Spatial Data Analysis in Stata An Overview - Darryl Mcleoddarrylmcleod.com/.../2016/06/Spatial-Data-Analysis-in-Stata.pdf · Visualizing spatial data ... Fitting spatial regression

IntroductionVisualizing spatial data

Exploring spatial point patternsMeasuring spatial proximity

Detecting spatial autocorrelationFitting spatial regression models

Spatial Data Analysis in StataAn Overview

Maurizio Pisati

Department of Sociology and Social ResearchUniversity of Milano-Bicocca (Italy)

[email protected]

2012 Italian Stata Users Group meetingBologna

September 20-21, 2012

Maurizio Pisati Spatial Data Analysis in Stata 1/65




Outline

1 IntroductionSpatial data analysis in StataSpace, spatial objects, spatial data

2 Visualizing spatial dataOverviewDot mapsProportional symbol mapsDiagram mapsChoropleth mapsMultivariate maps

3 Exploring spatial point patternsOverviewKernel density estimation





Outline








Outline








Outline

4 Measuring spatial proximity

5 Detecting spatial autocorrelationOverviewMeasuring spatial autocorrelationGlobal indices of spatial autocorrelationLocal indices of spatial autocorrelation

6 Fitting spatial regression models





Outline








Outline








Spatial data analysis in StataSpace, spatial objects, spatial data

Introduction






Spatial data analysis in Stata

• Stata users can perform spatial data analysis using avariety of user-written commands published in the StataTechnical Bulletin, the Stata Journal, or the SSC Archive

• In this talk, I will briefly illustrate the use of six suchcommands: spmap, spgrid, spkde, spatwmat, spatgsa,and spatlsa

• I will also mention a pair of Stata commands/suites forfitting spatial regression models: spatreg and sppack
























Spatial data: a discrete view

• For simplicity, let us represent space as a plane, i.e., as aflat two-dimensional surface

• In spatial data analysis, we can distinguish two conceptionsof space (Bailey and Gatrell 1995: 18):

• Entity view : Space as an area filled with a set of discreteobjects

• Field view : Space as an area covered with essentiallycontinuous surfaces

• Here we take the former view and define spatial data asinformation regarding a given set of discrete spatial objectslocated within a study area A




























Attributes of spatial objects

• Information about spatial objects can be classified into twocategories:

• Spatial attributes• Non-spatial attributes

• The spatial attributes of a spatial object consist of oneor more pairs of coordinates that represent its shapeand/or its location within the study area

• The non-spatial attributes of a spatial object consist ofits additional features that are relevant to the analysis athand


























Types of spatial objects

• According to their spatial attributes, spatial objects canbe classified into several types

• Here, we focus on two basic types:• Points (point data)• Polygons (area data)






Types of spatial objects

• According to their spatial attributes, spatial objects canbe classified into several types

• Here, we focus on two basic types:• Points (point data)• Polygons (area data)






Points

• A point si is a zero-dimensionalspatial object located withinstudy area A at coordinates(si1, si2)

• Points can represent several kindsof real entities, e.g., dwellings,buildings, places where specificevents took place, pollutionsources, trees

Washington D.C. (2009)

Homicides






Points




Homicides






Points




Homicides






Points




Homicides






Polygons

• A polygon ri is a region of studyarea A bounded by a closedpolygonal chain whose M ≥ 4vertices are defined by thecoordinate set {(ri1(1), ri2(1)),(ri1(2), ri2(2)), . . . , (ri1(m), ri2(m)),. . . , (ri1(M), ri2(M))}, whereri1(1) = ri1(M) and ri2(1) = ri2(M)

• Polygons can represent severalkinds of real entities, e.g., states,provinces, counties, census tracts,electoral districts, parks, lakes

First Sixth

Fourth

Fifth

Seventh

SecondThird

Washington D.C.

Police Districts






Polygons



First Sixth

Fourth

Fifth

Seventh

SecondThird

Washington D.C.

Police Districts






Polygons



First Sixth

Fourth

Fifth

Seventh

SecondThird

Washington D.C.

Police Districts






Polygons



First Sixth

Fourth

Fifth

Seventh

SecondThird

Washington D.C.

