Methods: Spatial autocorrelation Daniel A. Griffith, Ashbel Smith Professor University of Texas at Dallas, Richardson, Texas, USA

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1 Methods: Spatal autoorrelato Dael A. Grffth, Ashbel Smth Professor Uversty of Texas at Dallas, Rhardso, Texas, USA Address Dr. Dael A. Grffth, Ashbel Smth Professor of Geospatal Iformato See Shool of Soal Sees Uversty of Dallas PO Box , GR 3 60 North Floyd Road Rhardso, TX USA Key words: estmator propertes, Geary Rato, map patter, Mora Coeffet, Mora satterplot, egatve spatal autoorrelato, postve spatal autoorrelato, redudat formato, self-orrelato, spatal spllover

2 Glossary auto- A prefx lterally meag self; spatal autoorrelato meas self-orrelato, or values wth a gve varable are orrelated, resultg the varable beg orrelated wth tself. orrelato A desrpto of the ature ad degree of a relatoshp betwee a par of quattatve varables. Geary Rato A dex of spatal autoorrelato, volvg the omputato of squared dfferees of values that are geograph eghbors (.e., pared omparsos), that rages from 0 to for egatve, ad to approxmately for postve, spatal autoorrelato, wth a expeted value of for zero spatal autoorrelato. geograph oetvty/weghts matrx A -by- matrx wth the same sequee of row ad olum loato labels, whose etres date whh pars of loatos are eghbors. map patter The systemat orgazato of values for some varable aross a map resultg vsually ospuous texture that ossts of global, regoal, ad loal treds, gradets, swaths or mosas. Mora Coeffet A dex of spatal autoorrelato, volvg the omputato of rossproduts of mea-adjusted values that are geograph eghbors (.e., ovaratos), that rages from roughly (-, -0.5)0 to early 0 for egatve, ad early 0 to approxmately for postve, spatal autoorrelato, wth a expeted value of -/(-) for zero spatal autoorrelato, where deotes the umber of areal uts. Mora satterplot A satterplot of stadardzed versus summed earby stadardzed values whose assoated bvarate regresso slope oeffet s the ustadardzed Mora Coeffet. egatve spatal autoorrelato For the geograph dstrbuto of some varable aross a map, hgh values ted to be geograph eghbors of low values, termedate values ted to be geograph eghbors of termedate values, ad low values ted to be geograph eghbors of hgh values. postve spatal autoorrelato For the geograph dstrbuto of some varable aross a map, hgh values ted to be geograph eghbors of hgh values, termedate values ted to be geograph eghbors of termedate values, ad low values ted to be geograph eghbors of low values. redudat formato Iformato data that s duplated, ad hee ueeded for spatal statstal aalyses; for georefereed data, ths duplato arses from loatoal loseess that results mutually shared formato, allowg a awareess of earby values oe the value for a gve loato s kow. Syopss Spatal autoorrelato s the orrelato amog values of a sgle varable aross a two-dmesoal surfae that are loatoally refereed or ted together by a uderlyg spatal struture, trodug a volato of the depedet observatos assumpto of lassal

