A New Methodology of Spatial Crosscorrelation Analysis

Transcription

1 A Ne ethodolog of Spatal Crossorrelato Aalss aguag Che Departmet of Geograph, College of Urba ad Erometal Sees, Pekg Uerst, 0087, Beg, Cha. Emal: Abstrat: he dea of spatal rossorrelato as oeed of log ago. Hoeer, ulke the related spatal autoorrelato, the theor ad method of spatal rossorrelato aalss hae remaed udeeloped. hs paper presets a set of models ad orkg methods for spatal rossorrelato aalss. B aalog th ora s de el epressed a spatal quadrat form ad b meas of mathematal reasog, I dere a theoretal frameork for geographal rossorrelato aalss. Frst, to sets of spatal rossorrelato oeffets are defed, ludg a global spatal rossorrelato oeffet ad a set of loal spatal rossorrelato oeffets. Seod, a par of satterplots of spatal rossorrelato s proposed, ad dfferet satterplots sho dfferet relatoshps betee orrelated arables. Based o the spatal rossorrelato oeffet, Pearso s orrelato oeffet a be deomposed to to parts: dret orrelato partal rossorrelato ad dret orrelato spatal rossorrelato. As a eample, the aaltal proess s appled to the relatoshps betee Cha s urbazato ad eoom deelopmet. Spatal rossorrelato ad spatal autoorrelato a omplemet oe aother, ad the spatal rossorrelato satterplots a be used to reeal the ausalt sde a self-orgazed sstem. he spatal rossorrelato models ll pla a useful role future geographal spatal aalss. Ke ords: Spatal autoorrelato; Spatal rossorrelato; Crossorrelato oeffet; Salg of matres; Spatal aalss; Urbazato; Eoom deelopmet; Cha

2 Itroduto I geographal researh, spatal orrelato falls to to tpes: autoorrelato ad rossorrelato. he former reflets trasample orrelato, that s, a relato betee oe measure ad tself, hle the latter reflets tersample orrelato, amel, a relatoshp betee oe measure ad aother measure. Spatal autoorrelato s defed b oe sze measuremet e.g. t populato ad oe spatal otgut measuremet e.g., Euldea dstae, hle spatal rossorrelato a be defed b to sze measuremets e.g., t populato ad urba area ad oe spatal otgut measuremet. Based o the statstal measuremets of ora s de ad Gear s oeffet ora, 950; Gear, 954, a relatel mature theor has bee deeloped for spatal autoorrelato Asel, 995; Asel, 996; Bad et al, 009; Che, 0; Che, 03; Clff ad Ord, 969; Clff ad Ord, 973; Clff ad Ord, 98; Fsher ad Wag, 0; Gets, 009; Gets ad Ord, 99; Grffth, 003; Haggett et al, 977; Hag, 009; Jag ad ao, 00; et al, 007; Odlad, 988; Sokal ad Ode, 978; Sokal ad homso, 987; efelsdorf, 00; Wall, 004; Wag et al, 0; Weeks et al, 004. Spatal autoorrelato aalss has bee del appled to arous orrelatoal aalses of atural ad huma pheomea ma felds Bek ad Seber, 00; Beedett-Ceh et al, 00; Bzzarro et al, 04; Boot et al, 00; Brau et al, 0; Chu et al, 03; Deblaue et al, 0; Dore et al, 04; Impol et al, 0; Kumar et al; 0; hste et al, 00; ateo-omás ad Olea, 00; attsso et al, 03; Stark et al, 0; Wag, 006; Wag et al, 0, ad partular t has bee tegrated to the spatal aaltal teholog of geographal formato sstems GIS ogle et al, 0; Smth et al, 009. I otrast, the theor ad methodolog of spatal rossorrelato has ot et bee ell ostruted for geographal aalss, despte the oept of spatal rossorrelato emergg lterature Che, 009; Duff ad Hughes-Clarke, 005; Kleber ad Geto, 03; amb et al, 04; ogle ad Batt, 996; oth ad Baker, 03; Rak et al, 008. For a umber of geographal elemets th a regoal sstem, the relatoshp betee to measuremets used to be haraterzed th Pearso s orrelato oeffet, hh dates the smplest rossorrelato. Hoeer, Pearso s orrelato oeffet shos othg about teratos based o spatal dstaes. I ths paper, I preset a e graeful theoretal