Police Districts





OverviewDot mapsProportional symbol mapsDiagram mapsChoropleth mapsMultivariate maps

Visualizing spatial data






Thematic maps

• Most analyses of spatial data have their natural startingpoint in displaying the information of interest by one ormore maps

• If properly designed, maps can help the analyst to detectinteresting patterns in the data, spatial relationshipsbetween two or more phenomena, unusual observations,and so on

• Thematic maps represent the spatial distribution of aphenomenon of interest within a given study area (Slocumet al. 2005)






Thematic maps in Stata

• Stata users can generate thematic maps using spmap, auser-written command freely available from the SSCArchive (latest version: 1.2.0)

• spmap is a very flexible command that allows for creating alarge variety of thematic maps, from the simplest to themost complex

• While providing sensible defaults for most options andsupoptions, spmap gives the user full control over theformatting of almost every map element, thus allowing theproduction of highly customized maps






Thematic maps in Stata

• In the following, I will show some examples on using spmapfor creating common types of thematic maps:

• Dot maps• Proportional symbol maps• Diagram maps• Choropleth maps• Multivariate maps






Dot maps

• A dot map shows the spatial distribution of a set of pointspatial objects S ≡ {si; i = 1, . . . , N}, i.e., their locationwithin a given study area A

• If the point spatial objects have variable attributes, it ispossible to represent this information using symbols ofdifferent colors and/or of different shape






Dot maps: example 1

Spatial distribution of 359 cases of sexabuse, Washington D.C. (2009). Differentcolors are used to distinguish adultvictims from child victims

!"#$%&'()#*++,-./0%!$12#0'$3#4#'0/#$567$"$#$$3#4#'0/#$8(1/()$"$)#/9:.$'#1:.#$8(1/()$%&'(")*%)(')+",*%-".*$20;#2$.#<(4#$8(1/()$)$%=.!2/%$,$%&9(2.%$20;#2$802!#"$8(1/()$8(1/()$">)0>$!"(43$%?:!4.0'(#"-./0%!$(.%567*$<1:2:'%#33"9#22*$$$$$$'''$$$$>:(4/%@%@51::'.*$A%A51::'.*$"#2#1/%B##>$(<$:<<#4"#""/*$$$'''$$$$$$;A%8(1/()*$"(C#%-).,*$<1:2:'%'#.$408A*$$$$$$$$$$$$$$$$$'''$$$$$$:1:2:'%D9(/#$..*$:"(C#%-0.1$..*$2#3#4.0%:4*$$$$$$$$$$$$'''$$$$$$2#31:!4/*$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$'''$$$$2#3#4.%"(C#%-).+*$':D30>%).,**$$$$$$$$$$$$$$$$$$$$$$$$$$$'''$$$$/(/2#%%E#@$0;!"#"F$;A$8(1/()$03#%!$"(C#%-).,**$$$$$$$$$$$'''$$$$"!;/(/2#%%G0"9(43/:4$7-&-$H*++,I%$%$%!$"(C#%-).,**!

Adult (155)Child (204)


Sex abuses, by victim age






Dot maps: example 2

Spatial distribution of 359 cases of sexabuse, Washington D.C. (2009). Majorroads, watercourses and parks are addedto the map for reference

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


Sex abuses






Proportional symbol maps

• A proportional symbol map represents the values takenby a numeric variable of interest Y on a set of point spatialobjects S located within a given study area A

• Proportional symbol maps can be used with two types ofpoint data (Slocum et al. 2005: 310):

• True point data are measured at actual point locations• Conceptual point data are collected over a set of regions

R ≡ {ri; i = 1, . . . , N}, but are conceived as being locatedat representative points within the regions, typically attheir centroids

• The area of each point symbol is sized in direct proportionto the corresponding value of Y






Proportional symbol maps: example

Mean family income in the seven PoliceDistricts of Washington D.C. (2000)

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

26.1 12.7

19.6

14.0

8.1

51.522.7


Mean family income (in thousands of US dollars)






Diagram maps

• A diagram map follows the same logic as a proportionalsymbol map, but represents the values of the variable ofinterest using bar charts, pie charts, or other types ofdiagram

• The use of pie charts allows to display the spatialdistribution of compositional data, i.e., of two or morenumeric variables that represent parts of a whole






Diagram maps: example 1

Mean family income in the seven PoliceDistricts of Washington D.C. (2000).Data are represented by framed-rectanglecharts, with the overall mean income asthe reference value

!"#$%&'()*#+)",-)*,".+/,/01,/%!$*(#/-$"23/2$!")45$%&'()*#+)",-)*,".6''-1)4/,#"01,/%!$)1")1#$$$$$$$$$$$$$$7*'('-"#55"8#((#$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$1)/5-/3"9/-")4*'3#:3/#$-#7;#)58,"2'2,',#$7*'('-"5-##4#$$$$$$$$$$$$$<"<:*''-1#$="=:*''-1#$")>#"%&'##$$$$$$$$$$$$$$$$$$$$$$$$$$$$$,),(#"%?#/4$7/3)(=$)4*'3#%#$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$"!@,),(#"%A/"8)45,'4$+060$BCDDDE%$%$%#!


