Motivations & Tools for Spatial Data Analysis · Motivations & Tools for Spatial Data Analysis...

Motivations & Tools Motivations & Tools for Spatial Data Analysisfor Spatial Data Analysis

Katherine CurtisKatherine CurtisUniversity of WisconsinUniversity of Wisconsin--MadisonMadison

[email protected]@ssc.wisc.edu

Prepared for the Consortium for Education and Social Science ResPrepared for the Consortium for Education and Social Science Research 2010earch 2010--2011 2011 Workshop in Methods, Indiana University on 19 February, 2011.Workshop in Methods, Indiana University on 19 February, 2011.

Center for Demography & EcologyCenter for Demography & Ecology Applied Population LaboratoryApplied Population Laboratory

•• Workshop ObjectiveWorkshop Objective

Provide an introduction & overview of concepts & Provide an introduction & overview of concepts & techniques for spatial data analysistechniques for spatial data analysis

•• Workshop OutlineWorkshop Outline

Why Spatial is SpecialWhy Spatial is SpecialStatistical & Conceptual MotivationsStatistical & Conceptual MotivationsGlobal & Local Spatial AutocorrelationGlobal & Local Spatial AutocorrelationSpatial Error & Spatial Lag RegressionSpatial Error & Spatial Lag Regression

Applications of Foundational MethodsApplications of Foundational MethodsExploratory Data AnalysisExploratory Data AnalysisSpatial Regression AnalysisSpatial Regression Analysis

IntroductionsIntroductions

Why Spatial is SpecialWhy Spatial is Special

motivations: whatmotivations: what

•• What are What are ““spatial dataspatial data””??

Data where, in addition to attribute values relating to Data where, in addition to attribute values relating to the primary phenomenon or phenomena of interest, the primary phenomenon or phenomena of interest, the the relative spatial locationsrelative spatial locations of observations are also of observations are also recordedrecorded

Housing prices for city blocksHousing prices for city blocksChild poverty rates for countiesChild poverty rates for countiesAccident counts by intersectionAccident counts by intersectionCancer incidence for Cancer incidence for geocodedgeocoded addressesaddressesCountyCounty--toto--county migration streams for persons 65+county migration streams for persons 65+


GISGIS

Spatial Data Spatial Data AnalysisAnalysis

Spatial Spatial AnalysisAnalysis

““Spatial Data ProductionSpatial Data Production””

ExamplesExamples•• GeocodingGeocoding•• Proximity/Network Proximity/Network

distancedistance•• Zonal StatsZonal Stats•• SamplingSampling



GISGIS

Spatial Data Spatial Data AnalysisAnalysis

Spatial Spatial AnalysisAnalysis

““Spatial StatisticsSpatial Statistics””

GeostatisticalGeostatisticalDataData

LatticeLatticeDataData

PointPoint--PatternPatternDataData

Spatial InteractionSpatial InteractionDataData


GeostatisticalGeostatistical DataDataWisconsin ElevationWisconsin Elevation

175 - 259

260 - 343

344 - 427

428 - 511

512 - 596

Elevation (Meters)

N

Applied Population LaboratoryUW Extension Basin Educators In ServiceSource: USGS Digitial Elevation Model1 Degree DEM with 75m Cell Size

WisconsinElevation Model

PointPoint--Pattern (Event) DataPattern (Event) DataLocation of Deaths from Cholera in Central London for Location of Deaths from Cholera in Central London for

September 1854 (Snow)September 1854 (Snow)



Spatial Flows DataSpatial Flows DataMigration in the UK, Map 5 from Migration in the UK, Map 5 from RavensteinRavenstein 18851885


Lattice DataLattice DataWisconsin Latino Population, 1990Wisconsin Latino Population, 1990--2007 Estimated Percent Change2007 Estimated Percent Change


Lattice DataLattice DataPercent Seasonal Housing for Wisconsin Watersheds, 2000Percent Seasonal Housing for Wisconsin Watersheds, 2000

Percent Seasonal HUSSNL HU / TOT HU

0.1% - 6.3%

6.4% - 16%

16.1% - 30.6%

30.7% - 49.8%

49.9% - 78.6%


Classes of Problems in Spatial Data AnalysisClasses of Problems in Spatial Data Analysis

Source: Griffith and Layne (1999:6)Source: Griffith and Layne (1999:6)


•• Spatial vs. NonSpatial vs. Non--spatial Data Analysisspatial Data Analysis

In spatial data analysisIn spatial data analysis……

○○ the focus is on the focus is on modifications, extensions & additionsmodifications, extensions & additions to standard to standard statistical data analytical methods statistical data analytical methods

○○ the modifications, etc., consider explicitly the locations or ththe modifications, etc., consider explicitly the locations or the e spatial arrangementspatial arrangement of the objects being analyzedof the objects being analyzed

motivations: whymotivations: why

•• Why are we interested in spatial data?Why are we interested in spatial data?

