1

Eats4400.03Eats4400.03

GIS and Data IntegrationGIS and Data IntegrationGIS and Data IntegrationGIS and Data IntegrationGIS and Data IntegrationGIS and Data IntegrationGIS and Data IntegrationGIS and Data Integration

Lecture Eight Lecture Eight Lecture Eight Lecture Eight Lecture Eight Lecture Eight Lecture Eight Lecture Eight

Weights of Evidence Method Weights of Evidence Method Weights of Evidence Method Weights of Evidence Method Weights of Evidence Method Weights of Evidence Method Weights of Evidence Method Weights of Evidence Method

for for for for for for for for

Spatial Decision Support ModelingSpatial Decision Support ModelingSpatial Decision Support ModelingSpatial Decision Support ModelingSpatial Decision Support ModelingSpatial Decision Support ModelingSpatial Decision Support ModelingSpatial Decision Support Modeling

Cheng Cheng Cheng Cheng ESSEESSEESSEESSE Spatial Decision Support System Spatial Decision Support System Spatial Decision Support System Spatial Decision Support System Multiple Map ModelingMultiple Map ModelingMultiple Map ModelingMultiple Map Modeling

Decision Theory is concerned with the Decision Theory is concerned with the Decision Theory is concerned with the Decision Theory is concerned with the logic by which one arrives at a choice logic by which one arrives at a choice logic by which one arrives at a choice logic by which one arrives at a choice between alternatives. between alternatives. between alternatives. between alternatives.

Alternative ActionsAlternative ActionsAlternative ActionsAlternative ActionsAlternative hypothesesAlternative hypothesesAlternative hypothesesAlternative hypothesesAlternative objectsAlternative objectsAlternative objectsAlternative objects

so onso onso onso on

Potential ApplicationsPotential ApplicationsPotential ApplicationsPotential ApplicationsPotential ApplicationsPotential ApplicationsPotential ApplicationsPotential Applications

Site Selection Site Selection Site Selection Site Selection Site Selection Site Selection Site Selection Site Selection Suitability Assessment Suitability Assessment Suitability Assessment Suitability Assessment Suitability Assessment Suitability Assessment Suitability Assessment Suitability Assessment Favorability AssessmentFavorability AssessmentFavorability AssessmentFavorability AssessmentFavorability AssessmentFavorability AssessmentFavorability AssessmentFavorability Assessment

Probability Assessment Probability Assessment Probability Assessment Probability Assessment Probability Assessment Probability Assessment Probability Assessment Probability Assessment

Spatial Decision Support System (SDSS)Spatial Decision Support System (SDSS)Spatial Decision Support System (SDSS)Spatial Decision Support System (SDSS)Spatial Decision Support System (SDSS)Spatial Decision Support System (SDSS)Spatial Decision Support System (SDSS)Spatial Decision Support System (SDSS)GIS Data Integration for PredictionGIS Data Integration for PredictionGIS Data Integration for PredictionGIS Data Integration for PredictionGIS Data Integration for PredictionGIS Data Integration for PredictionGIS Data Integration for PredictionGIS Data Integration for Prediction

Remote SensingRemote SensingRemote SensingRemote Sensing

GeologicalGeologicalGeologicalGeological

GeochemicalGeochemicalGeochemicalGeochemical

GeophysicalGeophysicalGeophysicalGeophysical

.

. PotentialPotentialPotentialPotential

Evidential Layers (X)Evidential Layers (X)Evidential Layers (X)Evidential Layers (X)

Modeling Modeling Modeling Modeling (F) (F) (F) (F)

Output Data (S)Output Data (S)Output Data (S)Output Data (S)

ProcessingProcessingProcessingProcessing

GIS Data GIS Data GIS Data GIS Data SourcesSourcesSourcesSources

Data Data Data Data PreprocessingPreprocessingPreprocessingPreprocessingInterpretingInterpretingInterpretingInterpretingInformation Information Information Information ExtractionExtractionExtractionExtraction

IntegrationIntegrationIntegrationIntegration

DBMSDBMSDBMSDBMS

DBMSDBMSDBMSDBMS

DBMSDBMSDBMSDBMS

DBMSDBMSDBMSDBMS

GeographicalGeographicalGeographicalGeographical

General Model General Model General Model General Model General Model General Model General Model General Model

nnxwxwxwwS ++++= ...22110

S S S S S S S S –––––––– Index map showing Index map showing Index map showing Index map showing Index map showing Index map showing Index map showing Index map showing SuitabilitySuitabilitySuitabilitySuitabilitySuitabilitySuitabilitySuitabilitySuitabilityProbabilityProbabilityProbabilityProbabilityProbabilityProbabilityProbabilityProbability

xxxxxxxxiiiiiiii - maps or evidences - maps or evidences - maps or evidences - maps or evidences - maps or evidences - maps or evidences - maps or evidences - maps or evidenceswwwwwwwwiiiiiiii - weights - weights - weights - weights - weights - weights - weights - weights