3 frameork for spatal orrelato aalss. he oelt of ths frameork rests th three aspets. Frst, t s foud b aalog th a e epresso of ora s de Che, 03. herefore, the defto of spatal rossorrelato oeffet s eas to uderstad, ad the relatoshp betee spatal autoorrelato ad spatal rossorrelato s lear. Seod, t s formulated the smplest form of etors ad matres, so t s eas to alulate the orrelato oeffet usg S Eel or atlab. hrd, the proedure of alulatos ad aalss s ell deeloped. he methodologal frameork otas global des, loal des, ad rossorrelato satterplots. he rest of the artle s arraged as follos. I Seto Results, the global ad loal des of spatal rossorrelato are defed, ad a par of spatal rossorrelato satterplots s preseted b aalog th ora s satterplots. I Seto 3 Dsusso, Pearso s orrelato oeffet s deomposed to a dret orrelato oeffet ad a dret orrelato oeffet based o the dea from spatal rossorrelato ge Seto, ad a omparso s dra betee the spatal rossorrelato oeffet ad ora s de. I Seto 4 aterals ad ethods, as a ase stud, the aaltal proess of spatal rossorrelato s appled to the sstem of Cha s tes ad regos. Fall, the paper oludes th a summar of the ma pots of ths ork. odels. Global ad loal measuremets of spatal rossorrelato he theoretal frameork ossts of a set of models, ad the related mathematal reasog ll be sho frst. Suppose there are elemets e.g., tes a sstem e.g., a etork of tes hh a be measured b to arables e.g., t populato ad urba area, ad. A par of etors a be defed as belo: [ ], [ ], here ad are to sze measuremets of the th elemet,,,, ad the smbol deotes traspose, a proess of terhagg the ros ad olums of a ge matr. he meas of ad are ge as follos 3

4 , he etralzed arable a be alulated b. 3, here ad represet the aerage alues of the arables ad. he stadard deatos of the to arables are as follos σ, σ, 4 here σ ad σ are the populato araes PVs of ad, respetel. he results of a salg trasform of the etralzed arables form a par of stadardzed etors suh as σ, σ. 5 hh are termed stadard sores statsts. It a be sho that the orm of ad,.e., the legths of the etors, ad, eatl equals the dmesos of the sstem,.e., the umber of elemets the sstem,. hus e hae,. 6 he models of spatal orrelato, ludg autoorrelato ad rossorrelato, are based o spatal dstae or spatal otgut. Defe a -b- utar spatal eghts matr suh as σ σ W [ ]. 7 hh s atuall a utzed spatal eghts matr USW. he matr a be produed b a spatal otgut matr SC, ad t has three propertes as belo: Smmetr,.e., ; Zero dagoal elemets, amel, 0, hh mples that the etres the dagoal are all 0; 3 Utzato odto, that s. 8 he, b aalog th the mproed formula of ora s de for spatal autoorrelato Che, 03, a e measuremet for spatal rossorrelato aalss a be defed as R W, 9 4

5 here R deotes the oeffet of spatal rossorrelato, hh a be termed spatal rossorrelato de SCI. It s eas to proe that the SCI s a orrelato oeffet, ad ts alue falls betee - ad. Beause of smmetr of the spatal eghts matr, trasposg R elds aother epresso R W W W, 0 hh s equalet to equato 9. Hoeer, as dated the follog seto, from equatos 9 ad 0, e a dere dfferet models for dfferet uses of spatal aalss. A set of matr equatos a be ostruted, based o the SCI formulae. Equatos 9 ad 0 multpled left b or o both sdes of the equal sgs elds W R, W R, W R, 3 W R. 4 We a demostrate that W W, W W. I these equatos, there are to Ideal Spatal Correlato atres ISC for spatal autoorrelato as follos W, W there are to ISCs for spatal rossorrelato suh as W, W ; 5. 6 SCI s ust the egealue of the ISC of spatal rossorrelato. hs dffers from ora s de, hh s the haraterst alue of the ISC of spatal autoorrelato Che, 03. A mportat measuremet of spatal autoorrelato s alled oal Idators of Spatal Assoato ISA. ISA s also termed loal ora s de Asel, 995. Smlarl, a par of sets of loal spatal rossorrelato oeffets a be defed b R, 7 R, 8 5

6 6 here R ad R refer to the loal spatal rossorrelato de SCI of the th elemet ad the th elemet. Aordgl, R deotes the global spatal rossorrelato de GSCI, hh a be termed SCI for short. As, for arbtrar, equatos 7 ad 8 a be epressed th matr equatos suh as [ ] O O, 9 [ ] O O. 0 Comparg equatos 9 ad 0 th equatos 7 ad 8 shos that the elemets the dagoals of ad ge the SCI alues. he traes of or are equal to the GSCI alue. It s er oeet for us to ompute the SCIs b meas of matr operatos.. Pratal equatos for spatal rossorrelato I prate, the spatal rossorrelato oeffet a be defed aother form. he preodto for equato 9 s as follos R W, hh represets a pratal relato for SCI. I fat, aordg to equato 6, equato multpled left b elds W RR, hh results equato 9. Smlarl, the preodto equato 0 s as belo R W, hh multpled left b elds W RR, hh elds equato 0. A Real Spatal

