, the G statistic for x is defined to be: 2. x W x. (1 x)

Transcription

1 5. Tests of Sptl Cocetrto The bove testg procedures re ll motvted by the sptl utoregressve model of resdul errors. So before movg o to sptl regresso lyses of rel dt, t s pproprte to cosder cert ltertve mesures of sptl ssocto tht re lso bsed o sptl weghts mtrces. By fr the most mportt of these for our purposes re the so-clled G-sttstcs, developed by Gets d Ord (1992,1995). 1 These sttstcs focus o drect ssoctos mog (oegtve) sptl ttrbutes rther th sptl resduls from some uderlyg expltory model. For y gve set of oegtve dt, x ( x1,.., x ), ssocted wth rel uts, together wth pproprte sptl weghts mtrx, ( w :, j 1,.., ), the G sttstc for x s defed to be: 2 (5.1) xw 1 1 x j j ( x) G 1 j1 xx j xx 2 (1 x) As dscussed further below, the dgol elemets of re llowed to be ozero (sce o utoregressve-type reltos re volved). However, f oe s oly terested reltos betwee dstct rel uts, j, so tht the dgol elemets of re treted s zeros, the the resultg sttstc s clled smply the G sttstc, d s gve by: (5.2) xw 1 1 x j j ( x) G 1 j1 xx j (1 x) 0 x x 2 xx 0 where dg( ). However, our focus wll be lmost etrely o 5.1 A Probblstc Iterpretto of G G sttstcs. 3 hle the deftos (5.1) d (5.2) serve to clrfy the forml smlrtes betwee these dces d those of the prevous secto, there s ltertve represetto whch suggests more megful terpretto of these dces. Here we focus o G. Frst observe tht sce x 0, f we let (5.1.1) p x x x 1 x j1 1 The 1992 pper s Referece 7 the clss Referece Mterls. 2 hle our preset focus s o rel uts, t should be oted tht these G-sttstcs re eqully pplcble to sets of pot loctos, such s hosptls or supermrkets wth gve urb re. 3 It should be cler from these deftos tht better choce of otto would hve bee to use G wth 0 0 d G wth. But t ths pot, t s best to sty wth the stdrd otto the lterture. ESE 502 III.5-1 Toy E. Smth

2 deote the proporto (or frcto) of x ut, d let p ( p1,.., p ) deote the correspodg vector of proportos, the G c be rewrtte s (5.1.2) xw x x x G w p p w j j 2 j (1 x) 1 x 1x Next observe (from the ttle of ther 1992 pper) tht Gets d Ord re prmrly terested dstce-bsed mesures of proxmty or ccessblty. I prtculr, f we let d deote some pproprte oto of dstce betwee uts d j, d let d ( ) deote pproprte (ocresg) ccessblty fucto of dstce [such s d ( ) d or d ( ) exp( d) ], the we my ow terpret ech sptl weght s ccessblty mesure (5.1.3) w ( d ),, j 1,.., d wrte (5.1.4) G ( p p ) ( d ) j To gve cocrete terpretto to G, let us ssume for the momet tht x represets the populto rel ut, so tht p s the frcto of populto, d p ( p1,.., p ) s the populto dstrbuto mog rel uts. I ths cotext oe my sk: ht s the expected ccessblty betwee two rdomly smpled dvduls from ths dstrbuto? To swer ths questo, observe tht sce p s by defto the probblty tht rdomly smpled dvdul s from ut, t follows by depedece tht pp j must be the jot probblty tht these two rdom smples re from uts d j, respectvely. So f ccessblty s treted s rdom vrble wth vlues, d ( ), for ech pr of rel uts, the t follows from (5.1.4) tht G must be the expected vlue of ths rdom vrble,.e., (5.1.5) G E( ) Thus the vlue of G s precsely the swer to the questo bove,.e., the expected ccessblty betwee two rdomly smpled dvduls ths populto. I terms of ths prtculr exmple, there re severl ddtol fetures tht should be oted. Frst t should be cler tht two dvduls the sme rel ut re by defto mxmlly ccessble to oe other. So y mesure of overll ccessblty wll surely be dstorted f these reltos re omtted s G sttstcs. It s for ths reso tht our focus s lmost exclusvely o G sttstcs. Notce lso from the deftos of d p ESE 502 III.5-2 Toy E. Smth