Mean family income






Diagram maps: example 2

Race/Ethnic composition of thepopulation of the seven Police Districts ofWashington D.C. (2000). Data arerepresented by pie charts

!"#$%&'()*#+)",-)*,".+/,/01,/%!$*(#/-$2#3#-/,#$45),#67*,$"$7'7645),##7'7,',$%&&$2#3#-/,#$/8-'/967*,$"$7'76/8-'/9#7'7,',$%&&$2#3#-/,#$',5#-67*,$"$7'76',5#-#7'7,',$%&&$(/:#($;/-)/:(#$45),#67*,$%<5),#%$(/:#($;/-)/:(#$/8-'/967*,$%=8-)*/3$=9#-)*/3%$(/:#($;/-)/:(#$',5#-67*,$%>,5#-%$"79/7$!")32$%&'()*#+)",-)*,".?''-1)3/,#"01,/%!$)1')1($$$$$$$###$$$$8*'('-'",'3#($$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$###$$$$1)/2-/9';/-'45),#67*,$/8-'/967*,$',5#-67*,($@'@6*''-1($$$###$$$$$$A'A6*''-1($8*'('-'#22"5#(($-#1$'-/32#($")B#'%)*($$$$$$$###$$$$$$(#2#31/''3(($$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$###$$$$(#2#31'")B#'$%)+(($$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$###$$$$,),(#'%C/*#DE,53)*$*'97'"),)'3$'8$,5#$7'7!(/,)'3%($$$$$$$###$$$$"!:,),(#'%</"5)32,'3$+0?0$FGHHHI%$%$%(!

WhiteAfrican AmericanOther


Race/Ethnic composition of the population






Choropleth maps

• A choropleth map displays the values taken by a variableof interest Y on a set of regions R within a given studyarea A

• When Y is numeric, each region is colored or shadedaccording to a discrete scale based on its value on Y

• The number of classes k that make up the discrete scale,and the corresponding class breaks, can be based on severaldifferent criteria – e.g., quantiles, equal intervals, boxplot,standard deviates






Choropleth maps: example 1

Mean family income in the 188 CensusTracts of Washington D.C. (2000).Income is divided into six classes basedon the quantiles method

!"#$%&#'"!"()))*+,-,./-,%!$01#,2$3#'#2,-#$4$"$5'067#87,#$%%%$9627,-$4$&'(%9$":7,:$4$!"5'3$%&#'"!"()))*&662/5',-#"./-,%!$5/)5/*$$$$$$$$$$###$$$$01'!7;#2)+*$017#-<6/)=!,'-51#*$906162)>!?/*$$$$$$$$$$$$$$###$$$$'/906162)3"@*$'/1,;)%A5""5'3%*$$$$$$$$$$$$$$$$$$$$$$$$$$$###$$$$1#3#'/)"5B#),$(-**$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$###$$$$-5-1#)%A#,'$9,751C$5'067#$D5'$-<6!",'/"$69$EF$/611,2"G%*$###$$$$"!;-5-1#)%H,"<5'3-6'$+.&.$D()))G%$%$%*!

(93,341](54,93](40,54](33,40](28,33][6,28]Missing








Choropleth maps: example 2

Mean family income in the 188 CensusTracts of Washington D.C. (2000).Income is divided into six classes basedon the boxplot method

!"#$%&#'"!"()))*+,-,./-,%!$01#,2$3#'#2,-#$4$"$5'067#87,#$%%%$9627,-$4$&'(%9$":7,:$4$!"5'3$%&#'"!"()))*&662/5',-#"./-,%!$5/)5/*$$$$$$$$$$###$$$$01'!7;#2)+*$017#-<6/);6=:16-*$906162)>!?/*$$$$$$$$$$$$$$$###$$$$'/906162)3"@*$'/1,;)%A5""5'3%*$$$$$$$$$$$$$$$$$$$$$$$$$$$###$$$$1#3#'/)"5B#),$(-**$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$###$$$$-5-1#)%A#,'$9,751C$5'067#$D5'$-<6!",'/"$69$EF$/611,2"G%*$###$$$$"!;-5-1#)%H,"<5'3-6'$+.&.$D()))G%$%$%*!

(132,341](71,132](40,71](31,40](6,31][6,6]Missing








Multivariate maps

• A multivariate map combines several types of thematicmapping to simultaneously display the spatial distributionof multiple phenomena within a given study area A






Multivariate maps: example

The map shows the relationship betweenpct. white population (represented byframed-rectangle charts), mean familyincome (represented by the width offramed-rectangle charts) and robberyrate (represented by shades of color)across the seven Police Districts ofWashington D.C. (2000/2009)

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

Robberies per 1,000 pop.(10,12](8,10](6,8][2,6]

Washington D.C. (2000/2009)

Pct. white population, income and robberies





OverviewKernel density estimation

Exploring spatial point patterns






Two-dimensional spatial point patterns

• A two-dimensional spatial point pattern can bedefined as a set of N point spatial objects S located withina given study area A

• Usually, each point si ∈ S represents a real entity of somekind: people, events, sites, buildings, plants, cases of adisease, etc.

