Wye are rivers.4. City Eye belongs to Aland.5. City Jay belongs to Bland.6. City Cay belongs to Cland.7. City Ell belongs to Dland.8. Cities Eye, Ell, and Cay lie on the river Exe.9. City Jay lies on the river Wye.and rule:10. Each river passes through all countries to which the cities that lie

on it belong.

Spatial reasoning exampleSpatial reasoning example

Aland

B landCland

Dland

Eye

Ell

Cay

J ay

River Exe

River Wye

Assume that this representation is accurate.

There are truths expressed by the map but not deducible from the KB. e.g. ALand and BLand share a common boundary.

But, restrict attention to facts about countries, cities, rivers, cities in countries, cities on rivers, rivers through countries.

The KB is sound (all the statements in the KB are true in the map). The KB is not complete: e.g.”River Exe passes through countries Aland, Bland, Dland, Cland”, is true but not deducible in the KB.

Spatial reasoning exampleSpatial reasoning exampleHowever, if we add a further city Em, and facts to the KB:

13. Em is a city.14. Em belongs to the country Bland.15. The river Exe passes through city Em.

Aland

B landCland

Dland

Eye

Ell

Cay

J ay

River Exe

River WyeEm

Then the revised KB is sound and complete with respect to map, because we can now deduce: River Exe passes through the country Bland.

Spatial reasoning exampleSpatial reasoning example• UNO geologist: Video tells bin Laden's

hiding place Omaha World-Herald Posted on Tuesday,

October 16, 2001• “The image of Osama bin Laden that flickered on Jack Shroder'sTV was grainy and brief, but it was all he needed. JackShroder, a University of Nebraska at Omaha geologist who hasdone research in Afghanistan, says a videotape of Osama binLaden gives important clues to where he might be hiding…he iscertain that the type of sedimentary rock visible in the videotapeis found only in Paktia and Paktika, two provinces insoutheastern Afghanistan about 125 miles from Kabul.”• http://www.freerepublic.com/focus/f-news/549291/posts

Spatial reasoning exampleSpatial reasoning example• The Search for BlandingsBlandings Castle is a recurring fictionallocation in the stories of British comic writerP. G. Wodehouse, being the seat of LordEmsworth (Clarence Threepwood, 9th Earl ofEmsworth), home to many of his family, andsetting for numerous tales and adventures,written between 1915 and 1975• http://en.wikipedia.org/wiki/Blandings_Castle

Spatial reasoning exampleSpatial reasoning example• DARPA finder challenge• ESRI user conference finder challenge

Information and uncertaintyInformation and uncertainty

Information “flow”Information “flow”• Information source

produces a message consisting of an arrangement of symbols.

• Transmitter operates on message to produce a suitable signal to transmit.

• Channel the medium used to transmit the signal from transmitter to receiver.

• Receiver reconstructs the message from the signal.

• Destination for whom the message is intended.

UncertaintyUncertainty• Uncertainty

o May refer to state of mind: “I am unsure where the meeting will take place”

o May be applied directly to data or information about the world: “The depth of the sea at a particular location is uncertain”

• Uncertainty is an unavoidable property of the world, information about the world, and our cognition of the world

Spatial uncertainty exampleSpatial uncertainty example• Consider the capture of data about the boundary

of a lakeo Uncertain specifications: The lake’s boundary may not be

completely specified, e.g., • temporal variation in water’s edge• lack of clarity in definition of lake (vagueness)

o Uncertain measurements: The location of the lake’s boundary may be difficult to capture, e.g.,

• Incorrect instrument calibration (inaccuracy)• Mistakes in using the instruments• Lack of detail in measurement (imprecision)

o Uncertain transformations: Transformation of the data may introduce further uncertainty, e.g.,

• Measured points may be interpolated between to produce complete boundary

Typology of imperfectionTypology of imperfection

imperfection

error imprecision

vagueness

lack of correlation with reality

lack of specificity

“The Eiffel Tower is in

Lyons”

“The Eiffel Tower is in

France”

existence of borderline cases

“The Eiffel Tower is near the Arc de

Triomphe”

Granularity and indiscernibilityGranularity and indiscernibility• Granularity concerns the existence of

“clumps” or “grains” in data, where individual element cannot be discerned apart

• Indiscernibility is often assumed to be an equivalence relation (reflexive, symmetric, and transitive)

VaguenessVagueness• Vagueness concerns the existence of boundary cases• Vague predicates and objects admit borderline cases for

which it is not clear whether the predicate is true of false, e.g., “Mount Everest”o Some locations are definitely part of Mount Everest (e.g., the

summit)o Some locations are definitely not part of Mount Everest (e.g.,

Paris)o But for some locations it is indeterminate whether or not they

are part of Mount Everest• Vagueness is a pervasive feature of representations of the

real world.• Vagueness is not easy to handle using classical reasoning

approaches.