Data that are referenced to location bring extremely Data that are referenced to location bring extremely important additional, useable important additional, useable informationinformation to analysisto analysis

And it also brings some (possibly unfamiliar) And it also brings some (possibly unfamiliar) pitfallspitfalls that that require a new awareness for analysis to proceed require a new awareness for analysis to proceed


•• Why are we interested in spatial data?Why are we interested in spatial data?

Spatially referenced data bring special Spatially referenced data bring special problemsproblems to an to an analysis:analysis:

Heterogeneity of observational units Heterogeneity of observational units heteroskedasticityheteroskedasticitySpatial autocorrelation Spatial autocorrelation residual dependenceresidual dependence

Consequently, the Consequently, the assumptionassumption of of iidiid errors in a standard errors in a standard OLS regression specification is violatedOLS regression specification is violated

Meaning, statistical Meaning, statistical inferenceinference from such a model is not from such a model is not validvalid


•• Erroneous Statistical Inference & Substantive Erroneous Statistical Inference & Substantive ConclusionsConclusions

““If a naIf a naïïve researcher completes a standard statistical ve researcher completes a standard statistical analysis of analysis of georeferencedgeoreferenced data, it does not follow that data, it does not follow that the data analytic results have turned data into the data analytic results have turned data into meaningful information merely because to the inexpert meaningful information merely because to the inexpert eye they are indistinguishable from conventional eye they are indistinguishable from conventional statistical results!statistical results!””

Daniel Griffith and Larry Layne(Oxford University Press, 1999:vii)


•• Heterogeneity of Observational Units Heterogeneity of Observational Units (different sized units)(different sized units)

Brewster Co. TX:Area: 6,193 mi2

Fairfax City, VA:Area: 6.3 mi2

US Southern Counties

n = 1,387


•• Heterogeneity of Observational Units Heterogeneity of Observational Units (different sized units)(different sized units)

Harris Co. TX:Pop: 3,400,578Loving Co. TX:

Pop: 67

US Southern Counties

n = 1,387


•• HeteroskedasticityHeteroskedasticity


•• Spatial Autocorrelation Spatial Autocorrelation (neighbors are similar)(neighbors are similar)

ToblerTobler’’ss First Law of Geography:First Law of Geography:

““Everything is related to everything else, but near things Everything is related to everything else, but near things are more related than distant thingsare more related than distant things””

Waldo ToblerEconomic Geography (1970:236)


•• Spatial Autocorrelation Spatial Autocorrelation (neighbors are similar)(neighbors are similar)

County Child Poverty Rates, 2000Source:SF3 Table P87

Loudoun Co. VA: 0.028

Starr Co. TX: 0.595


•• Spatially Spatially AutocorrelatedAutocorrelated ResidualsResiduals

County Child Poverty Rates, 2000Source:SF3 Table P87


•• Classic Illustration of Spatial AutocorrelationClassic Illustration of Spatial Autocorrelation

Omer R. Galle, Walter R. Gove, & J. Miller McPherson. 1972. Omer R. Galle, Walter R. Gove, & J. Miller McPherson. 1972. ““Population Density and Pathology: What Are the Relations for Population Density and Pathology: What Are the Relations for ManMan”” ScienceScience 176(4030):23176(4030):23--3030

○○ Analyzed 75 community areas in Chicago for 1960Analyzed 75 community areas in Chicago for 1960○○ Used 5 measures of Used 5 measures of ““social pathologysocial pathology”” as function of crowding, as function of crowding,

controlling for social class & ethnicitycontrolling for social class & ethnicity○○ Found Found “…“…the greater the density, the greater the fertilitythe greater the density, the greater the fertility”” (p. 176)(p. 176)

