Comparison of Thematic Maps Using Symbolic Entropy

Transcription

1 Coparon of Theatc Map Ung Sybolc Entropy Manuel Ruz Marín (), Fernando A. López Hernández (), Antono Páez (2) () Techncal Unverty of Cartagena; (2) McMater Unverty Abtract Coparon of theatc ap an portant ta n a nuber of dcplne. Map coparon ha tradtonally been conducted ung cell-by-cell agreeent ndcator, uch a the Kappa eaure. More recently, other ethod have been propoed that tae nto account not only patally concdent cell n two ap, but alo ther urroundng or the patal tructure of ther dfference. The obectve of th paper to propoe a fraewor for ap coparon that conder ) the pattern of patal aocaton n two ap, n other word, the ap eleent n ther urroundng; 2) the equvalence of thoe pattern; and 3) the ndependence of pattern between ap. Two new tattc for the patal analy of qualtatve data are ntroduced. Thee tattc are baed on the ybolc entropy of the ap, and functon a eaure of ap copotonal equvalence and ndependence. A well, all nferental eleent to conduct hypothe tetng are developed. The fraewor llutrated ung real and ynthetc ap. Key word: Theatc ap, ap coparon, qualtatve varable, patal aocaton, ybolc entropy, hypothe tet

2 . Introducton It frequently the cae when coparng ap that a reearcher ntereted n deternng f two ap are gnfcantly dfferent, and f o, whether oberved dfference are rando. There are nuerou reaon why th type of nforaton can be valuable. A nuber of dtnct obectve have been dentfed n the lterature (Boot and Cllag 2006; Kuhnert et al 2005; Stehan 999). A reearcher ay be ntereted n accuracy aeent, that, n aeng the degree of ()atch between a reference ap and one or ore alternatve. For exaple, the ap could be of the ae regon but due to dfferent producer, and end uer need to undertand the nature of the dfference between ap (e.g., Wulder et al 2004). Alternatvely, ap can be copared a part of a odel valdaton exerce, when the coparon ade between a bae ap and the output of dfferent odelng technque or odel pecfcaton (e.g. dcrnant analy and neural networ; Foody 2004). In other tuaton, the coparon follow an nteret to detect change, n order to deterne whether the tuaton depcted n two ap of the ae regon rean unchanged (e.g., Ma 999). Lat but not leat, there a landcape ecology tradton that conder dfference wthn or between regon (e.g., Gutafon 998). A nuber of dfferent approache have been propoed n the lterature to copare ap, ncludng exanaton of the frequency of clae n each ap, the ue of concdence atrce, the appa eaure of agreeent, fuzzy appa, and other approache (e.g., Foody 2002; Hagen 2003; Reel 2009). Our obectve n th paper to propoe a novel fraewor for ap coparon that baed on the ue of ybolc entropy to detect patal pattern n theatc ap (Ruz et al 200). The fraewor follow a nuber of logcally content tep. Frt, the ap are eparately teted for patal aocaton of the qualtatve varable to deterne f a patal tructure preent or contrarwe, whether the value of the varable are patally rando. Secondly, f patal pattern are detected, whch would ndcate a yteatc proce n the generaton of the ap, then the copotonal equvalence of the ap aeed. Fnally, the ap are teted to deterne f they are ndependent, or rather a tranton rule could ext that ln the. The two queton that we poe wth regard to the coparon of ap are:

3 ) Are the ap gnfcantly dfferent? And f o: 2) Are the two ap ndependent? Our procedure ae ue of the Q() tattc of patal aocaton ntroduced by Ruz et al. (200) for the analy of apped qualtatve data. In addton, we ntroduce two new tattc needed for our coparon fraewor. The frt of thee new tattc degned wth the purpoe of coparng the overall correpondence (.e. the copotonal equvalence) of ap pattern. In broad ter, th lar to the quantty agreeent crteron, however conderng ere of patally ebedded ap eleent (.e. ap egent), a oppoed to par of ndvdual ap eleent. The econd tattc degned wth the purpoe of aeng whether two ap are ndependent. We llutrate the applcaton of the fraewor and all tet tattc ung real data collected fro the Canada Land Inventory, and ynthetc data. Varou cae that can be found n practce are dcued. In the concluon, we dentfy drecton for future reearch. 2. Bacground Two portant apect to conder when conductng ap coparon are the copotonal and confguratonal charactertc of the ap. The dtncton between thee two perhap bet captured by Pontu et al. (2004) a the quantty agreeent and the locaton agreeent between ap. A rough eaure of copotonal agreeent gven by the total nuber of eleent n a ap that belong to cla veru the nuber of eleent n a econd ap that belong to the ae cla. In th way, two ap could have dfferent copoton f the coverage by cla n one ap 70% of the urface whle n the other only 50%. Of coure, even f the proporton of cover by a pecfc cla reaned contant, t ay have hfted over pace, n whch cae the locaton agreeent between the two ap would be poor. Concdence atrce are ued to record the degree of agreeent between clae, and are the golden tandard for accuracy aeent (Reel 2009). Lewe, the appa tattc, wdely ued n reote enng applcaton, baed on an eleent-by-eleent coparon between the two ap. The coparon gve the nuber of patally concdent cell n the ap that belong to the ae or a dfferent cla. Th approach ntroduce a eaure of locaton agreeent wth repect to unchanged eleent. Alternatve have

