Motivations & Tools for Spatial Data Analysis · Motivations & Tools for Spatial Data Analysis...
Transcript of Motivations & Tools for Spatial Data Analysis · Motivations & Tools for Spatial Data Analysis...
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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
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•• Workshop ObjectiveWorkshop Objective
Provide an introduction & overview of concepts & Provide an introduction & overview of concepts & techniques for spatial data analysistechniques for spatial data analysis
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•• 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
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IntroductionsIntroductions
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Why Spatial is SpecialWhy Spatial is Special
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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+
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motivations: whatmotivations: what
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
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motivations: whatmotivations: what
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motivations: whatmotivations: what
GISGIS
Spatial Data Spatial Data AnalysisAnalysis
Spatial Spatial AnalysisAnalysis
““Spatial StatisticsSpatial Statistics””
GeostatisticalGeostatisticalDataData
LatticeLatticeDataData
PointPoint--PatternPatternDataData
Spatial InteractionSpatial InteractionDataData
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motivations: whatmotivations: what
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
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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)
motivations: whatmotivations: what
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Spatial Flows DataSpatial Flows DataMigration in the UK, Map 5 from Migration in the UK, Map 5 from RavensteinRavenstein 18851885
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Lattice DataLattice DataWisconsin Latino Population, 1990Wisconsin Latino Population, 1990--2007 Estimated Percent Change2007 Estimated Percent Change
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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%
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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)
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motivations: whatmotivations: what
•• 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
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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
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motivations: whymotivations: why
•• 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
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•• 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)
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motivations: whymotivations: why
•• 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
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motivations: whymotivations: why
•• 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
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•• HeteroskedasticityHeteroskedasticity
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motivations: whymotivations: why
•• 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)
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motivations: whymotivations: why
•• 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
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•• Spatially Spatially AutocorrelatedAutocorrelated ResidualsResiduals
County Child Poverty Rates, 2000Source:SF3 Table P87
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motivations: whymotivations: why
•• 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)
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motivations: whymotivations: why
•• 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)
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motivations: whymotivations: why
•• 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)
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motivations: whymotivations: why
•• 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
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motivations: whymotivations: why
•• 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
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•• 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.
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motivations: whymotivations: why
•• 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
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motivations: whymotivations: why
•• 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
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motivations: whymotivations: why
•• 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
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motivations: whymotivations: why
•• 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!)!)
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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
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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
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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
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•• Showing Interesting PatternsShowing Interesting Patterns
motivations: howmotivations: how
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))
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•• Finding Interesting PatternsFinding Interesting Patterns
Exploratory Spatial Data Analysis (ESDA)Exploratory Spatial Data Analysis (ESDA)
motivations: howmotivations: how
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motivations: howmotivations: how
•• 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
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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
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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?
global spatial autocorrelationglobal spatial autocorrelation
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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?
global spatial autocorrelationglobal spatial autocorrelation
% African American US South, Census 2000
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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
global spatial autocorrelationglobal spatial autocorrelation
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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?
global spatial autocorrelationglobal spatial autocorrelation
1. observed map
2. alternative map
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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
global spatial autocorrelationglobal spatial autocorrelation
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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
global spatial autocorrelationglobal spatial autocorrelation
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global spatial autocorrelationglobal spatial autocorrelation
n
i
n
iii
n
iii
xy
yyxx
yyxxr
1
2
1
2
1
)()(
))((
PearsonPearsonproductproduct--moment correlationmoment correlation
n
jj
n
ii
n
i
n
jjiij
n
i
n
jij
x
xxxx
xxxxw
w
nI
1
2
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1 1
1 1)()(
))((
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)
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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
i
n
ij i jj
n
i
n
ii
n
( )
( )( )
( )11
11
2
1
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spatial weights matricesspatial weights matrices
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?
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spatial weights matricesspatial weights matrices
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
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spatial weights matricesspatial weights matrices
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
Queen ContiguityQueen Contiguity
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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
jj
SimpleSimpleContiguityContiguity
MatrixMatrix{{ccijij}}
Queen Queen CriterionCriterion
(zeros implicit)(zeros implicit)
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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
w ccijij
ijj
RowRowStandardizedStandardized
WeightsWeightsMatrixMatrix
(zeros implicit)(zeros implicit)
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spatial weights matricesspatial weights matrices
17
26
35
44
54
65
74
84
95
106
113
124
133
144
151
163
Consider Consider yyii valuesvalues, , ii = 1,= 1,……,16,16
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spatial weights matricesspatial weights matrices
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
61
16
18
7 18
6 18
4 18
4 18
4 18
5 18
6 18
3
= 4.9 (zeros not shown)(zeros not shown)
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spatial weights matricesspatial weights matrices
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
i
1
16
1
16
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spatial weights matricesspatial weights matrices
Which takes us back to this Which takes us back to this (earlier slide)(earlier slide)……
I n
w
w y y y y
y yijj
n
i
n
ij i jj
n
i
n
ii
n
( )
( )( )
( )11
11
2
1
zzWzzI'
'
Assume Assume WW row standardized & row standardized & zzii = = yyii -- y
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spatial weights matricesspatial weights matrices
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
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spatial weights matricesspatial weights matrices
Expected valueExpected value of Moranof Moran’’s s I I value value (under H(under H00))……
E In
( ) 1
1
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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
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WXx w xi i ij j
j
for for j Ji && zi = (yi – y)/σy
local spatial associationslocal spatial associations
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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local spatial associationslocal spatial associations
17
26
35
44
54
65
74
84
95
106
113
124
133
144
151
163
RecallRecall……
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local spatial associationslocal spatial associations
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)
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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
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spatial regressionspatial regression
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
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spatial regressionspatial regression
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
E uE uu C
( )( ' )
0 E
E I( )( ' )
0
2
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spatial regressionspatial regression
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
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spatial regressionspatial regression
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
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Application of Foundational MethodsApplication of Foundational Methods(in (in GeoDaGeoDa))