Spatial Modeling and Predi tion under RangeAnisotropyMark D. E ker� and Alan E. Gelfand��July 2, 1999Abstra tFor modeling spatial pro esses, we propose ri h lasses of range anisotropi o-varian e stru tures that greatly in rease the s ope of variogram ontours in R2 andin lude geometri anisotropy and isotropy as spe ial ases. We demonstrate howthe lass of all ompletely monotoni isotropi variograms an be extended to ap-ture range anisotropy and illustrate with two examples, the Mat�ern and the generalexponential. We adopt a Bayesian perspe tive and �t these range anisotropi o-varian e models using sampling-based methods. In the presen e of measurementerror/mi ros ale e�e t, we develop the noiseless predi tive distribution. We ana-lyze a data set of s allop at hes, withholding ten sites, to ompare the a ura yand pre ision of the standard and noiseless predi tive distributions..Key Words & Phrases : Bayesian modeling, Bessel fun tion, importan e sampling, kriging,monotoni ity, variogram1

�Mark D. E ker is Assistant Professor in the Department of Mathemati s, University of North-ern Iowa, Cedar Falls, IA 50614-0506 and ��Alan E. Gelfand is Professor in the Depart-ment of Statisti s, University of Conne ti ut, Storrs 06269-3120. The work of the se ondauthor was supported in part by NSF DMS 96-25383.

2

1 Introdu tionFor spatial lo ations fsi : si 2 D � Rmg, suppose we observe responses Y (si); i =1; : : : ; N where Y = (Y (s1); Y (s2); : : : ; Y (sN))0 is viewed as a single (ve tor) observationfrom a random �eld. Under the intrinsi hypothesis of Matheron (1963),E(Y (si)� Y (sj)) = 0 and Var(Y (si)� Y (sj)) = 2 (si � sj) = 2 (hij) (1)where hij = si � sj is the separation ve tor between sites si and sj , 2 (h) is alled thevariogram and (h) is the semivariogram. A stronger assumption is that the pro essY (s) is se ond-order or weakly stationary, in whi h aseE(Y (si)) = � and Cov(Y (si); Y (sj)) = C(si � sj) = C(hij) <1: (2)The spatial pro ess Y (s) is said to be isotropi if hij is repla ed by the Eu lideandistan e, khijk = dij , in (1) and (2). Smith (1996) de�nes a pro ess to be homogeneousif it satis�es both isotropy and se ond-order stationarity.For a homogeneous pro ess, the variogram is bounded sin e C(dij) is �nite and thesill of the variogram is de�ned to be limd!1 2 (d). For a monotoni ally in reasingvariogram, the range is the distan e at whi h the variogram a hieves its sill or equivalentlythe distan e at whi h Corr(Y (si); Y (sj)) = 0. If a monotoni variogram rea hes itssill asymptoti ally, the e�e tive range is de�ned to be (M Bratney and Webster, 1986)the distan e, r, at whi h the orrelation drops to 5%. The range is not de�ned fornonmonotoni or hole-e�e t variograms. The nugget is de�ned to be limd!0 2 (d) = 2� 2whi h need not be zero due to measurement error and/or a mi ros ale e�e t.Sin e, under spe i� ation (1), no distributional assumption is given for Y , estimating an be viewed as a urve �tting exer ise. For ea h of the N(N�1)2 pairs of data valuesin Rm, a plot of 12(Y (si) � Y (sj))2 versus dij is termed the semivariogram loud. TheMatheron estimator of the semivariogram (1963) �k(dij) = 12NBk Xf(i;j):dij2Bkg(Y (si)� Y (sj))2 (3)smoothes the semivariogram loud by aggregating distan es into K sets B1; B2; : : : ; BKwhere ea h Bk = fdij : bk�1 � dij < bkg for onstants bk�1 and bk; k = 1; : : : ;K with2

b0 = 0. Also, NBk equals the number of dij in Bk with Pk NBk = N(N�1)2 . The onstantsbk are typi ally hosen to be k� where � is a spe i�ed lag. Plotting �k versus the midpointof the interval (bk�1; bk) results in the empiri al semivariogram. As de�ned in (3), �need not be valid or asso iated with a positive de�nite ovarian e matrix. Thus, astandard semivariogram model that is known to be asso iated with a positive de�nite ovarian e fun tion is typi ally �t to the Matheron empiri al semivariogram yieldingpoint estimation of the variogram parameters.The Matheron empiri al semivariogram (3) an be extended to R2 and is alled anempiri al semivariogram ontour (ESC) plot (E ker and Gelfand, 1999). For ea h of theN(N�1)2 pairs of points, al ulate hx and hy, the separation distan e along ea h respe tiveaxis. To remove ambiguity resulting from the order in whi h di�eren es are taken, wespe ify that hx � 0. We aggregate these (hx; hy) into bins Bij and al ulate �ij = 12NBij Xf(i;j):(si�sj)2Bijg(Y (si)� Y (sj))2 (4)where NBij equals the number of pairs of sites in bin Bij . Smoothing the plot of �ijversus the enter of ea h bin results in the ESC plot.If, under (2), we adopt a distributional spe i� ation for Y , full inferen e is availableusing the Bayesian paradigm. In fa t, as we show below, a hierar hi al model is naturallydeveloped. Here we work with a Gaussian �rst stage, though this is not ne essary. TheBayesian framework also enables us to avoid ergodi ity assumptions whi h are requiredto allow likelihood-based inferen e to pro eed with only the single repli ate Y . (Su hergodi ity is also used to justify spatial averaging su h as in (3) and (4).)The spatial pro ess Y (s) is said to exhibit anisotropy when C(h) depends uponboth the magnitude and orientation of the separation ve tor. Following Zimmerman(1993), anisotropy an take three forms: sill anisotropy, nugget anisotropy and/or rangeanisotropy. For a unit ve tor in h's dire tion, hkhk , onsider limÆ!1 (Æ hkhk). If thislimit depends upon h, we have sill anisotropy. If limÆ!0 (Æ hkhk) depends upon h thenwe have nugget anisotropy. Both are problemati with regard to (2), see Zimmerman(1993). Instead we on�ne ourselves to range anisotropy where r depends upon h.Zimmerman (1993, p 456) notes that su h anisotropy is most ommon in pra ti e.3