2

Suitability Map for Planning School Suitability Map for Planning School Suitability Map for Planning School Suitability Map for Planning School Suitability Map for Planning School Suitability Map for Planning School Suitability Map for Planning School Suitability Map for Planning School Model Constraints Model Constraints Model Constraints Model Constraints Model Constraints Model Constraints Model Constraints Model Constraints

Normalization: Normalization: Normalization: Normalization: Normalization: Normalization: Normalization: Normalization:

1. Convert maps into comparable unit1. Convert maps into comparable unit1. Convert maps into comparable unit1. Convert maps into comparable unit1. Convert maps into comparable unit1. Convert maps into comparable unit1. Convert maps into comparable unit1. Convert maps into comparable unit

⎩⎩⎩⎩⎨⎨⎨⎨⎧⎧⎧⎧

====noyes

xi ,0,1

2. Weights showing relative importance 2. Weights showing relative importance 2. Weights showing relative importance 2. Weights showing relative importance 2. Weights showing relative importance 2. Weights showing relative importance 2. Weights showing relative importance 2. Weights showing relative importance of mapsof mapsof mapsof mapsof mapsof mapsof mapsof maps

101...21

≤≤≤≤≤≤≤≤====++++++++++++

i

n

wwww

⎪⎪⎪⎪⎩⎩⎩⎩

⎪⎪⎪⎪⎨⎨⎨⎨

⎧⎧⎧⎧====

nounknown

yesxi

,0,5.0

,110...,,2,1====ix

Methods for Calculating Weights for Methods for Calculating Weights for Methods for Calculating Weights for Methods for Calculating Weights for Methods for Calculating Weights for Methods for Calculating Weights for Methods for Calculating Weights for Methods for Calculating Weights for Data IntegrationData IntegrationData IntegrationData IntegrationData IntegrationData IntegrationData IntegrationData Integration

Data Driven Methods:Data Driven Methods:Data Driven Methods:Data Driven Methods:Data Driven Methods:Data Driven Methods:Data Driven Methods:Data Driven Methods:Weights of evidenceWeights of evidenceWeights of evidenceWeights of evidenceWeights of evidenceWeights of evidenceWeights of evidenceWeights of evidenceLogistic regressionLogistic regressionLogistic regressionLogistic regressionLogistic regressionLogistic regressionLogistic regressionLogistic regression

Artificial Neural networkArtificial Neural networkArtificial Neural networkArtificial Neural networkArtificial Neural networkArtificial Neural networkArtificial Neural networkArtificial Neural network

Knowledge driven Methods:Knowledge driven Methods:Knowledge driven Methods:Knowledge driven Methods:Knowledge driven Methods:Knowledge driven Methods:Knowledge driven Methods:Knowledge driven Methods:Fuzzy logicFuzzy logicFuzzy logicFuzzy logicFuzzy logicFuzzy logicFuzzy logicFuzzy logic

Hybrid Methods:Hybrid Methods:Hybrid Methods:Hybrid Methods:Hybrid Methods:Hybrid Methods:Hybrid Methods:Hybrid Methods:Fuzzy weights of evidenceFuzzy weights of evidenceFuzzy weights of evidenceFuzzy weights of evidenceFuzzy weights of evidenceFuzzy weights of evidenceFuzzy weights of evidenceFuzzy weights of evidence

Model Types Model Types Model Types Model Types Model Types Model Types Model Types Model Types

1. Probabilistic1. Probabilistic1. Probabilistic1. Probabilistic1. Probabilistic1. Probabilistic1. Probabilistic1. Probabilistic

2. Deterministic2. Deterministic2. Deterministic2. Deterministic2. Deterministic2. Deterministic2. Deterministic2. Deterministic

S S S S S S S S –––––––– random variable showing random variable showing random variable showing random variable showing random variable showing random variable showing random variable showing random variable showing probability 0 probability 0 probability 0 probability 0 probability 0 probability 0 probability 0 probability 0 ≤≤≤≤≤≤≤≤ S S S S S S S S ≤≤≤≤≤≤≤≤ 1 with uncertainty1 with uncertainty1 with uncertainty1 with uncertainty1 with uncertainty1 with uncertainty1 with uncertainty1 with uncertainty

S S S S S S S S –––––––– Score 0 Score 0 Score 0 Score 0 Score 0 Score 0 Score 0 Score 0 ≤≤≤≤≤≤≤≤ S S S S S S S S ≤≤≤≤≤≤≤≤ 11111111

