global measures of spatial autocorrelation

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  • Global Measures ofSpatial AutocorrelationBriggs Henan University 2010*

    Briggs Henan University 2010

    *

  • Last TimeThe concept of spatial autocorrelation.Near things are more similar than distant thingsThe use of the weights matrix Wij to measure nearnessThe difficulty of measuring nearnessThis was a surprise!

    This TimeMeasures of Spatial AutocorrelationJoin Count StatisticMorans IGearys CGetis-Ord G statistic

    Briggs Henan University 2010*

    Briggs Henan University 2010

    *

  • Global Measures and Local MeasuresBriggs Henan University 2010*An equivalent local measure can be calculated for most global measuresGlobal MeasuresA single value which applies to the entire data setThe same pattern or process occurs over the entire geographic areaAn average for the entire areaLocal MeasuresA value calculated for each observation unit Different patterns or processes may occur in different parts of the region A unique number for each location

    Briggs Henan University 2010

    *

  • Briggs Henan University 2010*Join (or Joins or Joint) Count StatisticPolygons onlybinary (1,0) data onlyPolygon has or does not have a characteristicFor example, a candidate won or lost an electionBased on examining polygons which share a borderDo they have the same characteristic or not?Border same

    on each sideBorder not the same

    on each side Requires a contiguity matrix for polygons

    Briggs Henan University 2010*

  • Different numbers of BW, BB and WW joinsBriggs Henan University 2010*Join (or Joint or Joins) Count StatisticUses binary (1,0) dataShown here as B/W (black/white) Measures the number of borders (joins) of each type (1,1), (0,0), (1,0 or 0,1) relative to total number of bordersFor 6 x 6 matrix, border totals are:60 for Rook Case110 for Queen Case

    Small number of BW joins (6 only for rook)Large proportion of BB and WW joins

    Large number of BW joins Small number of BB and WW joins

    Briggs Henan University 2010*

  • Briggs Henan University 2010*Join Count: Test StatisticTest Statistic given by: Z= Observed - Expected SD of ExpectedExpected given by:Standard Deviation of Expected (standard error) given by:Where: k is the total number of joins (neighbors)pB is the expected proportion Black, if randompW is the expected proportion Whitem is calculated from k according to:Note: the formulae given here are for free (normality) sampling. Those for non-free (randomization) sampling are substantially more complex. See Wong and Lee 1st ed. p. 151 compared to p. 155. Se next slide for explanation.Expected = random pattern generated by tossing a coin in each cell.

    Briggs Henan University 2010*

  • A Note on Sampling Assumptions:applies to most tests for spatial autocorrelationTest results depend on the assumption made regarding the type of sampling:Free (or normality) sampling Analogous to sampling with replacementAfter a polygon is selected for a sample, it is returned to the population setThe same polygon can occur more than one time in a sample Non-free (or randomization) samplingAnalogous to sampling without replacementAfter a polygon is selected for a sample, it is not returned to the population setThe same polygon can occur only one time in a sample The formulae used to calculate the test statistic (particularly the standard error) are different for eachGenerally, the formulae are substantially more complex for free samplingunfortunately, it is also the more common situation!Assuming free sampling requires knowledge about larger trends from outside the region or access to additional information within the region in order to estimate parameters.

    *Briggs Henan University 2010

    Briggs Henan University 2010*

  • Briggs Henan University 2010*total number of joins = 109 = sum of neighbors/2 in the sparse contiguity matrix= number of 1s/2 in the full contiguity matrix for US States (see slides from SA Concepts lecture)Gore/Bush Presidential Election 2000 Is there evidence of clustering by State?Use Join Count to answer this question!Many BB joins

    Briggs Henan University 2010*

  • Queens Case Sparse Contiguity Matrix for US StatesNcount is the number of neighbors for each stateEquals number of 1s in a row of full contiguity matrix Sum of Ncount is 218Number of common borders (joins) =

    ncount / 2 = 109

    N1, N2 FIPS codes for neighbors

    *Briggs Henan University 2010

    Briggs Henan University 2010

    us49_q-Contig

    NameFipsNcountN1N2N3N4N5N6N7N8

    Alabama1428131247

    Arizona4535849632

    Arkansas56222848474029

    California6343241

    Colorado873542040314956

    Connecticut93443625

    Delaware103244234

    District of Columbia1125124

    Florida122131

    Georgia135124537147

    Idaho166324156493053

    Illinois1752921185519

    Indiana18426211739

    Iowa196293117552746

    Kansas2044029318

    Kentucky21747291839545117

    Louisiana22328485

    Maine23133

    Maryland2455110544211

    Massachusetts255449365033

    Michigan263183955

    Minnesota27419554638

    Mississippi284225147

    Missouri298540172147201931

    Montana30416563846

    Nebraska31629208195646

    Nevada32564491641

    New Hampshire333252350

    New Jersey343103642

    New Mexico35548408449

    New York365349425025

    North Carolina37445134751

    North Dakota383462730

    Ohio3952621544218

    Oklahoma4065354829208

    Oregon4146321653

    Pennsylvania426245410393634

    Rhode Island442259

    South Carolina4521337

    South Dakota466562719313830

    Tennessee47852813713512129

    Texas4842253540

    Utah4964835563216

    Vermont503362533

    Virginia516473724541121

    Washington5324116

    West Virginia5455121243942

    Wisconsin55426171927

    Wyoming56649163184630

    us49_q-JC-%vote

    Sparse Contiguity Matrix for US States -- obtained from Anselin's web site (see powerpoint for link)Bush/Gore 2000 Results