• Alternatively, each point si represents the centroid of aregion

• Points si are referred to as the data points







• In the analysis of spatial point patterns, we are ofteninterested in determining whether the observed data pointsexhibit some form of clustering, as opposed to beingdistributed uniformly within A

• To explore the possibility of point clustering, it may beuseful to describe the spatial point pattern of interest bymeans of its probability density function p(s) and/or itsintensity function λ(s) (Waller and Gotway 2004)







• In the analysis of spatial point patterns, we are ofteninterested in determining whether the observed data pointsexhibit some form of clustering, as opposed to beingdistributed uniformly within A

• To explore the possibility of point clustering, it may beuseful to describe the spatial point pattern of interest bymeans of its probability density function p(s) and/or itsintensity function λ(s) (Waller and Gotway 2004)







• The probability density function p(s) defines theprobability of observing an object per unit area at locations ∈ A

• The intensity function λ(s) defines the expected numberof objects per unit area at location s ∈ A

• The probability density function and the intensity functiondiffer only by a constant of proportionality






Kernel estimators

• Both the probability density function p(s) and the intensityfunction λ(s) of a two-dimensional spatial point patterncan be estimated by means of nonparametric estimators,e.g., kernel estimators (Waller and Gotway 2004)

• Kernel estimators are used to generate a spatiallysmooth estimate of p(s) and/or λ(s) at a fine grid of pointssg (g = 1, ..., G) covering the study area A

• In the context of spatial data analysis, a grid is a regulartessellation of the study area A that divides it into a set ofG contiguous cells whose centers are referred to as the gridpoints and denoted by sg






Kernel estimators









Kernel estimators









Kernel estimation in Stata

• Stata users can generate kernel estimates of the probabilitydensity function p(s) and the intensity function λ(s) usingtwo user-written commands freely available from the SSCArchive: spgrid and spkde

• spgrid (latest version: 1.0.1) generates several kinds oftwo-dimensional grids covering rectangular or irregularstudy areas

• spkde (latest version: 1.0.0) implements a variety of kernelestimators of p(s) and λ(s)

• spmap can then be used to visualize the kernel estimatesgenerated by spgrid and spkde




































Kernel estimation: example

Our purpose is to estimate the probability density function ofa set of 139 points representing the homicides committed inWashington D.C. in 2009







Step 1

We use spgrid to generate a gridcovering the area of Washington D.C. Wechoose a relatively fine grid resolution(grid cell width = 200 meters). spmap isused to display the grid

!"#$%&'(!%)#'*+,()&-$%.!/&0-*!'$.!,1(0%,)"2344#''''$$$''''&,0!'5,6"$.!!'()%0"6.0.$!#'5.11!"*50.6"/&0-*#'''$$$''''",%)0!"*"0.6"/&0-*#'$."1-5.''(!.'*"0.6"/&0-*!'51.-$'!"6-"'(!%)#'*50.6"/&0-*!'%&"!"#$%&7%&#!







Step 1


!"#$%&'(!%)#'*+,()&-$%.!/&0-*!'$.!,1(0%,)"2344#''''$$$''''&,0!'5,6"$.!!'()%0"6.0.$!#'5.11!"*50.6"/&0-*#'''$$$''''",%)0!"*"0.6"/&0-*#'$."1-5.''(!.'*"0.6"/&0-*!'51.-$'!"6-"'(!%)#'*50.6"/&0-*!'%&"!"#$%&7%&#!







Step 1


!"#$%&'(!%)#'*+,()&-$%.!/&0-*!'$.!,1(0%,)"2344#''''$$$''''&,0!'5,6"$.!!'()%0"6.0.$!#'5.11!"*50.6"/&0-*#'''$$$''''",%)0!"*"0.6"/&0-*#'$."1-5.''(!.'*"0.6"/&0-*!'51.-$'!"6-"'(!%)#'*50.6"/&0-*!'%&"!"#$%&7%&#!