Model Constraints Model Constraints Model Constraints Model Constraints Model Constraints Model Constraints Model Constraints Model Constraints

Normalization: Normalization: Normalization: Normalization: Normalization: Normalization: Normalization: Normalization:

1. Convert maps into comparable unit1. Convert maps into comparable unit1. Convert maps into comparable unit1. Convert maps into comparable unit1. Convert maps into comparable unit1. Convert maps into comparable unit1. Convert maps into comparable unit1. Convert maps into comparable unit

⎜⎜⎜⎜⎜⎜⎜⎜⎝⎝⎝⎝

⎛⎛⎛⎛====

enotsuitablabsentnosuitablepresentyes

xi ,,,0,,,1 10...,,2,1====ix

2. Assigning weights for each map as %2. Assigning weights for each map as %2. Assigning weights for each map as %2. Assigning weights for each map as %2. Assigning weights for each map as %2. Assigning weights for each map as %2. Assigning weights for each map as %2. Assigning weights for each map as %

101...21

≤≤≤≤≤≤≤≤====++++++++++++

i

n

wwww

Combine Maps Combine Maps Combine Maps Combine Maps Combine Maps Combine Maps Combine Maps Combine Maps

Grid calculator with equationGrid calculator with equationGrid calculator with equationGrid calculator with equationGrid calculator with equationGrid calculator with equationGrid calculator with equationGrid calculator with equation

S = 0.50 S = 0.50 S = 0.50 S = 0.50 S = 0.50 S = 0.50 S = 0.50 S = 0.50 rec_siterec_siterec_siterec_siterec_siterec_siterec_siterec_site +0.25 dist_school + +0.25 dist_school + +0.25 dist_school + +0.25 dist_school + +0.25 dist_school + +0.25 dist_school + +0.25 dist_school + +0.25 dist_school + 0.125rec_landuse + 0.125 0.125rec_landuse + 0.125 0.125rec_landuse + 0.125 0.125rec_landuse + 0.125 0.125rec_landuse + 0.125 0.125rec_landuse + 0.125 0.125rec_landuse + 0.125 0.125rec_landuse + 0.125 rec_sloperec_sloperec_sloperec_sloperec_sloperec_sloperec_sloperec_slope

Map S has values between 1 -10Map S has values between 1 -10Map S has values between 1 -10Map S has values between 1 -10Map S has values between 1 -10Map S has values between 1 -10Map S has values between 1 -10Map S has values between 1 -10

3

More Models for SDSSMore Models for SDSSMore Models for SDSSMore Models for SDSSMore Models for SDSSMore Models for SDSSMore Models for SDSSMore Models for SDSS

nS = F(xS = F(xS = F(xS = F(x1111, x, x, x, x2222,,,,…………, , , , xxxxnnnn))))

OOP

PPO

+=

−=

11

nnxwxwxwwPPLog ++++=−

...1 22110

LogisticLogisticLogisticLogisticLogisticLogisticLogisticLogistic --------Model Model Model Model Model Model Model Model

]...[ 2211011

nnxwxwxwweP ++++−+=

LogisticLogisticLogisticLogisticLogisticLogisticLogisticLogistic --------Model Model Model Model Model Model Model Model

Logistic regression methodLogistic regression methodLogistic regression methodLogistic regression methodLogistic regression methodLogistic regression methodLogistic regression methodLogistic regression method

Weights of evidence methodWeights of evidence methodWeights of evidence methodWeights of evidence methodWeights of evidence methodWeights of evidence methodWeights of evidence methodWeights of evidence method

nnxwxwxww

PPLog ++++=−

...1 22110

LogisticLogisticLogisticLogisticLogisticLogisticLogisticLogistic --------Model Model Model Model Model Model Model Model

]...|[ 21 nxxxDP

D - point event D - point event D - point event D - point event D - point event D - point event D - point event D - point event P[D] - probability of point event

P[D|...] - conditional probability of D given ...

Spatial analysis of wild animals habitat Spatial analysis of wild animals habitat Spatial analysis of wild animals habitat Spatial analysis of wild animals habitat

In this example, one will investigate the In this example, one will investigate the In this example, one will investigate the In this example, one will investigate the association between wild animal habitat and association between wild animal habitat and association between wild animal habitat and association between wild animal habitat and two types of patterns: 1. treed area, and 2. two types of patterns: 1. treed area, and 2. two types of patterns: 1. treed area, and 2. two types of patterns: 1. treed area, and 2. park. The study is about 100kmpark. The study is about 100kmpark. The study is about 100kmpark. The study is about 100km2222, samples of , samples of , samples of , samples of locations of 10 wild animals were collected locations of 10 wild animals were collected locations of 10 wild animals were collected locations of 10 wild animals were collected using GPS. using GPS. using GPS. using GPS.

