, are assumed to fluctuate around zero, with E( i) 0. Now imagine that this overall random effect, , is composed of many independent factors,

Transcription

1 Part II. Contnuous Spatal Data Analyss 3. Spatally-Dependent Rando Effects Observe that all regressons n the llustratons above [startng wth expresson (..3) n the Sudan ranfall exaple] have reled on an plct odel of unobserved rando effects (.e., regresson resduals) as a collecton ( :,.., n) of ndependently and dentcally dstrbuted noral rando varables [where for our purposes, ndvdual saple ponts are taken to represent dfferent spatal locatons, s ]. But recall fro the ntroductory dscusson n Secton. above that for ore realstc spatal statstcal odels we ust allow for possble spatal dependences aong these resduals. Hence the an objectve of the present secton s to extend ths odel to one that s suffcently broad to cover the types of spatal dependences we shall need. To do so, we begn n Secton 3. by exanng rando effects at a sngle locaton, and show that noralty can be otvated by the classcal Central Lt Theore. In Secton 3., these results wll be extended to rando effects at ultple locatons by applyng the Multvarate Central Lt Theore to otvate ultvarate noralty of such jont rando effects. Ths ult-noral odel wll for the statstcal underpnnng for all subsequent analyses. Fnally n Secton 3.3 we ntroduce the noton of spatal statonarty to odel covarances aong these spatal rando effects ( :,.., n). 3. Rando Effects at a Sngle Locaton Frst recall that the unobserved rando effects,, at each locaton (or saple pont), s, are assued to fluctuate around zero, wth E( ) 0. Now agne that ths overall rando effect,, s coposed of any ndependent factors, (3..) e e e e k k, where n typcal realzatons soe of these factors, e k, wll be postve and others negatve. Suppose oreover that each ndvdual factor contrbutes only a very sall part of total. Then no atter how these ndvdual rando factors are dstrbuted, ther cuulatve effect,, ust eventually have a bell shaped dstrbuton centered around zero. Ths can be llustrated by a sple exaple n whch each rando coponent, e k, assues the values / and / wth equal probablty, so that Ee ( k ) 0 for all k,..,. Then each s dstrbuted as shown for the case n Fgure 3.(a) below. Now even though ths dstrbuton s clearly flat, f we consder the case (3..) e e then t s seen n Fgure 3.(b) that the dstrbuton s already startng to be bell shaped around zero. In partcular the value 0 s uch ore lkely than ether of the extrees, - and. The reason of course s that ths value can be acheved n two ways, naely ( e, e ) and ( e, e ), whereas the extree values can each occur n ESE 50 II.3- Tony E. Sth

2 Part II. Contnuous Spatal Data Analyss only one way. Ths sple observaton reveals a fundaental fact about sus of ndependent rando varables: nteredate values of sus can occur n ore ways than extree values, and hence tend to be ore lkely. It s ths property of ndependent sus that gves rse to ther bell shaped dstrbutons, as can be seen n parts (c) and (d) of Fgure (a) = (b) = (c) = 0 (d) = 0 Fgure 3. Cuulatve Bnary Errors But whle ths basc shape property s easly understood, the truly aazng fact s that the ltng for f ths bell shape always corresponds to essentally the sae dstrbuton, naely the noral dstrbuton. To state ths precsely, t s portant to notce frst that ESE 50 II.3- Tony E. Sth

3 Part II. Contnuous Spatal Data Analyss whle the dstrbutons n Fgure 3. start to becoe bell shaped, they are also startng to concentrate around zero. Indeed, the ltng for of ths partcular dstrbuton ust necessarly be a unt pont ass at zero, and s certanly not norally dstrbuted. Here s turns out that the ndvdual values of these factors, ( ek /, or ek / ), becoe too sall as ncreases, so that eventually even ther su,, wll alost certanly vansh. At the other extree, suppose that these values are ndependent of, say ( ek, or ek ). Then whle these ndvdual values wll eventually becoe sall relatve to ther su,, the varance of tself wll ncrease wthout bound. In a slar anner, observe that f the coon eans of these ndvdual factors were not dentcally zero, then the ltng ean of would also be unbounded. 3 So t should be clear that precse analyss of ltng rando sus s rather delcate. 3.. Standardzed Rando Varables The te-honored soluton to these dffcultes s to rescale these rando sus n a anner whch ensures that both ther ean and varance rean constant as ncreases. To do so, we begn by observng that for any rando varable, X wth ean, E( X ), and varance, var( X ) the transfored rando varable, X (3..3 ) Z X necessarly has zero ean snce (by the lnearty of expectatons), (3..4 ) EZ ( ) EX ( ) 0 Moreover, Z also has unt varance, snce by (3.4), (3..5 ) X E[( X ) ] var( Z) E( Z ) E E ( X ) Sply observe that f x s a bnary rando varable wth Pr( x ).5 Pr( x ) then by k k k defnton, e x /, so that ( x k k )/ x s seen to be the average of saples fro ths bnary dstrbuton. But by the Law of Large Nubers, such saple averages ust eventually concentrate at the populaton ean, E( x) 0. k In partcular snce var( e ) E( e ).5( ).5( ) for all k, t would then follow fro the k k ndependence of ndvdual factors that var( ) var( e ) var( e ), and hence that k k k var( ) as. 3 Snce E ( ) ( ) ( ) E e E e ples E( ) E( e ), t follows that f Ee ( ) 0 then k k E( ) as. ESE 50 II.3-3 Tony E. Sth