Colin Colin LoftinLoftin & Sally K. Ward. 1983. & Sally K. Ward. 1983. ““A Spatial Autocorrelation A Spatial Autocorrelation Model of the Effects of Population Density on FertilityModel of the Effects of Population Density on Fertility”” American American Sociological ReviewSociological Review 48(1):12148(1):121--128128

○○ Reanalysis found Reanalysis found “…“…the GGM findings with regard to fertility are an the GGM findings with regard to fertility are an artifact of the failure to recognize the presence of disturbanceartifact of the failure to recognize the presence of disturbance variables variables which are spatially which are spatially autocorrelatedautocorrelated”” (p. 127)(p. 127)


•• A closer look at OLSA closer look at OLS……

GaussGauss--Markov Theorem asserts that is a Markov Theorem asserts that is a ““Best Linear Best Linear Unbiased EstimatorUnbiased Estimator”” ((BLUEBLUE) of , provided the following ) of , provided the following assumptions are met:assumptions are met:

○○ LinearityLinearity○○ Mean independence Mean independence E[E[εεi |xii |xi] = 0 ] = 0 (implies E[(implies E[εε] = 0)] = 0)

○○ HomoskedasticityHomoskedasticity & uncorrelated disturbances & uncorrelated disturbances CovCov[[εε] = E[] = E[εε εε’’ ] = ] = II

○○ XX is of rank is of rank k+k+11 ((kk = no. of = no. of ““independentindependent”” vars.)vars.)

○○ XX is nonis non--stochastic stochastic (or stochastic with finite second moments, & (or stochastic with finite second moments, & E[E[XX’’εε] = 0 for ] = 0 for unbiasednessunbiasedness))

○○ Normal disturbance Normal disturbance (non(non--random errors)random errors)


•• Defining termsDefining terms……

Bias:Bias: An estimation method is unbiased if it produces An estimation method is unbiased if it produces estimates that have a statistical expectation equal to the estimates that have a statistical expectation equal to the true (population) valuetrue (population) value

Efficiency: Efficiency: Efficient estimates are those that have Efficient estimates are those that have smaller standard errors than estimates produced by smaller standard errors than estimates produced by some competing estimatorsome competing estimator

Consistency:Consistency: Estimates converge toward the quantity Estimates converge toward the quantity being estimated as the sample size increases being estimated as the sample size increases [[plimplim(1/(1/nn))XX’’εε = = 0 ] 0 ] (central limit (central limit theorumtheorum——unbiased within limits)unbiased within limits)


•• Regarding OLS AssumptionsRegarding OLS Assumptions……

Linearity & mean independence support Linearity & mean independence support unbiasedunbiasedestimatesestimates

HomoskedasticityHomoskedasticity & uncorrelated disturbances support & uncorrelated disturbances support efficiencyefficiency

Normality of disturbances means we can do statistical Normality of disturbances means we can do statistical inferenceinference using using tt tablestables

Normality also means we can estimate the model by Normality also means we can estimate the model by MLEMLE


•• So, why are we interested in spatial data?So, why are we interested in spatial data?

The The assumptionassumption of of iidiid errors in a standard OLS errors in a standard OLS regression specification is violatedregression specification is violated

Statistical Statistical inferenceinference from such a model is not validfrom such a model is not valid

Moral:Moral: Highly useful to know something about the Highly useful to know something about the rudiments of spatial data analysis (i.e., some rudiments of spatial data analysis (i.e., some understanding of why understanding of why ““spatial is specialspatial is special””) when analyzing ) when analyzing spatial dataspatial data


•• Statistical ImportanceStatistical Importance

○○ A potentialA potential problemproblemHeteroskedasticityHeteroskedasticity & dependent residuals& dependent residuals

○○ A means of A means of data integrationdata integrationVariable creationVariable creation

•• Theoretical ImportanceTheoretical Importance

○○ A means of A means of organizing human activitiesorganizing human activitiesHuman Ecology, Contextual Models, etc.Human Ecology, Contextual Models, etc.


•• Theoretical importanceTheoretical importance

Map 1Map 1 Map 2Map 2

Exploratory data analysis (EDA) that does not utilize the Exploratory data analysis (EDA) that does not utilize the spatial arrangement of the data spatial arrangement of the data

will lead to will lead to identical resultsidentical results for the two mapsfor the two maps


•• Theoretical importanceTheoretical importance

Traditional data analysis not utilizing Traditional data analysis not utilizing location & spatial arrangement will location & spatial arrangement will produce identical results for the 2 maps produce identical results for the 2 maps

observed female-headed hh

permutation


•• So, the importance?So, the importance?