3 statsts. Its may terprets lude: a usae parameter, self-orrelato, map patter, a dagost tool, a mssg varables surrogate, redudat formato, a spatal proess mehasm, a spatal spllover, ad the outome of areal ut demarato. It a be quatfed wth varous des, ludg the Geary Rato ad the Mora Coeffet, the statstally most powerful of the two measures. These two des are egatvely related. It a be portrayed wth a Mora satterplot. Most soo-eoom/demograph varables are measured at a geograph resoluto that results moderate, postve spatal autoorrelato. Satellte remotely sesed mages ted to have strog to marked postve spatal autoorrelato. To date, few empral examples of egatve spatal autoorrelato have bee reported, although t relates to stuatos of spatal ompetto. At the froters of spatal autoorrelato work s meagful mplemetato wth massvely large georefereed datasets. Itroduto Spatal autoorrelato s the orrelato amog values of a sgle varable strtly attrbutable to ther relatvely lose loatoal postos o a two-dmesoal surfae, trodug a devato from the depedet observatos assumpto of lassal statsts. Spatal autoorrelato exsts beause real world pheomea are typfed by orderless, patter, ad systemat oetrato, rather tha radomess. Tobler s Frst Law of geography eapsulates ths stuato: "everythg s related to everythg else, but ear thgs are more related tha dstat thgs." To ths maxmum should be added the qualfer: but ot eessarly through the same mehasms. I other words, spatal autoorrelato meas a depedey exsts betwee values of a varable eghborg or proxmal loatos, or a systemat patter values of a varable aross the loatos o a map due to uderlyg ommo fators. Seleted physal models portray the exstee of spatal autoorrelato. Frst, ad foremost, ptures (aalogous to map patter) would ot be dserble o a televso/omputer sree wthout spatal autoorrelato. Nor ould sldg tle or jgsaw puzzles be solved. Maget sulptures ostruted from separate pees of metal pled upo a magetzed base more ompletely llustrates spatal autoorrelato: whe the pees of metal are plaed o ther maget base, they a be sulpted to a 3-dmesoal fgure; whe the pees of metal are moved far from ther maget base (as well as ay other maget soure), they smply are depedet pees of metal that oly a be brushed to a ple. The spatal autoorrelato mehasm here s the maget feld that s trasferred from pee to pee by metal pees touhg. Meawhle, a umber of real world examples also portray the exstee of spatal autoorrelato. Most meral deposts luster at relatvely few loatos o the Earth s surfae; they are ot ubqutous. House pres ad house value assessmets are establshed the real estate market by omparsos betwee a house ad smlar earby houses. Zog ordaes fore smlar lad use types to group together otermous loatos. Dseases, suh as West Nle Vrus, reep aross a ladsape through otago. Fally, uabated water (wd) polluters geerate egatve osequees for those dow-stream (dow-wd) of ther loatos; a smlar but more fully -dmesoal example s furshed by defut smelter superfud ste. Coeptual meags of spatal autoorrelato Spatal autoorrelato has may terprets: a usae parameter, self-orrelato, map patter, a dagost tool, a mssg varables surrogate, redudat formato, a spatal proess mehasm, a spatal spllover, ad the outome of areal ut demarato. 3

4 The most dsmssve s as a usae parameter. Spatal autoorrelato s aptured by a model spefato beause ts presee s eessary for a good desrpto, but t s ot truly of terest ad oly terferes wth the estmato of other model parameters that are of true terest. Ths terferee teds to be for parameters suh as varaes, rather tha meas. Setsts reasgly are dedg that spatal autoorrelato should ot be treated as a usae parameter. Iterpretg spatal autoorrelato as self-orrelato s lteral. Correlato arses from the geograph otext wth whh attrbute values our, ad as suh a be expressed terms of the Pearso produt momet orrelato oeffet (r) formula, but wth eghborg values of varable Y replag those of varable X. Here the orrelato beg asertaed s a average of that betwee loato-spef tme seres of a varable for all possble pars of loatos. But these tme seres are ot observable, ad hee the assumpto voked s exhageablty (.e., the set of tme seres a be permute wthout affetg results the order whh a tme seres mehasm geerates values aross a map s rrelevat). Ths stuato s smlar to the equorrelato matrx for grouped ormal data tradtoal statstal aalyss. Iterpretg spatal autoorrelato as map patter emphaszes ospuous treds, gradets, swaths or mosas aross a map. Cosder a ostat, whh s the degeerate ase (.e., a ostat has o varae) of perfet postve spatal autoorrelato: oe the value of a ostat s kow at a sgle loato, t s kow at all loatos. Next, osder a varable that portrays a orth-south (or east-west) lear tred aross a map. If ths varable has a mea of zero, the t s geometrally orthogoal to ad uorrelated wth the ostat. These orth-south ad east-west oreted lear tred varables also are orthogoal ad uorrelated. A varable wth mea zero whose values magtudes form a 3-dmesoal symmetr moud the eter of a map osttutes yet aother mutually orthogoal ad uorrelated map patter. These three varables dsplay maxmum levels of postve spatal autoorrelato whe geograph varae s preset, ad may be desrbed as global geograph patters. Alteratg sequees of moderately large mouds ad bass wth ether a east-west or a orth-south oretato portray moderate postve spatal autoorrelato, ad osttute regoal map patters. Alteratg sequees of small mouds ad bass wth ether a east-west or a orth-south oretato portray weak postve spatal autoorrelato, ad osttute loal map patters. Ths fragmetato otues through radomess (zero spatal autoorrelato) to arragemets of reasgly alteratg values (.e., sgle value mouds ad bass), whh portrays reasg egatve spatal autoorrelato. Most substatve varables have geograph dstrbutos that a be desrbed by lear ombatos of some subset of these mutually orthogoal ad uorrelated varygszed moud-bas map patters. As a dagost tool, spatal autoorrelato plays a rual role model-based feree, whose foudato s a set of vald assumptos rather tha a setf samplg desg. Deteted spatal autoorrelato a sgfy model msspefato, ludg treatg olear relatoshps wth a lear spefato, suh as a expoetal relatoshp betwee a predtor varable, X, ad a respose varable, Y, that s desrbed wth a straght tred le. It also a sgfy mssg varables for a regresso equato, ad as suh serves as a surrogate for varato otherwse uaouted for beause these varables are mssg. Ths surrogate role learly materalzes whe autoorrelato effets latet predtor varables alg wth autoorrelato effets Y. Moreover, the same set of dstt map patters s ommo to both a 4