4 Part II. Contnuous Spatal Data Analyss Ths fundaental transforaton procedure s called the standardzaton of X. We shall use ths devce to study the lts of sus. But ore generally, t s portant to observe that f one wants to copare the dstrbutonal shapes of any two rando varables, say, X and Y, t s uch ore convenent to copare ther standardzatons, Z X and Z Y. Snce these new varables always have the sae ean and varance, a coparson of Z X and Z Y thus allows one to focus on qualtatve dfferences n ther shape. In partcular, we can n prncple use ths standardzaton procedure to study the ltng dstrbutonal shape of any su of rando varables, say k (3..6) S X X X k As n our exaple, let us assue for the present that these varables are ndependently and dentcally dstrbuted (d), wth coon ean,, and varance, [so that ( X,.., X ) can be vewed as a rando saple of sze fro soe coon dstrbuton]. Then the ean and varance of S are gven respectvely by (3..7) (3..8) k k k ES ( ) EX ( ) k k k var( S ) var( X ) So as above, we ay construct the assocated standardzed su, (3..9) Z S E( S) S var( S ) n whch by defnton ples that EZ ( ) 0 and var( Z) for all. The key property of these standardzed sus s that for large the dstrbuton of Z s approxately norally dstrbuted. 3.. Noral Dstrbuton To state ths precsely, we ust frst defne the noral dstrbuton. A rando varable, X, wth ean and varance s sad to be norally dstrbuted, wrtten, X ~ N(, ), f and only f X has probablty densty gven by (3..0) ( x ) x f( x) e e x f ( x ) ESE 50 II.3-4 Tony E. Sth

5 Part II. Contnuous Spatal Data Analyss [where the frst verson shows f ( x ) as an explct functon of (, ) and the second shows the ore standard verson of f ( x ) n ters of (, )]. Ths s the classcal bellshaped curve, centered on the ean,, as shown on the rght. A key property of noral rando varables (that we shall ake use of any tes) s that any lnear functon of a noral rando varable s also norally dstrbuted. In partcular, snce the standardzaton procedure n (3..3) s seen to be a lnear functon, t follows that the standardzaton, Z, of any noral rando varable ust be norally dstrbuted wth ean, EZ ( ) 0, varance, var( Z), and wth densty (3..) ( z) z exp 0 z ( z) For obvous reasons, ths s called the standard noral dstrbuton (or densty), and s generally denoted by. The portance of ths partcular dstrbuton s that all probablty questons about noral rando varables can be essentally answered by standardzng the and applyng the standard noral dstrbuton (so that all noral tables are based entrely on ths standardzed for). Next, f the cuulatve dstrbuton functon (cdf ) of any rando varable, X, s denoted for all values, x, by F( x) Prob( X x), then for any standard noral rando varable, Z ~ N (0,), the cdf of Z s denoted by (3..) ( z) Prob( Zz) ( z) dz z Agan s usually reserved for ths portant cdf (that fors the bass of all noral tables) Central Lt Theores Wth these prelnares, we can now gve a precse stateent of the ltng noral property of standardzed sus stated above. To do so, t s portant to note frst that the dstrbuton of any rando varable s copletely defned by ts cdf. [For exaple, n the standard noral case above t should be clear that the standard noral dstrbuton,, s recovered by sply dfferentatng.] Hence, lettng the cdf of the standardzed su, Z, n (3..9) be denoted by F Z, we now have the followng classcal for of the Central Lt Theore (CLT): Central Lt Theore (Classcal). For any sequence of d rando varables ( X,.., X ) wth standardzed su, Z, n (3..9), (3..3) l F ( z) ( z) for all z. Z ESE 50 II.3-5 Tony E. Sth

6 Part II. Contnuous Spatal Data Analyss In other words, the cdf of d standardzed sus, Z, converges to the cdf of the standard noral dstrbuton. The advantage of ths cdf forulaton s that one obtans an exact lt result. But n practcal ters, the plcaton of the CLT s that for suffcently large, the dstrbuton of such standardzed sus s approxately norally dstrbuted. 4 Even ore to the pont, snce (3..3) ples that d sus, S, are lnear functons of ther standardzatons, Z, and snce lnear functons of noral rando varables are agan noral, t ay also be concluded that these sus are approxately noral. If for convenence we now use the notaton, X d N(, ), to ndcate that a rando varable X s approxately dstrbuted noral wth ean,, and varance,, and f we recall fro (3..7) and (3..8) that the ean and varance of S are gven by and, respectvely, then we have the follows ore useful for of the CLT : Central Lt Theore (Practcal). For all sus, S, of d rando varables wth suffcently large, (3..4) S N d (, ) Ths result can n prncple be used to otvate the fundaental noralty assupton about rando effects,. In partcular, f s a su of d rando coponents as n (3..), wth zero eans, then by (3..4) t follows that wll also be approxately noral wth zero ean for suffcently large. However, t should be ephaszed here that n practcal exaples (such as the one dscussed n Secton 3. below) the ndvdual coponents, e k, of ay not be fully ndependent, and are of course not lkely to be dentcally dstrbuted. Hence t s portant to ephasze that the CLT s actually uch ore general that the classcal asserton above for d rando varables. Whle such generalzatons requre condtons that are too techncal to even be stated n a precse anner here, 5 t s nonetheless useful to gven a very rough stateent of the general verson as follows: 6 4 Recall fro footnote 5 n Secton 3.. of Part I that suffcently large s usually taken to ean 30, as long as the coon dstrbuton of the underlyng rando varables ( X k ) n (3..6) s not too skewed. 5 For further detals about such generalzatons, an excellent place to start s the Wkpeda dscusson of the CLT at 6 The followng verson of the Central Lt Theore (and the ultvarate verson of ths theore n secton 3..3 below) based on Theore 8. n Brean (969). The advantage of the present verson s that t drectly extends the d condtons of the classcal CLT. ESE 50 II.3-6 Tony E. Sth