““What makes the methods of modern [spatial data What makes the methods of modern [spatial data analysis] different from many of their predecessors is analysis] different from many of their predecessors is that they have been developed with the recognition that that they have been developed with the recognition that spatial data have spatial data have unique propertiesunique properties and that these and that these properties make the use of methods borrowed from properties make the use of methods borrowed from aspatialaspatial disciplines highly questionable.disciplines highly questionable.””

FotheringhamFotheringham, , BrunsdonBrunsdon & Charlton& CharltonGeographically Weighted RegressionGeographically Weighted Regression

Wiley, 2003 p. 4Wiley, 2003 p. 4


•• Given the unique propertiesGiven the unique properties……

if we blithely carry out a standard analysis of aggregated if we blithely carry out a standard analysis of aggregated geographic datageographic data

some large subset of the following some large subset of the following undesirable horrorsundesirable horrorsalmost certainly awaits us almost certainly awaits us (the curse of (the curse of ToblerTobler’’ss 1st Law)1st Law)::

1. Estimated regression coefficients are 1. Estimated regression coefficients are biased & inconsistentbiased & inconsistent, or, or……

2. Estimated regression coefficients are 2. Estimated regression coefficients are inefficientinefficient

3. 3. RR22 statistic is statistic is exaggeratedexaggerated

4. Made 4. Made incorrectincorrect inferencesinferences

5. Will 5. Will nevernever get it published get it published ((or or shouldnshouldn’’tt!)!)

motivations: whomotivations: who

•• EconomistsEconomists

•• GeographersGeographers

•• EpidemiologistsEpidemiologists

•• CriminologistsCriminologists

•• Political ScientistsPolitical Scientists

•• Demographers!Demographers!○○ Migration: spatial flowsMigration: spatial flows○○ Fertility: diffusion of innovationFertility: diffusion of innovation○○ Mortality: contagion & diseaseMortality: contagion & disease○○ Urbanization: uneven developmentUrbanization: uneven development

•• Even SociologistsEven Sociologists

motivations: whomotivations: who

•• Some important textsSome important texts……

Spatial Data Analysis by ExampleSpatial Data Analysis by Example, , Upton & Upton & FingletonFingleton 19851985

Spatial Econometric: Methods and ModelsSpatial Econometric: Methods and Models, , AnselinAnselin 19881988

Statistics for Spatial DataStatistics for Spatial Data, , CressieCressie 1991 (rev. 1993)1991 (rev. 1993)

Interactive Spatial Data AnalysisInteractive Spatial Data Analysis, , Bailey & Bailey & GatrellGatrell 19951995

Geographically Weighted RegressionGeographically Weighted Regression, , FotheringhamFotheringham et al. 2002et al. 2002

Hierarchical Modeling and Analysis for Spatial DataHierarchical Modeling and Analysis for Spatial Data, , BanerjeeBanerjee et al. 2004et al. 2004

Applied Spatial Statistics for Public Health DataApplied Spatial Statistics for Public Health Data, , Waller & Waller & GotwayGotway 20042004

Spatial and Spatiotemporal EconometricsSpatial and Spatiotemporal Econometrics, , LeSageLeSage & Pace 2004& Pace 2004

motivations: howmotivations: how

•• Components of Spatial Data AnalysisComponents of Spatial Data Analysis

○○ VisualizationVisualizationShowingShowing interesting patternsinteresting patterns

○○ Exploratory Spatial Data AnalysisExploratory Spatial Data AnalysisFindingFinding interesting patternsinteresting patterns

○○ Spatial ModelingSpatial ModelingExplainingExplaining interesting patternsinteresting patterns

•• Showing Interesting PatternsShowing Interesting Patterns


Social connection (Social connection (redred) through railroad lines ) through railroad lines overlayedoverlayed with with spatial connection (spatial connection (greygrey) through county adjacency, ) through county adjacency,

1880 Northern Great Plains 1880 Northern Great Plains (Curtis & Slez (Curtis & Slez ndnd))

•• Finding Interesting PatternsFinding Interesting Patterns

Exploratory Spatial Data Analysis (ESDA)Exploratory Spatial Data Analysis (ESDA)