Step 2

We use spkde to generate kernelestimates of the probability distributionof homicides in Washington D.C. Wechoose a quartic kernel function withfixed bandwidth equal to 1,000 metersand edge correction. spmap is used todisplay the results

!"#$%&'()#*++,-./0%!$12#0'$3##4$(5$655#7"#""#$"43.#$!"(78$%4/#)4-./0%!$9$9:166'.%$;$;:166'.%$$$&&&$$$$3#'7#2$<!0'/(1%$=07.>(./?$5=>%$5=>$'(((%$$$$$$&&&$$$$#.8#16''#1/$.6/"$"0@(78$%3.#-./0%!$'#4201#%$$!"#$%3.#-./0%!$12#0'$"4)04$4$!"(78$%1/#)4-./0%!$(.$"48'(.:(.%$12)#/?6.$<!07/(2#%$&&&$$$$127!)=#'$)(%$51626'$A0(7=6>%$61626'$767#$**%$2#8#7.$655%$&&&$$$$/(/2#$%B6)(1(.#"%!$"(C#$+'*)%%$$$$$$$$$$$$$$$$$$$$$$$$$$$&&&$$$$"!=/(/2#$%D0"?(78/67$E-&-$F*++,G%$%$%!$"(C#$+'*)%%!


Homicides







Step 2


!"#$%&'()#*++,-./0%!$12#0'$3##4$(5$655#7"#""#$"43.#$!"(78$%4/#)4-./0%!$9$9:166'.%$;$;:166'.%$$$&&&$$$$3#'7#2$<!0'/(1%$=07.>(./?$5=>%$5=>$'(((%$$$$$$&&&$$$$#.8#16''#1/$.6/"$"0@(78$%3.#-./0%!$'#4201#%$$!"#$%3.#-./0%!$12#0'$"4)04$4$!"(78$%1/#)4-./0%!$(.$"48'(.:(.%$12)#/?6.$<!07/(2#%$&&&$$$$127!)=#'$)(%$51626'$A0(7=6>%$61626'$767#$**%$2#8#7.$655%$&&&$$$$/(/2#$%B6)(1(.#"%!$"(C#$+'*)%%$$$$$$$$$$$$$$$$$$$$$$$$$$$&&&$$$$"!=/(/2#$%D0"?(78/67$E-&-$F*++,G%$%$%!$"(C#$+'*)%%!


Homicides







Step 2


!"#$%&'()#*++,-./0%!$12#0'$3##4$(5$655#7"#""#$"43.#$!"(78$%4/#)4-./0%!$9$9:166'.%$;$;:166'.%$$$&&&$$$$3#'7#2$<!0'/(1%$=07.>(./?$5=>%$5=>$'(((%$$$$$$&&&$$$$#.8#16''#1/$.6/"$"0@(78$%3.#-./0%!$'#4201#%$$!"#$%3.#-./0%!$12#0'$"4)04$4$!"(78$%1/#)4-./0%!$(.$"48'(.:(.%$12)#/?6.$<!07/(2#%$&&&$$$$127!)=#'$)(%$51626'$A0(7=6>%$61626'$767#$**%$2#8#7.$655%$&&&$$$$/(/2#$%B6)(1(.#"%!$"(C#$+'*)%%$$$$$$$$$$$$$$$$$$$$$$$$$$$&&&$$$$"!=/(/2#$%D0"?(78/67$E-&-$F*++,G%$%$%!$"(C#$+'*)%%!


Homicides







Step 2


!"#$%&'()#*++,-./0%!$12#0'$3##4$(5$655#7"#""#$"43.#$!"(78$%4/#)4-./0%!$9$9:166'.%$;$;:166'.%$$$&&&$$$$3#'7#2$<!0'/(1%$=07.>(./?$5=>%$5=>$'(((%$$$$$$&&&$$$$#.8#16''#1/$.6/"$"0@(78$%3.#-./0%!$'#4201#%$$!"#$%3.#-./0%!$12#0'$"4)04$4$!"(78$%1/#)4-./0%!$(.$"48'(.:(.%$12)#/?6.$<!07/(2#%$&&&$$$$127!)=#'$)(%$51626'$A0(7=6>%$61626'$767#$**%$2#8#7.$655%$&&&$$$$/(/2#$%B6)(1(.#"%!$"(C#$+'*)%%$$$$$$$$$$$$$$$$$$$$$$$$$$$&&&$$$$"!=/(/2#$%D0"?(78/67$E-&-$F*++,G%$%$%!$"(C#$+'*)%%!


Homicides





Measuring spatial proximity





Spatial weights matrix

• Most spatial data analyses require that the degree ofspatial proximity among the spatial objects of interestbe expressed in some way

• Typically, the degree of spatial proximity among a givenset of N spatial objects is represented by a N ×N matrixcalled spatial weights matrix and denoted by W

• Each element (i, j) of W – which we denote by wij –expresses the degree of spatial proximity between the pairof objects i and j

• Depending on the application, the N main diagonalelements of W are assigned value wii = 0 or value wii > 0

































• A common variant of W is the row-standardized spatialweights matrix Wstd, whose elements are defined asfollows:

wstdij =

wij

N∑j=1

wij





Spatial weights matrices in Stata

• Stata users can generate several kinds of spatial weightsmatrices using spatwmat, a user-written commandpublished in the Stata Technical Bulletin (Pisati 2001)