7 Part II. Contnuous Spatal Data Analyss Central Lt Theore (General). For any su, S X X, of rando varables wth eans,,..,, and varances,,..,, f () the dstrbutons of these rando varables are not too dfferent, and ( ) the dependences aong these rando varables s not too strong, then for suffcently large, the dstrbuton of S s approxately noral,.e., (3..5) S d N (, ) wth and. So for rando effects, e e, wth total varance,, t follows that as long as condtons () and () are reasonable and s suffcently large, rando effects,, wll be approxately norally dstrbuted as (3..6) N(0, ) d 3..4 CLT for the Saple Mean Whle the an applcaton of the CLT for our present purposes s to otvate the noralty assupton about resduals n a host of statstcal odels (ncludng lnear regresson), t s portant to add that perhaps the sngle ost portant applcaton of the CLT s for nference about populaton eans. In partcular, f one draws a d rando saple, ( X,.., X ) fro a populaton wth unknown ean,, and constructs the assocated saple ean: (3..7) X X S, k k then by (3..7) the dentty, (3..8) EX ( ) ES ( ) ( ) ples that X s the natural unbased estator of. Moreover, by (3..8), the second dentty, (3..9) var( ) var( ) ( ) / X S ples that for large ths estate has a sall varance, and hence should be close to (whch s of course precsely the Law of Large Nubers). But one can say even ore by the CLT. To do so, note frst that the standardzed saple ean, ESE 50 II.3-7 Tony E. Sth

8 Part II. Contnuous Spatal Data Analyss (3..0) Z X X E( X ) X ( X ) / can equvalently be wrtten as (3..) Z X S S S / / Z and hence satsfes exactly the sae ltng propertes as the saple su. In partcular ths yelds the follows verson of the practcal CLT n (3..4) above for saple eans: Central Lt Theore (Saple Means). For suffcently large d rando saples, ( X,.., X ), fro any gven statstcal populaton wth ean,, and varance,, the saple ean, X, s approxately noral,.e., (3..) X N d (, / ) Note n partcular that rando saples fro the sae populaton are by defnton dentcally dstrbuted. So as long as they are also ndependent, Corollary s always applcable. But the Clark-Evans test n Secton 3.. of Part I provdes a classc exaple where ths latter assupton ay fal to hold. More generally, the types of dependences nherent n spatal (or teporal) data requre ore careful analyss when applyng the CLT to saple eans. 3. Mult-Locaton Rando Effects Gven the above results for rando effects, at ndvdual locatons, s, we now consder the vector,, of such rando effects for a gven set of saple locatons, { s :,.., n} R,.e., (3..) ( :,.., n) [ ( s ) :,.., n] As a parallel to (3..) we agan assue that these rando effects are the cuulatve su of ndependent factors, (3..) e e e e k k where by defnton each ndependent factor, e k, s tself a rando vector over saple locatons,.e., (3..3) e ( e :,.., n) [ e ( s ) :,.., n] k k k ESE 50 II.3-8 Tony E. Sth

9 Part II. Contnuous Spatal Data Analyss As one llustraton, recall the Calforna ranfall exaple n whch annual precptaton, Y, at each of the n 30 saple locatons n Calforna was assued to depend on four explanatory varables ( x alttude, x lattude, x dstance to coast, and ran shadow, as follows x4 (3..4) 4 Y x,,.., n 0 j j j Here the unobserved resduals,, are the rando effects we wsh to odel. If we wrte (3..4) n vector for as (3..5) Y 4 0 n j x j j [where n (,..,) s the unt colun vector], then the resdual vector,, n (3..5) s an nstance of (3..) wth n 30. Ths rando vector by defnton contans all factors nfluencng precptaton other that the four an effects posted above. So the key assupton n (3..) s that the nfluence of each unobserved factor s only a sall addtve part of the total resdual effect,, not accounted for by the four an effects above. For exaple, the frst factor, e, ght be a cloud cover effect. More specfcally, the unobserved value, e e ( s) at each locaton, s, ght represent fluctuatons n cloud cover at s [where hgher (lower) levels of cloud cover tend to contrbute postvely (negatvely) to precptaton at s ]. Slarly, factor e ght be an atospherc pressure effect, where e e ( s) now represents fluctuatons n baroetrc pressure levels at s [and where n ths case hgher (lower) pressure levels tend to contrbute negatvely (postvely) to precptaton levels]. The key pont to observe s that whle fluctuatons n factors lke cloud cover or atospherc pressure wll surely exhbt strong spatal dependences, the dependency between these factors at any gven locaton s uch weaker. In the present nstance, whle there ay ndeed be soe degree of negatve relaton between fluctuatons n pressure and cloudness ( e, e ) at any gven locaton, s, ths tends to be uch weaker than the postve relatons between ether fluctuatons n cloud cover ( e, e j), or atospherc pressure ( e, e j), at locatons, s and s j, that are n close proxty. Hence whle the rando vectors, e and e, can each exhbt strong nternal spatal dependences, t s not unreasonable to treat the as utually ndependent. More generally, as a parallel to secton (3..3) above, t wll turn out that f () the ndvdual dstrbutons of the rando coponent vectors, e,.., e, n (3..) are not too dfferent, and () the statstcal dependences between these coponents are not too strong, then ther su,, wll be approxately noral for suffcently large. ESE 50 II.3-9 Tony E. Sth