5 set of mssg varables ad Y. Aoutg for spatal autoorrelato ths otext a do a surprsgly good job of represetg mssg varables a equato spefato. I otrast, spatal autoorrelato that remas uaouted for teds to dstort lassal orrelato oeffet terpretatos. The term orrelato alludes to the oto of redudat formato. If X ad Y are perfetly orrelated, the kowg X meas exatly kowg Y. I other words, the formato otet of Y s perfetly duplated X; ths degree of duplato dereases as the orrelato oeffet moves toward 0. Spatal autoorrelato exteds ths oto of redudat formato to georefereed data. The value at a gve loato a be predted wth some degree of auray from the values at earby loatos; ths spatal data feature osttutes the foudato of artograph terpolato. Reallg the example of housg pres, suh redudat formato results from spatal spllover: house values at oe loato spll over to mpat upo house values at earby loatos. Buldg a expesve house ear a expesve house teds to redue the value of the expesve house whle reasg the value of the expesve house, all other thgs beg equal. Ths mehasm s smlar to a rver that floods ts baks to ts flood pla. Aother example, at the mro-level of geograph resoluto, s seod-had smoke geerated by smokers. Ad yet aother example s the oxous smell geerated by suh faltes as sewage treatmet or rederg plats that permeates ther surroudg eghborhoods. Spatal autoorrelato a materalze from some ourse of ato operatg over a geograph ladsape, suh as otagous dffuso of a dsease, resultg t beg terpreted as a spatal proess mehasm. The dffuso of West Nle Vrus aross the otermous Uted States (US) llustrates ths terpretato. Ths dsease has qukly beome a serous problem the US, extremely rapdly dffusg from Log Islad 999 throughout the remader of the outry followg ortheast to west ad south paths. A weather frot movg aross a geograph ladsape a be vewed a smlar way beause t results hghly spatally autoorrelated loal weather odtos. Fally, terpretg spatal autoorrelato as a outome of areal ut demarato relates t to the modfable areal ut problem (MAUP), whereby results from statstal aalyses of georefereed data a be vared at wll smply by hagg the surfae parttog used to demarate areal uts. For example, a stadard 8-by-8 hekerboard patter exhbts egatve spatal autoorrelato betwee the red ad blak olors of ts squares. But f these squares are aggregated to ompat lusters of four (.e., -by- groupgs), ad the red ad blak olors averaged (resultg a ostat dark red olor aross the hekerboard), the the spatal autoorrelato beomes maxmally postve. Poltal redstrtg volvg gerrymaderg (.e., eletoral dstrt or osttuey boudares are mapulated order to aheve a prespefed geograph aggregato result, suh as some poltal advatage) exemplfes ths geeral stuato prate. Illustratos of spatal autoorrelato Most empral spatal autoorrelato ases volve moderate, postve relatoshp tedees betwee earby values o a map. Remotely sesed satellte mages are oe exepto, almost always dsplayg a very strog postve relatoshp. Most soo-eoom/demograph data dsplay a moderate postve relatoshp. Ad, egatve spatal autoorrelato rarely s eoutered prate. 5