• spatwmat (latest version: 1.0) imports or generates fromscratch the spatial weights matrices required by othercommands for spatial data analysis (see below)





Spatial weights matrices in Stata

• Stata users can generate several kinds of spatial weightsmatrices using spatwmat, a user-written commandpublished in the Stata Technical Bulletin (Pisati 2001)

• spatwmat (latest version: 1.0) imports or generates fromscratch the spatial weights matrices required by othercommands for spatial data analysis (see below)





OverviewMeasuring spatial autocorrelationGlobal indices of spatial autocorrelationLocal indices of spatial autocorrelation

Detecting spatial autocorrelation






Spatial autocorrelation

• Forty years ago, the geographer and statistician WaldoTobler formulated the first law of geography : “Everythingis related to everything else, but near things are morerelated than distant things” (Tobler 1970: 234)

• This “law” defines the statistical concept of (positive)spatial autocorrelation, according to which two or moreobjects that are spatially close tend to be more similar toeach other – with respect to a given attribute Y – than arespatially distant objects

• In general, spatial autocorrelation implies spatialcustering, i.e., the existence of sub-areas of the study areawhere the attribute of interest Y takes higher than averagevalues (hot spots) or lower than average values (cold spots)
























Indices of spatial autocorrelation

• We consider measures of spatial autocorrelation that applyto area data

• Measures of spatial autocorrelation can be classified intotwo broad categories:

• Global indices of spatial autocorrelation• Local indices of spatial autocorrelation


















• Global indices of spatial autocorrelation

• Local indices of spatial autocorrelation















Global indices of spatial autocorrelation

• A global index of spatial autocorrelation expressesthe overall degree of similarity between spatially closeregions observed in a given study area A with respect to anumeric variable Y (Pfeiffer et al. 2008)

• Since global indices of spatial autocorrelation summarizethe phenomenon of interest in a single value, they areintended not so much for identifying specific spatialclusters, as for detecting the presence of a general tendencyto clustering within the study area






Global indices of spatial autocorrelation

• A global index of spatial autocorrelation expressesthe overall degree of similarity between spatially closeregions observed in a given study area A with respect to anumeric variable Y (Pfeiffer et al. 2008)

• Since global indices of spatial autocorrelation summarizethe phenomenon of interest in a single value, they areintended not so much for identifying specific spatialclusters, as for detecting the presence of a general tendencyto clustering within the study area






Global indices of spatial autocorrelation in Stata

• Stata users can compute global indices of spatial auto-correlation using spatgsa, a user-written commandpublished in the Stata Technical Bulletin (Pisati 2001)

• spatgsa (latest version: 1.0) computes three global indicesof spatial autocorrelation: Moran’s I, Getis and Ord’s G,and Geary’s c. For each index and each numeric variable ofinterest, spatgsa computes and displays in tabular formthe value of the index itself, the expected value of the indexunder the null hypothesis of no global spatial auto-correlation, the standard deviation of the index, thez -value, and the corresponding one- or two-tailed p-value






Global indices of spatial autocorrelation: example

• Study area: Ohio

• Regions: 88 counties

• Variables of interest:• Pct. population aged 18+ with poor-to-fair health status

(pct poorhealth)• Pct. population aged 18+ currently smoking

(pct currsmoker)• Pct. population aged 18+ ever diagnosed with high blood

pressure (pct hibloodprs)• Pct. population aged 18+ obese (pct obese)







Step 1

We use spatwmat to import an existing binary spatial weights matrix– stored in the Stata dataset Counties-Contiguity.dta – andconvert it into a properly formatted row-standardized spatial weightsmatrix Ws

!"#$%&#$'(!)*+',-.(*$)/!0-.*$)+()$123$#,!'''"""''''*#&/#4!$'!$#*3#53)6/!







Step 1

We use spatwmat to import an existing binary spatial weights matrix– stored in the Stata dataset Counties-Contiguity.dta – andconvert it into a properly formatted row-standardized spatial weightsmatrix Ws

!"#$%&#$'(!)*+',-.(*$)/!0-.*$)+()$123$#,!'''"""''''*#&/#4!$'!$#*3#53)6/!







domenica 27 maggio 2012 20.48 Page 1

User: Maurizio

The following matrix has been created:

1. Imported binary weights matrix Ws (row-standardized)

Dimension: 88x88







Step 2

We use spatgsa with the spatial weights matrix Ws to computeMoran’s I on the variables of interest

!"#$%&'!()*#"+,-)-./)-%!$01#-2$"3-)4"-$30)53''26#-1)6$30)50!22"7'8#2$30)56*91''/32"$$$"""$$$$30)5'9#"#!$:#;"$$7'2-(!