4 been developed that prove the entvty of concdence atrce and appa to locaton and confguraton dfference (Pontu et al 2004; Reel 2009). Overall, there ha been over the pat few year an ncreaed recognton that the patal tructure of ap portant n any coparatve tuaton, and th reflected n oe recent and not-o-recent developent n the lterature. An early exaple where the patal tructure of ap wa dentfed a an portant eleent n ap coparon a paper by Congalton (988), where a dfference age wa generated baed on reotely ened dataet that clafed pxel n a par of ap a dentcal (=0) or dfferent (=). The on-count tattc (Clff and Ord 98; Dacey 968) wa then appled to the reultng age to dcover that the dfference were not patally rando, but rather followed a yteatc dtrbuton uggetve of underlyng topographcal factor or yteatc data proceng error. A lar approach wa ued by Wulder et al. (2004) to detect whether ap obtaned fro dfferent producer are dfferent n yteatc way. There, two ap were plfed by collapng all clae nto foret and non-foret, and dfference age were obtaned. Two cenaro condered the dfference between foret n the frt ap and non-foret n the econd, and vce-vera. The reult were helpful to dentfy patal cluter (wthn a gven level of tattcal gnfcance) where dfference were coon. The ethod ued by Congalton and Wulder et al. baed on the par-we coparon of ap eleent. Other approache have been propoed that copare ap egent ntead, by ebeddng a ap eleent a part of a neghborhood, notably the fuzzy appa ndex (Hagen 2003; Hagen-Zaner et al 2005; Hagen-Zaner 2006) and ovng wndow approache (Kuhnert et al 2005). The fuzzy appa copare patally concdent ap egent accordng to ther degree of larty. The nnovaton n the cae of th ethod the ue of fuzzy concept, o that ntead of a crp all-or-nothng decon rule for codng dfference, two ap egent can be perfectly, oewhat, or not at all dfferent. Recently Hagen-Zaner (2009) ha odfed the fuzzy appa tattc to account for the effect of patal autocorrelaton on expected agreeent. Kuhnert et al. (2005) decrbe a ovng wndow approach for calculatng larty ndcator. Accordng to th approach, a dfference age canned by ean of a ovng wndow that regter the degree of dlarty for a ap egent f dfference are coded a and agreeent a 0 and there perfect concdence wthn the wndow, the ndex would be zero. On the other hand, f the ap egent are

5 copletely dfferent at the locaton, the ndex would be one. The ndex aggregated to produce a uary eaure for the dfference age. Our approach hare oe feature wth the fuzzy appa and the ovng wndow approache, n partcular the way ap eleent are patally ebedded. In the cae of Hagen (2003) the ebeddng provde the ratonal for the ue of fuzzy concept. The ovng wndow approach n contrat ha uch n coon wth varou local tattc n patal analy (Fotherngha and Brundon 999) depte the fact that n the end t aggregated to produce a unque ndcator for a ap. In our cae, the ebeddng dctated by the atheatcal fraewor of ybolc entropy. The fuzzy appa eant to be ued a an nteractve and exploratory tool to nfor ubectve nterpretaton of ap (Hagen-Zaner et al 2005, p.784; Hagen-Zaner 2009, p. 7). Our fraewor, n contrat, taunchly nferental, and we derve all eleent needed for conductng hypothe tetng. Wth regard to tattcal gnfcance n the coparon of theatc ap, both Foody (2004) and Boot and Cllag (2006) propound the portance of conductng gnfcance tetng. On the other hand, Pontu et al. (2004) cauton agant the teptaton of conductng hypothe tet. Of the reaon advanced by Pontu et al., we agree that tattcally gnfcant dfference hould not be confued wth practcally portant dfference. We expect that th pont wll be farly evdent to ot potental uer of our approach. Fnally, our approach explctly ncorporate patal aocaton conderaton, and thu drectly addree an portant charactertc of ap that n need of attenton (Hagen-Zaner 2009). 3. A Fraewor for Map Coparon Coparon of ap n our fraewor done accordng to a equence of logcally content tep. The techncal detal of each tep wll be elaborated n ubequent ecton of the paper. To undertand the bac concept of the fraewor, conder the ap hown n Fgure. We wll llutrate the procedure n reference to thee ap.