6 Strog postve spatal autoorrelato. Remotely sesed mages are oe exepto to the georefereed data orm of moderate postve spatal autoorrelato, frequetly dsplayg very strog postve spatal autoorrelato. Ths feature partly results from lght refletae satterg, rather tha beg eatly otaed pxel boudares, hee spllg over to earby pxels measured by a satellte s sesors. A 8.5-meter resoluto Ladsat 7 Ehaed Themat Mapper Plus (ETM+) mage of the Florda Everglades for Jauary, 00, was orthoretfed ad overted to the UTM 7- projeto (Fgure.a). Ths mage omprses 8,58 orth-south olums, ad 7,649 east-west rows; the org s the orthwest orer of ths mage. Of the 65,636,069 total pxels, 4,6,007 have ozero spetral refletae values, ad 4,05,06 have all 0 refletae values; these latter pxels are the blak areas the mage. A,000-by-,000 pxel subset of ths mage was extrated for llustratve purposes (Fgure.a). The amout of gree vegetato ths rego a be quatfed wth a ormalzed dfferee vegetato dex (NDVI), whh s alulated wth the spetral refletae values for the ear-frared (B5) ad vsble red (B3) B5 B3 bads as follows:. Postve values beyod about 0.3 ted to represet gree vegetato, B5 + B3 whereas egatve values ted to represet swamp areas; postve values loser to zero ted to represet sol. The,000,000 seleted pxels yelded 3,033 dfferet NDVI values. Ths measuremet was subjeted to a Box-Cox type of power trasformato amely, r / B5 B3 sg +.09, where r s the desedg rak ad sg retas the orgal B5 + B3 -/+ of a NDVI value order to better alg t wth a bell-shaped urve (Fgure.b). Although the Kolmogorov-Smrov statst dereases from to 0.04, the trasformed measuremets stll oteably devate from a ormal dstrbuto the tals. The mage appearg Fgure.a dsplays very strog postve spatal autoorrelato. The outle of the Everglades s apparet, as s the southeast oast of Florda, south of Mam. The absee of autoorrelato would result ths mage appearg as a shufflg of the set of olors, smlar to the pture-dstortg mometary whte speks that appear o a televso sree whe, for example, atmospher stat s domat. Moderate postve spatal autoorrelato. Maps of populato desty ted to dsplay moderate postve spatal autoorrelato, part due to urbazato at a regoal or atoal sale, ad zog at a loal sale. Data from the 993 esus of Peru fursh populato outs by dstrts aross the Cuso Departmet; a ArGIS shapefle furshes area measures for these 08 dstrts. Populato desty teds to be skewed, wth a atural lower boud of 0, ad few areal uts wth relatvely szeable oetratos. Aordgly, aalyses based upo the ormal probablty model requre a applato of Box-Cox power trasformatos to better alg empral populato desty frequey dstrbutos wth a bell-shaped urve. Populato desty by dstrt the Cuso Departmet rages from 0.8 to 579. per ut area. It better mms a bell-shaped urve after beg subjeted to the followg trasformato (a power of -0.5; see Fgure ): 6