•• Explaining Interesting PatternsExplaining Interesting Patterns

○○ SpatialSpatial HeterogeneityHeterogeneity: First Order Effects: First Order EffectsExists when the mean, and/or variance, and/or covariance Exists when the mean, and/or variance, and/or covariance structure structure ““driftsdrifts”” over the study regionover the study regionDue to unmeasured or Due to unmeasured or unmeasurableunmeasurable exogenous exogenous factor(sfactor(s))

○○ Spatial DependenceSpatial Dependence: Second Order Effects: Second Order Effects““Spatial dependence can be considered to be the existence Spatial dependence can be considered to be the existence of a functional relationship between what happens at one of a functional relationship between what happens at one point in space and what happens elsewhere.point in space and what happens elsewhere.”” (Luc (Luc AnselinAnselin1988:11)1988:11)Due to interactive, diffusive relationshipDue to interactive, diffusive relationship

RecallRecall……

When correlated errors arise from a specification with When correlated errors arise from a specification with missing variablesmissing variables, OLS estimates of , OLS estimates of tt--test values are test values are unreliableunreliable

○○ OLS estimates are not efficientOLS estimates are not efficient

○○ SE of parameter estimates are biased downward SE of parameter estimates are biased downward (under (under positive spatial autocorrelation)positive spatial autocorrelation)

○○ Informally, arises because OLS model Informally, arises because OLS model ““thinksthinks”” itit’’s getting s getting more information from the observations than it is more information from the observations than it is (redundancy)(redundancy)

○○ Correlated errors inflate value of Correlated errors inflate value of RR22 statisticstatistic

When correlated errors result from When correlated errors result from endogeneityendogeneity, OLS , OLS regression parameter estimates are biased & inconsistentregression parameter estimates are biased & inconsistent

global spatial autocorrelationglobal spatial autocorrelation

Where do we go from here?Where do we go from here?……

How do we develop a means to How do we develop a means to (statistically)(statistically) differentiate differentiate among different kinds of maps?among different kinds of maps?

○○ That is, can we That is, can we quantifyquantify different kinds of map patterns?different kinds of map patterns?

Once we develop a statistic for describing Once we develop a statistic for describing (quantifying)(quantifying) different different kinds of map patternskinds of map patterns……

○○ Can we derive the sampling distribution for this statistic & Can we derive the sampling distribution for this statistic & thus make thus make inferential claimsinferential claims about one map vs. another?about one map vs. another?


The question for us isThe question for us is……

Is this observed spatial distribution of African Americans in USIs this observed spatial distribution of African Americans in US southern southern counties a likely outcome of a random allocation procedure?counties a likely outcome of a random allocation procedure?

But what does this question even mean? Does it even make sense?But what does this question even mean? Does it even make sense?


% African American US South, Census 2000

Points us toward concept of a Points us toward concept of a spatial random processspatial random process or a or a spatial random fieldspatial random field……

○○ Theoretical notion that our data represent Theoretical notion that our data represent one realizationone realizationof a large number of possible outcomes of a large number of possible outcomes


Under nonUnder non--free sampling, or randomizationfree sampling, or randomization……

Question becomes:Question becomes: How unusual is the pattern How unusual is the pattern (Moran(Moran’’s s II)) in in map 1 given the 1,387! possible permutations?map 1 given the 1,387! possible permutations?


1. observed map

2. alternative map

So, spatial autocorrelationSo, spatial autocorrelation……

(Positive)(Positive) spatial autocorrelation is the coexistence of attribute spatial autocorrelation is the coexistence of attribute similarity & location similaritysimilarity & location similarity

○○ Confirmation of Confirmation of ToblerTobler’’ss First LawFirst Law


2 classes of tests for spatial autocorrelation2 classes of tests for spatial autocorrelation……

GlobalGlobal spatial autocorrelation measuresspatial autocorrelation measures

○○ Assess whether data Assess whether data as a wholeas a whole exhibit a spatial patternexhibit a spatial pattern

○○ Most common statistic: Most common statistic: MoranMoran’’s s II

LocalLocal indicators of spatial association (LISA) statisticsindicators of spatial association (LISA) statistics

○○ Identifies Identifies which unitswhich units are significantly spatially are significantly spatially autocorrelatedautocorrelated with neighboring unitswith neighboring units