Step 2

We use spatgsa with the spatial weights matrix Ws to computeMoran’s I on the variables of interest

!"#$%&'!()*#"+,-)-./)-%!$01#-2$"3-)4"-$30)53''26#-1)6$30)50!22"7'8#2$30)56*91''/32"$$$"""$$$$30)5'9#"#!$:#;"$$7'2-(!







domenica 27 maggio 2012 21.08 Page 1

User: Maurizio

Measures of global spatial autocorrelation

Weights matrix

Name: Ws

Type: Imported (binary)

Row-standardized: Yes

Moran's I

Variables I E(I) sd(I) z p-value*

pct_poorhealth 0.399 -0.011 0.065 6.337 0.000

pct_currsmoker 0.339 -0.011 0.065 5.367 0.000

pct_hibloodprs 0.126 -0.011 0.065 2.119 0.017

pct_obese 0.167 -0.011 0.065 2.730 0.003

*1-tail test






Local indices of spatial autocorrelation

• A local index of spatial autocorrelation expresses, foreach region ri of a given study area A, the degree ofsimilarity between that region and its neighboring regionswith respect to a numeric variable Y (Pfeiffer et al. 2008)

• The local indices of spatial autocorrelation are derivedfrom the corresponding global indices and share theirfundamental properties






Local indices of spatial autocorrelation

• A local index of spatial autocorrelation expresses, foreach region ri of a given study area A, the degree ofsimilarity between that region and its neighboring regionswith respect to a numeric variable Y (Pfeiffer et al. 2008)

• The local indices of spatial autocorrelation are derivedfrom the corresponding global indices and share theirfundamental properties






Local indices of spatial autocorrelation in Stata

• Stata users can compute local indices of spatial auto-correlation using spatlsa, a user-written commandpublished in the Stata Technical Bulletin (Pisati 2001)

• spatlsa (latest version: 1.0) computes four indices ofspatial autocorrelation: Moran’s Ii, Getis and Ord’s Gi andG?

i , and Geary’s ci. For each index and each region in theanalysis, spatlsa computes and displays in tabular formthe value of the index itself, the expected value of the indexunder the null hypothesis of no local spatial auto-correlation, the standard deviation of the index, thez -value, and the corresponding one- or two-tailed p-value






Local indices of spatial autocorrelation in Stata

• Stata users can compute local indices of spatial auto-correlation using spatlsa, a user-written commandpublished in the Stata Technical Bulletin (Pisati 2001)

• spatlsa (latest version: 1.0) computes four indices ofspatial autocorrelation: Moran’s Ii, Getis and Ord’s Gi andG?

i , and Geary’s ci. For each index and each region in theanalysis, spatlsa computes and displays in tabular formthe value of the index itself, the expected value of the indexunder the null hypothesis of no local spatial auto-correlation, the standard deviation of the index, thez -value, and the corresponding one- or two-tailed p-value






Local indices of spatial autocorrelation: example

• Study area: Ohio

• Regions: 88 counties

• Variable of interest: Pct. population aged 18+ withpoor-to-fair health status (pct poorhealth)







We use spatlsa with the standardized spatial weights matrix Ws –previously generated by spatwmat – to compute Moran’s Ii on thevariable of interest. In the output, counties are sorted by z -value

!"#$%&#$'(!)*+',-.(*$)/!0-.*$)+()$123$#,!'*#&/"4!#'!$#*3#53)6/'(!/',-.(*$)/!07#$#23$#,!'89/#5'!"#$9!#'"8$:"..5;/#9$;!'%"4!#'&.5#*')3"*#&/#'!.5$!







We use spatlsa with the standardized spatial weights matrix Ws –previously generated by spatwmat – to compute Moran’s Ii on thevariable of interest. In the output, counties are sorted by z -value

!"#$%&#$'(!)*+',-.(*$)/!0-.*$)+()$123$#,!'*#&/"4!#'!$#*3#53)6/'(!/',-.(*$)/!07#$#23$#,!'89/#5'!"#$9!#'"8$:"..5;/#9$;!'%"4!#'&.5#*')3"*#&/#'!.5$!







lunedì 28 maggio 2012 17.22 Page 1

User: Maurizio

Measures of local spatial autocorrelation

(Output omitted)

Moran's Ii (Poor-to-fair health status (pct. pop 18+))

name Ii E(Ii) sd(Ii) z p-value*

Knox -0.816 -0.011 0.358 -2.246 0.012

Hardin -0.760 -0.011 0.358 -2.090 0.018

Paulding -1.089 -0.011 0.560 -1.924 0.027

Licking -0.457 -0.011 0.358 -1.244 0.107

(Output omitted)