3 tht G must cheve ts mxmum vlue whe ll populto s cocetrted the smllest of these rel uts. Ths suggests tht G s more ccurtely descrbed s mesure of sptl cocetrto th ssocto. More geerlly, these terprettos crry over to essetlly y oegtve dt. For exmple, f x deotes come or crme levels, the G represets the sptl cocetrto of come or crme. But here oe must be creful to dstgush betwee extesve d tesve quttes. For exmple, whle proporto of totl come (dollrs) rel ut s strghtforwrd, the proporto of per cpt come s less cler. Hece oe must tret such tesve quttes terms of desty uts tht c be dded. So for exmple, f per cpt come s twce s hgh s j, ths would here be tke to me tht the come desty s twce tht j. So better terpretto of G ths cse would be terms of the sptl cocetrto of come desty. I y cse t s certly megful to sk whether cert sptl ptters of per cpt come re more cocetrted th others Flly, we should dd tht eve for sptl weghts mtrces,, tht re ot dstce bsed (such s sptl cotguty mtrces), such weghts c stll be vewed s mesures of closeess pproprte sese. So the lyses to follow, we shll cotue to terpret 1 G (5.1.2) s mesurg the degree of sptl cocetrto of quttes, x ( x,.., x ). 5.2 Globl Tests of Sptl Cocetrto To test whether populto (come, crme, etc.) s sgfctly cocetrted spce, t s turl to g cosder permutto tests volvg G, where w s mplctly terpreted s mesure of ccessblty,, s (5.1.3) bove. The detls of such testg procedure re essetlly detcl to the sc_perm test bove. The oly dfferece s tht the relevt test sttstc, S, Secto bove s ow G rther th sy the Mor sttstc, I. Ths procedure opertolzed the MATLAB progrm, g_perm.m. As oe pplcto of ths testg procedure, we g cosder the Eglsh Mortlty dt Fgure 1.9 bove (p.iii.1-5). For purposes of llustrto, we here cosder ew type of sptl weghts mtrces, mely expoetl-dstce weghts [expresso (2.1.13)] whch s lso costructed by usg the MATLAB progrm, dst_wts.m. Strtg wth expoetl-dstce weghts, sy (5.2.1) w ( d ) exp( d ) we frst ote tht sce the egtve expoetl fucto pproches zero very rpdly, t s ofte dvsble to ormlze dstce dt to the ut tervl to vod vshgly ESE 502 III.5-3 Toy E. Smth

4 smll vlues. 4 To do so we frst detfy the lrgest possble cetrod dstce, d mx, betwee ll prs of Helth Dstrcts, d the covert cetrod dstces, d, to the ut tervl by settg (5.2.2) d d / d,, j 1,.., ( 199) mx so tht 0d 1. Usg ths ormlzto, we c the desg expoetl dstce weghts to yeld some pproprte effectve bd wdth by smply plottg the fucto exp( d), 0 d 1, for vrous choces of. For our preset purposes, the vlue 10 yelds the plot show Fgure 5.1 below, 5 whch s see to yeld effectve bdwdth of bout d 1/2 (show by the red rrow). I terms of our ormlzto (5.2.2) ths yelds the fmlr vlue, d /2: mx exp( 10 d) d Fgure 5.1. Negtve Expoetl Fucto Usg the workspce, eg_mort.mt, the correspodg sptl weghts mtrx, 1, s costructed by usg dst_wts.m wth the commds: >> fo.type = [4,10,1]; >> 1 = dst_wts(l,fo); Here L s the 199x2 mtrx of cetrod coordtes, 4 dctes tht expoetldstce weghts re opto 4 dst_wts.m, 10 deotes the expoet vlue, d (most mporttly) 1 deotes the opto to leve ll dgol elemets s clculted [ ths cse, exp(0) 1]. Note lso tht sce these weghts re lredy gurteed to le the ut tervl (s Fgure 5.1), there s o eed to cosder y ddtol ormlztos (s provded by the fo.orm opto). Flly, deotg the myocrdl frcto rtes 4 For exmple, f dstce were meters, the whle dstce of 800 meters s ot very lrge, you wll dscover tht MATLAB yelds the egtve expoetl vlue, exp(-800) = 0. Moreover, ths s ot rouded to zero, but s ctully so smll umber tht t s beyod the lmts of double precso rthmetc to detect. 5 Ths plot s obted wth the commds: x = [0:.01:1]; y = exp(-10x); plot(x,y,'k','lewdth',5); ESE 502 III.5-4 Toy E. Smth