We will create a map to show probability of We will create a map to show probability of We will create a map to show probability of We will create a map to show probability of occurrence of wild animals in the area. occurrence of wild animals in the area. occurrence of wild animals in the area. occurrence of wild animals in the area.

5.0)|( =DABP

Probability mappingProbability mappingProbability mappingProbability mapping

not A not B

A not B

not A B

A B

Treed area and park Treed area and park Treed area and park Treed area and park –––– wild animals habitat wild animals habitat wild animals habitat wild animals habitat

4

Probability of point event D in an area Probability of point event D in an area Probability of point event D in an area Probability of point event D in an area

P[D]P[D]P[D]P[D] = #of = #of = #of = #of boxes occupied by boxes occupied by boxes occupied by boxes occupied by points/ points/ points/ points/ # of boxes in an # of boxes in an # of boxes in an # of boxes in an areaareaareaarea

Density of points = 10/100 = 0.10Density of points = 10/100 = 0.10Density of points = 10/100 = 0.10Density of points = 10/100 = 0.10 prior probability prior probability prior probability prior probability

Area = 100 cellsArea = 100 cellsArea = 100 cellsArea = 100 cellsPoints = 10Points = 10Points = 10Points = 10

Assume each point Assume each point Assume each point Assume each point occupies one celloccupies one celloccupies one celloccupies one cell

5.0)|( =DABP

3

7

point

0.0648not B

0.1352B

densityarea

0.06

0.13

density of points density of points density of points density of points

B

not B

0.30

0.05

A

not A

density of points density of points density of points density of points 4

6

point

0.0580not A

0.3020A

densityarea

3124Points

4535137Area

0.06BnotA40.02not BnotA30.15notBA 20.57BA 1points/areaPolyBPolyAID

A B

A

not A

not B

B

not A not B

A not B

not A B

A B

not A not B

A not B

not A B

0.57

0.15

0.02

0.06

Prior probability total number of point / total area 10/100 = 0.10 (10%)

Posterior probability: number of point /pattern area

5

BayesBayesBayesBayes’’’’ssss Rule: Rule: Rule: Rule:

Probability map Probability map Probability map Probability map P(D|A) = P(D)P(A|D)/P(A)P(D|A) = P(D)P(A|D)/P(A)P(D|A) = P(D)P(A|D)/P(A)P(D|A) = P(D)P(A|D)/P(A)P(D|notAP(D|notAP(D|notAP(D|notA) = ) = ) = ) = P(D)P(notA|D)/P(notDP(D)P(notA|D)/P(notDP(D)P(notA|D)/P(notDP(D)P(notA|D)/P(notD))))

P(D|AB) = P(D) P(AB|D)/P(AB)P(D|AB) = P(D) P(AB|D)/P(AB)P(D|AB) = P(D) P(AB|D)/P(AB)P(D|AB) = P(D) P(AB|D)/P(AB)P(D|AnotBP(D|AnotBP(D|AnotBP(D|AnotB) = ) = ) = ) = P(D)P(AnotB|D)/P(AnotBP(D)P(AnotB|D)/P(AnotBP(D)P(AnotB|D)/P(AnotBP(D)P(AnotB|D)/P(AnotB))))........

Prior probability: total number of point / total area 10/100 = 0.10 (10%) Posterior probability: number of point /pattern area (density of point/area) - P(D|AB)

Concept of Prior probability and Posterior probability

110.40.40.60.6

0.30.30.10.1(0.12)(0.12)

0.20.2(0.18)(0.18)

not Bnot B

0.70.70.30.3(0.28)(0.28)

0.40.4(0.42)(0.42)

BB

not Anot AAA

Percentage of points Percentage of points Percentage of points Percentage of points

P(AB|D) P(notA B|D)

P(A notB|D) P(notA notB|D)

P(B|D)

P(notB|D)

P(notA|D)P(A|D)

marginal probabilitymarginal probabilitymarginal probabilitymarginal probability

Joint probabilityJoint probabilityJoint probabilityJoint probability

Percentage of points of independent eventsPercentage of points of independent eventsPercentage of points of independent eventsPercentage of points of independent events

P(AB|D) = P(A|D) P(B|D)P(AB|D) = P(A|D) P(B|D)P(AB|D) = P(A|D) P(B|D)P(AB|D) = P(A|D) P(B|D)

marginal probabilitymarginal probabilitymarginal probabilitymarginal probabilityJoint probabilityJoint probabilityJoint probabilityJoint probability

Percentage of points on AB = Percentage of points on AB = Percentage of points on AB = Percentage of points on AB = % points on A % points on A % points on A % points on A **** % points on B % points on B % points on B % points on B