    NameFipsNcountN1N2N3N4N5N6N7N8Winnerk*(k-1)

    Alabama1428131247B04312

    Arizona4535849632B05420

    Arkansas56222848474029B06530

    California6343241G1326

    Colorado873542040314956B07642

    Connecticut93443625G1326

    Delaware103244234G1326

    District of Columbia1125124G1212

    Florida122131B0212

    Georgia135124537147B05420

    Idaho166324156493053B06530

    Illinois1752921185519G15420

    Indiana18426211739B04312

    Iowa196293117552746G16530

    Kansas2044029318B04312

    Kentucky21747291839545117B07642

    Louisiana22328485B0326

    Maine23133G1100

    Maryland2455110544211G15420

    Massachusetts255449365033B05420

    Michigan263183955G1326

    Minnesota27419554638G14312

    Mississippi284225147B04312

    Missouri298540172147201931B08756

    Montana30416563846B04312

    Nebraska31629208195646B06530

    Nevada32564491641B05420

    New Hampshire333252350B0326

    New Jersey343103642G1326

    New Mexico35548408449G15420

    New York365349425025G15420

    North Carolina37445134751B04312

    North Dakota383462730B0326

    Ohio3952621544218B05420

    Oklahoma4065354829208B06530

    Oregon4146321653G14312

    Pennsylvania426245410393634G16530

    Rhode Island442259G1212

    South Carolina4521337B0212

    South Dakota466562719313830B06530

    Tennessee47852813713512129B08756

    Texas4842253540B04312

    Utah4964835563216B06530

    Vermont503362533G1326

    Virginia516473724541121B06530

    Washington5324116G1212

    West Virginia5455121243942B05420

    Wisconsin55426171927G14312

    Wyoming56649163184630B06530

    Expected Values (number of joins)Totals19218880

    (% of popular vote)from election resultsFormulae used for expected value calculation

    Bush % (Pb)0.49885E(Jbb)=KPg^227.1248=Q55*B55^2k=total number of joins=218/2109

    Gore % (Pg)0.50115E(Jgg)=KPb^227.3755=Q55*B56^2=880/2440

    E(Jbg)=2KPbPg54.4997=2*Q55*B55*B56

    Formulae used for Standard deviation calculation

    StDev (bb)8.6673=SQRT(Q55*B55^2+2*Q56*B55^3-(Q55+2*Q56)*B55^4)

    StDev (gg)8.7036=SQRT(Q55*B56^2+2*Q56*B56^3-(Q55+2*Q56)*B56^4)

    StDev (bg)5.2203=SQRT(2*(Q55+Q56)*B55*B56-4*(Q55+2*Q56)*B55^2*B56^2)

    Actual values (number of joins)z-scores= (actual - expected) / standev

    from contiguity matrix aboveFormulae used for z-scores

    BushActual(Jbb)603.7930=(F67-F55)/F60

    GoreActual(Jgg)21-0.7325=(F68-F56)/F61

    Bush/GoreActual(Jbg)28-5.0763=(F69-F57)/F62

    (actually, I copied these from O&U, p 195)

    The formulae here assume free (normality) sampling.

    % of VotesNumber of Joins

    ExpectedStan DevActualZ-score

    Bush % (Pb)0.49885Jbb27.1258.667603.7930

    Gore % (Pg)0.50115Jgg27.3758.70421-0.7325

    Jbg54.5005.22028-5.0763

    us49_q-JC-%state

    Sparse Contiguity Matrix for US States -- obtained from Anselin's web site (see powerpoint for link)Bush/Gore 2000 Results

    NameFipsNcountN1N2N3N4N5N6N7N8Winnerk*(k-1)

    Alabama1428131247AlabamaB04312

    Arizona4535849632ArizonaB05420

    Arkansas56222848474029ArkansasB06530

    California6343241CaliforniaG1326

    Colorado873542040314956ColoradoB07642

    Connecticut93443625ConnecticutG1326

    Delaware103244234DelawareG1326

    District of Columbia1125124District of ColumbiaG1212

    Florida122131FloridaB0212

    Georgia135124537147GeorgiaB05420

    Idaho166324156493053IdahoB06530

    Illinois1752921185519IllinoisG15420

    Indiana18426211739IndianaB04312

    Iowa196293117552746IowaG16530

    Kansas2044029318KansasB04312

    Kentucky21747291839545117KentuckyB07642

    Louisiana22328485LouisianaB0326

    Maine23133MaineG1100

    Maryland245

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