Nonseparable Space-Time Covariance Models: Some Parametric Families 1

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1 Mathematial Geology, ol. 34, No. 1, January 2002 ( C 2002) Nonseparale Spae-Time Covariane Models: Some Parametri Families 1 S. De Iao, 2 D. E. Myers, 3 and D. Posa 4,5 By extending the produt and produt sum spae-time ovariane models, new families are generated as integrated produts and produt sums. These inlude nonintegrale spae-time ovariane models not otainale y the Cressie Huang representation. It is shown how to fit the spatial and temporal omponents of the models as well as the proaility density funtion. The methods are illustrated y a ase study. KEY WORDS: produt sum models, integrated models, separaility, admissiility. INTRODUCTION While there are no diffiulties in extending the various kriging estimators and the kriging equations to the spae-time setting, there has een a lak of known valid spae-time ovarianes and variograms. The ovious possiility for extending to spae-time involves the use of a zonal anisotropy: the diffiulties assoiated with this method are disussed in Myers and Journel (1990) and Rouhani and Myers (1990). A reent review of geostatistial spae-time models was given y Kyriakidis and Journel (1999). In order to estimate the orrelation of a spae-time proess, the main questions are as follows: Is it useful and does it make sense to define a spatio-temporal metri, suh as d(u 1, u 2 ) = (a(x 1 x 2 ) 2 + (y 1 y 2 ) 2 + (t 1 t 2 ) 2 ) 1/2, 1 Reeived 11 May 2000; aepted 10 January Faoltà di Eonomia, G. d Annunzio, via dei estini, Chieti, Italy; sdeiao@ tisalinet.it 3 Department of Mathematis, University of Arizona, Tuson, Arizona 85721; myers@math. arizona.edu 4 Dipartimento di Sienze Eonomihe e Matematio-Statistihe, ia per Monteroni, Eotekne, Lee, Italy; posa@eonomia.unile.it 5 IRMA-CNR, via Amendola 122/I, Bari, Italy; irmadp08@area.a.nr.it /02/ /1 C 2002 International Assoiation for Mathematial Geology

2 24 De Iao, Myers, and Posa with u 1 = (x 1, y 1, t 1 ), u 2 = (x 2, y 2, t 2 ), where (x 1, y 1 ), (x 2, y 2 ) D R 2, and t 1, t 2 T R, where D and T are the spatial and temporal domains, respetively. In general the units for spae and time will e disparate, e.g., meters and hours. How to hoose a spae-time ovariane or variogram model and how to hoose parameters to ensure that the est fit to data is ahieved? One of the ojetives of this paper is to furnish answers to the aove questions; moreover, starting from the produt sum ovariane model (De Cesare, Myers, and Posa, 2001) and y using the staility properties, some nonseparale parametri families of spae-time ovariane funtions have een derived. It is important to point out that this new lass of models annot e otained, in general, from the Cressie Huang representation (Cressie and Huang, 1999). SOME PARAMETRIC FAMILIES OF SPACE-TIME STATIONARY COARIANCES Most spae-time ovariane or variogram models, in literature, have een derived y utilizing the following theoretial results, sine ovarianes or variograms in R n an e otained, in general, from other valid funtions. 1. From the onvexity property of the family of ovarianes, if C 1 (h) and C 2 (h) are ovarianes in R n and > 0, then oth C 1 (h) + C 2 (h) and C 1 (h) are ovarianes in R n. The same results hold for the family of variograms. 2. From the first staility property (Chilès and Delfiner, 1999, p. 60), if C 1 (h) and C 2 (h) are ovarianes in R n, then their produt, C 1 (h) C 2 (h), is still a ovariane in R n. 3. From the seond staility property (Chilès and Delfiner, 1999), if µ(a) is a positive measure in U Rand C(x, y; a) is a ovariane funtion in R n for eah a U, whih is integrale over the suset of U for every pair (x, y), then C(x, y), defined as C(x, y) = C(x, y; a) dµ(a), is a ovariane in R n (Matern, 1980, p. 10). 4. Representing the random field in R n as a omination of independent omponents in the separate domains, one an utilize the following separale models: a) the fatorized or separale ovariane (Chilès and Delfiner, 1999): n C(h) = C i (h i ), (1) i=1