Reasoning with vaguenessReasoning with vagueness• Portland is definitely in

“southern Maine”• Presque Isle is definitely not

in “southern Maine”• Because “southern Maine”

has no precise boundary, a person’s single step cannot take you over the boundary

• Therefore, a hiker walking from Portland to Presque Isle would (eventually) conclude that Presque Isle is in “southern Maine”

• The sorites paradox

Dimensions of data qualityDimensions of data quality• Data quality refers to the characteristics of a

data set that may influence the decision based on that data set

Element Concise definitionaccuracy Closeness of the match between data and the things to which data

refersbias Existence of systematic distortions within data

completeness Exhaustiveness of data, in terms of the types of features that are represented in data

consistency Level of logical contradictions within datacurrency How “up-to-date” data is

format Structure and syntax used to encode datagranularity Existence of clumps or grains within data

lineage Provenance of data, including source, age, and intended use

precision Level of detail or specificity of datareliability Trustworthiness of degree of confidence a user may have in data

timeliness How relevant data is to the current needs of a user

ConsistencyConsistency• Consistency is violated when information is self-

contradictoryBangor, Maine has a population of 31, 000 inhabitants.Only cites with more than 50,000 inhabitants are large.Bangor is a large city.

• Inconsistency can arise with:o Inaccuracyo Imprecisiono vagueness

• Action prompted by inconsistency:o Resolve inconsistencyo Retain inconsistencyo Initiate dialog

RelevanceRelevance• Relevance: the connection of a data set to a

particular application• Relevance helps to assess fitness for use of a

data set for a particular application o Study of habitat change in a national park o Tourist map to help inform and educate visitors

• Role of metadata

Quantitative approaches to Quantitative approaches to uncertaintyuncertainty

ProbabilityProbabilityRandom experiments

If X denotes the set of possible outcomes, we can specify a chance function

ch : X -> [0,1]

ch(x) gives the proportion of times that a particular outcome x in X might occur

• Frequency analysis

• The nature of the experiment

ch should satisfy the constraint that the sum of chances of all possible outcomes is 1

For a subset S belonging to X, ch(S) is the chance of an outcome from set S

RulesRules• ch(0) = 0• ch(X) = 1• If A and B = 0 ;, then ch(A or B) = ch(A) = ch(B)Also, given n independent trials of a random

experiment, the chance of the compound outcome chn (x1,…,xn) is given by:

• chn(x1, …, xn) = ch(x1)*…*ch(xn)

Conditional ProbabilityConditional Probability• Suppose a random experiment has been

partly completedo Set V belongs to X

• If U belonging to X is the outcome set under consideration, the chance of U given V is written:o ch(U|V)

• Then:

Bayesian probabilityBayesian probability• A degree of belief with respect to a set X

of possibilitieso Bel : X ! [0,1]

• Suppose we begin with the above belief function and then learn that only a subset of possibilities V µ X is the case

Bayesian probabilityBayesian probability• We can manipulate the equation to get:

Posterior belief Bel(U|V) is calculated by multiplying our prior belief Bel(U) by the likelihood that V will occur if U is the case.

Bel(V) acts as a normalizing constant that ensures that Bel(U|V) will lie in the interval [0,1]

Dempster-Shafter theory of evidenceDempster-Shafter theory of evidence• Takes account of evidence both for and against a

belief• Take the statement: p: “Region A is forested”• Credibility: the amount of evidence we have in

its favoro credibility (p) = Bel (p)

• Plausibility: the lack of evidence we have against ito plausibility (p) = 1 - Bel(: p)

Applications of uncertainty in GISApplications of uncertainty in GIS

Uncertainty in GISUncertainty in GIS• GIS databases built from maps are not

necessarily objective, scientific measurements of the world it is impossible to create a perfect representation of the world in a GIS database therefore all GIS data are subject to uncertaintyo Uncertainty arise in every state of map production processeso uncertainty regarding what the data tell us about the real world a

range of possible truthso that uncertainty will affect the results of analysiso all GIS results should have confidence limits, "plus or minus"

Uncertain in GISUncertain in GIS

Uncertain in GISUncertain in GIS• It is an example of positional errors in two

commercial street centerline databases of Goletao the background fill is darkest where errors are smallest