5.0)|( =DABP

0.07(0.10)

110.480.480.520.52

0.80.80.350.35(0.38)(0.38)

0.450.45(0.42)(0.42)

not Bnot B

0.20.20.130.13(0.(0.1010))

0.070.07(0.10)(0.10)

BB

not Anot AAA

0.45(0.42)

0.35(0.38)

0.13 (0.10)

not A not B

A not B

not A B

A B

Percentage of Areas Percentage of Areas Percentage of Areas Percentage of Areas

110.480.480.520.52

0.800.800.350.35(0.38)(0.38)

0.450.45(0.42)(0.42)

not Bnot B

0.200.200.130.13(0.(0.1010))

0.0.0707(0.10)(0.10)

BB

not Anot AAA

Percentage of areas Percentage of areas Percentage of areas Percentage of areas

P(AB) P(notA B)

P(A notB) P(notA notB)

P(B)

P(notB)

P(notA)P(A)

marginal probabilitymarginal probabilitymarginal probabilitymarginal probability

Joint probabilityJoint probabilityJoint probabilityJoint probability

6

Percentage of areas of independent eventsPercentage of areas of independent eventsPercentage of areas of independent eventsPercentage of areas of independent events

P(AB) = P(A) P(B)P(AB) = P(A) P(B)P(AB) = P(A) P(B)P(AB) = P(A) P(B)

marginal probabilitymarginal probabilitymarginal probabilitymarginal probabilityJoint probabilityJoint probabilityJoint probabilityJoint probability

% Area of AB = % Area of AB = % Area of AB = % Area of AB = % Area of A % Area of A % Area of A % Area of A **** % Area of B % Area of B % Area of B % Area of B

BayesBayesBayesBayes’’’’ssss Rule: Rule: Rule: Rule:

Probability map Probability map Probability map Probability map

P(D|A) = P(D)P(A|D)/P(A)P(D|A) = P(D)P(A|D)/P(A)P(D|A) = P(D)P(A|D)/P(A)P(D|A) = P(D)P(A|D)/P(A)

P(D|notAP(D|notAP(D|notAP(D|notA) = ) = ) = ) = P(D)P(notA|D)/P(notDP(D)P(notA|D)/P(notDP(D)P(notA|D)/P(notDP(D)P(notA|D)/P(notD))))

P(D|AB) = P(D) P(A|D)/P(A) P(B|D)/P(B)P(D|AB) = P(D) P(A|D)/P(A) P(B|D)/P(B)P(D|AB) = P(D) P(A|D)/P(A) P(B|D)/P(B)P(D|AB) = P(D) P(A|D)/P(A) P(B|D)/P(B)

P(D|AnotBP(D|AnotBP(D|AnotBP(D|AnotB) = P(D)P(A|D)/P(A) ) = P(D)P(A|D)/P(A) ) = P(D)P(A|D)/P(A) ) = P(D)P(A|D)/P(A) P(notB|D)/P(notBP(notB|D)/P(notBP(notB|D)/P(notBP(notB|D)/P(notB))))

P(D|notP(D|notP(D|notP(D|not AB) = AB) = AB) = AB) = P(D)P(notA|D)/P(notAP(D)P(notA|D)/P(notAP(D)P(notA|D)/P(notAP(D)P(notA|D)/P(notA) P(B|D)/P(B)) P(B|D)/P(B)) P(B|D)/P(B)) P(B|D)/P(B)P(D|notP(D|notP(D|notP(D|not A A A A notBnotBnotBnotB) = ) = ) = ) = P(D)P(notA|D)/P(notAP(D)P(notA|D)/P(notAP(D)P(notA|D)/P(notAP(D)P(notA|D)/P(notA) ) ) )

P(notP(notP(notP(not B|D)/B|D)/B|D)/B|D)/P(notP(notP(notP(not B) B) B) B)

BayesBayesBayesBayes’’’’ssss Rule: Rule: Rule: Rule:

Log (Probability) Log (Probability) Log (Probability) Log (Probability)

log[P(D|Alog[P(D|Alog[P(D|Alog[P(D|A)] = )] = )] = )] = log[P(D)]+log[P(A|Dlog[P(D)]+log[P(A|Dlog[P(D)]+log[P(A|Dlog[P(D)]+log[P(A|D)/P(A)])/P(A)])/P(A)])/P(A)]

= = = = Log[P(DLog[P(DLog[P(DLog[P(D)] + W)] + W)] + W)] + WAAAA++++

Log[P(D|notALog[P(D|notALog[P(D|notALog[P(D|notA)] = )] = )] = )] = log[P(Dlog[P(Dlog[P(Dlog[P(D)] + )] + )] + )] + log[P(notA|D)/P(notDlog[P(notA|D)/P(notDlog[P(notA|D)/P(notDlog[P(notA|D)/P(notD)])])])]