7 Correlato atr RSC for spatal rossorrelato a be defed as W W W W W. 3 It a be proed that R s ust the egealue of, ad the orrespodg egeetor s +. Atuall, equato plus equato elds + W + R +. 4 hs suggests that orrespods to ad. he relatoshp betee equato 3 ad equato ges a error equato - W W U, 5 hh U represets a error etor. he relatoshp betee equato 4 ad equato ges aother error equato W W V, 6 hh V represets aother error etor. Emprall, there are alas errors betee W ad W, also there are errors betee W ad W. hs suggests a approah to testg the goodess of ft a spatal rossorrelato aalss. If the spatal rossorrelato s er strog, ll be a er lose to, ad ll be a er lose to..3 Spatal rossorrelato satterplots Spatal rossorrelato a be suall dsplaed th to satterplots, hh are smlar to ora s satterplot of spatal autoorrelato. Hoeer, the rossorrelato satterplots ome pars. I order to reate the satterplots, s arables based o the spatal orrelato matr SC are defed as belo: f f f W, 7 W, 8 W, 9 f W. 30 f W. 3 7

8 f W. 3 Usg these equatos, e a geerate a set of satterplots omprsg four graphs th obseratoal data ad alulatos. he arables a be mathed to make rossorrelato satterplots as follos. he relatoshp betee ad f ge the frst satterplot, the relatoshp betee ad f ge the seod satterplot, the relatoshp betee ad f ge the thrd satterplot, ad the relatoshp betee ad f ge the fourth satterplot able. I fat, the frst plot s the same as the seod oe, hle the thrd plot s detal form to the fourth oe. herefore, e atuall eed to satterplots for spatal rossorrelato aalss empral studes. able he futoal relatoshps of to pars of satterplots defed for spatal rossorrelato aalss Satterplot Absssa -as Ordate -as Effet Sattered pots red le he frst plot f W f W ats o he seod plot f W f W ats o he thrd plot f W f W reats o he fourth plot f W f W reats o he rossorrelato satterplots a be easl elded b the spreadsheet, rosoft Eel, or the matr programmg laguage, atlab. akg or as a absssa -as ad f or f as a ordate -as, e a reate a satterplot. he usg the relatoshps betee or ad f or f or f or f, e a produe a tredle. Eah satterplot ludes to parts: sattered pots ad a straght le. he relatoshp betee or ad f or f take o sattered pots, but the relatoshp betee or ad f or f or f or f ehbt a tredle, hh s fat a regresso le. I other ords, the plot of f or f s. or presets a set of radoml sattered data pots, hle the plot of f or f or f or f s. or shos a set of ordered data pots, hh make a straght le. Supermposg the tredle oto the sattered data pots elds a satter dagram for spatal rossorrelato aalss. 8

9 3 Dsusso 3. Geographal meags of spatal rossorrelato measuremets he geographal meag of the spatal rossorrelato a be llumated b larfg the mathematal relatoshp betee Peasro s orrelato oeffet ad the SCI. eag out spatal dstaes, e a re-epress equatos 9 ad 0 as follos R0 0 0 W W, 33 here R 0 s the smple orrelato oeffet SCC, hh a be treated as a speal ase of SCI, ad W 0 E 34 represets a utzed dett matr, hh takes the plae of the USW, ad E deotes a dett matr, hh s a square matr th umerals,, alog the dagoal from upper left to loer rght ad - umerals, 0, all other postos. It s eas to proe that R 0 s ust a Pearso s orrelato oeffet: R 0 E E, 35 hh dates smple rossorrelato betee ad. A partal orrelato oeffet a be defed as R p R R W W W W, 36 0 here R p refers to the partal spatal rossorrelato oeffet PSCC. 0 No, the meags of the spatal orrelato oeffets a be eplaed as follos. he SCI, R, deotes the dret orrelato betee ad through the spatal dstae ad other elemets a geographal sstem; the PSCC, R p, represets the dret rossorrelato betee ad ; Pearso s orrelato oeffet, R 0, s a smple rossorrelato oeffet refletg the summato of spatal orrelato, ludg both the dret rossorrelato ad the dret rossorrelato. he SCI has to futos. Frst, t presets the dret orrelato betee ad, hh s based o spatal dstae. Seod, b meas of dret spatal rossorrelato oeffet, e a estmate the dret rossorrelato oeffet. hus, the smple spatal 0 9