3 Nonseparale Spae-Time Covariane Models 25 where Z(u) = n i=1 Z i(u i ) and the Z i s are unorrelated random fields in R; ) the nested struture variogram: γ (h) = n γ i (h i ), (2) i=1 where Z(u) = n i=1 Z i(u i ) and the Z i s are unorrelated random fields in R. In oth ases h i are the omponents of the vetor h and C i (h i ) and γ i (h i ) are, respetively, ovarianes and variograms in R, i = 1,...,n. Hene, under the onvenient assumption of treating spae and time separately, the fatorized ovariane (1) and the nested variogram (2) represent one of the first attempts to generate parametri families of spae-time ovarianes and variograms. The produt model (De Cesare, Myers, and Posa, 1997; Posa, 1993; Rodriguez- Iture and Mejia, 1974) and the nested model (Rouhani and Hall, 1989) elong to this ategory. The nested model will in general not e stritly positive definite ut only semidefinite (Myers and Journel, 1990). The Produt Sum Covariane Model An extension of these two simple models (produt model and nested model) is given y the lass of produt sum ovariane models, introdued in De Cesare, Myers, and Posa (2001): C s,t (h s, h t ) = k 1 C s (h s )C t (h t ) + k 2 C s (h s ) + k 3 C t (h t ), (3) where C t and C s are valid temporal and spatial ovariane models, respetively. Note that these models are generally nonseparale. Cressie Huang Models Cressie and Huang (1999) have reently shown how to onstrut some nonseparale lasses of integrale spae-time ovarianes. They used the representation: C s,t (h s, h t ) = e ih R n s ω ρ(ω; h t )k(ω) dω,

4 26 De Iao, Myers, and Posa where ρ(ω; ) is a ontinuous autoorrelation funtion for eah ω R n, ρ(ω;h t )dh t <, R + k(ω) > 0 and k(ω) dω <. R n These spatial temporal ovarianes are generated y using Bohner s theorem and y hoosing an appropriate spetral density. NEW PARAMETRIC FAMILIES OF SPACE-TIME COARIANCE MODELS Both the produt sum and the Cressie Huang onstrutions will now e extended. Theorem 1. Let µ(a) e a positive measure over U R,let C s (h s ; a) and C t (h t ; a) e ovarianes, respetively, in D R n and T R +, for eah a U. (a) If C s (h s ; a) C t (h t ; a) is integrale with respet to the measure µ over for eah h s and h t, given k > 0, then C s,t (h s, h t ) = kc s (h s ;a)c t (h t ;a) dµ(a) (4) is a ovariane in D T. () Likewise, if k 1 C s (h s ; a)c t (h t ; a) + k 2 C s (h s ; a) + k 3 C t (h t ; a) is integrale with respet to the measure µ over for eah h s and h t, given k 1 > 0, k 2 0, and k 3 0, then C s,t (h s, h t ) = [k 1 C s (h s ; a)c t (h t ; a) + k 2 C s (h s ; a) + k 3 C t (h t ; a)] dµ(a) (5) is a ovariane in D T. This result follows from the seond staility property and the previous models, suh as the produt and produt sum ovariane models. Sine the produt and the produt sum ovariane models an e written in terms of the variograms (De Cesare, Myers, and Posa, 1997, 2001), from Theorem 1 it follows that γ s,t (h s, h t ) = k[c t (0; a)γ s (h s ; a) + C s (0; a)γ t (h t ; a) γ s (h s ; a)γ t (h t ; a)] dµ(a), (6)

5 Nonseparale Spae-Time Covariane Models 27 and γ s,t (h s, h t ) = [(k 2 + k 1 C t (0; a))γ s (h s ; a) + (k 3 + k 1 C s (0; a))γ t (h t ; a) k 1 γ s (h s ; a)γ t (h t ; a)] dµ(a), (7) where γ s (h s ; a) and γ t (h t ; a) are valid spatial and temporal variogram models for eah hoie of a, while C s (0; a) and C t (0; a) are the orresponding sill values. Remarks. In general produt sum ovariane models are not integrale over h s and h t, are not separale, and do not orrespond to the use of a spae-time metri. Models otained y an integrated produt sum representation will have the same harateristis. Although the produt ovariane models are separale and integrale, the integrated produt representation (4) an also produe nonseparale and nonintegrale models, as will e shown in the following examples. Hene, this new lass of models annot e otained, in general, from the Cressie Huang representation. Sine the omplex exponential an e written as e ih ω = os(h ω) + i sin(h ω) if ρ(ω; h t )k(ω) is symmetri aout the origin in R n, then Cressie Huang representation an e viewed as a speial ase of (4). Hene, C s,t (h s, h t ) = e ih R n s ω ρ(ω; h t )k(ω) dω = R n + C s (h s ; ω)c t (h t ; ω)k(ω) dω, where k(ω) is defined, positive, and integrale over R n + =R + R + n times; C s (h s ; ω) is only a positive semidefinite spatial ovariane funtion for eah ω R n + ; and C t(h t ; ω) is a temporal ovariane. In this speial ase only, the nonseparale Cressie Huang ovariane models ould e rewritten in terms of variograms as in (6) and take advantage of the property that variograms are zero at zero lag. All the ovariane models otained y Cressie and Huang satisfy the symmetry property; moreover, γ s,t (h s, h t ), γ s,t (h s, 0), and γ s,t (0, h t ) have the same sill values. By using nonseparale Cressie Huang models, one ould take into aount separate spae and time omponents only y adding them to the model onsidered, while in the aove produt sum and the integrated produt sum