6 Map [] Map [2] Map [3] Map [4] Map [5] Map [6] Map [7] Fgure. Map n exaple. Map [8] Map [9]

7 Each ap n Fgure copoed of three dfferent color (clae) n varou patal arrangeent. We tae Map [] a the bae ap. Th ap wa extracted fro the Canada Land Inventory, and depct land ue n a regon n the provnce of Ontaro. The pecfc land ue are not relevant for our dcuon here, and the ethod decrbed below can be appled to any theatc ap. In addton to Map [], we alo generate a nuber of ynthetc ap a follow. Frt, we generate three rando ap that lac an ordered patal pattern: Map [2] generated ung a rando equence of value drawn fro the tandard noral dtrbuton and ubequently dcretzed; Map [3] derved fro Map [2] ung a ple rule whereby each ap cla change to one other color pecfcally, all becoe 2, all 2 becoe 3, and all 3 turn nto the ap rando, but not ndependent fro Map [2]; Map [4] another copletely rando dtrbuton of value, unrelated to Map [2] and [3]. Map [6] derved fro the bae ap baed on rando tranton, o that a cell can, wth equal probablty, tay n t current tate, or change to one of the other two clae In addton, to the bae ap and a ere of rando ap, we alo nclude a et of patterned ap: Map [5] derved fro Map [] ung tructured tranton, wth cla havng a 30% chance of trantonng to 2, cla 2 havng a 0% of trantonng to 3, and 3 havng a 30% chance of trantonng to cla 2; Map [7] the ae a Map [], but rotated 80 degree; Map [8] the rror age of Map []; and fnally Map [9] ulated baed on a pure autocorrelaton odel to produce a ap pattern th ap ha a hgh degree of patal tructure, but the data generatng proce eparate fro any of the other patterned ap n the exaple. The relatonhp between the ap are hown n Fgure 2 below.

8 Fgure 2. Relatonhp between ap n exaple. Broadly peang, a ap can dplay a patal pattern of aocaton, or n other word, a patally coherent dtrbuton of value of the qualtatve varable. The alternatve that the dtrbuton of value patally rando. Accordngly, we can dentfy the followng cae for par-we coparon of ap: Cae. If one of the ap patally aocated (.e. there patal tructure) and the other rando, t can be concluded that the ap are dfferent and ndependent, nce no organzed tranton rule can be ued to convert one ap nto the other (.e. Map [], [5], [7], [8], [9] : Map [2], [3], [4], [6]). The preence of patal aocaton n two ap not by telf ndcatve of ap agreeent. When two ap are non-rando, t becoe neceary to ae the overall larty of the ap, that, ther copotonal equvalence. If the overall copoton of the ap gnfcantly dfferent, the ap are of necety dfferent, and the queton whether the ap are ndependent, or a tranton rule could pobly ext between the. If the overall copoton of the ap not gnfcantly dfferent, th would ndcate copotonal agreeent, but not necearly confguratonal agreeent, and lewe t would be neceary to nvetgate whether the two ap are ndependent. Th lead to the followng cae: Cae 2. If ap and ap 2 are each patally aocated, two ue need to be reolved. The frt queton whether there copotonal agreeent between the two ap.

9 Cae 2.. The copoton of the ap agree, and therefore the ap could ether be dentcal n all repect or dentcal n copoton but not n confguraton (Map [] : Map [7] : Map [8]). Deterne whether the ap are ndependent or a tranton rule could ext. Cae 2.2. The copoton of the ap doe not agree, and therefore t can be concluded that the ap are dfferent (Map [] : Map [5] : Map [9]). Deterne whether the ap are ndependent or a tranton rule could ext. A thrd cae of conderably le practcal nteret, but we enton t for copletene. Th when the two ap are patally rando (e.g., Map [2] : Map [3] : Map [4] : Map [6]). The eleent-we dfference could be rando or not. It traghtforward to verfy whether the tranton are rando by loong at the frequency dtrbuton of clae. 4. Tet tattc In th ecton we ntroduce the techncal tool needed to pleent our ap coparon fraewor. We begn by defnng oe portant ter (a nuber of thee concept are decrbed n detal n Ruz et al 200). We wll conder a ap to be the repreentaton of a dcrete patal proce { X }, where S a et of coordnate. The patal proce can tae a nuber of dcrete value or clae A= { a, a 2,, a } that could be for exaple dfferent land ue clafcaton, type of foret cover, etc. The proce can be condered to be patally ebedded, f we defne a trng that contan the clae found n a neghborhood of ze, called an -urroundng. The - urroundng lar to the neghborhood vector N ued n the fuzzy appa eaure of agreeent (Hagen 2003; Hagen-Zaner et al 2005): X ( ) = ( X ) 0, X 0,, X () S ap ( 0 The trng defned for a pecfc coordnate correpondng to a locaton n the S) where, 2,, are the nearet neghbor to 0. In other word, X ( ) 0 a lt of value that repreent a ap egent of ze centered on locaton 0. The value n the trng are the clae found n a neghborhood of ze. We wll denote the et of the -nearet neghbor by N = {, 2,, }. A an exaple,