7 populato / + ; area ths trasformato results the Shapro-Wlk ormalty dagost statst probablty reasg from < to , mplyg a trasformed frequey dstrbuto that oforms well to the ormal dstrbuto. Ths trasformed populato desty dsplays moderate postve spatal autoorrelato, formg a elogated moud map patter wth a sgle peak. The hghest desty s the ty of Cuso, wth the ext-hghest destes strethg alog a eoom orrdor formed by the Vlaota Rver valley; the lowest destes are the most rural areas of ths Departmet. Moderate egatve spatal autoorrelato. Few empral examples of egatve spatal autoorrelato are reported the lterature. Usually ths pheomeo s dsussed oeptually terms of geograph ompetto. I other words, f a fte amout of lad s avalable, gas lad sze of oe terrtory a our oly through the loss of lad sze earby terrtores. The World Wars fought o the Europea otet llustrate ths stuato. Cotetal Europe prpally terms of the Europea Uo s parttoed to outres. Treatg the aptal of eah outry as ts seat of power, ad hee ts foal pot, ths porto of the otet also a be parttoed to Thesse polygos (Fgure 3.b). The rato of eah outry s atual to orrespodg Thesse polygo lad sze gves a dex of loal ompetto. The Shapro-Wlk ormalty dagost statst probablty for ths areas rato s (ts quatle plot appears Fgure 3.a), mplyg a frequey dstrbuto that oforms well to the ormal dstrbuto. Ths areas rato dsplays egatve spatal autoorrelato: Swtzerlad, Luxembourg, Slovea ad the Czeh Republ are slads of very low rato values that are ompletely surrouded by outres that have the hghest rato values (Fgure 3.). Estmators of spatal autoorrelato Oe the spatal autoorrelato oept was establshed ad wdely reogzed, spatal setsts beame terested quatfyg t, ad the testg hypotheses about t, wth a ultmate ad ofte today s goal of orporatg t to models. The two most ommoly used quattatve des are the Mora Coeffet (MC) ad the Geary Rato (GR). From r to MC. Explotg the terpretato of self-orrelato, spatal autoorrelato a be expressed terms of the formula for r, but wth eghborg values of varable Y replag those of X: = (x = (x x) x) / = y) / / beomes = = j j = j= = j= / / / j. The left-had expresso overts to the rght-had oe by replag ys wth xs the rght-had sde, by omputg the umerator term oly whe areal uts ad j are earby ( j s a dator varable whose value s ths ase, ad 0 otherwse), ad by averagg the umerator ross- 7

8 produt terms over the total umber of pars deoted as beg earby. The deomator of ths revsed expresso s the sample varae of Y, s. The MC s a ovarato (.e., parwse produts of devatos from the mea) dex, ad s defed by the preedg rght-had expresso. Its samplg dstrbuto a be ostruted oe of two ways: radomly samplg from a hypothetal probablty dstrbuto, or, takg the set of observed values as gve, ostrutg all possble permutatos of ther values aross a gve surfae parttog. I the former ase, preset dstrbutoal results assume that a ormal probablty model uderles varable Y. I ether ase, the expeted value of the samplg dstrbuto s -/(-). Y The stadard error formulae for the MC are rather omplated, but a be well approxmated by whe s at least 0; the sum j outs the umber of s the dator j = j= = j= varables, whh equals twe the umber of eghbors. Ths stadard error result s ot surprsg, reallg that the stadard error of a sample orrelato oeffet relates to, ad that all eghbors are double outed (both to j ad j to ). Furthermore, ulke the values ± for r, the extreme values of the MC are determed by sophstated mathematal quattes alled egevalues that are omputed whe the set of dator varables s orgazed to a table, muh lke a Mrosoft Exel spreadsheet, alled a matrx. These values rarely are ± ; more ofte, the lower boud s betwee - ad -0.5, ad the upper boud s betwee ad.. The MC values for the preedg empral examples are 0.78 for the Everglades NDVI, 0.5 for the Cuso populato desty, ad for the Europea areas rato. Ther respetve stadard devatos are: , , ad The GR, the other popular spatal autoorrelato dex, s based upo pared omparsos (.e., parwse squared dfferees), ad may be defed as follows: j y j) / = j= = j= = /( -) j. Ths expresso, whh volves a average squared dfferees term, also volves the ubased varae estmate for varable Y. A appears the deomator beause of double outg: y j) = y j + y = + j j. The expeted value of GR s. It rages betwee roughly 0 ad, wth 0- sgfyg postve spatal autoorrelato [.e., y j) goes to 0], ad wth - sgfyg egatve spatal autoorrelato [.e., y j) s reasg magtude]. Aga, the atual extreme values are a futo of egevalues afflated wth the matrx ostruted wth ther dator varables. I 8