Range anisotropy an be spe i�ed as either geometri (ellipti al) range anisotropyor non-geometri range anisotropy. Under geometri anisotropy for D, 2 (si � sj) =2 (kh�ijk) = 2 (kAhijk) where h�ij = Ahij for a nonsingular A. Equivalently, D exhibitsgeometri anisotropy if and only if 2 (kh�k) = 2 ((h0Bh) 12 ) where B = A0A is positivede�nite. Zonal anisotropy (Journel and Huijbregts, 1978, p 181) refers to any type ofanisotropy ex ept geometri .The re ent literature onsiders methods for �tting geometri ally anisotropi models.Likelihood based te hniques have been suggested by Zimmerman (1993) and Ve hia(1988). E ker and Gelfand (1999) present a Bayesian approa h while Borgman and Chao(1994) and Krajewski, Molinska and Molinska (1996) present distribution free te hniquesfor estimating the variogram parameters and the linear transformation matrix A. Fornon-geometri range anisotropi models, the urrent methodology is to model nestedvariograms with di�erent range parameters in arbitrarily hosen dire tions of interest(Zimmerman, 1993, p 466).In this paper, we propose ri h lasses of variogram models that allow for general rangeanisotropy, and in ludes geometri anisotropy and isotropy as spe ial ases. We dis usshow our approa h di�ers from the distribution free method developed by Sampson andGuttorp (1992) and the likelihood based te hnique of Smith (1996). Within the Bayesianframework, we �t these models using simulation, estimating the mean omponent andthe variogram parameters simultaneously. Moreover, we provide omplete inferen e inthe form of a posterior distribution for any feature of our model in luding the nugget,sill, range as a fun tion of dire tion and the posterior mean of the variogram itself. Wealso larify both standard and noiseless spatial predi tion. We illustrate using a dataset of s allop at hes in the Atlanti O ean in 1993. These data were olle ted by theNational Marine Fisheries Servi e (NMFS) in Woods Hole, Massa husetts. A similardata set but sampled at di�erent sites in 1990 (see Figure 1) helped to develop the priordistributions.In se tion 2, we develop two lasses of range anisotropi orrelation stru tures. Inse tion 3, we examine Bayesian modeling issues in luding predi tion. Se tion 4 onsidersprior spe i� ations and model �tting, while se tion 5 dis usses the s allop data.4

2 Range Anisotropi Correlation Spe i� ationsCurrently, range anisotropi variogram models are formulated using di�erent range pa-rameters for a set of distin t dire tions (Zimmerman, 1993, p 466). The number ofdire tions and the a tual angles for whi h one models the range anisotropi variogramsare hosen arbitrarily. For example, a variogram model without the measurement er-ror/mi ros ale omponent in the north-south dire tion under range anisotropy is 2 N�S =2�2(1� �(hy; �y)) and in the east-west dire tion is 2 E�W = 2�2(1� �(hx; �x)) where �is a valid orrelation fun tion and the ommon �2 avoids sill anisotropy.Another method that an apture range anisotropy, developed by Sampson and Gut-torp (1992), sets 2 (si � sj;�) = 2 Æ(f(si)� f(sj);�) (5)where 2 Æ is any valid isotropi variogram model, f is a smooth nonlinear mapping ofRm ! Rm over the region D and � is a set of variogram parameters. Their idea is to �nda smooth, inje tive transformation, f , that maps the original geographi al spa e or the Gplane to a spa e that is homogeneous, the H plane, in two steps. First, a multidimensionals aling algorithm �nds a mapping of the G plane to the H plane by minimizing a stress riterion over all monotoni ally in reasing fun tions. Then, the mapping is smoothedby thin-plate splines to produ e f . The hoi e of the smoothing parameter is riti albe ause the algorithm need not produ e an inje tive transformation, i.e., f may fold overitself. When f is not inje tive, over�tting o urs yielding poor predi tion. The variogram2 Æ is then �t by distribution free methods to the transformed lo ations. Smith (1996)extends these ideas by allowing 2 Æ to be nonmonotoni and employing likelihood basedinferen e. He onstru ts f to be a linear ombination of radial basis fun tions. As withthe Sampson and Guttorp method, f need not be inje tive; hen e, Smith advo ates usinga small number of radial basis fun tions. The Sampson and Guttorp method and theSmith extension provide a valid variogram or ovarian e stru ture in the H plane, butneed not indu e a valid variogram in the G plane, thus limiting their usefulness.We seek a ri h lass of valid variogram models within whi h to investigate generalrange anisotropy in the G plane. Re all that, in the isotropi ase, the range is only5