10 conder the ple cae of a lneal tranect wth =2 (a =Whte=0, and a 2 =Blue=; ee Fgure 3). Fgure 3. Lneal tranect wth =2. If the ze of the -urroundng elected a 3, then X(3) for the econd cell fro the left {Blue,Whte, Whte} or equvalently {, 0, 0}. X(3) for the ffth cell fro the left {Whte, Blue, Whte} or {0,, 0}. It eay to ee that nce the nuber of dfferent clae n the ap, the nuber of all poble value (the cobnaton of clae) that an -urroundng M ( ) can tae. In the exaple above, wth =2 and =3, there are eght poble cobnaton of clae (ee Table ). Each unque cobnaton of clae called a ybol, whch we denote by (=, 2,..., ) Table : Lt of ybol for =2, =3. = {0,0,0} 5 = {,,0} 2 = {,0,0} 6 = {,0,} 3 = {0,,0} 7 = {0,,} 4 = {0,0,} 8 = {,,} Let: Γ= {,,, } (2) 2 be the et of all of thee poble value. For a gven ap { X } we ay that a locaton of -type f and only f X ( ) =. In the exaple, the econd cell fro the left ad to be of 2 -type, and the ffth cell fro the left of 3 -type. Each cell can be agned a unque ybol, out of the fnte lt of poble ybol. The cell on the edge would ue the two nearet neghbor, and would therefore be of type 3 (on the left) and of type 6 (on the rght). Prece rule for electng the neghbor are decrbed n ore detal below. Let Γ and denote by: n = { S X ( ) = }, (3) S

11 the ze of the ubet of S fored by all the eleent of -type. Gven th, t eay to copute the eprcal relatve frequency of a ybol Γ a: { S of type} p( ):= p = S (4) where by S we denote the cardnalty of the et S (the nuber of ybolzed locaton). Now, under th ettng, we can defne the ybolc entropy of a theatc ap { } X S for an ebeddng denon 2. Th entropy defned a the Shanon entropy of the dtnct ybol a follow: h ( ) = p ln( p ) (5) X = Sybolc entropy, h ( ), the nforaton contaned n coparng the - urroundng n the ap. Note that 0 h ( ) ln( ), where the lower bound attaned when only one ybol occur, and the upper bound for a copletely rando yte (..d. patal equence) where all poble ybol appear wth the ae probablty. In th ecton we ntroduce the tet that allow the reearcher to ae the degree of patal aocaton (tructure) of a theatc ap, and copare two ap n ter of () ther copotonal equvalence and (2) ther ndependence. In order to do o, we wll conder two ap { X } and { Y } wth the ae zonng yte, and denote the S S ybolc frequence n each ap, and ther cobnaton, by (the tattc can be expanded to conder ore than two ap ultaneouly): n n = { S X ( ) } = ; p = S = { S Y ( ) } = ; q =, (6) S c c = n g + ; = 2 S 4.. Spatal tructure: The Q() tattc Tetng for patal tructure (.e. patal aocaton) of a qualtatve varable baed on ybolc entropy dcued n Ruz et al. (200). In th ecton, we lt ourelve to

12 revewng oe ey concept of the relevant tattc. The reader drected to the reference for further nforaton. The Q() tattc defned a follow: n Q S q h = S = where α the nuber of te that cla a appear n ybol ( ) = 2 α ln( ) + ( ) (7) and q = P( X = a ). The tattc eentally a lelhood rato tet that copare the ybolc entropy of the ap to the expected ybolc entropy under the null hypothe of patal ndependence (.e. where all ybol appear wth lar probablty contngent on the frequency of clae). The tattc can be hown to be χ 2 dtrbuted wth degree of freedo (Ruz et al 200), and the decon rule at a 00( α)% confdence level a follow: 2 Reect null hypothe f Q ( ) > χα Otherwe do not reect the null hypothe (8) In order to pleent the tet, the reearcher ut ae a nuber of decon. Frt, the reearcher ut decde the ze of for the -urroundng; there a certan degree of flexblty n dong o, however lted by oe tattcal and nterpretve conderaton. In order to acheve a uffcent approxaton of the 2 χ dtrbuton, t trongly advable to wor wth at leat 5 ybolzed obervaton (Rohatg 976, chapter 0); n other word, there ut be at leat fve te a any ybolzed obervaton a ybol ext. Thu, the ze of ay be contraned by the ze of the ap, wth larger urroundng only practcable n large ap. In ter of nterpretaton, a large nuber of ybol capturng a hgh level of pecfcty ay alo be dffcult to nterpret. On the other hand, f patal aocaton detected for a urroundng of ze, the tattc guarantee that the aocaton ext for the full urroundng or even only part of t. The preence of patal aocaton at hgher cale rean ndeternate. Once a decon ha been ade regardng the ze of the -urroundng, a rule needed to dentfy - nearet neghbor for each locaton that wll be ybolzed. The rule pleented n Ruz et al. (200) a follow: (a) The dtance of the - neghbour fro 0 atfe the condton that ρ ρ2 ρ ; and