5 by z = mort(:,3), the test of sptl cocetrto usg g_perm.m s performed wth the commd: >> g_perm(z,1,999); The results of ths test (wth 999 rdom permuttos of Helth Dstrcts) s show below: SPATIAL CONCENTRATION RESULTS INDEX VALUE PROB G G Notce frst tht both G d G vlues re reported, eve though G s of prmry terest for our purposes. Next observe tht, ot surprsgly, these myocrdl frcto rtes re mxmlly sgfct gve 999 permuttos, d tht ths cse there s very lttle dsgreemet betwee G d G. For purposes of comprso, we lso try the more locl sptl weghts mtrx, _5, lredy employed Secto bove to test for sptl utocorrelto the regresso resduls for ths sme dt. Here the results of usg >> g_perm(z,_5,999); re see to be prctclly the sme: SPATIAL CONCENTRATION RESULTS INDEX VALUE PROB G G As wth sptl utocorrelto, t s lwys good de to use severl sptl weght mtrces to check the robustess of the results. Here t s cler from the very dfferet (mplct) bdwdths used these two exmples tht the sgfcce of sptl cocetrto ths cse s frmly estblshed. Before movg o to the more terestg locl tests of sptl cocetrto, t s of terest to ote tht such tests c lso be doe ARCMAP. Here ARCMAP hs for some reso chose to use oly G-sttstcs rther th G -sttstcs. 6 But the more mportt cse of locl sptl cocetrto below, they do use G -sttstcs. So we shll ot sped much tme o ths prtculr pplcto, other th to ote tht t c be ccessed by 6 To see ths, smply Google How Hgh/Low Clusterg (Gets-Ord Geerl G) works. ESE 502 III.5-5 Toy E. Smth

6 ArcToolbox > Sptl Sttstcs Tools > Alyzg Ptters > Hgh/Low Clusterg (Gets-Ord Geerl G) For ske of comprso wth the MATLAB results bove, we hve used exctly the sme procedure developed Secto bove for testg sptl utocorrelto terms of _5. Here the oly dfferece s tht Geerl G s used rther th Mor s I. The grphcl output for ths pplcto s show Fgure 5.2 below: Observed Geerl G: z-score: p-vlue: Gve the z-score of 9.05, there s less th 1% lkelhood tht ths hgh-clustered ptter could be the result of rdom chce. Fgure 5.2. Applcto of the G Sttstc Notce from the vlue of G = tht ths s the sme vlue (whe rouded) s tht obted MATLAB bove. Notce lso tht the result here s terms of the symptotc orml pproxmto of ths G sttstc (obted by Gets-Ord, 1992, uder the sme rdom permutto hypothess s bove), d s thus reported s z-score (9.0538) wth extremely smll p-vlue. Ths g suggests tht the MATLAB results would cotue to obt mxml sgfcce for my more permuttos th Locl Tests of Sptl Cocetrto Observe tht both G d bout ech locto s follows. Let the locl G re decomposble to locl mesures of cocetrto G vlue t be defed by ESE 502 III.5-6 Toy E. Smth

7 (5.3.1) wx G p w j1 j () j1 j x j1 j d smlrly, let the locl (5.3.2) G () j j x j j G vlue t be defed by wx where, g, our terest focuses lmost etrely o G (). Note prtculr from (5.1.2) tht these locl mesures of cocetrto re relted to G by the detty, j j 1 (5.3.3) G p pw pg () Thus G c be vewed s weghted verge of these locl cocetrto mesures, where the weghts, p, re smply the proportos of x ech rel ut. I terms of the probblty terpretto bove, f we g cosder ccessblty weghts of the form, w ( d ), the G s precsely the expected ccessblty from rdomly () smpled ut of x to y other rdomly smpled ut,.e., the codtol expected ccessblty (5.3.4) j1 j G () p ( d ) E( ) I these terms, t follows from (5.1.5) together wth (5.3.4) tht the decomposto (5.3.3) s smply stce of the stdrd codtol-expectto detty: (5.3.5) E ( ) pe ( ) But the rel terest these locl mesures s tht they provde formto bout where cocetrto s d s ot occurrg. 8 I prtculr, by ssgg p-vlues dctg the sgfcce of locl cocetrto t ech rel ut, oe c mp the results d vsulze the ptter of these sgfcce levels. Those res of hgh cocetrto re geerlly referred to s hot spots ( mer completely logous to strog clusters pot ptters). 7 It s of terest to ote tht ths decomposto s stce of wht Asel (1995) hs clled Locl Idctors of Sptl Assocto (LISA). 8 Ideed, the orgl pper by Gets d Ord (1992) strts wth these locl dces, d oly groups them to Geerl G sttstc lter secto of the pper. ESE 502 III.5-7 Toy E. Smth