10 orrelato, Peaso s orrelato, a be separated to to parts: a dret orrelato thout dstae effet ad a dret orrelato based o a dstae dea effet. 3. A omparso betee spatal autoorrelato ad spatal rossorrelato I spatal aalss, autoorrelato ad rossorrelato represet to dfferet sdes of the same o. I fat, the oept of autoorrelato omes from the smplest rossorrelato,.e. oe thout a tme lag. he autoorrelato oeffet defed the -dmesoal spae proeeds from the autoorrelato futo defed the -dmesoal tme or spae Fgure. he -dmesoal rossorrelato oeffet s ostruted b aalog th the -dmesoal autoorrelato oeffet,.e., ora s de, hh as re-epressed a e mathematal form Che, 03. A omparso a be dra betee spatal autoorrelato ad spatal rossorrelato as sho able. I short, the spatal autoorrelato s the trasample spatal orrelato, hle the spatal rossorrelato s the tersample spatal orrelato. he former s based o oe sze measuremet, hle the latter s based o to sze measuremets. able he smlartes ad dfferees betee spatal autoorrelato ad spatal rossorrelato Item Spatal autoorrelato Spatal rossorrelato Correlato propert Itrasample orrelato Itersample orrelato Correlato oeffet I W, I W R W, R W ISW W, W W, W RSW W W W W W W Satterplot Oe plot o plots he -dmeso spatal orrelato aalses, ludg spatal autoorrelato ad spatal rossorrelato, are based o spatal eght matres. A spatal eght matr omes from a spatal otgut matr SC. We hae at least four approahes to make a SC Che, 0. For a geographal sstem th spatal elemets, A SC a be epressed as 0

11 [ ] V O, 37 here V deotes the SC, ad s a measure used to ompare ad udge the degree of otgut betee plae ad plae,,,,. he elemets o the dagoal are zeros, otherse the must be tured to zero.e., for, 0. A USW a be defed as /, here deotes the sum of SC etres, that s. 38 hus, based o the populato stadard deato PSD, the SCI formulae, equatos 9 ad 0, a be deeloped a sophstated form as follos W R, 39 W R, 40 hh bear a aalog th the tradtoal epresso of ora s de. If our spatal aalss s based o a sample rather tha a populato uerse, the PSD should be replaed b the sample stadard deato SSD. I ths ase, equatos 39 ad 40 should be resed as belo W R ] [, 4 W R ] [. 4 For the omparablt of the spatal rossorrelato measuremet th ora s de, a empral aalss ll be made usg PSD rather tha SSD ths paper.

12 Zero-lag rossorrelato Pearso orrelato -D temporal autoorrelato futo -D spatal autoorrelato futo -D Spatal autoorrelato ora s de -D Spatal rossorrelato Ne frameork Fgure he paths from smple rossorrelato to the -dmesoal spatal rossorrelato b a of autoorrelato Note: I the blok dagram, the sold le represets dret relatos or paths, hle the dashed le deotes the dret relatos or paths. -D refers to -dmesoal, ad -D to -dmesoal. 4 aterals ad ethods 4. Stud area, measuremets, ad aaltal proess he e frameork of spatal rossorrelato a be emploed to stud the relatoshp betee urbazato ad eoom deelopmet of a outr. It proed that there s orrelato betee populato urbazato ad regoal eoom deelopmet Zhou, 989. Hoeer, the relatoshp betee ause ad effet s ot et lear. he spatal rossorrelato aalss a be emploed to reeal the ausalt betee urbazato ad eoom deelopmet. As a eample, the spatal rossorrelato models ad methods ll be appled to alad Cha s regos ad tes. he spatal obets are the 3 proes, autoomous regos, ad mupaltes dretl uder the Cetral Goermet of Cha ad the aptal tes of these regos. he leel of urbazato s measured b the proporto of urba populato to total populato a rego,

13 hle the leel of eoom deelopmet s measured b the per apta gross regoal produt GRP. As for the spatal eght matr, the dstaes b tra betee a to aptal tes a be used to quatf the spatal otgut. he statstal data of urbazato leels ad per apta GRP are aalable from the ebste of Natoal Bureau of Statsts NBS of the People's Republ of Cha ad the ralroad dstae matr a be foud ma Chese road atlases. Beause the tes of Hakou ad hasa ere ot oeted to the etork of Chese tes b rala from 000 to 0, ol 9 regos ad ther aptal tes are take to aout, ad thus the sze of eah spatal sample s 9 able 3. able 3 he GRP, leel of urbazato, ad the SCI alues of 9 Chese regos 0 Rego Ct Orgal arables Stadard arables SCI p GRP Urbazato leel W W Beg Beg a a Hebe Shazhuag Sha aua Ier Hohehot ogola aog Sheag Jl Chaghu Helogag Harb Shagha Shagha Jagsu Nag Zheag Hagzhou Ahu Hefe Fua Fuzhou Jag Nahag Shadog Ja Hea Zhegzhou Hube Wuha Hua Chagsha Guagdog Guagzhou Guag Nag Chogqg Chogqg Shua Chegdu