13 (b) In the cae of a te n ter of the dtance fro 0, (.e. f ρ 0 0 = ρ + ) then 0 0 precedence goe to the aller angle (.e. θ < θ + ). Planly, neghbor are elected n order of proxty, and f there are te accordng to th crteron, electon proceed by drecton. Th enure the unquene of X ( ) for all S. Other rule are certanly poble, for exaple by changng the crtera to elect neghbor by drecton f anotropc procee are upected; however, we do not purue th dea further n the preent paper Copotonal equvalence: The Q E tet After tetng the ap for patal tructure, f the two ap are found to be non-rando, we are ntereted n tetng whether the copoton of the ap equvalent. In other word, we now wh to acertan whether ybol appear wth lar frequence n both ap. Hence, the null hypothe that we want to tet whether the ybolc dtrbuton for both ap are equvalent, or: H 0 : { X } S and { Y } Sfollow the ae dtrbuton. The null hypothe can be retated n ter of ybol a follow: H0 : pη = qη for all =, 2,, (9) that, the frequency of ybol dentcal n the two ap, for all ybol. In order to develop a tet tattc, we defne, for a ybol varable: Γ, the followng rando f X( ) = f Y( ) = Z ( ) ( ) X = Z Y = 0 otherwe, 0 otherwe, (0) Then Zη ( X ) (repectvely Zη ( Y ) ) a Bernoull varable wth probablty of ucce p η (rep. q η ), where ucce ean that of -type. Now aue that et S fnte and of order R. Then we are ntereted n nowng how any are of -type for all ybol S n each of the ap. In order to anwer the queton, we contruct the followng varable () T ( X) = Z T ( Y) = Z ( X) ( Y) S S

14 The varable T can tae the value {0,, 2, R, }. Notce that not all the varable are ndependent (due to the overlappng of oe -urroundng), and therefore T not exactly a Bnoal rando varable. Neverthele, the u of dependent ndcator can be approxated to a Bnoal rando varable whenever the dependence aong the ndcator are wea (Soon 996). A procedure to obtan a good approxaton to a Bnoal dtrbuton decrbed n Ruz et al. (Ruz et al 200). Brefly, we can conder a a et of locaton the ubet S fored by thoe coordnate n S uch that for any two coordnate, S the et of nearet neghbor of and have a all (or even epty) nterecton, that : N N = r (2) for a all enough potve nteger r. We wll call the nteger r a the overlappng degree. Now we propoe a contructon of the et S. Frt choe a locaton 0 S at rando Z and fx an nteger r wth 0 r <. Let N = {, 2,, } be the et of nearet 0 neghbor to 0, where the defne 0 are ordered by dtance to 0. Let u call 0 = r and 0 0 A0 = { 0,,, r 2}. Tae the et of nearet neghbor to, naely N = {, 2,, }, n the et of locaton S A0 and defne = 2 r. Now for > we defne = where r, n the et of nearet neghbor to r N = {,,, }, n the et 2 S { A }. Contnue th proce whle there are = 0 locaton to be ybolzed. Therefore we have contructed a et of locaton: S = { 0,,, M} (3) uch that the varable: (4) T ( X) = Z = B( S, p ) T ( Y) = Z = B( S, q ) ( X) ( Y) S S baed on S can be approxated to a bnoal dtrbuton for a utable choce of r ( r all enough) for all =, 2,,. Moreover, when r = 0 (.e. no overlap allowed) we exactly have Bnoal rando varable. Note that the axu nuber of R locaton that can be ybolzed wth an overlappng degree r M = r +, where

15 [ x ] denote the nteger part of a real nuber x, and therefore reducng the degree of overlap alo ple that the nuber of ybolzed locaton wll be aller than the nuber of obervaton n the aple. Now, the dtrbuton of the 2 rando varable: T = ( T ( X ), T, ( X ), T ( Y ), T, ( Y )) (5) a ultnoal dtrbuton and t lelhood functon L ( p,, p, q,, q ) gven by: 2M! n n p p q q n! n!!! (6) In order to obtan the axu lelhood etator pˆ and qˆ of repectvely and tang nto account that: p and q n p = = q (7) = = n we olve the followng yte of equaton: ln( L ( p )) ln( ( )),, p, q,, q L p p q q,,,,, = 0 = 0 p q (8) for all =,, to obtan: n p = q M = M. (9) Denote by (0) p and (0) q the probablte under the null for all ybol Γ Then the lelhood rato tattc (ee for exaple Lehann 986): λ ( T ) = (0) n (0) n (0) (0) p p q q n n p p q q (20) Now, under the null, we have that Therefore: p = q and hence p = g for all =, 2,,. 2

16 λ( T ) = 2M n + ( ) g 2 g n n n + p p q q (2) We defne the tructural equvalence tet a Q ( E ) 2ln ( λ ( T) ) =, whch nown to ayptotcally follow a χ 2 dtrbuton wth degree of freedo (Lehann 986). Hence, the etator Q ˆ E ( ) of QE ( ) can be hown to be: ˆ c n ( ) 2 2 ln ln ln ln QE = M + c n 2 = 2M = M = M c c n n = 4M ln + ln ln( ) ln( ) 2 = 2M 2M = 2M M = 2M M = 4M ln(2) + h( ) h ( ) ( ) X h 2 2 Y (22) where h ( ) = g ln( g ) the total ybolc entropy. We have proved the followng reult. = Theore. Let { X } and { Y } be two ap wth S = M. Let S S Γ= {,,, } be the et of ybol a defned n (2) If 2 are tructural equvalent then: ˆ ( ) 4 ln(2) ( ) ( ) QE = M + h h X hy ( ) 2 2 { X } and { } S Y S (23) 2 ayptotcally χ dtrbuted. The QE ( ) -tet can be generalzed to the cae of N ap, { X },{ X },,{ X } a t can be een n the next corollary. The proof S 2 S N S traghtforward followng the tep of the proof of Theore. Corollary 2. Let { X } = 2,, N be N ap wth S = M. Let S Γ= {,,, } be the et of 2 equvalent for all =, 2,, N then ybol a defned n (2) If { X } are tructural S N ˆ Q ( ) 2 ln( ) ( ) E = NM N + h h X ( ) N (24) =