de�ned for a monotoni variogram. Extension to range anisotropy in Rm suggests thatwe on�ne ourselves to valid m-dimensional orrelation fun tions, �(h), su h that, forany unit ve tor hkhk , �(Æ hkhk) is stri tly de reasing in Æ. Then, for ea h ve tor h withasso iated angle �, �(Æ hkhk) = 0:05 has a unique solution whi h we denote as r�, i.e., therange in h's dire tion. A onvenient lass whi h ensures su h monotoni ity takes theform �(h1; h2; : : : ; hm) = �1(h1) � �2(h2) � : : : � �m(hm) (6)where �j(�) is a positive, valid, stri tly de reasing, one-dimensional orrelation fun tion.Thus, � arises as the multivariate hara teristi fun tion asso iated with m independentrandom variables, ea h symmetri about 0.In the plane, i.e., m = 2, we note that if �1(�) and �2(�) yield almost everywhere ontinuous sample paths for asso iated pro esses W1(�) and W2(�), then almost ev-erywhere ontinuous realizations of the pro ess W (�) asso iated with �(h1; h2) result.This is evident by noting that, sin e Corr(W (si; sj);W (si + h1; sj)) = �1(jh1j) andCorr(W (si; sj);W (si; sj + h2)) = �2(jh2j); we have ontinuity along all verti al andhorizontal lines and therefore, by rotation, over R2. Sample fun tion ontinuity in one-dimension is well studied (Cram�er and Leadbetter, 1967, hapter 4). This argument isreadily extended to general m in (6). It an also be extended if �1 and �2 are smoother.For instan e, if �1(�) and �2(�) yield almost surely di�erentiable sample paths then almosteverywhere di�erentiable realizations are asso iated with �(h1; h2).How shall we hoose the �j's in (6)? Consider the lass of all isotropi orrelationfun tions valid in m dimensions for every m, denoted by D1. From S hoenberg (1938,p 822), all � in D1 are ompletely monotone. We shall onsider, in parti ular, two lasses of orrelation fun tions within D1; see Yaglom (1987, pp 361-366) for additionalmembers. The �rst is the Mat�ern lass (Mat�ern, 1986, pp 17-18) whi h is de�ned by�(d; �; �) = 12��1�(�) (�d)�K�(�d) (7)where � � 0, � > 0 and K�(�) is a modi�ed Bessel fun tion of the third kind of or-der � (Abramowitz and Stegun, 1965, Chapter 10). Here, the parameter � ontrols thesmoothness of the asso iated random �eld. Note that � � 0 ensures ontinuous real-6

izations. In general, realizations are d� � 1 times (mean square) di�erentiable whered is the integer eiling fun tion (Hand o k and Stein, 1993, p 406). The ase � = 12provides the exponential orrelation fun tion and when � !1, we obtain the Gaussian orrelation fun tion. The latter has almost surely analyti realizations, dis ouraging itsuse in pra ti e.The general exponential orrelation fun tion is a se ond lass in D1. Its form is�(d;�; �) = exp(��jdj�) (8)where 0 < � � 2 and � > 0. Obviously, � = 1 provides the usual exponential orrelationfun tion while � = 2 gives the Gaussian. Though (8) is omputationally easier to workwith than (7), less appealing sample fun tion behavior results. For � < 2, we obtain ontinuous, but nondi�erentiable realizations. Abruptly, when � = 2, we obtain analyti realizations. In other words, ex luding � = 2, (8) pla es no mass on di�erentiablerealizations.Note that adopting (7) in preferen e to (8) due to the ri her s ope of sample be-havior is a me hanisti de ision. Di�erentiability is a property of the spatial pro essthat produ ed the data. The observed sample provides little insight into the underlyingsmoothness of the spatial pro ess. Moreover, in pra ti e, the data will not easily distin-guish, in parti ular near the variogram origin, the respe tive two parameter orrelationspe i� ations in (7) and (8).In working with (7) and (8), for ea h j we an enri h the lass by using mixtures.For instan e with (8), let �j(hj) = njX̀=1 pj` exp(��j`jhjj�j`);where 0 < �j` < 2 and Pnj`=1 pj` = 1; j = 1; 2. Hen e, �1 and �2 provide ontinuouspro ess realizations. We an introdu e a linear transformation h0 = Ah whi h yields thegeneral exponential range anisotropi family�GE(h1; h2;�1;�2;�1;�2; A) = n1X̀=1 p1` exp(��1`ja11h1 + a12h2j�1`)!� n2X̀=1 p2` exp(��2`ja21h1 + a22h2j�2`)! (9)7