14 Guzhou Guag ua Kumg Shaa a Gasu azhou Qgha g Nga hua ag Urumh Note: he ut of the leel of urbazato s %, ad the ut of GRP s ua of Remb RB. Aordg to the theoretal model Results, the aaltal proess of spatal rossorrelato omprses three prpal steps. Step: global aalss of spatal rossorrelato. he bas measuremet s the GSCI, hh a be ge b equatos 9 ad 0. Step: loal aalss of spatal rossorrelato. he bas measuremets are the SCIs, hh a be alulated oe b oe usg equatos 7 ad 8, or proessed as bathes usg equatos 9 ad 0. I fat, to etors of SCIs a be suall dsplaed th a satterplot. Step3: eplaato of spatal rossorrelato satterplots. o pars of satterplots a be dra usg equatos 7 to 3. Amog them e eed ol oe par of satterplots. able has sho the orrespodg relatoshps betee dfferet equatos. 4. Calulatos ad aalses he e alulato method for ora s de preseted b Che 03 a be adapted to the SCI. Based o the stadardzed etor, ad the utzed eghts matr W, the SCI a be omputed easl usg Eel or atlab. he method omprses three steps as follos. Step : stadardze the arables. I other ords, oert the tal etors, equato to the stadardzed etors equato 5. As dated aboe, the PSD stead of the SSD ll be emploed to stadardze the data so that the results are omparable th ora s de. he results of 0 are sho able 3. Step : utze the spatal eght matr. Usg the matr of rala dstaes, e a ompute the spatal otgut matr th the dstae dea futo /, here deotes the rala dstae betee a to aptal tes. Note that the dagoal elemets of the matr should be tured to zeros. he utze the otgut matr b usg the sum of the hole etres to dde eah etr. he fal eghts matr a be 4

15 epressed th equatos 7 ad 8. Step 3: ompute SCI. Aordg to equato 9, the USW s frst left multpled b the traspose of, ad the the produt of ad W s rght multpled b ; Aordg to equato 0, the utzed eghts matr s frst left multpled b the traspose of, ad the the produt of ad W s rght multpled b. he fal produt of the otued multplato elds the alue of the SCI, ad the to results are equalet to oe aother. For eample, 0, the de of spatal rossorrelato betee the leel of urbazato ad per apta GRP s R W 0.566, R W A par of satterplots of spatal rossorrelato a be dra usg to approahes. he frst approah s to make use of the arables,, W, W, W, ad W. Oe satterplot s based o the relatoshp betee -as ad W as ell as W -as, hh reflet the ato of per apta GRP o leel of urbazato. he relatoshp betee ad W ges the satterpots, hle the relatoshp betee ad W elds the tredle Fgure a. he other satterplot s based o the relatoshp betee horzotal as ad W as ell as W ertal as, hh reflet the reato of leel of urbazato o per apta GRP. he relatoshp betee ad W elds the satterpots, hle the relatoshp betee ad W ges the tredle Fgure b. he seod approah s to utlze the arables,, W, W, W, ad W. Compared th the frst approah, W s replaed b W, ad W s substtuted b W. he results ad effets are same as those from the frst approah, ad the satterplots are the same as those dsplaed Fgure. I the satterplots, the slopes of the tred les equal the SCI alue. hs suggests that e a emplo regresso aalss based o the least squares method to estmate the SCI b usg equatos ad. If the depedet arable s, the depedet arable ll be W. For 0, the SCI alue s about R 0.566, ad the oeffet of determato s appromatel R 0.6. hs suggests that the per apta GRP a epla about 6.% of the spatal hage of the leel of urbazato. If the depedet arable s, the depedet arable ll be W. For 0, the SCI alue s stll about R 0.566, but the determato oeffet s appromatel R hs suggests that the leel of urbazato a epla about 7.73% of the spatal hage of the per apta GRP. Note that the terept should be set to 0 as there s o ostat term the regresso equatos aboemetoed. A dsoer s that equato ad equato ge the same SCI alue, but the alues of 5

16 goodess of ft are dfferet. For our eample, the ato of o R 0.6 s stroger tha the reato of o R hs seems to suggest that the fluee of eoom deelopmet o urbazato s greater tha the mpat of urbazato o eoom deelopmet W, W W, W W R² W R² a ats o b reats o Fgure he satterplots of spatal rossorrelato betee the per apta GDP ad the leel of urbazato tes of Cha 0 he oeffet of smple orrelato betee the leel of urbazato ad that of eoom deelopmet of alad Cha a be deomposed b usg the SCI alue. For 0, the smple orrelato oeffet a be alulated th equato 35, ad the result s about R I fat, t s eas to obta the R 0 alue S Eel usg the futo orrel or pearso, ad the epresso s orrel, or pearso,. hus, aordg to equato 36, the PSCC s appromatel R p A oluso a be dra from these alues of orrelato oeffets that the dret orrelato de of the 9 regos s about 0.789, ad the dret orrelato de s about he former has o relato to dstaes betee dfferet tes ad a be regarded as tragroup orrelato, ad the latter s related to spatal terato of dfferet regos based o dstaes ad a be treated as tergroup orrelato. Further, the SCI a be separated to SCI, hh reflet the spatal orrelato betee a rego or t ad all other regos or tes. Usg equato 9, e a alulate the frst etor of the loal spatal orrelato oeffet, hh reflets the ato eoom deelopmet o urbazato; usg equato 0, e a ompute the seod etor of SCI, hh reflets 6