8 5.3.1 Rdom Permutto Test I ths settg, oe my test for the presece of such hot spots wth respect to dt set, ( x : 1,.., ) by employg essetlly the sme rdom permutto test s bove. I prtculr, for y rdom permutto, ( 1,.., ), of the rel ut dces (1,.., ), oe my compute for ech ut the ssocted sttstc, G (), d compre ths observed vlue wth the dstrbuto of vlues, G (, k ) for N rdom permuttos, k ( k 1,.., k ), k 1,.., N. Here t s mportt to ote tht the dex s tself cluded ths permutto. For f the vlue of x s reltvely lrge, the to reflect the sgfcce of ths locl cocetrto t t s mportt to llow smller vlues to pper t other rdom permuttos. If the observed vlue of G () hs rk k mog ll vlues [ G(), G(,1),.., G(, N )] (wth rk 1 deotg the hghest vlue), the the sgfcce of cocetrto t s g represeted by the p-vlue, k (5.3.6) P, 1,..,. N 1 It s these vlues tht re plotted to revel vsul ptters of cocetrto Eglsh Mortlty Exmple Ths testg procedure s mplemeted for locl G -sttstcs the MATLAB progrm, g_perm_loc.m. Here t s ssumed tht tests for ll rel uts, 1,..,, re to be doe. Hece the outputs cot the locl G -sttstc d P-vlue for ech rel ut. To llustrte the use of ths locl-testg procedure, t s coveet to cotue wth the Eglsh Mortlty exmple bove. For the expoetl-dstce weghts mtrx, 1, costructed bove, together wth the myocrdl frcto dt, z, the commd: >> GP1 = g_perm_loc(z,1,999); yelds (190 x 2) output mtrx GP1 [( G, P) : 1,..,190] cotg the locl G - sttstc, G [ G1( )] d P-vlue, P, for ech of the 190 dstrcts, bsed o 999 rdom permuttos. These vlues were mported to ARCMAP d dsplyed the mp documet, Eg_mort.mxd, s show Fgure 5.3 d 5.4 below. Fgure 5.3 plots the ctul vlues of G ech rel ut,, wth drker gree res deotg hgher vlues. The correspodg P-vlues re show Fgure 5.4, where drker red shows the re of most sgfcce (d where oly the leged for P-vlues s show). As expected, there s see to be rough correspodece betwee hgh locl G vlues d more sgfct res of cocetrto. ESE 502 III.5-8 Toy E. Smth

9 P-VALUES Fg.5.3. Expoetl G-Vlues Fg.5.4. Expoetl P-vlues Notce prtculr tht the locl G -vlues reflect the geerl cocetrto of myocrdl frcto rtes the orth tht s see the orgl dt set [Fgure 1.9 (p.iii.1-5)], but ow re smoothed by the expoetlly weghted verges the locl G sttstcs. However ths orth-south dvde ([B-G], p.279) s see to be much more drmtc the ssocted P-vlues, where the drkest rego, deotg P-vlues less th.01, ow covers ll of Norther Egld. Turg ext to the erest-eghbor weghts mtrx, _5, the test results re ow obted wth the commd, >> GP2 = g_perm_loc(z,_5,999); whch g yelds (190 x 2) output mtrx GP2 [( G, P) : 1,..,190] cotg the locl G -sttstcs d P-vlue for ths cse. By g mportg these vlues to ARCMAP, we obt the comprble dsplys show Fgures 5.5 d 5.6 below. Notce tht key dfferece betwee these two sets of results s the ddtol locl vrto vlues creted by the smller umbers of eghbors used by _5. For exmple, whle ech rel ut hs oly 5 eghbors _5, f we pproxmte the bdwdth expoetl mtrx, 1, by coutg oly weghts, w.01, the some rel uts stll hve more th 70 eghbors. So the degree of smoothg s much greter the ssocted G vlues. But stll, the hghest vlues of both G d P cotue to be the orth, d fct re see to gree more closely wth those cocetrtos of myocrdl frcto rtes see the orgl dt, such s the cocetrto see roud Lcshre couty [compre Fgure 1.6 (p.i.1-3) wth Fgure 1.9 (p.iii.1-5)]. So t would pper tht 5 erest eghbor yelds more pproprte scle for ths lyss. ESE 502 III.5-9 Toy E. Smth

10 P-VALUES Fg.5.5. Nerest Neghbor G-Vlues Fg.5.6. Nerest Neghbor P-Vlues Asymptotc G Test ARCMAP A ltertve test usg G s vlble ARCMAP. Ths procedure c be foud t: ArcToolbox > Sptl Sttstcs Tool > Mppg Clusters > Hot Spot Alyss (Gets-Ord G) To employ ths procedure, we wll g use the Eglsh Mortlty dt wth the eresteghbor sptl weghts mtrx, _5, lredy costructed for ARCMAP Secto I the Hot Spot wdow tht opes, type: ESE 502 III.5-10 Toy E. Smth