10 Part II. Contnuous Spatal Data Analyss But n order to ake sense of ths stateent, we ust frst extend the noral dstrbuton n (3..0) to ts ultvarate verson. Ths s done n the next secton, where we also develop ts correspondng nvarance property under lnear transforatons. Ths wll be followed by a developent of the ultvarate verson of the Central Lt Theore that underscores the portance of ths dstrbuton. 3.. Multvarate Noral Dstrbuton To otvate the ultvarate noral (or ult-noral) dstrbuton observe that there s one case n whch we can deterne the jont dstrbuton of a rando vector, X ( X,.., Xn), n ters of the argnal dstrbutons of ts coponent, X,.., X n, naely when these coponents are ndependently dstrbuted. In partcular, suppose that each X s ndependently norally dstrbuted as n (3..0) wth densty (3..6) ( x) f ( x) e,,.., n Then lettng, and usng the exponent notaton, a a densty, f ( x,.., x n), of X s gven by the product of these argnals,.e., (3..7) f ( x,.., xn) f( x) f( x) fn( xn) /, t follows that the jont ( x) ( x) ( x) nn e e e nn ( x) ( xnn) n/ / nn nn e ( ) ( ) a a an a a an where the last lne uses the dentty, ( e )( e ) ( e ) e. To wrte ths n atrx for, observe frst that f x ( x,.., xn) now denotes a typcal realzaton of rando vector, X ( X,.., Xn), then by (3..6) the assocated ean vector of X s gven by (,.., ) [as n expresson (..4)]. Moreover, snce ndependence ples that n cov( X, X ) 0 for j, t follows that the covarance atrx of X now takes the j j for [as n expresson (..7)], ESE 50 II.3-0 Tony E. Sth

11 Part II. Contnuous Spatal Data Analyss (3..7) cov( X ) nn But snce the nverse of a dagonal atrx s sply the dagonal atrx of nverse values, (3..8) nn t follows that (3..9) ( x ) ( x ) x x ( x, x,.., xn n) x nn n n ( x )/ ( x )/ ( x, x,.., xn n) ( xn n)/ nn ( x ) ( x ) ( x ) n n nn whch s precsely the exponent su n (3..7). Fnally, snce the deternant,, of a dagonal atrx,, s sply the product of ts dagonal eleents,.e., (3..0) nn nn, we see fro (3..9) and (3..0) that (3..7) can be rewrtten n atrx for as (3..) f( x) ( ) n/ / e ( x) ( x) ESE 50 II.3- Tony E. Sth

12 Part II. Contnuous Spatal Data Analyss Ths s n fact an nstance of the ult-noral densty (or ultvarate noral densty). More generally, a rando vector, X ( X,.., Xn), wth assocated ean vector, (,.., n), and covarance atrx, ( j :, j,.., n), s sad to be ult-norally dstrbuted f and only f ts jont densty s of the for (3..) for ths choce of and. As a generalzaton of the unvarate case, ths s denoted sybolcally by X ~ N(, ). Whle t s not possble to vsualze ths dstrbuton n hgh densons, we can gan soe nsght by focusng on the -densonal case, known as the b-noral (or bvarate noral) dstrbuton. If X ( X, X) s b-norally dstrbuted wth ean vector, (, ) and covarance atrx, (3..) then the basc shape of the densty functon n (3..) s largely deterned by the correlaton between X and X,.e., by cov( X, X) (3..3) ( X, X) ( X ) ( X ) Ths s ost easly llustrated by settng 0 and so that the only paraeter of ths dstrbuton s covarance,, whch n ths case s seen fro (3..3) to be precsely the correlaton,, between X and X. The ndependence case ( 0) s shown n Fgure 3. below, whch s sply a -densonal verson of the standard noral dstrbuton n (3..) above. Indeed both of ts argnal dstrbutons are dentcal wth (3..). Fgure 3.3 depcts a case wth extree postve correlaton (.8) to ephasze the role of correlaton n shapng ths dstrbuton. In partcular, ths hgh correlaton ples that value pars ( x, x ) that are slar n agntude (close to the 45 lne) are ore lkely to occur, and hence have hgher probablty densty. Thus the densty s ore concentrated along the 45 lne, as shown n the fgure. These propertes persst n hgher densons as well. In partcular, the bell-shaped concentraton of densty around the orgn contnues to hold n hgher densons, and s ore elongated n those drectons where correlatons between coponents are ore extree. ESE 50 II.3- Tony E. Sth