17 the reato urbazato o eoom deelopmet. All the results are dsplaed able 3, hh shos that the sum of the SCI equals the GSCI. A satterplot of loal spatal rossorrelato a be dra b usg the to sets of SCI alues Fgure 3. he satterplot a be used to ategorze Chese regos terms of spatal rossorrelato. All the 9 regos a be lassfed to 4 groups aordg to the quadrats of a Cartesa oordate sstem. For eample, Beg, a, ad Shagha are the frst quadrat, Helogag s the seod quadrat, Guagdog, Hebe, Sha are the thrd quadrat, ad Chogqg ad Shadog are the fourth quadrat. For Beg ad Shagha, the ato of eoom deelopmet o urbazato s eaker tha the reato of urbazato o eoom deelopmet, but for a, Jagsu, Zheag, ad Fua, the ato of eoom deelopmet o urbazato s stroger tha the reato of urbazato o eoom deelopmet W W Fgure 3 he satterplot of loal spatal rossorrelato betee the per apta GRP ad the leel of urbazato regos of Cha 0 he aaltal proess of spatal rossorrelato a be appled to the datasets of the ears from 000 to 0. he alulato results lude SCC R 0, SCI R, ad PSCC R p, ad the goodess of ft for the regresso aalses of spatal rossorrelato hae bee estmated able 4. From these alulatos, e a get useful spato-temporal formato for Cha s urbazato ad eoom deelopmet. Frst, the spatal rossorrelato betee per apta GRP ad the leel of urbazato beame stroger ad stroger. he smple orrelato s 7

18 relatel stable, ad the SCC alues flutuate aroud Hoeer, the SCI alues et up ad up, hle the PSCC alues et do graduall Fgure 4. hs suggests that the spatal terato betee dfferet regos ad tes beame more ad more sgfat the proess of spato-temporal eoluto of regoal sstems. Seod, the ato of eoom deelopmet o urbazato s stroger tha the reato of urbazato o eoom deelopmet. he goodess of ft for the regresso of W depedg o, R -, s all greater tha that for the regresso of W depedg o, R -. hs suggests that eoom deelopmet s a ause of urbazato, ad urbazato s a effet of eoom deelopmet. Both the alues of R - ad R - go up ad up from 000 to 0. hs leds further support to the feree that the spatal terato of the 9 regos beame more ad more sgfat oer tme. he absolute groth rate of R - s less tha that of R -. Hoeer, the relate groth rate of R - s greater tha that of R -, hh a be sho b the allometr relatoshp betee R - ad R - Fgure 5. he allometr salg epoet of R - depedg o R - s about.655, hh s sgfatl greater tha. hs leds further support that the leel of urbazato a geographal rego s determed b the leel of eoom deelopmet ad tur reats to t. able 4 he alues of SCC, SCI, PSCC ad determato oeffets of the 9 Chese regos ear SCC R SCI R PSCC R p R R R -+ R Note: he statstal data of the leel of urbazato from 00 to 004 are abset the ebste of Cha s NBS. he statst R - deotes the goodess of ft for the regresso of W depedg o, ad R - refers to the goodess of ft for the regresso of W depedg o. 8

19 SCI PSCC ear ear a SCI b PSCC Fgure 4 Hstograms of SCI ad PSCC of the spatal rossorrelato of 9 Chese regos R R R² R R - Fgure 5 he allometr relatoshp betee to kds of determato oeffets for spatal rossorrelato aalss Colusos he theor of spatal orrelato aalss eeds both spatal autoorrelato ad spatal rossorrelato measuremets; the latter prodes a e frameork for geographal researh. Based o the theoretal results ad empral studes, three bas olusos a be dra as follos. 9