= = = = Log[P(DLog[P(DLog[P(DLog[P(D)] + W)] + W)] + W)] + WAAAA----

Where Where Where Where WWWWAAAA++++ = = = = log[P(A|Dlog[P(A|Dlog[P(A|Dlog[P(A|D)/P(A)])/P(A)])/P(A)])/P(A)]

W W W WAAAA---- = = = = log[P(notA|D)/P(notAlog[P(notA|D)/P(notAlog[P(notA|D)/P(notAlog[P(notA|D)/P(notA)])])])]

Log[P(D|ABLog[P(D|ABLog[P(D|ABLog[P(D|AB)] = )] = )] = )] = log[P(Dlog[P(Dlog[P(Dlog[P(D)] +W)] +W)] +W)] +WAAAA+++++ W+ W+ W+ WBBBB

++++

Log[P(D|ALog[P(D|ALog[P(D|ALog[P(D|A notBnotBnotBnotB)] = )] = )] = )] = log[P(Dlog[P(Dlog[P(Dlog[P(D)] +W)] +W)] +W)] +WAAAA+++++ W+ W+ W+ WBBBB

----

Log[P(D|notABLog[P(D|notABLog[P(D|notABLog[P(D|notAB)] = )] = )] = )] = log[P(Dlog[P(Dlog[P(Dlog[P(D)] +W)] +W)] +W)] +WAAAA----+ W+ W+ W+ WBBBB

++++

Log[P(D|notALog[P(D|notALog[P(D|notALog[P(D|notA not B)] = not B)] = not B)] = not B)] = log[P(Dlog[P(Dlog[P(Dlog[P(D)] +W)] +W)] +W)] +WAAAA----+ W+ W+ W+ WBBBB

----

WWWWAAAA++++ = = = = log[P(A|Dlog[P(A|Dlog[P(A|Dlog[P(A|D)/P(A)] = Log[ ])/P(A)] = Log[ ])/P(A)] = Log[ ])/P(A)] = Log[ ]

= Log[ ]= Log[ ]= Log[ ]= Log[ ]

P(A|D)P(A|D)P(A|D)P(A|D)

P(A) P(A) P(A) P(A)

% points on A% points on A% points on A% points on A

% Area of A % Area of A % Area of A % Area of A

WWWWAAAA+ + + + > 0 positive correlation between A and points> 0 positive correlation between A and points> 0 positive correlation between A and points> 0 positive correlation between A and points

WWWWAAAA+ + + + = 0 no correlation between A and points= 0 no correlation between A and points= 0 no correlation between A and points= 0 no correlation between A and points

WWWWAAAA+ + + + < 0 negative correlation between A and points< 0 negative correlation between A and points< 0 negative correlation between A and points< 0 negative correlation between A and points

Spatial Association Index Spatial Association Index Spatial Association Index Spatial Association Index

Contrast Contrast Contrast Contrast

C = WC = WC = WC = WAAAA++++ - W - W - W - WAAAA

----

(1) -(1) -(1) -(1) -∞∞∞∞ < C < < C < < C < < C < ∞∞∞∞

(2) C = 0 A and D are independent(2) C = 0 A and D are independent(2) C = 0 A and D are independent(2) C = 0 A and D are independent

(3) C > 0 positive correlation between D and A(3) C > 0 positive correlation between D and A(3) C > 0 positive correlation between D and A(3) C > 0 positive correlation between D and A

(4) C < 0 negative correlation between D and A(4) C < 0 negative correlation between D and A(4) C < 0 negative correlation between D and A(4) C < 0 negative correlation between D and A

7

not A not B C

A not B C

not A B C

Three patterns: trees, lake and road buffer

A B CA notB notC

A B

not

CnotA notB notC

notA

B n

otC

Log[P(D|ABLog[P(D|ABLog[P(D|ABLog[P(D|ABCCCC)] = )] = )] = )] = log[P(Dlog[P(Dlog[P(Dlog[P(D)] +W)] +W)] +W)] +WAAAA+++++ W+ W+ W+ WBBBB

+++++W+W+W+WCCCC++++

Log[P(D|ALog[P(D|ALog[P(D|ALog[P(D|A notBnotBnotBnotBCCCC)] = )] = )] = )] = log[P(Dlog[P(Dlog[P(Dlog[P(D)] +W)] +W)] +W)] +WAAAA+++++ W+ W+ W+ WBBBB

----++++WWWWCCCC++++

............