with obvious de�nitions for �1, �2, �1 and �2. Note that with A unknown, only nj � 1of the �j`'s are identi�able. In the sequel, we illustrate (9) with n1 = n2 = 1, yieldingthe six parameter range anisotropi general exponential lass�GE(h1; h2) = exp(�(ja11h1 + a12h2j�1 + ja21h1 + a22h2j�2)): (10)We note that (10) redu es to geometri anisotropy, i.e., ellipti al variogram ontours,when a12 = a21 and �1 and �2 ! 2 and isotropy, i.e., ir ular variogram ontours, if, inaddition, a12 = 0 = a21 and a11 = a22 = q�2 , say. General ontours of (10) arise fromja11h1 + a12h2j�1 + ja21h1 + a22h2j�2 = k:Figure 2 exhibits the exibility of (10) using di�erent hoi es for A;�1 and �2 with k = 1.For (10), the range in dire tion h with asso iated angle �, r�, is shown in the Appendixto be the solution to the equation 1r�1� + 2r�2� = 3 (11)for �xed 1; 2; �1; �2 where 1 = ja11 os � + a12 sin �j�1 and 2 = ja21 os � + a22 sin �j�2.Ex ept in the spe ial ase when �1 = �2, evaluation of (11) requires a �xed point solution(easily found using a bise tion algorithm). When �1 = �2 = �, we �nd r� = � 3 1+ 2 �1=�.Analogous to (9), we obtain the Mat�ern range anisotropi family,�M (h1; h2;�1;�2;�1;�2; A) = n1X̀=1 p1` 12�1`�1 1�(�1`)(�1`ja11h1 + a12h2j)�1`� K�1`(�1`ja11h1 + a12h2j))� n2X̀=1 p2` 12�2`�1 1�(�2`)(�2`ja21h1 + a22h2j)�2`� K�2`(�2`ja21h1 + a22h2j)) (12)for Pnj`=1 pj` = 1; j = 1; 2. Again, ontinuous pro ess realizations result, but additionalsmoothness will vary with dire tion. As with the general exponential lass, we illustrate(12) with n1 = n2 = 1 whi h yields the six parameter range anisotropi Mat�ern family�M (h1; h2) = 12�1+�2�2 1�(�1)�(�2)(ja11h1 + a12h2j)�1(ja21h1 + a22h2j)�2� K�1(ja11h1 + a12h2j)K�2(ja21h1 + a22h2j): (13)8

We note that (13) redu es to geometri anisotropy, i.e., ellipti al variogram ontours,when a12 = a21 and �1 and �2 ! 1 and isotropy if, in addition, A = �I. General ontours of (13) arise from setting �M = k. Figure 3 exhibits the exibility of (13) usingdi�erent ombinations of A; �1 and �2 for various hoi es of k. In parti ular, noti e how lose to isotropy Figure 3b is with �1 = 3 = �2. The range in dire tion h with angle �,r�, solves �M (r�) = 0:05 and is al ulated using a bise tion algorithm.Lastly, an alternative onstru tion of range anisotropi lasses in Rm follows from theobservation that if �k(h), k = 1; : : : ; p is set of p validm-dimensional isotropi orrelationfun tions, ea h positive and monotoni , then Qpk=1 �k(h0Akh), where Ak = B 0kBk, is avalid orrelation fun tion in Rm. (It is the hara teristi fun tion of the random variablePpk=1BkXk where theXk are independent with hara teristi fun tion �k.) It is obviouslymonotoni in ea h dire tion. Smoothness of the asso iated pro ess ould be investigatedusing Theorem 1 in Kent (1989).3 Spatial Modeling IssuesAssumeY follows an N dimensionalmultivariate normal distribution with mean� = X�and ovarian e matrix �(�). The mean stru ture has ith omponent,Pj X(si)j�j whereX(si)0 = 1. In the absen e of site level ovariates, the mean be omes a polynomial inthe spatial lo ations and is alled a trend surfa e (Upton and Fingleton, 1985, pp 322-329). �(�) is expressed as � 2I + �2H(�) where H(�)ij = �(si � sj;�) and � is a valid orrelation fun tion parameterized by �.The multivariate normal model Y � N(�;�(�)) an be expressed as a random e�e tsmodel (Diggle, Liang and Zeger, 1994, p 87)Y = X� +W + � (14)where W = (w(s1); w(s2); : : : ; w(sN ))0 � N(0; �2H(�)) is a ve tor of spatial e�e tsindependent of � = (�(s1); �(s2); : : : ; �(sN))0 � N(0; � 2I) whi h orresponds to the mea-surement error and/or the mi ros ale e�e t.The marginal model in (14) an be viewed as a hierar hi al spe i� ation. LettingV = X� +W , the �rst stage is Y jV ; � 2 � N(V ; � 2I), i.e., onditionally independent9

Y 's given V . The se ond stage is V j�; �2;� � N(X�; �2H(�)) des ribing the spatialasso iation. Obvious extension would allow a non-Gaussian �rst stage for the Y 's withV still the mean ve tor, as in Diggle, Tawn and Moyeed (1998). It is often onvenient touse the hierar hi al form to �t the model, parti ularly if simulation-based model �ttingsu h as Markov Chain Monte Carlo is employed.An important obje tive of spatial modeling is predi tion at sampled or unsampledlo ations or at a holdout set to he k model performan e. Let s01; s02; : : : ; s0L denotesites at whi h we want to predi t the response Y 0 = (Y (s01); Y (s02); : : : ; Y (s0L))0. Thenthe joint distribution of the observed sites Y and unsampled sites Y 0 is0B� YY 0 1CA � N 0B�264 X�X0� 375 ;264 �11 �10�010 �00 3751CA (15)where X� and �11 are the mean ve tor and ovarian e matrix of the data given above.The predi tive mean X0� and ovarian e matrix �00 parallel these expressions. The ross- ovarian e matrix �10 has entries Cov(Y (si); Y (s0j)) = � 21(si = s0j)+�2H10(si�s0j ;�) where 1(si = s0j) indi ates whether the predi ted site oin ides with a sampledsite, i.e., 1(si = s0j) = 8><>: 1 if si = s0j0 if si 6= s0j :Then from (15), the onditional distribution of Y 0jY ;�;� is N(�0jY ;�0jY ) where�0jY = X0� + �010��111 (Y �X�)and �0jY = �00 � �010��111 �10:If � and � are known, then the optimal predi tor is �0jY . If not, ustomary kriging(Kitanidis, 1986, p 500) repla es them with estimates and then treats these estimates asknown parameters; however in doing so, we ignore their un ertainty. The fully Bayesianapproa h obtains the standard posterior predi tive distributionp(Y 0jY ) = Z� Z� p(Y 0jY ;�;�)�(�;�jY )d�d� (16)10