9 addto, 0 a ever be reahed, beause t s assoated wth the degeerate ase of a ostat aross a geograph ladsape, ad hee yelds a dvso by 0. The GR values for the preedg empral examples are 0. for the Everglades NDVI, 0.4 for the Cuso populato desty, ad.67 for the Europea areas rato. Relatoshps betwee the MC ad GR. The MC ad GR values ted to gve osstet mplatos whe Y s a ormal radom varable. I ths ase, a useful rule of thumb s that the two oeffets should sum to approxmately. Dramat devatos from by ths sum suggest that Y s o-ormal. Meawhle, most but ot all empral ases yeld roughly the same MC ad GR values for raw ad Box-Cox power-trasformed data. Ad, the MC s the statstally most powerful of the two des: t does the better job, o average, of dfferetatg betwee ull ad alteratve hypotheses. The algebra relatoshp betwee the MC ad GR hghlghts ths feature: GR = = j= j = y j ) / / = j= j MC Ths equato reveals that the GR orporates loatoal formato addto to that luded the MC. Ths addtoal formato s the rato of squared devatos tmes ther umber of eghbors, dvded by the sample varae. If outlers are preset, the ths umerator a beome exessvely large; f a areal ut has a large umber of eghbors, the ths umerator a be markedly flueed by the orrespodg devato. If varable Y oforms to a bellshaped urve, the ths addtoal formato term s approxmately ; as beomes reasgly large, the rato overges o. Therefore, addg MC to both sdes of ths equato results (MC + GR). The sg for the MC term ths preedg equato s egatve, emphaszg the egatve relatoshp betwee the MC ad GR: MC values approahg orrespod to GR values approahg 0 (postve spatal autoorrelato), ad MC values approahg - orrespod to GR values approahg (egatve spatal autoorrelato). Zero spatal autoorrelato orrespods to a expeted value for MC of -/(-), whh asymptotally overges o 0, resultg the orrespodg GR value overgg o, ts expeted value for zero spatal autoorrelato. Ths egatve relatoshp a be see smply by ostrutg a satterplot of MC ad GR values for the three preedg empral examples. Graphal portrayals of spatal autoorrelato Spatal autoorrelato s a oept that leds tself to vsualzato, ludg satterplots ad mappgs. Varous graphal portrayals of ths oept a be ostruted, ludg a Mora satterplot, a semvarogram plot (based upo the GR ad ter-areal ut dstaes), ad a spatal orrelogram (based upo MC ad GR values for st, the d, the 3 rd, ad so forth, earest 9