Log[Log[Log[Log[OOOO(D|AB(D|AB(D|AB(D|ABCCCC)] = )] = )] = )] = log[log[log[log[OOOO(D(D(D(D)] +W)] +W)] +W)] +WAAAA

+++++ W+ W+ W+ WBBBB+++++W+W+W+WCCCC

++++

Log[Log[Log[Log[OOOO(D|A(D|A(D|A(D|A notBnotBnotBnotBCCCC)] = )] = )] = )] = log[log[log[log[OOOO(D(D(D(D)] +W)] +W)] +W)] +WAAAA+++++ W+ W+ W+ WBBBB

----++++WWWWCCCC++++

............

]|[]|[log

notDAPDAPW A =+

]|[]|[log

notDnotAPDnotAPW A =−

Prediction of Potential Flowing Prediction of Potential Flowing Prediction of Potential Flowing Prediction of Potential Flowing Wells in the ORMWells in the ORMWells in the ORMWells in the ORM

Flowing Wells and SpringsFlowing Wells and SpringsFlowing Wells and SpringsFlowing Wells and SpringsFlowing Wells and SpringsFlowing Wells and SpringsFlowing Wells and SpringsFlowing Wells and Springs Flowing Wells vs. Distance from Flowing Wells vs. Distance from Flowing Wells vs. Distance from Flowing Wells vs. Distance from Flowing Wells vs. Distance from Flowing Wells vs. Distance from Flowing Wells vs. Distance from Flowing Wells vs. Distance from ORMORMORMORMORMORMORMORM

-8

-4

0

4

8

12

0 5000 10000 15000

Spatial CorrelationSpatial CorrelationSpatial CorrelationSpatial Correlation

DistanceDistanceDistanceDistance

8

Flowing Wells vs. Distance from Flowing Wells vs. Distance from Flowing Wells vs. Distance from Flowing Wells vs. Distance from Flowing Wells vs. Distance from Flowing Wells vs. Distance from Flowing Wells vs. Distance from Flowing Wells vs. Distance from ORMORMORMORMORMORMORMORM

Spatial CorrelationSpatial CorrelationSpatial CorrelationSpatial Correlation

DistanceDistanceDistanceDistance

Flowing Wells vs. Distance Flowing Wells vs. Distance Flowing Wells vs. Distance Flowing Wells vs. Distance Flowing Wells vs. Distance Flowing Wells vs. Distance Flowing Wells vs. Distance Flowing Wells vs. Distance From High Slope ZoneFrom High Slope ZoneFrom High Slope ZoneFrom High Slope ZoneFrom High Slope ZoneFrom High Slope ZoneFrom High Slope ZoneFrom High Slope Zone

-8

-4

0

4

8

0 2000 4000 6000 8000

Spatial CorrelationSpatial CorrelationSpatial CorrelationSpatial Correlation

DistanceDistanceDistanceDistance

Flowing Wells vs. Thickness of Flowing Wells vs. Thickness of Flowing Wells vs. Thickness of Flowing Wells vs. Thickness of Flowing Wells vs. Thickness of Flowing Wells vs. Thickness of Flowing Wells vs. Thickness of Flowing Wells vs. Thickness of DriftDriftDriftDriftDriftDriftDriftDrift

Flowing vs. Distance from Thick Flowing vs. Distance from Thick Flowing vs. Distance from Thick Flowing vs. Distance from Thick Flowing vs. Distance from Thick Flowing vs. Distance from Thick Flowing vs. Distance from Thick Flowing vs. Distance from Thick DriftDriftDriftDriftDriftDriftDriftDrift

-4

0

4

8

0 5000 10000 15000 20000

Spatial CorrelationSpatial CorrelationSpatial CorrelationSpatial Correlation

DistanceDistanceDistanceDistance

Flowing vs. Distance from Thick Flowing vs. Distance from Thick Flowing vs. Distance from Thick Flowing vs. Distance from Thick Flowing vs. Distance from Thick Flowing vs. Distance from Thick Flowing vs. Distance from Thick Flowing vs. Distance from Thick DriftDriftDriftDriftDriftDriftDriftDrift

9

Potential Locations of Flowing WellsPotential Locations of Flowing WellsPotential Locations of Flowing WellsPotential Locations of Flowing WellsPotential Locations of Flowing WellsPotential Locations of Flowing WellsPotential Locations of Flowing WellsPotential Locations of Flowing Wells by by by by by by by by Weights of Evidence Weights of Evidence Weights of Evidence Weights of Evidence Weights of Evidence Weights of Evidence Weights of Evidence Weights of Evidence (Cheng, 2001) (Cheng, 2001) (Cheng, 2001) (Cheng, 2001) (Cheng, 2001) (Cheng, 2001) (Cheng, 2001) (Cheng, 2001)

ThemeThemeThemeTheme Area Area Area Area %%%% Points%Points%Points%Points% CoCoCoContrastntrastntrastntrast tttt----valuevaluevaluevalue

LR LR LR LR CoeffCoeffCoeffCoeff....