○○ Identifies clusters Identifies clusters ((““hot spotshot spots”” &/or &/or ““cold spotscold spots””))

○○ Commonly measured as Commonly measured as local Moranlocal Moran’’s s II



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PearsonPearsonproductproduct--moment correlationmoment correlation

n

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MoranMoran’’ss IIcoefficientcoefficient

feasible range: feasible range: --1 to +11 to +1 feasible range: feasible range: --1 to +1 1 to +1 (sort of)(sort of)

Calculating a MoranCalculating a Moran’’s s II requires a requires a spatial weights matrixspatial weights matrix……

spatial weights matricesspatial weights matrices

I n

w

w y y y y

y yijj

n

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n

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n

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1 2 3 4

5 6 7 8

9 10 11 12

13 14 15 16

Which cells are Which cells are ““neighborsneighbors”” of cell 6?of cell 6?


Depends on our definition of a Depends on our definition of a ““neighborneighbor””……

○○ Queen contiguity, rook contiguity, & (occasionally) bishop contiQueen contiguity, rook contiguity, & (occasionally) bishop contiguityguity


1 2 3 4

5 6 7 8

9 10 11 12

13 14 15 16

Queen ContiguityQueen Contiguity

11 22 33 44 55 66 77 88 99 1010 1111 1212 1313 1414 1515 1616 ΣΣ

11 11 11 11 3322 11 11 11 11 11 5533 11 11 11 11 11 5544 11 11 11 3355 11 11 11 11 11 5566 11 11 11 11 11 11 11 11 8877 11 11 11 11 11 11 11 11 8888 11 11 11 11 11 5599 11 11 11 11 11 551010 11 11 11 11 11 11 11 11 881111 11 11 11 11 11 11 11 11 881212 11 11 11 11 11 551313 11 11 11 331414 11 11 11 11 11 551515 11 11 11 11 11 551616 11 11 11 33

ii

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SimpleSimpleContiguityContiguity

MatrixMatrix{{ccijij}}

Queen Queen CriterionCriterion

(zeros implicit)(zeros implicit)

11 22 33 44 55 66 77 88 99 1010 1111 1212 1313 1414 1515 1616 ΣΣ

11 1/31/3 1/31/3 1/31/3 1122 1/51/5 1/51/5 1/51/5 1/51/5 1/51/5 1133 1/51/5 1/51/5 1/51/5 1/51/5 1/51/5 1144 1/31/3 1/31/3 1/31/3 1155 1/51/5 1/51/5 1/51/5 1/51/5 1/51/5 1166 1/81/8 1/81/8 1/81/8 1/81/8 1/81/8 1/81/8 1/81/8 1/81/8 1177 1/81/8 1/81/8 1/81/8 1/81/8 1/81/8 1/81/8 1/81/8 1/81/8 11

88 1/51/5 1/51/5 1/51/5 1/51/5 1/51/5 1199 1/51/5 1/51/5 1/51/5 1/51/5 1/51/5 111010 1/81/8 1/81/8 1/81/8 1/81/8 1/81/8 1/81/8 1/81/8 1/81/8 111111 1/81/8 1/81/8 1/81/8 1/81/8 1/81/8 1/81/8 1/81/8 1/81/8 111212 1/51/5 1/51/5 1/51/5 1/51/5 1/51/5 111313 1/31/3 1/31/3 1/31/3 111414 1/51/5 1/51/5 1/51/5 1/51/5 1/51/5 111515 1/51/5 1/51/5 1/51/5 1/51/5 1/51/5 111616 1/31/3 1/31/3 1/31/3 11

ii

jj

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RowRowStandardizedStandardized

WeightsWeightsMatrixMatrix

(zeros implicit)(zeros implicit)


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26

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54

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74

84

95

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113

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Consider Consider yyii valuesvalues, , ii = 1,= 1,……,16,16


ForFor ii = 6, the = 6, the spatial lag operatorspatial lag operator ww66jj yyjj is given by:is given by:

w y6 j j

w yj jj

j

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16

18

7 18

6 18

4 18

4 18

4 18

5 18

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Spatial lag operatorSpatial lag operator expressed in matrix notationexpressed in matrix notation……

where W is a (16 x 16) weights matrix &

y is a (16 x 1) column vector

Wy

w yj jj

j

ii

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1

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Which takes us back to this Which takes us back to this (earlier slide)(earlier slide)……