10 eghbors). The frst of these s the smplest to ostrut, ad explots the preedg MC equatoal form: = = j j = j= = j= / / / j = = = j= = j= = j= j j j = j= j / = z ( z ) j = j / I other words, a Mora satterplot a be ostruted by: overtg georefereed data values to z-sores (.e., subtrat the mea ad the dvde by the stadard devato), summg the surroudg z-sores for eah areal ut (.e., jz j ), ad the plottg these pars of z-sores j= (horzotal axs) versus sums of surroudg z-sores (vertal axs). The slope of the resultg tred le s proportoal to the MC (.e., t eeds to be dvded by j ). Mora satterplots for eah of the three preedg empral examples appear Fgure 4. = j= Ispeto of a map ofte suggests the ature ad degree of the spatal autoorrelato t otas. Ths omposte map patter a be vewed terms of global, regoal, ad loal map patter ompoets. For example, the omposte map patter assoated wth the geograph dstrbuto of populato desty aross the Cuso Departmet appears Fgure.d, ad omprses 4 dvdual dstt map patters, of whh are global ad regoal (strog-tomarked postve spatal autoorrelato), ad three are loal (weak-to-moderate postve spatal autoorrelato). The most promet of these s a ortheast-southwest tred, followed by the moud-shaped map patter; the most fragmeted of these map patters (.e., a loal patter) ludes e vsually detetable lusters sattered aross the Departmet, may of whh are partly merged. Ths omposte map patter aouts for roughly 6% of the geograph varato (.e., redudat formato), ad for all but a trae amout of the postve spatal autoorrelato populato desty aross the Departmet. Smlar desrptos a be furshed for the Everglades NDVI ad the Europea areas rato maps. Theoretal statstal propertes of spatal autoorrelato Classal statsts furshes rtera to dstgush useful from useless spatal autoorrelato measures, ludg ubasedess, effey, suffey, ad osstey. The arthmet mea of samplg dstrbuto for ubased estmator equals ts orrespodg populato parameter. Ths property has bee used extesvely to evaluate the ase of zero spatal autoorrelato measures, ad the mpat of o-zero spatal autoorrelato o ovetoal sample statsts. I geeral, sample meas ad regresso oeffets ted to be ubased, whereas sample varaes ad orrelato oeffets ted to be based. Thus, the prpal mpat of spatal autoorrelato s o stadard error alulatos. Beause a orrelato 0

11 oeffet s the rato of a ovarae ad two stadard devatos, spatal autoorrelato mpats both of these alulatos ted to dsappear whe the rato s alulated. A effey estmator s both ubased ad has the smallest possble stadard error. Beause spatal autoorrelato mostly mpats upo varae alulatos, whh tur are used to ompute stadard errors, ths statstal property s the oe most affeted by o-zero spatal autoorrelato. I the presee of postve spatal autoorrelato, a varae teds to be flated. The et result s that the orrespodg samplg dstrbuto s flatter tha tradtoal statstal theory dates. Redudat formato trodued by spatal autoorrelato results, for example, sample szes beg msleadgly large. Cosequetly, more geograph samples are eeded to aqure a gve level of preso. A suffet estmator utlzes all formato otaed a sample that s relevat to a partular parameter. Covetoal statsts appled to georefereed data overlook loatoal formato otaed earby values. Reogto of ths formato s requred for suffey to be preserved. For example, a arthmet mea ot oly eeds y, but also the ross-produt term y = j= j y j =, whh s used to measure spatal autoorrelato. Fally, a osstet estmator's samplg dstrbuto oetrates at the orrespodg parameter value as reases. The effey rtero mples that ths wll our for ubased statsts, suh as the arthmet mea, but at a slower rate tha s suggested by ovetoal statsts. Asymptot oetrato wll our eve wth varae alulatos, although ot eessarly at the orret value. If geograph samplg tesfes a gve rego (fll samplg), the sample pots reasgly beome loser as reases, resultg spatal autoorrelato reasg, ad hee oetrato reasgly slowg dow. If geograph samplg volves expadg a rego whle matag the same average spag betwee sample pot (reasg doma samplg), the the rate of overgee wll be ostat rather tha dereasg, ad thus osstey reles oly o the fteess of all or part of the globe. Summary ad otemporary ssues Spatal autoorrelato has may faes, wth ts most ommo terpretatos beg expressed terms of self-orrelato, map patter, ad redudat formato. Its preferred measure s the Mora Coeffet, ad oe of ts two most popular graphal portrayals s the assoate Mora satterplot. Few empral examples of egatve autoorrelato have bee foud, wth most empral examples volvg moderate postve spatal autoorrelato; remotely sesed mages ted to dsplay strog postve spatal autoorrelato. Impats of spatal autoorrelato o ovetoal statsts a be assessed wth stadard mathematal statsts rtera, suh as ubasedess, effey, suffey, ad osstey. Today spatal setsts routely ompute measures of spatal autoorrelato, ad rather tha test hypotheses about ts presee, automatally lude t ther model spefatos. Dog so ofte osts oly degree of freedom. Spatal autoorrelato wth lear models s well-uderstood, ad has yelded spatal autoregressve tools used spatal statsts ad spatal eoometrs. Spatal autoorrelato wth geeralzed lear (mxed) models s ot well-