13 Part II. Contnuous Spatal Data Analyss x x x x Fgure 3.. B-noral Dstrbuton (ρ = 0) Fgure 3.3. B-noral Dstrbuton (ρ =.8) 3.. Lnear Invarance Property For purposes of analyss, the sngle ost useful feature of ths dstrbuton s that all lnear transforatons of ult-noral rando vectors are agan ult-noral. To state ths precsely, we begn by calculatng the ean and covarance atrx for general lnear transforatons of rando vectors. Gven a rando vector, X ( X,.., Xn), wth ean vector, EX ( ) (,.., n) and covarance atrx, cov( X ), together wth any copatble ( n) atrx, A ( aj :,..,, j,.., n), and n -vector, b ( b,.., bn ) of coeffcents, consder the lnear transforaton of X defned by (3..4) Y AX b Followng standard conventons, f then the ( n) atrx, A, s usually wrtten as the transpose of an n-vector, a ( a,.., a n ), so that (3..4) takes the for, (3..5) Y ax b where b s a scalar. If b 0 then the rando varable, Y ax, s called a lnear copound of X. For exaple, each coponent of X can be dentfed by such a lnear copound as follows. If the coluns of the n -square dentty atrx, I n, are denoted by ESE 50 II.3-3 Tony E. Sth

14 Part II. Contnuous Spatal Data Analyss (3..6) I n,,..., [ e, e,..., en ] then by settng a e and b 0 n (3..5), we see that (3..7) X ex,,.., n So lnear transforatons provde a very flexble tool for analyzng rando vectors. Next recall fro the lnearty of expectatons that by takng expectatons n (3..4) we obtan (3..8) E( Y) E( AX b) AE( X) b A b By usng ths result, we can obtan the covarance atrx for Y as follows. Frst note that by defnton the expected value of a atrx of rando varable s sply the atrx of ther expectatons,.e., (3..9) Z Z n EZ ( ) EZ ( n) E Z Z E( Z ) ( Z ) n n So the defnton of cov( Y ) n (..7) can equvalently be wrtten n atrx ters as (3..0) cov( Y ) E[( Y)( Y)] E[( Y)( Yn n)] E[( Yn n)( Y)] E[( Yn n)( Yn n)] ( Y)( Y) ( Y)( Yn n) E ( Yn n)( Y) ( Yn n)( Yn n) Y E Y,..., Yn n Yn n E[( Y )( Y ) ] ESE 50 II.3-4 Tony E. Sth

15 Part II. Contnuous Spatal Data Analyss By applyng ths to (3..5) we obtan the followng very useful result: (3..) cov( Y) E[( Y )( Y ) ] E{([ AX b] [ A b])([ AX b] [ A b])} E[( AX A)( AX A) ] E[ AX ( )( X ) A] AE[( X )( X ) ] A Acov( X) A cov( AX) A A So both the ean and covarance atrx of AX b are drectly obtanable fro those of X. We shall use these propertes any tes n analyzng the ultvarate spatal odels of subsequent sectons. But for the oent, the key feature of these results s that the dstrbuton of any lnear transforaton, AX b, of a ult-noral rando vector, X ~ N(, ), s obtaned by sply replacng the ean and covarance atrx of X n (3..) wth those of AX b. The only requreent here s that the resultng covarance atrx, A A, be nonsngular so that the nverse covarance atrx, ( A A), n (3..) exsts. Ths n turn s equvalent to the condton that the rows of A be lnearly ndependent vectors, so that A s sad to be of full row rank. Wth ths stpulaton, we have the followng result [establshed n Secton A3..3 of the Appendx to Part III n ths NOTEBOOK]: 7 Lnear Invarance Theore. For any ult-noral rando vector, X ~ N(, ), and lnear transforaton, Y AX b, of X wth A of full row rank, Y s also ult-norally dstrbuted as (3..) Y ~ N( A b, A A) What ths eans n practcal ters s that f a gven rando vector, X, s known (or assued) to be ult-norally dstrbuted as X ~ N(, ), then we can edately wrte down the exact dstrbuton of essentally any lnear functon, AX b, of X Multvarate Central Lt Theore We are now ready to consder ultvarate extensons of the unvarate central lt theores above. Our objectve here s to develop only those aspects of the ultvarate 7 For an alternatve developent of ths portant result, see for exaple Theore.4.4 n Anderson (958). ESE 50 II.3-5 Tony E. Sth

16 Part II. Contnuous Spatal Data Analyss case that are relevant for our present purposes. The frst objectve s to show that the ultvarate case relates to the unvarate case n a rearkably sple way. To do so, recall frst fro (3..7) above that for any rando vector, X ( X,.., X n ), each of ts coponents, X, can be represented as a lnear transforaton, X ex, of X. So each argnal dstrbuton of X s autoatcally the dstrbuton of ths lnear copound. More generally, each lnear copound, ax, can be sad to defne a generalzed argnal dstrbuton of X. 8 Now whle the argnal dstrbutons of X only deterne ts jont dstrbuton n the case of ndependence [as n (3..7) above], t turns out that the jont dstrbuton of X s always copletely deterned by ts generalzed argnal dstrbutons. 9 To apprecate the power of ths result, recall fro the Lnear Invarance Theore above that f X s ult-noral wth ean vector,, and covarance atrx,, then all of ts lnear copounds, ax, are autoatcally unvarate norally dstrbuted wth eans, a, and varances, a a. But snce these argnals n turn unquely deterne the dstrbuton of X, t ust necessarly be ult-noral. Thus we are led to the followng fundaental correspondence: Unvarate-Multvarate Correspondence. A rando vector, X, wth ean vector,, and covarance atrx,, s ult-norally dstrbuted as (3..3) X ~ N(, ) f and only f every lnear copound, ax, s unvarate noral,.e., (3..4) ax ~ N( a, a a) In vew of ths correspondence, t s not surprsng that there s an ntate relaton between unvarate and ultvarate central lt theores. In partcular, f any of the unvarate condtons n the central lt theores above hold for all generalzed argnal dstrbutons of X, then X wll autoatcally be asyptotcally ultvarate noral. For exaple, f as an extenson of (3..5) one consders a su of d rando vectors, (3..5) S X X then t follows at once that the ters n each lnear copound, (3..6) as ax ax ust necessarly be d as well. Hence we obtan an edate extenson of the Practcal Central Lt Theore n (3..4) above 8 Snce each argnal copound, ex, has a coeffcent vector of unt length,.e., e, t s forally ore approprate to restrct generalzed argnals to lnear copounds, a, of unt length ( a ). But for our present purposes we need not be concerned wth such scalng effects. 9 For a developent of ths dea (due to Craer and Wold), see Theore 9.4 n Bllngsley (979). ESE 50 II.3-6 Tony E. Sth