where �(�;�jY ) is the posterior distribution, taking, e.g., E(Y 0jY ) as the predi tor.Kitanidis (1986) provides analyti expressions for (16) when �11 is known up to a s ale onstant, but otherwise, (16) is unavailable expli itly. However, given samples from theposterior, simulation of realizations from (16) is straightforward.A substantial body of literature fo uses on predi tion from a Bayesian perspe tive.Hand o k and Stein (1993), Hand o k and Wallis (1994), Gaudard et. al (1999) andDe Oliveira, Kedem and Short (1997) all predi t with (16) using �11 = �2H(�). Le andZidek (1992) and Brown, Le and Zidek (1994) use a hierar hi al model for the ovarian ematrix by pla ing a Wishart prior on �11, hen e, transferring the spe i� ation of thefun tional form for the ovarian e matrix to the se ond stage. Other work in ludes Cui,Stein and Myers (1995) who repla e the variogram in the usual kriging equations (seeCressie, 1993) with the posterior mean of the variogram. The work of Woodbury (1989),Abrahamsen (1993) and Omre (Omre (1987), Omre (1988), Omre and Halvorsen (1989)Omre, Halvorsen and Berteig (1989) and Hjort and Omre (1994)) is partially Bayesianin the sense that prior spe i� ation of the mean parameters � and ovarian e fun tionare eli ited; however, no distributional assumption for Y is made.All of the forgoing work ex ept Diggle, Tawn and Moyeed (1998) fun tionally assumes� 2 = 0, i.e., no measurement error/mi ros ale e�e t is modeled. Under our model, (14),one might want to analogously predi t only the signal (without the noise). Then, anoiseless predi tive distribution, �((X0� +W (s0))jY ), ould be employed rather than(16). With regard to predi tion, both have the same mean, but the latter is more dis-persed. Letting W 0 =W (s0), simulation-based examination of the noiseless predi tivedistribution requires posterior samples from p(�;W 0jY ), whi h an be obtained throughp(�;W 0jY ) = Z p(�;W 0j�;W ;Y )�(�;W jY )d�dW= Z p(W 0j�;W )�(�;�;W jY )d�dW= Z p(W 0j�2;�;W )p(W j�;�;Y )�(�;�jY )d�dW : (17)In (17), p(W 0j�2;�;W ) � N(H 010H�1W ; �2(H00 �H 010H�1H10)). Using, e.g., Lindleyand Smith (1972), p(W j�;�;Y ) is multivariate normal with mean ( 1�2 I + 1�2H�111 )�1 �1�2 (Y � X�) and ovarian e matrix ( 1�2 I + 1�2H�111 )�1. Again, given posterior samples,11

draws from (17) are straighforward.4 Prior Spe i� ations and Model FittingWith (10) and a given mean spe i� ation, the likelihood L(�;�jY ) is determined. We hoose the fun tional form for the prior to be�(�;�) = �(� 2; �2; A; �1; �2)�(�) = �(� 2)�(�2)�(A)�(�1)�(�2)�(�) (18)where we have olle ted the a's into a nonsingular matrix A = 264 a11 a12a21 a22 375 : Althoughwe do not believe that the ovarian e parameters, �, are truly independent, spe ifyinga prior dependen e stru ture is diÆ ult to justify. Rather, we let the data modify ourprior independen e assumption through the posterior.In (18), it is onvenient to assume that both � 2 and �2 have inverse gamma priorswith, in the interest of vagueness, shape parameter equal to two (whi h implies an in�niteprior varian e). For �1 and �2, we hoose uniform(0,2) priors. Under (13), we repla e �1,�2 in (18) with �1, �2, again taking inverse gamma priors with shape parameter equal totwo. The inverse gamma s ale parameter an be hosen to provide any desired a priori(mean) smoothness for the spatial pro ess. We model A through the positive de�niteA0A as Wishart (�;R) where�(A) / jA0Aj��2�12 exp��12tra e(A0A�R�1)� :In this formulation, E(A0A) = R and � � 2 is a pre ision parameter. We set R = I for an appropriate s ale and � = 2, re e ting entering around isotropy withminimal pre ision, i.e, large varian e of �(A). Finally, � would re eive a suitably vaguemultivariate normal prior.The model in (14) with prior (18) an be �tted using noniterative Monte-Carlo witha suitable importan e sampling density (ISD) denoted by g(�;�). We draw (�1;�1),(�2;�2); : : : ; (�V ;�V ) � g(�;�) to form weights vi = f(Y j�i;�i)�(�i;�i)=g(�i;�i).After al ulating qi = viPvi ; i = 1; : : : ; V , Monte-Carlo integration for any posteriorexpe tation, E(b(�;�)jY ), takes the form P qib(�i;�i) while resampling the (�i;�i)12