12 uderstood, wth oly a few umbersome tools avalable to hadle t. Spatal flterg, whh s ts fay ad explots the map patter terpretato, offers a approah that spas both lear ad olear statstal models wth tools that aout for the presee of spatal autoorrelato. Ths methodology deomposes a uderlyg omposte map patter to global, regoal, ad loal ompoets of spatal autoorrelato. The froters of spatal autoorrelato researh etal fuller developmet of otemporary tehques suh as spatal flterg, ad effet extesos of exstg tehques to massvely large datasets. For example, the Everglades remotely sesed mage otas 4,6,007 lad overage pxels, whereas the smple preedg aalyss was halleged by dealg wth oly,000,000 of these pxels. Furthermore, a eed stll exsts for developmet of qualty spatal autoorrelato measures, espeally robust oes, for o-ormal data. Further readg Asel, L Loal dators of spatal assoato LISA. Geographal Aalyss 7, Besag, J Spatal terato ad the statstal aalyss of latte systems. J. of the Royal Statstal Soety 36 (Seres B), 9-5. Grffth, D Spatal Autoorrelato: A Prmer. Washgto, DC: Assoato of Amera Geographers Resoure Publato. Grffth, D. 99. What s spatal autoorrelato? refletos o the past 5 years of spatal statsts. l'espae Géographque : Grffth, D Spatal autoorrelato ad egefutos of the geograph weghts matrx aompayg geo-refereed data. The Caada Geographer 40: Marda, K., ad R. Marshall Maxmum lkelhood estmato of models for resdual ovarae spatal regresso. Bometrka 7: Rhardso, S., ad D Hémo. 98. O the varae of the sample orrelato betwee two depedet latte proesses. J. of Appled Probablty 8: Tefelsdorf, M., ad B. Boots The exat dstrbuto of Mora's I. Evromet ad Plag A 7:

13 Fgure. A satellte mage of the Florda Everglades for Jauary, 00. Top left: (a) a omposte mage ostruted usg all spetral bads, wth the subrego of aalyss outled wth a whte retagle ad the o-mage areas whted out. Top rght: (b) a quatle plot of the,000,000 trasformed NDVI values for the subrego. Bottom left: () vsble red spetral bad [wavelegth mros (.e., aometers/000)] for the subrego, wth data values the rage 3 to 75. Bottom eter: (d) md-fra-red spetral bad (wavelegth mros) for the subrego, wth data values the rage to 73. Bottom rght: (e) trasformed NDVI, wth values the rage to.89. 3

14 Fgure. The 993 populato desty aross the Departmet of Cuso, Peru. Top left: (a) a hstogram ostruted wth the 08 raw populato destes. Top rght: (b) a hstogram ostruted wth the 08 power-trasformed populato destes. Bottom left: () a qutle map of the geograph dstrbuto of trasformed populato desty values. Bottom rght: (d) a qutle map of the omposte spatal autoorrelato map patter latet the geograph dstrbuto of trasformed populato desty values. NOTE: the gray sale for the maps s dretly proportoal to populato desty. 4

15 Fgure 3. The geograph dstrbuto of hypothetal lad sze ompetto aross otetal Europe. Top left: (a) a quatle plot of the rato betwee atual ad Thesse polygo areas, by outry. Top rght: (b) The Thesse polygo parttog of the otet (gray les), wth outry aptals as foal pots. Bottom left: a qutle map of the geograph dstrbuto of the areas rato values. () Bottom rght: (d) a qutle map of the omposte spatal autoorrelato map patter latet the geograph dstrbuto of the areas rato values. NOTE: the gray sale for the maps s dretly proportoal to the sze of the values mapped. Fgure 4. Mora satterplots for the three empral examples. Left: (a) Everglades NDVI subrego. Mddle: (b) Cuso populato desty. Rght: () Europea areas rato. 5