17 2 ayptotcally χ dtrbuted. ( N )( ) 4.3. Map ndependence: The Q I tet Two ap nown to have patal aocaton a deterned by Q() are organzed and therefore n prncple nteretng. The ap can be copotonally equvalent a deterned by the Q E () tattc. Th doe not yet ndcate whether the ap are related n a non-rando fahon. We are therefore ntereted n aeng whether the ap, naely { X } and { Y }, are dependent or ndependent ap. In S th cae, ndependence ean that a non-rando tranforaton rule doe not ext that could be ued to convert one ap nto the other. Note that two ndependent ap can be tructurally equvalent and vcevera. Then next tep to tet for ndependence. Frt we are gong to ntroduce oe defnton and notaton that wll be needed n order to contruct the tattc. Conder the ap overlay { W = ( X, Y)} S. Let Ω =Γ Γ be the drect product of the et of ybol Γ= {, 2,, }. We wll call the eleent n Ω ybol for the 2- denonal ap. Next we defne the -urroundng aocated to the 2-denonal x y ap { W = ( X, Y)} S a W( ) = ( X( ), Y( )). Now gven a ybol (, ) Ω x y we wll ay that a locaton of (, )-type for W f and only f the locaton of S x -type for the ap X and of y -type for the ap Y. Let d = { S W ( ) = (, )} and denote by p d = the probablty of the ybol S x y (, ) to occur. Then we can defne the ybolc entropy of the ap overlay a W =. = = h ( ) p ln( p ) Defne the ndcator functon I x y f of (, )-type for W = 0 otherwe. (25) Hence we have that I = B( p ) a Bernoull rando varable wth probablty of ucce p. A n the prevou ecton we can defne a ubet of

18 locaton S uch that I can be approxated to a bnoal dtrbuton for a S utable choce of r (r all enough) for all, =, 2,,. Therefore F = I = B( M, p ), a bnoal rando varable where M denote the S cardnalty of the et S. Then: F = ( F2, F 3,, F ) (26) a ultnoal dtrbuton. Now we are ntereted n tetng for the followng null: H { } { Y} : The ap X and are ndependent. 0 S S (27) Th null hypothe can be retated n ter of ybol a: H : p = p q (28) 0 Followng the ae tep a n the prevou ecton we get that under the null the lelhood rato tattc rean a: λ( F) = = = d p q p d d (29) We then defne Q λ ( F) I ( ) = 2ln( ), whch ayptotcally follow a χ 2 dtrbuton wth ( 2) + degree of freedo ee (ee Lehann 986). Hence, and tang nto account that d = d = n and = d = d = and that = p q n = we obtan that the etator ˆ ( ) I M M Q of Q ( ) : I ˆ n d ( ) 2 ln ln ln QI = d + d d M M = = = = M n d d d d = 2M ln ln ln = M M = M M = = M M = 2 M hˆ ( ) ˆ ( ) ˆ X + hx hw( ). (30) Therefore we have proved the followng reult.

19 Theore 3. Let { X } S and { Y } S be two ap and let { W = ( X, X2)} S be a 2-denonal ap wth S = M. Let Γ = {, 2,, } be the et of ybol a defned n (2). If { X } and { Y } are ndependent then S [ ] Q ( ) = 2 M h ( ) + h ( ) h ( ) (3) I X Y W S ayptotcally χ + dtrbuted. 2 ( 2) The QI ( ) -tet can be generalzed to the cae of N ap, { X },{ X },,{ X } a t can be een n the next corollary. The proof S 2 S N S traghtforward followng the tep of the proof of Theore 3. Corollary 4. Let { X } = 2,, N be N ap and S let{ W = ( X, X2,, XN)} S be a N-denonal ap wth S = M. Let Γ= {,,, } be the et of ybol a defned n (2) If 2 ndependent for all =, 2,, N then { X } are S N QI( ) = 2 M hx ( ) h ( ) W (32) = ayptotcally χ dtrbuted. 2 N N( ) 5. Illutraton The fraewor propoed n the precedng ecton llutrated wth reference to the ap n Fgure. The frt tep to tet the ap for patal aocaton. We calculate the Q() tattc for all ap ung an -urroundng of 4 and ybolze the obervaton ung an overlap degree r=. The reult of applyng the tattc to the ap appear n Table 2. The table how the value of the tattc for each ap, the probablty value, and the decon (at a 95% level of confdence). In addton, we alo nclude a fgure howng the frequency dtrbuton of equvalent ybol n each ap. The equvalent ybol are a reduced for of the ybol that dplay only the nuber of cae of each cla n an -urroundng, ntead of all proxty nforaton; th reducton of nforaton facltate nterpretaton. The fgure are llutratve only, and the tet are baed on the tandard ybolzaton. The vertcal ax n the fgure the proporton of the total, and n the horzontal ax the equvalent ybol appear. For