using the probabilities qi provides an approximate sample from the posterior (Smithand Gelfand, 1992). g is obtained using West's (1993) adaptive mixture method aftertransforming all parameters to R1.5 Analysis of the S allop DataSin e 1982, the NMFS has been annually sampling s allops o� the Northeastern UnitedStates' oastline from the Delmarva Peninsula to the Georges Bank. We analyze s allop ounts at the 146 sites olle ted in 1993 in the New York Bight region whi h is lo atedo� the New Jersey and Long Island oast. Counts range from 0 to 726. A data set onsisting of 148 s allop at hes in 1990 in the New York Bight (E ker and Gelfand,1997) was used to formulate some of the hyperparameters for the prior distributions in(18), as des ribed in the previous se tion. Figure 1 displays the sites for both the 1990and 1993 s allop data. We randomly sele t L = 10 of the 146 sites for the 1993 data toform a holdout set whi h is used to ompare the predi tive performan e of the two rangeanisotropi models.After transforming the geographi al oordinates using a Universal Transverse Mer a-tor (UTM) proje tion, we adopt a onstant mean spe i� ation, i.e., X� = �1 in (14). InFigure 4, we al ulate the ESC plot using (4), overlaid on the bins with their respe tive ounts. Inspe ting Figure 4, it appears that isotropy should not be assumed althoughgeometri anisotropy might model the ESC plot reasonably well.Table 1 gives posterior means, standard deviations and 95% interval estimates for thegeneral exponential and Mat�ern range anisotropi models, (10) and (13), respe tively.For the general exponential model, the posterior mean for �1 is 1.322 and 0.948 for �2,illustrating the departure from geometri anisotropy. For the Mat�ern model, the posteriormean for �1 is 1.305 and 1.438 for �2 whi h indu es a fairly rough, non-di�erentiablespatial pro ess. The similarity of these two values suggests that the smoothness of thepro ess is insensitive to dire tion.Figure 5 provides ontours of the posterior mean of the general exponential semivar-iogram whi h models the ESC plot (Figure 4). Contours for the Mat�ern model reveal a13

Table 1: Posterior means and 95% interval estimates for varian e omponent parameters.General Exponential Mat�ernMean St. Dev. Interval Mean St. Dev. Interval� 2 1.076 0.451 (0.41,2.14) � 2 1.372 0.293 (0.89,2.08)�2 2.547 1.378 (0.89,6.34) �2 2.479 1.263 (1.01,5.88)Sill 3.624 1.356 (1.91,7.42) Sill 3.851 1.213 (2.43,7.09)�1 1.322 0.419 (0.42,1.90) �1 1.305 0.954 (0.37,4.05)�2 0.948 0.457 (0.17,1.79) �2 1.438 1.370 (0.31,5.47)similar shape in Figure 6. The posterior mean and intervals for the range as a fun tionof angle for the general exponential semivariogram are presented in Figure 7 with themaximum range (strongest orrelation) roughly parallel to the oastline (� 60 � 70Æ).The analogous plot for the Mat�ern model (Figure 8) provides similar information.Table 2 reveals, for the holdout set, the predi tions of the noiseless and the standardpredi tive distributions for both the general exponential and the Mat�ern models. Thenoiseless and standard predi tive means should agree in ea h respe tive ase. They are lose, with the di�eren e providing an indi ation of the a ura y of the simulation-based�tting. As expe ted, the varian es of the noiseless predi tion are smaller than those ofthe standard predi tive distribution. Comparing the standard predi tive distributionsfor the Mat�ern and the general exponential models, they provide essentially equivalentpredi tion. However, the noiseless Mat�ern has a more pre ise (smaller) predi tion vari-an e than that of the noiseless general exponential, while their respe tive means areapproximately equal. This smaller varian e may possibly be attributed to the nuggetpoint estimate for the Mat�ern model being slightly larger than that for the general ex-ponential (although their respe tive interval estimates greatly overlap). Assessing theperforman e of all four predi tions, the observed holdout value falls within a two stan-dard deviation predi tion interval, ex ept for the noiseless Mat�ern, whi h requires a threestandard deviation interval to over the true value for all ten holdout sites.14

Table 2: A tual (log s aled) responses together with means and standard deviations forthe noiseless predi tive distribution and the standard predi tive distribution for the twomodels evaluated at the ten holdout sites.General Exponential Mat�ernNoiseless �(Y0jY ) Noiseless �(Y0jY )Site Y (s0`) Mean St. Dev. Mean St. Dev. Mean St. Dev. Mean St. Dev.1 1.946 2.194 0.804 2.181 1.331 2.161 0.487 2.169 1.3322 1.792 2.802 0.913 2.745 1.372 2.862 0.662 2.867 1.3623 4.007 3.604 0.905 3.666 1.369 3.801 0.617 3.845 1.3744 4.331 4.338 0.785 4.318 1.325 4.359 0.523 4.396 1.2805 5.501 4.430 0.928 4.463 1.330 4.298 0.637 4.284 1.3116 5.645 4.387 0.725 4.456 1.309 4.263 0.528 4.238 1.2567 5.620 4.142 0.820 4.131 1.369 4.227 0.619 4.199 1.3378 4.394 3.659 0.808 3.718 1.374 3.633 0.534 3.565 1.2859 3.332 2.744 0.781 2.756 1.240 2.766 0.508 2.748 1.28010 0 1.215 0.841 1.216 1.304 1.484 0.653 1.456 1.30115