17 Part II. Contnuous Spatal Data Analyss Multvarate Central Lt Theore (Practcal). For all sus of d rando vectors, S X X, wth coon ean vector,, and covarance atrx,, f suffcently large then (3..7) S N(, ) d But snce ultvarate noralty wll alost always arse as a odel assupton n our spatal applcatons, the ost useful extenson s the General Central Lt Theore n (3..5), whch ay now be stated as follows: 0 Multvarate Central Lt Theore (General). For any su, S X X, of rando vectors wth ndvdual eans,,..,, and covarance atrces,,..,, f () the dstrbutons of these rando vectors are not too dfferent, and ( ) the dependences aong these rando vectors are not too strong, then for suffcently large, the dstrbuton of S s approxately ult-noral,.e., (3..8) S N(, ) d wth and. Fnally, t s approprate to restate ths result explctly n ters of ult-locaton rando effects, whch for the central focus of ths secton. Spatal Rando Effects Theore. For any rando vector of ult-locaton effects, ( :,.., n), coprsed of a su of ndvdual rando factors, ee e, wth zero eans and covarance atrces,,..,, f () the dstrbutons of these rando factors are not too dfferent, and ( ) the dependences aong these rando factors are not too strong, then for suffcently large, the dstrbuton of s approxately ult-noral,.e., (3..9) N (0, ) wth. d It s ths verson of the Central Lt Theore that wll for the bass for essentally all rando-effects odels n the analyses to follow. 0 For a slar (nforal) stateent of ths general verson of the Multvarate Central Theore, see Theore 8. n Brean (969). ESE 50 II.3-7 Tony E. Sth

18 Part II. Contnuous Spatal Data Analyss 3.3 Spatal Statonarty Gven the Spatal Rando Effects Theore above, the task reanng s to specfy the unknown covarance atrx,, for these rando effects. Snce s n turn a su of ndvdual covarance atrces, k, for rando factors k,...,, t ght see better to specfy these ndvdual covarance structures. But rather than attept to dentfy such factors, our strategy wll be to focus on general spatal dependences that should be coon to all these covarance structures, and hence should be exhbted by. In dong so, t s also portant to ephasze that such statstcal dependences often have lttle substantve relaton to the an phenoena of nterest. In ters of our basc odelng fraework, Ys () () s () s, n (..) above, we are usually uch ore nterested n the global structure of the spatal process, as represented by () s, than n the specfc relatons aong unobserved resduals { ( s ) :,.., n} at saple locatons { s :,.., n}. Indeed, these relatons are typcally regarded as second-order effects n contrast to the frst-order effects represented by () s. Hence t s desrable to odel such secondorder effects n a anner that wll allow the analyss to focus on the frst-order effects, whle at the sae te takng these unobserved dependences nto account. Ths general strategy can be llustrated by the followng exaple Exaple: Measurng Ocean Depths Suppose that one s nterested n appng the depth of the sea floor over a gven regon. Typcally ths s done by takng echo soundngs (sonar easureents) at regular ntervals fro a vessel traversng a syste of paths over the ocean surface. Ths wll yeld a set of depth readngs, { D D( s) :,.., n}, such as the set of easureents s shown n Fgure 3.4 below: s s s n D D D n Fgure 3.4. Pattern of Depth Measureents However, the ocean s not a hoogeneous edu. In partcular, t s well known that such echo soundngs can be nfluenced by the local concentraton of zooplankton n the regon of each soundng. These clouds of zooplankton (llustrated n Fgure 3.5 below) create nterference called ocean volue reverberaton. ESE 50 II.3-8 Tony E. Sth