20 exaple {0 0 4} the ybol ndcatng an -urroundng of 4 where all ap eleent are of cla a 3 ; {2 } the ybol ndcatng an -urroundng of 4 where two eleent are of cla a, one of cla a 2, and one of cla a 3. Table 2: Q() tet for patal aocaton (non-rando ap n bold, p-value n parenthee). Tet conducted for =4 and r=. Map # Map Q (p-value) Decon (ndependence) Frequency dtrbuton of ybol (0.000) Reect Map (0.7668) Do not reect Map (0.7668) Do not reect Map (0.9480) Do not reect Table 2 (contnued): Q() tet for patal aocaton (non-rando ap n bold, p- value n parenthee). Tet conducted for =4 and r=.

21 Map # Map 5 Q (p-value) Decon (ndependence) Frequency dtrbuton of ybol (0.000) Reect Map (0.669) Do not reect Map (0.000) Reect Map (0.000) Reect Map (0.000) Reect A een n the table, the null hypothe of patal ndependence reected for Map [], thu confrng the vual preon of a hghly tructured patal dtrbuton of varable value. A expected, the null hypothe reected alo for Map [5], [7], and [8], nce thee ap (ee Fgure 2) were derved fro Map [] accordng to oe

22 organzed prncple (.e., non-rando tranton, rotaton, and reflecton, repectvely). A well, the tet alo reect the hypothe of ndependence for Map [9], whch wa ulated ung a trongly autocorrelated patal proce. The null hypothe not reected for Map [2], [3], and [4]. Lewe, the hypothe not reected for Map [6], whch wa orgnally baed on Map [], but ubected to a proce of rando tranton. At th pont, our nteret n the coparon of ap would practcally be lted to par of ap that dplay a gnfcant degree of patal aocaton. Th would ean a par-we coparon of Map [] : [5] : [7] : [8] : [9]. For llutratve purpoe, we calculate the tattc for all par of ap, a th erve to deontrate the appropratene of the tattc n a wde range of tuaton. The next tep n our coparon fraewor to deterne f two ap are copotonally equvalent, or n other word, to evaluate f the ybol appear wth lar or gnfcantly dfferent frequency. The reult of applyng the Q E tet to each par of ap appear n Table 3 (calculaton of the tattc baed on the ae paraeter ued before, naely =4 and r=). Frt, note that a ap dentcal to telf, and therefore the tattc alway fal to reect the null hypothe of copotonal equvalence when the two ap n the nput are the ae. When two ap are rando, the expectaton that they wll be equvalent n ther copoton, nce by defnton n a rando ap all ybol appear wth lar frequence (contngent on the frequency of clae; ee frequency plot n Table 3 for Map [2], [3], [4], and [6]). The tet confr th, by falng to reect the null hypothe for the coparon of Map [2] : [3] : [4] : [6]. Next, the expectaton that any patally aocated ap wll have a dfferent copoton when copared to any rando ap, whch would lead to the concluon that the two ap are dfferent (and the coparon relatvely unnteretng, nce one of the two ap rando). Th ntuton agan confred by the reult n the table, where t can be een that the null hypothe of copotonal equvalence reected for Map [], [5], [7], [8], and [9] : [2], [3], [4], and [6]. In other word, any two ap n thee par are gnfcantly dfferent fro each other. The ore nteretng cae are when the two ap for the coparon dplay gnfcant patal aocaton, or n other word, patternng. The reult are agan n full agreeent wth the etup of the exaple. Frt, we fal to reect the null hypothe when coparng Map [] : [7] : [8]. Th logcal, nce the nput ap n each cae

23 are eentally the ae ap, only wth dfferent orentaton. In contrat, the null hypothe reected when coparng Map [], [7], [8] : [5] : [9]. In the cae of Map [], [7], and [8] : [5], th to be expected nce Map [], [7], and [8] have already been deterned to be equvalent, and Map [5] wa obtaned fro Map [] ung a tranforaton rule that changed t copoton. A a reult, Map [5] gnfcantly dfferent fro the other. Fnally, Map [9] wa generated ung a eparate patally tructured proce that wa not contraned to generate the ae copoton a any of the precedng ap. Table 3: Q E tet for copotonal equvalence (non-rando ap n bold, and p-value n parenthee). Decon rule to reect the hypothe of equvalence. Shaded cell ndcate reect. Map Map 2 Map 3 Map 4 Map 5 Map 6 Map 7 Map 8 Map 9 Map Map 2 Map 3 Map 4 Map 5 Map 6 Map 7 Map 8 Map (.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.94) (0.989) (0.000) (.000) (0.933) (0.670) (0.000) (0.858) (0.000) (0.000) (0.000) (.000) (0.75) (0.000) (0.580) (0.000) (0.000) (0.000) (.000) (0.000) (0.660) (0.000) (0.000) (0.000) (.000) (0.000) (0.000) (0.000) (0.000) (.000) (0.000) (0.000) (0.000) (.000) (.000) (0.000) (.000) (0.000) 0.0 (.000) The fnal tep n our coparon fraewor to tet the hypothe of ndependence between two ap. The obectve to deterne whether two ap could be lned by a rule other than rando tranton. A een n Table 4, the Q I tattc (calculated wth =) correctly reect the null hypothe when the two ap copared are the ae: a ap not ndependent fro telf. We begn our dcuon of thee reult wth Map []. In th cae, we oberve that the null hypothe not reected when the ap ued n the coparon rando (.e. Map [2], [3], [4], and [6]); a expected, an organzed rule doe not ext that could tranfor one ap nto the other, and the ap are ndependent. The null hypothe