11 where the specfc pth mes wll of course vry. Clck OK, d shpefle wll be costructed d dded to the Tble of Cotets your mp documet. The result dsplyed s show Fgure 5.7 below (where the leged from the Tble of Cotets hs bee dded). < Std. Dev Std. Dev Std. Dev Std. Dev Std. Dev Std. Dev. > 2.58 Std. Dev. Fgure 5.7. Asymptotc G Test Output As wth the Geerl G test Fgure 5.2 bove, ths test s bsed o the symptotc orml pproxmto of the locl G -sttstcs uder the sme rdom permutto hypothess s bove. So the vlues show the leged bove re ctully terms of the z-scores obted for ech test. For exmple, the fmlr vlued the secod to lst red etry dctes tht myocrdl frcto rtes for dstrcts wth ths color re sgfctly cocetrted t betwee the.05 d.01 level. (The ctul p-vlues re lsted the Attrbute Tble for ths mp). Here t s mportt to ote tht two-sded tests re beg performed. So for correspodg oe-sded test (s doe bove), these vlues re ctully twce s sgfct (.e., wth oe-sded p-vlues betwee.025 d.005). So eve though the red res look slghtly smller th those Fgure 5.6, the results re ctully more sgfct th those of MATLAB, mer cosstet wth ll of the symptotc tests we hve see so fr. Notce lso tht becuse two-sded tests re beg doe, t s lso pproprte show res wth sgfctly less cocetrto th would be expected uder the ull hypothess. These dstrcts re show blue The Advtge of G over G for Alyzg Sptl Cocetrto Before levg ths topc, t s structve to cosder ddtol exmple tht llustrtes the dvtge of locl G -sttstcs over G-sttstcs for the lyss of sptl cocetrto. Here we costruct fcttous populto dstrbuto for the cse of Ere whch t s ssumed tht there s sgle mjor cocetrto of populto oe couty (FID 18 = Offly Couty), s show Fgure 5.8 below. 9 9 I prtculr, bout 25% of the populto hs bee plced ths couty, d the rest hs bee dstrbuted rdomly (uder the ddtol codto tht o other couty cotg more th 5% of the populto). ESE 502 III.5-11 Toy E. Smth

12 Fg.5.8. Fcttous Dt Fg.5.9. Expoetl G-Vlues Here expoetl-dstce mtrx hs bee costructed smlr to 1 bove (to esure smooth represetto), d the locl G -sttstcs for ths cse re show Fgure 5.9. Notce these these G -vlues roughly pproxmte the cocetrto of the orgl dt, but re somewht smoother (s ws lso see for the myocrdl frcto dt bove usg 1). The correspodg P-vlues (g for 999 smultos) re show Fgure 5.10 below. Fg P-Vlues for G Fg P-Vlues for G These results cofrm tht Offly Couty s the overwhelmgly most sgfct cocetrto of populto ( P-Vlue.02 ), wth severl of the surroudg coutes ESE 502 III.5-12 Toy E. Smth

13 gg sgfcce from ther proxmty to Offly. However, f oe crres out the sme test procedure usg locl G -sttstcs, the substtlly dfferet pcture mmerges. Here Offly Couty s ot the lest sgfct but two of ts mmedte eghbors re. The reso of course s tht by settg the mtrx dgol to zero, the populto of Offly tself s gored the locl G -test for ths couty. Moreover, sce ts eghbors do ot exhbt uusully hgh populto cocetrtos, the locl G -vlue for Offly wll ot be uusully hgh compred to the correspodg vlues for rdom permuttos of couty popultos. However, ts eghbors re stll lkely to exhbt sgfctly hgh vlues, becuse ther proxmty to the populto cocetrto Offly yelds uusully hgh locl G -vlues compred to those for rdom permuttos. Hece the tcpted result here s somethg lke dout hot spot, wth the dout hole correspodg to Offly. Ths s bsclly wht s see Fgure 5.10, except tht some eghbors re closer ( expoetl proxmtes) to Offly th others. Ths extreme exmple serves to uderscore the dfferece betwee these two locl sttstcs, d shows tht locl G -sttstcs re fr more pproprte for detfyg sgfct locl cocetrtos. ESE 502 III.5-13 Toy E. Smth

14 ESE 502 III.5-14 Toy E. Smth