20 Frst, spatal rossorrelato ad spatal autoorrelato a omplemet oe aother. Both spatal autoorrelato ad spatal rossorrelato aalses a be emploed to stud dfferet geographal elemets a regoal sstem or dfferet subsstems th a geographal sstem. he to methods are dfferet, but the a ombe to make a tegrated frameork. he spatal autoorrelato aalss shos the smultaeous hage alue of oe umerall alued radom arable, hle the spatal rossorrelato aalss dsplas the smultaeous hage alues of to radom arables. If e use oe arable to measure a umber of spatal ettes, e a utlze spatal autoorrelato aalss; o the other had, f e use to or more arables to measure a umber of spatal ettes, e a utlze both spatal autoorrelato aalss ad spatal rossorrelato aalss. Seod, the spatal rossorrelato oeffet represets the dret relatoshps betee spatal arables. Usg SCI, e a aalze the ell-ko smple orrelato oeffet spatal statsts. Pearso s orrelato betee to spatal arables ludes to ompoets: dret orrelato ad dret orrelato. he spatal orrelato oeffet reflets the dret orrelato based o the spatal otgut betee a to geographal ettes. Pearso s orrelato oeffet mus the spatal rossorrelato oeffet leaes the dret orrelato oeffet. he dret orrelato s atuall a kd of partal orrelato, hh s depedet of spatal patters. I ths sese, spatal rossorrelato aalss a reeal the mportae of the part plaed b geographal dstaes or spatal relatoshps. hrd, the satterplots of spatal rossorrelato a be used to reeal the ausalt betee to arables. Pearso s orrelato oeffet ad spatal rossorrelato oeffet a reflet the orrelato betee to arables, but the aot dstgush betee ause ad effet. Fortuatel, the satterplots of spatal rossorrelato a be used to dfferetate betee the ause ad the effet. he spatal rossorrelato plots appear b tos, ad the to plots are of asmmetr. herefore, the a sho us hh arable s the leadg posto ad hh s the subordate posto. I setf researh, determg ausalt ma be more mportat tha desrbg orrelato a geographal sstem. I ths sese, the e frameork of spatal rossorrelato a be useful future spatal aalses. 0

21 Akoledgemet: hs researh as sposored b the Natoal Natural See Foudato of Cha Grat No See: Referees Asel 995. oal dators of spatal assoato ISA. Geographal Aalss, 7: 93-5 Asel 996. he ora satterplot as a ESDA tool to assess loal stablt spatal assoato. I:. Fsher, H.J. Sholte, D. U Eds. Spatal Aaltal Perspetes o GIS. odo: alor & Fras, pp-5 Bek J, Seber A 00. Is the spatal dstrbuto of makd s most bas eoom trats determed b lmate ad sol aloe? PoS ONE, 55: e046 Beedett-Ceh, Ike K, Koar B, Cruz-otta J, Kolto A, et al. 00. Spatal relatoshps betee polhaete assemblages ad erometal arables oer broad geographal sales. PoS ONE, 59: e946 Bad RS, üller W, Reder 009. Poer alulatos for global ad loal ora s I. Computatoal Statsts ad Data Aalss, 53: Bzzarro JJ, Broms K, ogsdo G, Ebert DA, oklah, et al 04. Spatal segregato easter orth Paf skate assemblages. PoS ONE, 90: e09907 Boot N, Gallard J-, Coulo A, Gala, Cosso J-F, et al. 00. No dfferee betee the sees fe-sale spatal geet struture of roe deer. PoS ONE, 5: e4436 Brau A, Auersald K, Gest J 0. Drers ad spato-temporal etet of hporhe path arato: Implatos for samplg. PoS ONE, 77: e4046 Che G 009. Urba grat model based o ross-orrelato futo ad Fourer aalses of spato-temporal proess. Chaos, Soltos & Fratals, 4: Che G 0. O the four tpes of eght futos for spatal otgut matr. etters Spatal ad Resoure Sees, 5: 65-7 Che G 03. Ne approahes for alulatg ora s de of spatal autoorrelato. PoS ONE, 87: e68336 Chu HJ, Wu CF, P 03. Iorporatg spatal autoorrelato th eural etorks empral

22 lad-use hage models. Eromet ad Plag B: Plag ad Desg, 403: Clff AD, Ord JK 969. he problem of spatal autoorrelato. I: A.J. Sott Ed.. odo Papers Regoal See, Studes Regoal See. odo: Po, pp5-55 Clff AD, Ord JK 973. Spatal Autoorrelato. odo: Po Clff AD, Ord JK 98. Spatal Proesses: odels ad Applatos. odo: Po Deblaue V, Keel P, Coutero P 0. estg parse assoato betee spatall autoorrelated arables: A e approah usg surrogate latte data. PoS ONE, 7: e48766 Dore S, Fr D, Stephes S 04. Spatal heterogeet of sol CO efflu after harest ad presrbed fre. Forest Eolog ad aagemet, 39: Duff GP, Hughes-Clarke JE 005. Applato of spatal ross orrelato to deteto of mgrato of submare sad dues. Joural of Geographal Researh: Earth Surfae, 0, F04S Fsher, Wag JF 0. Spatal Data Aalss: odels, ethods ad ehques. Ne ork: Sprger Gear RC 954. he otgut rato ad statstal mappg. he Iorporated Statsta, 53: 5 45 Gets A 009. Spatal eghts matres. Geographal Aalss, 4 4: Gets A, Ord JK 99. he aalss of spatal assoato b use of dstae statst. Geographal Aalss, 43:89-06 Grffth DA 003. Spatal Autoorrelato ad Spatal Flterg: Gag Uderstadg hrough heor ad Setf Vsualzato. Ne ork: Sprger-Verlag Haggett P, Clff AD, Fre A 977. oatoal Aalss Huma Geograph. odo: Edard Arold Hag RP 009. Spatal autoorrelato ad the quattate reoluto. Geographal Aalss, 4 4: Impol DE, Solomo, Shluter WW, Raamah A, Bhha RP, et al. 0. he spatal heterogeet betee Japaese eephalts dee dstrbuto ad erometal arables Nepal. PoS ONE, 67: e9 Jag B, ao 00. Geospatal Aalss ad odelg of Urba Struture ad Dams. Ne ork: Sprger Kleber W, Geto G 03. Spatall arg ross-orrelato oeffets the presee of