AppendixTo al ulate the range as a fun tion of angle for the general exponential model, let~h� = (~hx�; ~hy�) = ( os �; sin�) be a unit ve tor in R2 in dire tion �. Following M Bratneyand Webster (1986), the range r in dire tion � or r� solves �(~hx�r�; ~hy�r�;�) = 0:05. Thusfrom (10), solveexp(�(ja11(~hx�r�) + a12(~hy�r�)j�1 + ja21(~hx�r�) + a22(~hy�r�)j�2)) = 0:05:Simplifying, we see thatja11(r� os �) + a12(r� sin �)j�1 + ja21(r� os �) + a22(r� sin �)j�2 = � log(0:05) � 3:Sin e r� � 0,r�1� ja11 os � + a12 sin �j�1 + r�2� ja21 os � + a22 sin �j�2 = � log(0:05) � 3is of the form 1r�1� + 2r�2� = 3where 1 = ja11 os � + a12 sin �j�1 and 2 = ja21 os � + a22 sin �j�2.Referen esAbrahamsen, N. (1993). Bayesian Kriging for Seismi Depth Conversion of a Multi-Layer Reservoir. In A. Soares, editor, Geostatisti s Troia, `92. Boston: KluwerA ademi Publishers. pp 385-398.Abramowitz, M and Stegun, I.A. (1965). Handbook of Mathemati al Fun tions. NewYork: Dover.Borgman, L. and Chao, L. (1994). Estimates of a Multidimensional Covarian e Fun tionin Case of Anisotropy. Math Geology. 26(2), pp 161-179.Brown, P., Le, N. and Zidek, J. (1994). Multivariate Spatial Interpolation and Exposureto Air Pollutants. The Canadian Journal of Statisti s. 22(4), pp 489-509.16

Cram�er, H. and Leadbetter, M.R. (1967). Stationary and Related Sto hasti Pro esses. NewYork: John Wiley and Sons.Cressie, N. (1993). Statisti s for Spatial Data. New York: John Wiley and Sons.Cui, H., Stein, A. and Myers, D. (1995). Extension of Spatial Information, BayesianKriging and Updating of Prior Variogram Parameters. Environmetri s. 6, pp373-384.De Oliveira, V., Kedem, B. and Short, D. (1997). Bayesian Predi tion of TransformedGaussian Random Fields. Journal of the Ameri an Stat. Asso . 92, 1422-1433.Diggle, P., Liang, K-L. and Zeger, S. (1994). Analysis of Longitudinal Data. New York:Oxford S ien e Publi ations.Diggle, P.J., Tawn, J.A. and Moyeed, R.A. (1998). Model-based Geostatisti s (withdis ussion). Applied Statisti s. 47, pp 299-350.E ker, M. and Gelfand, A. (1997). Bayesian VariogramModeling for an Isotropi SpatialPro ess. Journal of Agri ultural, Biologi al and Environmental Statisti s. 2, pp347-369.E ker, M. and Gelfand, A. (1999). Bayesian Modeling and Inferen e for Geometri allyAnisotropi Spatial Data. Math Geology. 31(1), pp 67-83.Gaudard, M., Karson, M., Linder, E. and Sinha, D. (1999). Bayesian Spatial Predi tion.Environmental and E ologi al Statisti s. (to appear).Hand o k, M. and Stein, M. (1993). A Bayesian Analysis of Kriging. Te hnometri s.35(4), pp 403-410.Hand o k, M. and Wallis, J. (1994). An Approa h to Statisti al Spatial-Temporal Mod-eling of Meteorologi al Fields (with dis ussion). Journal of the Ameri an Stat. As-so . 89(426), pp 368-390.Hjort, N. and Omre, H. (1994). Topi s in Spatial Statisti s (with dis ussion). S andi-navian Journal of Statisti s. 21, pp 289-357.17

Journel, A. and Huijbregts, C. (1978).Mining Geostatisti s. New York: A ademi Press.Kent, J. (1989). Continuity Properties for Random Fields. The Annals of Probability.17(4), pp 1432-1440.Kitanidis, P. (1986). Parameter Un ertainty in Estimation of Spatial Fun tions: BayesianAnalysis. Water Resour es Resear h. 22(4), pp 499-507.Krajewski, P., Molinska, A. and Molinska, K. (1996). Ellipti al Anisotropy in Pra ti e- A Study of Air Monitoring Data. Environmetri s. 7, pp 291-298.Le, N. and Zidek, J. (1992). Interpolation with Un ertain Spatial Covarian es: ABayesian Alternative to Kriging. Journal of Multivariate Analysis. 43, pp 351-374.Lindley, D. and Smith A. (1972). Bayes Estimates for the Linear Model (with dis us-sion). J. R. Statist. So . B. 34, pp 1-41.Mat�ern, B. (1986). Spatial Variation. New York: Springer.Matheron, G. (1963). Prin iples of Geostatisti s. E onomi Geology. 58, pp 1246-1266.M Bratney, A. and Webster, R. (1986). Choosing Fun tions for Semi-variograms of SoilProperties and Fitting them to Sampling Estimates. Journal of Soil S ien e. 37,pp 617-639.Omre, H. (1987). Bayesian Kriging - Merging Observations and Quali�ed Guesses inKriging. Math Geology. 19(1), pp 25-39.Omre, H. (1988). A Bayesian Approa h to Surfa e Estimation. In C.F. Chung et al. ed-itors, Quantitative Analysis of Mineral and Energy Resour es. Boston: D. ReidelPublishing Co. pp 289-306.Omre, H. and Halvorsen, K.B. (1989). The Bayesian Bridge Between Simple and Uni-versal Kriging. Math Geology. 21(7), pp 767-786.18