Hancock 0.545 -0.011 0.358 1.555 0.060

Williams 0.927 -0.011 0.560 1.675 0.047

Delaware 0.677 -0.011 0.389 1.769 0.038

Mercer 0.908 -0.011 0.482 1.906 0.028

Putnam 0.949 -0.011 0.358 2.682 0.004

Henry 1.087 -0.011 0.358 3.067 0.001

Vinton 1.246 -0.011 0.389 3.230 0.001

Gallia 2.433 -0.011 0.482 5.069 0.000

Pike 2.197 -0.011 0.429 5.150 0.000

Adams 3.578 -0.011 0.482 7.442 0.000

Lawrence 5.503 -0.011 0.560 9.844 0.000

Jackson 3.911 -0.011 0.389 10.077 0.000

Scioto 5.400 -0.011 0.482 11.220 0.000

*1-tail test







Williams

Henry

PauldingPutnam

HardinMercer

KnoxDelaware

Vinton

JacksonPike

Adams GalliaScioto

Lawrence





Fitting spatial regression models





Spatial regression

• The aim of spatial regression is to estimate therelationship between an outcome variable of interest Y andone or more predictors X, taking into proper account thespatial dependence among observations

• Two types of spatial dependence are most commonlyconsidered (Ward and Gleditsch 2008):

• A spatial autoregressive process in the error term• A spatial autoregressive process in the outcome variable





Spatial regression

• The aim of spatial regression is to estimate therelationship between an outcome variable of interest Y andone or more predictors X, taking into proper account thespatial dependence among observations

• Two types of spatial dependence are most commonlyconsidered (Ward and Gleditsch 2008):

• A spatial autoregressive process in the error term• A spatial autoregressive process in the outcome variable





Spatial error model

• The first type of spatial dependence is represented by thespatial error model:

Y = Xβ + λWξ + ε

where Y denotes an N × 1 vector of observations on theoutcome variable; X denotes an N × j matrix ofobservations on the predictor variables; β denotes a j × 1vector of regression coefficients; λ denotes the spatialautoregressive parameter; W denotes the N ×N spatialweights matrix; ξ denotes an N × 1 vector of spatial errors;and ε denotes an N × 1 vector of normally distributed,homoskedastic, and uncorrelated errors





Spatial lag model

• The second type of spatial dependence is represented bythe spatial lag model:

Y = Xβ + ρWY + ε

where ρ denotes the spatial autoregressive parameter; andall the other terms are defined as above





Spatial error vs. spatial lag models

• The spatial lag model treats spatial dependence assubstance, assuming that the value taken by Y in eachregion is affected by the values taken by Y in theneighboring regions

• On the other hand, the spatial error model treats spatialdependence as nuisance





Spatial error vs. spatial lag models

• The spatial lag model treats spatial dependence assubstance, assuming that the value taken by Y in eachregion is affected by the values taken by Y in theneighboring regions

• On the other hand, the spatial error model treats spatialdependence as nuisance





Spatial regression in Stata

• Stata users can fit spatial error and spatial lag modelsusing spatreg, a user-written command published in theStata Technical Bulletin (Pisati 2001)

• An excellent alternative to spatreg is represented bysppack, a suite of Stata commands – freely available fromthe SSC Archive – written by David M. Drukker, HuaPeng, Ingmar Prucha, and Rafal Raciborski

• sppack is faster and more flexible than spatreg. Moreover,while spatreg is limited to the analysis of small sets ofobservations, sppack can deal with very large Ns





















References





References

• Bailey, T.C. and A.C. Gatrell. 1995. Interactive Spatial Data Analysis.Harlow: Longman.

• Pfeiffer, D., Robinson, T., Stevenson, M., Stevens, K., Rogers, D. and A.Clements. 2008. Spatial Analysis in Epidemiology. Oxford: OxfordUniversity Press.

• Pisati, M. 2001. sg162: Tools for spatial data analysis. Stata TechnicalBulletin 60: 21–37. In Stata Technical Bulletin Reprints, vol. 10, 277–298.College Station, TX: Stata Press.

• Slocum, T.A., McMaster, R.B., Kessler, F.C. and H.H. Howard. 2005.Thematic Cartography and Geographic Visualization. 2nd ed. Upper SaddleRiver, NJ: Pearson Prentice Hall.

• Tobler, W.R. 1970. A computer movie simulating urban growth in theDetroit region. Economic Geography 46: 234–240.

• Waller, L.A. and C.A. Gotway. 2004. Applied Spatial Statistics for PublicHealth Data. Hoboken, NJ: Wiley.

• Ward, M.D. and K.S. Gleditsch. 2008. Spatial Regression Models. ThousandOaks, CA: Sage.


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Spatial Data Analysis in Stata An Overview - Darryl Mcleoddarrylmcleod.com/.../2016/06/Spatial-Data-Analysis-in-Stata.pdf · Visualizing spatial data ... Fitting spatial regression