19 Part II. Contnuous Spatal Data Analyss Fgure 3.5. Zooplankton Interference These nterference patterns tend to vary fro locaton to locaton, and even fro day to day (uch n the sae way that sunlght s affected by cloud patterns). So actual readngs are rando varables of the for, (3.3.) Ds ( ) ds ( ) ( s),,.., n where n ths case the actual depth at locaton s s represented by ds ( ) EDs [ ( )], and ( s ) represents easureent error due to nterference. Moreover these errors are statstcally dependent, snce plankton concentratons at nearby locatons wll tend to be ore slar than at locatons wdely separated n space. Hence to obtan confdence bounds on the true depth at locaton s, t s necessary to postulate a statstcal odel of these jont nterference levels, [ ( s ) :,.., n]. Now one could n prncple develop a detaled odel of zooplankton behavor, ncludng ther patterns of ndvdual oveent and clusterng behavor. However, such odels are not only hghly coplex n nature, they are very far reoved fro the present target of nterest, whch s to obtan accurate depth easureents. 3 Actual varatons n the dstrbuton of zooplankton are ore dffuse than the clouds depcted n Fgure 3.5. Vertcal oveent of zooplankton n the water colun s governed anly by changes n sunlght, and horzontal oveent by ocean currents. In actualty, such easureent errors nclude any dfferent sources, such as the reflectve propertes of the sea floor. Moreover, depth easureents are actually ade ndrectly n ters of the transsson loss, L L( s ), between the sgnal sent and the echo receved. The correspondng depth, D, s obtaned fro L by a functonal relaton, D ( L, ), where s a vector of paraeters that have been calbrated under dealzed condtons. For further detals, see Urck, R.J. (983) Prncples of Underwater Sound, 3 rd ed., McGraw-Hll: New York, and n partcular the dscusson around p Here t portant to note that such detaled odels can be of great nterest n other contexts. For exaple, acoustc sgnals are also used to estate the volue of zooplankton avalable as a food source for sea creatures hgher n the food chan. To do so, t s essental to relate acoustc sgnals to the detaled behavor of such croscopc creatures. See for exaple, Stanton T.K. and D. Chu (000) Revew and recoendatons for the odelng of acoustc scatterng by flud-lke elongated zooplankton: euphausds and copepods, ICES Journal of Marne Scence, 57: ESE 50 II.3-9 Tony E. Sth

20 Part II. Contnuous Spatal Data Analyss So what s needed here s a statstcal odel of spatal resduals that allows for local spatal dependences, but s sple enough to be estated explctly. To do so, we wll adopt the followng basc assuptons of spatal statonarty: (3.3.) [Hoogenety] Resduals, ( s ), are dentcally dstrbuted at all locatons s. (3.3.3) [Isotropy] The jont dstrbuton of dstnct resduals, ( s ) and ( s j ) depends only on the dstance between locatons s and s j. These assuptons are loosely related to the noton of sotropc statonarty for pont processes dscussed n Secton.5 of Part I. But here we focus on the jont dstrbuton of rando varables at selected locatons n space rather than pont counts n selected regons of space. To otvate the present assuptons n the context of our exaple, observe frst that whle zooplankton concentratons at any pont of te ay dffer between locatons, t can be expected that the range of possble concentraton levels over te wll be qute slar at each locaton. More generally, the Hoogenety assupton asserts that the argnal dstrbutons of these concentraton levels are the sae at each locaton. To apprecate the need for such an assupton, observe frst that whle t s n prncple possble to take any depth easureents at each locaton and eploy these saples to estate locaton-specfc dstrbutons of each rando varable, ths s generally very costly (or even nfeasble). Moreover, the sae s true of ost spatal data sets, such as the set of total ranfall levels or peak daly teperatures reported by regonal weather statons on a gven day. So n ters of the present exaple, one typcally has a sngle set of depth easureents [ Ds ( ) :,.., n], and hence only a sngle jont realzaton of the set of unobserved resduals [ ( s) :,.., n]. Thus, wthout further assuptons, t s possble to say anythng statstcally about these resduals. Fro ths vewpont, the fundaental role of the Hoogenety assupton s to allow the jont realzatons, [ ( s ) :,.., n], to be treated as ultple saples fro a coon populaton that can be used to estate paraeters of ths populaton. The Isotropy assupton s very slar n sprt. But here the focus s on statstcal dependences between dstnct rando varables, ( s ) and ( s j ). For even f ther argnal dstrbutons are known, one cannot hope to say anythng further about ther jont dstrbuton on the bass of a sngle saple. But n the present exaple t s reasonable to assue that f a gven cloud of zooplankton (n Fgure 3.5) covers locaton, s, then t s very lkely to cover locatons s j whch are suffcently close to s j. Slarly for locatons that are very far apart, t s reasonable to suppose that clouds coverng s have lttle to do wth those coverng s j. Hence the Isotropy assupton asserts ore generally that slartes between concentraton levels at dfferent locatons depend only on the dstance between the. The practcal plcaton of ths assupton s that all ESE 50 II.3-0 Tony E. Sth

21 Part II. Contnuous Spatal Data Analyss pars of resduals, ( s ) and ( s j ), separated by the sae dstance, h s sj, ust exhbt the sae degree of dependency. Thus a collecton of such pars can n prncple provde ultple saples to estate the degree of statstcal dependency at any gven dstance, h. A second advantage of ths Isotropy assupton s that t allows sple odels of local spatal dependency to be forulated drectly n ters of ths sngle dstance paraeter. So t should be clear that these two assuptons of spatal statonarty do ndeed provde a natural startng pont for the desred statstcal odel of resduals. But before proceedng, t should also be ephaszed that whle these assuptons are conceptually appealng and analytcally useful they ay of course be wrong. For exaple, t can be argued n the present llustraton that locatons n shallow depths (Fgure 3.5) wll tend to experence lower concentraton levels than locatons n deeper waters. If so, then the Hoogenety assupton wll fal to hold. Hence ore coplex odels nvolved nonhoogeneous resduals ay be requred n soe cases. 4 As a second exaple, suppose that the spatal oveent of zooplankton s known to be largely governed by prevalng ocean currents, so that clouds of zooplankton tend to be ore elongated n the drecton of the current. If so, then spatal dependences wll depend on drecton as well as dstance, and the Isotropy assupton wll fal to hold. Such cases ay requre ore coplex ansotropc odels of spatal dependences Covarance Statonarty In any cases the assuptons above are stronger than necessary. In partcular, recall fro the Spatal Rando Effects Theore (together wth the ntroductory dscusson n Secton 3.3) that such rando effects are already postulated to be ult-norally dstrbuted wth zero eans. So all that s requred for our purposes s that these hoogenety and sotropy assuptons be reflected by the atrx,, of covarances aong these rando effects. To do so, t wll be convenent for our later purposes to forulate such covarance propertes n ters of ore general spatal stochastc processes. A spatal stochastc process, { Y( s) : s R}, s sad to be covarance statonary f and only f the followng two condtons hold for all s, s, v, v R: (3.3.4) EY [ ( s)] EY [ ( s)] (3.3.5) ss vv Y s Y s Y v Y v cov[ ( ), ( )] cov[ ( ), ( )] These condtons can be stated ore copactly by observng that (3.3.4) ples the exstence of a coon ean value,, for all rando varables. Moreover, (3.3.5) 4 For exaple, t ght be postulated that the varance of () s depends on the unknown true depth, () d s, at each locaton, s. Such nonstatonary forulatons are coplex, and beyond the scope of these notes. 5 Such odels are dscussed for exaple by Gotway and Waller (004, Secton.8.5). ESE 50 II.3- Tony E. Sth