Omre, H., Halvorsen, K.B. and Berteig, V. (1989). A Bayesian Approa h to Kriging.In M. Armstrong, editor, Geostatisti s. Boston: Kluwer A ademi Publishers. pp109-126.Sampson, P. and Guttorp, P. (1992). Nonparametri Estimation of Nonstationary Spa-tial Covarian e Stru ture. Journal of the Ameri an Stat. Asso . 87(417), pp 108-119.S hoenberg, I. (1938). Metri Spa es and Completely Monotone Fun tions. Annals ofMathemati s. 39, pp 811-841.Smith, A.F.M. and Gelfand, A.E. (1992). Bayesian statisti s without tears: A sampling-resampling perspe tive. Ameri an Statisti ian. 46(2), pp 84-88.Smith, R. (1996). Estimating Nonstationary Spatial Correlations. Te hni al Report,Department of Statisti s. University of North Carolina.Upton, G. and Fingleton, B. (1985). Spatial Data Analysis by Example. New York:John Wiley and Sons.Ve hia, A. (1988). Estimation and Model Identi� ation for Continuous Spatial Pro- esses. J. R. Statist. So . B. 50(2), pp 297-312.West, M. (1993). Approximating Posterior Distributions by Mixtures. J. R. Statist.So . B. 55(2), pp 563-586.Woodbury, A. (1989). Bayesian Updating Revisited. Math Geology. 21(3), pp 285-308.Yaglom, A. (1987). Correlation Theory of Stationary and Related Random Fun tionsI. New York: Springer-Verlag.Zimmerman, D. (1993). Another Look at Anisotropy in Geostatisti s. Math Geol-ogy. 25(4), pp 453-470. 19

••••••

••••••••• •• •••••••••

••••••••

•••••••

•••••

••••••••••

•••••••• ••

•••••••

••••••••

•••• • ••• •• •

•••••• ••

•••••• •••••••• •

••••••••••••••••••••

••••••••• ••

••

1990 Scallop Sites

••••••

•••••••

• ••••••

••••••••• •• •••••••••••••• •••••••••••••••

••••••••

• ••••• •• •

••• ••••• • ••••• •••••••••• ••••

•••••••••• ••

•••• • •

• • • ••• ••••••

•••••••

••••

1993 Scallop Sites

Figure 1: Sites sampled in the Atlanti O ean for 1990 and 1993 s allop at h data.20

h, E-W

h,

N-S

-1.0 -0.5 0.0 0.5 1.0

0.0

0.2

0.4

0.6

0.8

1.0

a)

h, E-W

h,

N-S

-1.0 -0.5 0.0 0.5 1.00

.00

.20

.40

.60

.81

.0

b)

h, E-W

h,

N-S

-1.0 -0.5 0.0 0.5 1.0

0.0

0.2

0.4

0.6

0.8

1.0

c)

h, E-W

h,

N-S

-1.0 -0.5 0.0 0.5 1.0

0.0

0.2

0.4

0.6

0.8

1.0

d)

Figure 2: Illustrative general exponential range anisotropi ontours in R2 under (10). a)has (A;�1; �2) = (2;�0:75;�0:75; 1:5; 1; 0:5), b) (1; 0; 0; 2; 1:5; 0:5), ) (1; 0; 0; 2; 0:5; 0:5)and d) (1; 1;�1; 0:75; 0:8; 1:25). 21

h, E-W

h,

N-S

-1.0 -0.5 0.0 0.5 1.0

0.0

0.2

0.4

0.6

0.8

1.0

a)

h, E-W

h,

N-S

-1.0 -0.5 0.0 0.5 1.00

.00

.20

.40

.60

.81

.0

b)

h, E-W

h,

N-S

-1.0 -0.5 0.0 0.5 1.0

0.0

0.2

0.4

0.6

0.8

1.0

c)

h, E-W

h,

N-S

-1.0 -0.5 0.0 0.5 1.0

0.0

0.2

0.4

0.6

0.8

1.0

d)

Figure 3: Illustrative Mat�ern range anisotropi ontours in R2 under (13). a) has(A; �1; �2) = (1;�0:75;�0:75; 1; 0:5; 0:5), b) (2; 0; 0; 2; 3; 3), ) (2; 0:75; 0:75; 2; 0:75; 0:25)and d) (1; 1;�1; 0:75; 0:8; 1:25). 22

• • • • • • • •

• • • • • • • •

• • • • • • •

• • • • • • •

• • • • • • •

• • • • • • •

• • • • • • •

• • • • • •

• • • • •

h, E-W

h,

N-S

-50 0 50 100

02

04

06

08

01

00

12

0

1.5

22.52.5

3 3

3

3

3

3.5

3.5

3.5

3.5

4

4

4

4.5

55.5

66.5

6.5

6.5

6.57

7

7

77.5

7.58

25 90 265 449 299 129 41 24

23 69 253 377 336 174 65 27

69 204 289 333 218 87 40

65 155 243 314 247 105 53

48 82 185 288 254 132 55

43 57 135 229 261 162 61

27 42 85 195 243 172 62

27 72 133 190 179 68

58 86 173 157 71

Figure 4: ESC plot for the 1993 s allop data.23

hx (km)

hy (

km

)

-50 0 50 100

02

04

06

08

01

00

12

0

1.51.7522.252.52.7533.25 3.253.25

3.5

3.5Figure 5: Posterior mean ontours of the semivariogram for the general exponentialmodel. 24

hx (km)

hy (

km

)

-50 0 50 100

02

04

06

08

01

00

12

0

1.51.7522.252.52.753

3

3.25

3.5

3.5

3.5

3.75

Figure 6: Posterior mean ontours of the semivariogram for the Mat�ern model.25

Angle (degrees)

Ra

ng

e (

km

)

0 50 100 150

02

00

40

06

00

80

01

00

01

20

0

MeanLowerUpper

Figure 7: Posterior mean and interval estimates of range by angle for the general expo-nential model. 26

Angle (degrees)

Ra

ng

e (

km

)

0 50 100 150

01

00

20

03

00

40

05

00

MeanLowerUpper

Figure 8: Posterior mean and interval estimates of range by angle for the Mat�ern model.27