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Assume Assume WW row standardized & row standardized & zzii = = yyii -- y


Feasible Feasible rangerange of Moranof Moran’’s s I I valuesvalues……

○○ Function of Function of nn○○ Function of Function of weights matrixweights matrix usedused

○○ Function of structure of Function of structure of tesselationtesselation

○○ Minimum/maximum Minimum/maximum theoreticaltheoretical values |values |1|1|

○○ Minimum Minimum empiricalempirical value for an irregular lattice generally value for an irregular lattice generally around around --0.6 0.6


Expected valueExpected value of Moranof Moran’’s s I I value value (under H(under H00))……

E In

( ) 1

1

local spatial associationslocal spatial associations

Broad class of spatial association statistics can be based on Broad class of spatial association statistics can be based on a general index of matrix association a general index of matrix association ((AnselinAnselin 1995)1995)……

WX

w x

x w x

ijji

ij

i i ij ijj

Global measure

Local measure

WXx w xi i ij j

j

for for j Ji && zi = (yi – y)/σy


For exampleFor example……

Local MoranLocal Moran I z w zi i ij jj

Use this equation, not the ones shown in Use this equation, not the ones shown in AnselinAnselin (1995)(1995)

_


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RecallRecall……


Calculate Local Moran Statistic for Cell 6Calculate Local Moran Statistic for Cell 6

Global = 4.1875Standard deviation = 2.2958

Attrib. Standardized wij timesCell wij value attribute value std. attrib.

1 0.125 7 1.2250 0.15312 0.125 6 0.7895 0.09873 0.125 4 -0.0817 -0.01025 0.125 4 -0.0817 -0.01026 5 0.35397 0.125 4 -0.0817 -0.01029 0.125 5 0.3539 0.0442

10 0.125 6 0.7895 0 .098711 0.125 3 -0.5172 -0.0647

Sum = 0.2995

Local Moran: Local Moran: II66 = 0.3539 x 0.2995 = 0.1060= 0.3539 x 0.2995 = 0.1060Source for calculations: Anselin (1995) Equation (7)

spatial regressionspatial regression

•• RecallRecall……

○○ SpatialSpatial HeterogeneityHeterogeneity: First Order Effects: First Order EffectsExists when the mean, and/or variance, and/or covariance Exists when the mean, and/or variance, and/or covariance structure structure ““driftsdrifts”” over the study regionover the study regionDue to unmeasured or Due to unmeasured or unmeasurableunmeasurable exogenous exogenous factor(sfactor(s))

○○ Spatial DependenceSpatial Dependence: Second Order Effects: Second Order Effects““Spatial dependence can be considered to be the existence Spatial dependence can be considered to be the existence of a functional relationship between what happens at one of a functional relationship between what happens at one point in space and what happens elsewhere.point in space and what happens elsewhere.”” (Luc (Luc AnselinAnselin1988:11)1988:11)Due to interactive, diffusive relationshipDue to interactive, diffusive relationship


Spatial Error ModelSpatial Error Model……(mixed regressive spatial autoregressive error model)(mixed regressive spatial autoregressive error model)

Spatial Lag ModelSpatial Lag Model……(mixed regressive spatial autoregressive lag model)(mixed regressive spatial autoregressive lag model)

WuuuXy

XWyy


Spatial Error ModelSpatial Error Model……

○○ FirstFirst--order variation order variation comes only through comes only through XXββ○○ SecondSecond--order variation is represented as an autoregressive, order variation is represented as an autoregressive,

interactive effect throughinteractive effect through λλWuWu

WuuuXy

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( )( ' )

0 E

E I( )( ' )

0

2


Spatial Lag ModelSpatial Lag Model……

○○ FirstFirst--order variation order variation comes only through comes only through XXββ○○ SecondSecond--order variation is represented as an autoregressive, order variation is represented as an autoregressive,

interactive effect throughinteractive effect through ρWyWy

○○ Analogous to a distributed lag model in timeAnalogous to a distributed lag model in time--series analysisseries analysis

XWyy


Reduced formReduced form……

Spatial Error ModelSpatial Error Model

Spatial Lag ModelSpatial Lag Model

y X [ ...]I W W W 2 2 3 3

...

...3322

3322

WWWIXWWWIy

Application of Foundational MethodsApplication of Foundational Methods(in (in GeoDaGeoDa))

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