23 ugget effets. Bometrka, 00: 3-0 Kumar C, Sgh PK, Ra RK 0. Uder-fe mortalt hgh fous states Ida: A dstrt leel geospatal aalss. PoS ONE, 75: e3755 amb AP, Wk K a, bert, kesell D 04. he spatal ross-orrelato method for dsperse surfae aes. Geophsal Joural Iteratoal, 99 : -0 H, Calder CA, Cresse N 007. Beod ora s I: estg for spatal depedee based o the spatal autoregresse model. Geographal Aalss, 394: hste JW, Smos R, Shrer SA, Frazreb KE 00. Spatal autoorrelato ad autoregresse models eolog. Eologal oographs, 73: ogle PA, Batt 996. Spatal Aalss: odellg a GIS Eromet. Cambrdge: Pearso Professoal ogle PA, Goodhld F, agure DJ, Rhd DW 0. Geograph Iformato Sstems ad See 3rd Edto. Hoboke, NJ: Wle oth C, Baker JW 03. A spatal ross-orrelato model of spetral aeleratos at multple perods. Earthquake Egeerg ad Strutural Dams, 43: ateo-omás P, Olea PP 00. Atpatg koledge to form spees maagemet: predtg spatall eplt habtat sutablt of a oloal ulture spreadg ts rage. PoS ONE, 58: e374 attsso BJ, Zpk EF, Garder B, Blak PJ, Sauer JR, et al 03. Eplag loal-sale spees dstrbutos: relate otrbutos of spatal autoorrelato ad ladsape heterogeet for a aa assemblage. PoS ONE, 8: e55097 ora PAP 950. Notes o otuous stohast pheomea. Bometrka, 37-: 7-33 Odlad J 988. Spatal Autoorrelato. odo: SAGE Publatos Rak A, Helfe, Baumbah, Krste S, Bahart J, Shladtz K, Ohser J 008. Aalss of spatal ross-orrelatos mult-osttuet olume data. Joural of rosop, 3: 8-9 Smth J de, Goodhld F, ogle PA 009.Geospatal Aalss: A Comprehese Gude to Prples, ehques ad Softare ools 3rd Edto. eester: roubador Sokal RR, Ode N 978. Spatal autoorrelato bolog.. ethodolog. Bologal Joural of the ea Soet, 0: 99-8 Sokal RR, homso JD 987. Applatos of spatal autoorrelato eolog. I: P. egedre,. 3

24 egedre Eds. Deelopmets Numeral Eolog, NAO ASI Seres, Vol. G4. Berl: Sprger-Verlag, pp Stark JH, Sharma R, Ostroff S, Cummgs DA, Ermetrout B, et al. 0. oal spatal ad temporal proesses of flueza Peslaa, USA: PoS ONE, 73: e3445 efelsdorf 00. he saddle pot appromato of ora s I ad loal ora s I referee dstrbutos ad ther umeral ealuato. Geographal Aalss, 343: Wall 004. A lose look at the spatal struture mpled b the CAR ad SAR models. Joural of Statstal Plag ad Iferee, : 3-34 Wag J 006. Spatal Aalss. Beg: Setf Press [I Chese] Wag JF, Res B, Hu G, Chrstakos G, ag WZ, et al 0. Area dsease estmato based o setel hosptal reords. PoS ONE, 68: e348 Wag JF, Ste A, Gao BB, Ge 0. A ree of spatal samplg. Spatal Statsts, : -4 Weeks JR, Gets A, Hll AG, Gadalla S, Rashed 004. he fertlt trasto Egpt: Itraurba patters Caro. Aals of the Assoato of Amera Geographers, 94: Zhou 989. O the relatoshp betee urbazato ad gross atoal produt. Chese Soolog & Athropolog, : 3-6 4