22 Part II. Contnuous Spatal Data Analyss ples that covarance depends only on dstance, so that for each dstance, h, and par of locatons sv, R wth sv h there exsts a coon covarance value, Ch, ( ) such that cov[ Ys ( ), Yv ( )] Ch ( ). Hence, process { Ys ( ): s R} s covarance statonary f and only f (ff) the followng two condtons hold for all sv, R, (3.3.6) EY [ ( s)] (3.3.7) sv h cov[ Y( s), Y( v)] C( h) Note n partcular fro (3.3.7) that snce var[ Ys ( )] cov[ Ys ( ), Ys ( )] by defnton, and snce ss 0, t follows that these rando varables ust also have a coon varance, gven by (3.3.8) var[ ( )] (0), Y s C s R Whle these defntons are n ters of general spatal stochastc processes, { Y( s) : s R}, our ost portant applcatons wll be n ters of spatal resduals (rando effects). Wth ths n nd, notce that (3.3.6) together wth (..) ply that every covarance statonary process can be wrtten as (3.3.9) Y( s) ( s) so that each such process s assocated wth a unque resdual process, { ( s) : s R}. Moreover, snce cov[ Y( s), Y( v)] cov[ ( s), ( v)] E[ ( s) ( v)] E[ ( s)] E[ ( v)], we see that { ( s) : s R} ust satsfy the followng ore specalzed set of condtons for all sv, R: (3.3.0) E[ ( s)] 0 (3.3.) sv h E[ ( s) ( v)] C( h) These are the approprate covarance statonarty condtons for resduals that correspond to the stronger Hoogenety (3.3.) and Isotropy (3.3.3) condtons n Secton 3.3. above. Note fnally that even these assuptons are too strong n any contexts. For exaple (as entoned above) t s often convenent to relax the sotropy condton plct n (3.3.7) and (3.3.) to allow drectonal varatons n covarances. Ths can be done by requrng that covarances dependent only on the dfference between locatons,.e., that for all h ( h, h), svh cov[ Y( s), Y( v)] C( h). Ths weaker statonarty condton s often called ntrnsc statonarty. See for exaple [BG] (p.6), Cresse (993, Sectons ESE 50 II.3- Tony E. Sth

23 Part II. Contnuous Spatal Data Analyss.. and.3) and Waller and Gotway (004, p.73). However, we shall treat only the sotropc case [(3.3.7),(3.3.)], and shall use these assuptons throughout Covarogras and Correlogras Note that snce the above covarance values, Ch, ( ) are unque for each dstance value, h, n regon R, they defne a functon, C, of these dstances whch s desgnated as the covarogra for the gven covarance statonary process. 6 But as wth all rando varables, the values of ths covarogra are only eanngful wth respect to the partcular unts n whch the varables are easured. Moreover, unlke ean values, the values of the covarogra are actually n squared unts, whch are dffcult to nterpret n any case. Hence t s often ore convenent to analyze dependences between rando varables n ters of (densonless) correlaton coeffcents. For any statonary process, { Y( s): s R}, the (product oent) correlaton between any Y( s ) and Yv ( ) wth sv h s gven by the rato: (3.3.) cov[ Y( s), Y( v)] C( h) C( h) [ Y( s), Y( v)] var[ Y( s) var[ Y( v) C(0) C(0) C(0) whch s sply a noralzed verson of the covarogra. Hence the correlatons at every dstance, h, for a covarance statonary process are suarzed by a functon called the correlogra for the process: (3.3.3) Ch ( ) ( h), s R C(0) Probably the ost portant applcaton of correlogras s to allow coparsons between covarogras that happen to be n dfferent unts. One such applcaton s llustrated n Secton below. 6 To be ore precse, f the set of all dstances assocated wth pars of locatons n regon R s denoted by hr ( ) { h: sv hfor soe sv, R}, then the covarogra, C, s a nuercal functon on hr ( ). Note also that for the weaker for of ntrnsc statonarty dscussed above, the covarogra depends on the dfferences n both coordnates, h ( h, h ), and hence s a two-densonal functon n ths case. ESE 50 II.3-3 Tony E. Sth