24 correctly reected when coparng Map []: [5], [7], [8]. Snce an underlyng rule ext that can be ued to tranfor the ap (e.g., fro [] to [5] and vce vera f the rule nverted), the ap are not ndependent. The tet cannot dentfy the underlyng rule, but can pont to t extence. Wth repect to Map [2], the tattc correctly reect the hypothe of ndependence when the coparon ade wth repect to Map [3]. A een n Fgure 2, the two ap are rando; however, Map [3] wa derved fro Map [2] ung a et of yteatc tranton rule. Th reult ndcate that the tattc able to detect pattern of dependency between ap, even f the two ap are rando. When coparon ade to any of the other ap n the exaple, on the other hand, the tet fal to reect the null hypothe. Th precely a expected, nce the ap rando and therefore ndependent fro the other cae. The tuaton the ae for rando ap [3], [4], and [6] when copared to any other ap (wth the excepton of the coparon between Map [2] : [3] a per the precedng dcuon). Table 4: Q I tet for ndependence (non-rando ap n bold, and p-value n parenthee). Decon rule to reect hypothe of ndependence. Shaded cell ndcate reect. Map Map 2 Map 3 Map 4 Map 5 Map 6 Map 7 Map 8 Map 9 Map Map 2 Map 3 Map 4 Map 5 Map 6 Map 7 Map 8 Map (0.000) (0.880) (0.880) (0.83) (0.000) (0.893) (0.000) (0.000) (0.000) (0.000) (0.000) (0.642) (0.559) (0.86) (0.52) (0.560) (0.629) (0.000) (0.642) (0.559) (0.86) (0.52) (0.560) (0.629) (0.000) (0.732) (0.389) (0.637) (0.645) (0.687) (0.000) (0.34) (0.000) (0.000) (0.000) (0.000) (0.744) (0.534) (0.92) (0.000) (0.000) (0.000) (0.000) (0.000) 427. (0.000) Applcaton of the tet to Map [5] : [7] : [8] lead to reecton of the null hypothe, for the evdent reaon that underlyng tranton rule ext aong thee ap through ther connecton wth Map [].

25 An ntrgung cae Map [9]. The tattc reect the hypothe of ndependence when the coparon ade wth repect to Map [], [5], [7], and [8]. A prevouly explaned, th ap wa generated ung a eparate patal autocorrelaton proce, and Map [9] therefore not drectly derved fro any of thee ap. Conderng the hgh degree of patal aocaton n the ap, on the other hand, t perfectly plauble that oe et of non-rando tranforaton could ext to ln Map [], [5], [7], [8] : [9]. Thee reult offer a cautonary counter-exaple n the applcaton of the tattc: qute ply, lac of ndependence doe not necearly ply caualty. Th pont hould be clear to anyone falar wth eleentary tattc and the bac tenet that correlaton not cauaton. Whle the tattc appear to do a reaonable ob of dentfyng lac of ndependence due to caual procee, the teptaton hould be avoded to nfer caualty baed on th tet. 6. Concluon In th paper we have ntroduced a new approach for the coparon of theatc ap. Our approach equentally tre to dentfy dfference between ap by aeng the degree of patal aocaton of each ap and the copotonal equvalence between ap baed on ap egent (a oppoed to ap cell/eleent), and by tetng for ndependence between the two ap. The fraewor cover a wde range of cae of practcal nteret, and provde a coplete nferental fraewor wth clearly defned null hypothee at each tep of the equence. Whle the ue of gnfcance tetng n ap coparon appear to be oewhat controveral, we would argue that tetng for well defned hypothee, uch a ntroduced n th paper, can enhance exploratory analy by clarfyng the relatonhp between ap. In order to pleent our ap coparon fraewor, we have ntroduced two new tattc for the patal analy of qualtatve data. The tattc are eental to our approach, but wll lely be of nteret n a broader varety of ettng: for ntance, Q E can be appled to ap n dfferent regon and/or wth dfferent zonng yte to ae larte n copoton. One ltaton of our approach that ap can be teted for dfference at a gven level of confdence. However, unle the ethod of Wulder et al. (2004) that can detect cluter of dfference between ap, when dfference are detected n our fraewor, the tattc do not drectly ndcate where they happen. Therefore, our approach tll lacng tool for the dentfcaton of cluter of concdence or dfference. Th a atter for future reearch.

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