• note how the errors are often up to 100mo a problem if someone reports the location of a fire using one map, and

a response is dispatched using the other mapo the response vehicle could be sent to the wrong streetnote also how

many streets are not in both databaseso notice how errors persist over large areas

• the error at one point is not independent of error at neighboring points

• this is a general characteristic of error in GIS databases

Uncertain in GISUncertain in GIS• It is an example of positional errors in two

commercial street centerline databases of Goletao the background fill is darkest where errors are smallest

• note how the errors are often up to 100mo a problem if someone reports the location of a fire using one map, and

a response is dispatched using the other mapo the response vehicle could be sent to the wrong streetnote also how

many streets are not in both databaseso notice how errors persist over large areas

• the error at one point is not independent of error at neighboring points

• this is a general characteristic of error in GIS databases

Uncertainty in DEMsUncertainty in DEMs• USGS quality description

o a DEM provides measurements of the elevation of the land surface at each grid point

o errors are due to:• measurement of the wrong elevation at the grid pointmeasurement

of the right elevation at the wrong location• any combination of these• it is impossible to determine which case applies

• the USGS provides simple quality statements for its DEMso given as "root mean square error"this is the square root of the average

squared difference between recorded elevation and the trutho roughly interpreted as the average differenceo e.g. many DEMs have RMSE of 7mo an error of 7m is common and errors of 10m, even 20m occur

sometimes

Uncertainty in DEMsUncertainty in DEMs

Uncertainty in area-class mapsUncertainty in area-class maps• Nature of errors

o area class maps show a class at every pointo Typical examples include vegetation cover maps, soil maps, land use

mapso they imply that class is uniform within areas, changes abruptly

between areas• in fact both assumptions are not right• there should be variation within areas (heterogeneity)• there should be blurring across boundaries

o area class maps have been described as "maps showing areas that have little in common, surrounded by lines that do not exist"

Uncertainty in area-class mapsUncertainty in area-class maps

Uncertainty in area-class mapsUncertainty in area-class maps• For yellow regions

o let's assume the legend says this class is "80% sand, with 20% inclusions of clay"this map is used for many purposes

o some involve land use regulationo some involve taxation, compensationo in principle, all of these are uncertain if the map is uncertaino GIS applications are in deep trouble in court if it can be shown that

regulations, taxes were based on uncertainty and that no effort was made to deal with that uncertainty

Uncertainty PropagationUncertainty Propagation

General Strategy for General Strategy for

Uncertainty EvaluationUncertainty Evaluation

Image courtesy: Phaedon C. Kyriakidis and Jennifer L. DunganImage courtesy: Phaedon C. Kyriakidis and Jennifer L. Dungan

Uncertain viewshedsUncertain viewsheds• Viewshed: a region of terrain visible from a point

or set of points• Probable viewshed:

o Uncertainty arising through imprecision and inaccuracy in measurements of the elevation

o Boundary will be crisp but its position uncertain

• Fuzzy viewshed:o Uncertainty arising from atmospheric conditions, light refraction, and

seasonal and vegetation effectso Boundary is broad and gradedo Fuzzy regions are often used

Spatial relations experimentSpatial relations experiment• Sketch map of significant

location on Keele University campus

• Experimento Human subjects were

divided into two equal groups

• Truth group: when is it true to say that place x is near place y

• Falsity group: when is it false to say that place x is near place y

Responses to questionnaireResponses to questionnaire• Amalgamated responses to questionnaires

concerning nearness to the library

Location T F

12. Horwood Hall 4 10

13. Keele Hall 8 2

14. Lakes 1 11

15. Leisure Center 0 11

16. Library 11 0

17. Lindsay Hall 2 8

18. Observatory 0 11

19. Physics 5 5

20. Reception 4 4

21. Student Union 10 0

22. Visual Arts 1 10

Location T F

1. Academic Affairs 5 2

2. Barnes hall 0 11

3. Biological Sciences 5 4

4. Chancellors Building 4 6

5. Chapel 10 0

6. Chemistry 4 6

7. Clock house 4 6

8. Computer science 1 10

9. Earth Sciences 7 0

10. Health Centre 1 11

11. Holy Cross 1 11

Significance testSignificance test

• Statistical significance test o Possible to

evaluate the extent to which the pooled responses indicate whether each location is considered near to the other locations.

o Three valued logic

Three-valued logicThree-valued logic• A three valued nearness relation could be used

to describe the nearness of campus locations to one anothero For two places x and y, xy will evaluate to

• T if x is significantly near to y• F if x is significantly not near to y• ? if xy > and xy ?

• End of this topic