LRLRLRLR StdStdStdStd

Buffer zone Buffer zone Buffer zone Buffer zone (1~1km)(1~1km)(1~1km)(1~1km) aroundaroundaroundaround steep slopesteep slopesteep slopesteep slope 63636363 80808080 0.890.890.890.89 6.546.546.546.54 0.880.880.880.88 0.140.140.140.14 Buffer zone Buffer zone Buffer zone Buffer zone (1~5km) (1~5km) (1~5km) (1~5km) around the ORMaround the ORMaround the ORMaround the ORM 40.340.340.340.3 55.355.355.355.3 0.620.620.620.62 5.685.685.685.68 0.280.280.280.28 0.120.120.120.12 Buffer zone around steep slope of lower sand Buffer zone around steep slope of lower sand Buffer zone around steep slope of lower sand Buffer zone around steep slope of lower sand / gravel t/ gravel t/ gravel t/ gravel top op op op 10.110.110.110.1 52.752.752.752.7 0.620.620.620.62 5.685.685.685.68 1.771.771.771.77 0.120.120.120.12

Ratio of sand/gravel unit cumulative Ratio of sand/gravel unit cumulative Ratio of sand/gravel unit cumulative Ratio of sand/gravel unit cumulative thickness in well depththickness in well depththickness in well depththickness in well depth (6~25%)(6~25%)(6~25%)(6~25%) 43.543.543.543.5 65.165.165.165.1 0.900.900.900.90 7.887.887.887.88 0.490.490.490.49 0.120.120.120.12

Buffer zone Buffer zone Buffer zone Buffer zone (1~2.5km)(1~2.5km)(1~2.5km)(1~2.5km) of thick drift area of thick drift area of thick drift area of thick drift area 16.216.216.216.2 29.329.329.329.3 0.790.790.790.79 6.566.566.566.56 0.450.450.450.45 0.130.130.130.13 Elevation of the upper confined Elevation of the upper confined Elevation of the upper confined Elevation of the upper confined aquaquaquaqu ifersifersifersifers at at at at 356~375356~375356~375356~375 (m a. s. l.) (m a. s. l.) (m a. s. l.) (m a. s. l.) 16.816.816.816.8 38.638.638.638.6 1.181.181.181.18 10.4410.4410.4410.44 0.470.470.470.47 0.130.130.130.13

Elevation of the lower confined aquifers at Elevation of the lower confined aquifers at Elevation of the lower confined aquifers at Elevation of the lower confined aquifers at 311~347311~347311~347311~347 (m a. s. l.)(m a. s. l.)(m a. s. l.)(m a. s. l.) 40.540.540.540.5 58.258.258.258.2 0.730.730.730.73 6.606.606.606.60 0.280.280.280.28 0.130.130.130.13

Steep slope of confined aquifer surfaceSteep slope of confined aquifer surfaceSteep slope of confined aquifer surfaceSteep slope of confined aquifer surface 22.922.922.922.9 54.754.754.754.7 1.451.451.451.45 13.1913.1913.1913.19 0.970.970.970.97 0.120.120.120.12 Buffer zone (Buffer zone (Buffer zone (Buffer zone ( 0~2km) 0~2km) 0~2km) 0~2km) aroundaroundaroundaround the small ponds the small ponds the small ponds the small ponds 61.161.161.161.1 74.974.974.974.9 0.670.670.670.67 5.315.315.315.31 0.430.430.430.43 0.130.130.130.13

Intercept constantIntercept constantIntercept constantIntercept constant ----11.6211.6211.6211.62 0.520.520.520.52

Results obtained by Logistic Regression Results obtained by Logistic Regression Results obtained by Logistic Regression Results obtained by Logistic Regression Results obtained by Logistic Regression Results obtained by Logistic Regression Results obtained by Logistic Regression Results obtained by Logistic Regression MethodMethodMethodMethodMethodMethodMethodMethod

Multivariate Logistic RegressionMultivariate Logistic RegressionMultivariate Logistic RegressionMultivariate Logistic Regression Spatial Data Modeler Extension: Spatial Data Modeler Extension: Spatial Data Modeler Extension: Spatial Data Modeler Extension: Arc-SDMArc-SDMArc-SDMArc-SDM

Weights of EvidenceWeights of EvidenceWeights of EvidenceWeights of Evidence

Logistic RegressionLogistic RegressionLogistic RegressionLogistic Regression

Fuzzy LogicFuzzy LogicFuzzy LogicFuzzy Logic

Neural Network Neural Network Neural Network Neural Network