6 28 De Iao, Myers, and Posa models, separate spatial and temporal strutures are a part of the models y onstrution. Oviously there are pratial prolems in hoosing among the parametri families of variograms that an e generated from (6) and (7), one that is losest to the empirial spae-time variogram. Both of these prolems are onsidered later. In the following setion, Theorem 1 is used to generate examples of parametri families of spae-time ovariane funtions. Some Examples By applying Theorem 1, a wide lass of parametri families of spae-time ovariane models are otained. Suppose that the measure µ is generated y an asolutely ontinuous funtion, then there exists a funtion φ suh that d (a) = φ(a) da almost everywhere. The following examples, ased on partiular hoies of isotropi ovarianes and funtions φ, show that one an otain other families y using the same hypothesis and riteria. The integrals onsidered in the following examples are easily evaluated (Gradshteyn and Ryzhik, 2000). Example 1. Given the following funtions: C s (h s ; a,,α) = e a hs α, 1 α 2, a >0, >0, C t (h t ;a,,δ) = e ahδ t, 1 δ 2, > 0, φ(a,n,) = n+1 Ɣ(n+1) an e a, n 0, >0, sine C s (h s ; a,,α) and C t (h t ; a,,δ) are, respetively, valid spatial and temporal ovariane models for eah hoie of a over the interval = [0; + [, the integraility onditions of Theorem 1 are satisfied and two new lasses of nonseparale spae-time ovarianes an e otained: C s,t (h s, h t ; Θ 1 ) = ke a hs α = kn+1 Ɣ(n + 1) = ( hs α e ahδ t n+1 Ɣ(n + 1) an e a da k n+1 a n e hs α a( + hδ t +) da + hδ t + ) n+1, (8)

7 Nonseparale Spae-Time Covariane Models 29 where Θ 1 = (,, n, k,α,,δ); C s,t (h s, h t ; Θ 2 ) = [ k 1 e a hs α e ahδ t + k 2 e a hs α + k 3 e ahδ t ] n+1 Ɣ(n + 1) an e a da = k 1 n+1 ( hs α + hδ t + ) n+1 + k 2 n+1 ( hs α + ) n+1 + k 3 n+1 ( h δ t + ) n+1, where Θ 2 = (,, n, k 1, k 2, k 3,α,,δ). Note that, when α = δ = 1 and n = 0, ( h s + h t ) in (8) and (9) might orrespond to a spae-time metri and would elong to a well known ( hs α + hδ t +) family of ovariane models, namely, C(h; w 1,w 2 )= w 1 w 2 + h. (9) Example 2. Given the funtions C s (h s ; a,,α) = e a2 hs α, 1 α 2, a >0, >0, C t (h t ;a,,δ) = e a2 hδ t, 1 δ 2, >0, φ(a,n,) = 2(n+1)/2 Ɣ((n + 1)/2) an e a2, n 0, >0, sine C s (h s ; a,,α) and C t (h t ; a,,δ) are, respetively, valid spatial and temporal ovariane models for eah hoie of a over the interval = [0; + [, the integraility onditions of Theorem 1 are satisfied and two new lasses of nonseparale spae-time ovarianes an e otained: C s,t (h s, h t ; Θ 1 ) = ke a2 hs α = 2k(n+1)/2 Ɣ((n + 1)/2) = ( hs α e a2 h δ t 2 (n+1)/2 Ɣ((n + 1)/2) an e a2 da k (n+1)/2 a n e a2 ( hs α + hδ t +) da + hδ t + ) (n+1)/2, (10)

8 30 De Iao, Myers, and Posa where Θ 1 = (,, n, k,α,,δ); [ C s,t (h s, h t ; Θ 2 ) = k 1 e a hs α e ahδ t + k 2 e a hs α + k 3 e ahδ t ] 2(n+1)/2 Ɣ((n + 1)/2) an e a2 da = k 1 (n+1)/2 ( hs α where Θ 2 = (,, n, k 1, k 2, k 3,α,,δ). + hδ t + ) (n+1)/2 + k 2 (n+1)/2 ( hs α + ) (n+1)/2 + k 3 (n+1)/2 ( h δ t + ) (n+1)/2, (11) Example 3. Given the funtions C s (h s ; a,ω) = os[a(ω h s )], a > 0, ω R, C t (h t ;a,,δ) = e ahδ t, 1 δ 2, > 0, φ(a,) = e a, > 0, sine the hypotheses of Theorem 1 are satisfied, two new lasses of nonseparale spae-time ovarianes are otained: C s,t (h s, h t ; Θ 1 ) = = k = ke ahδ t e a( hδ t +) os[a(ω h s )] e a da os[a(ω h s )] da k ( h δ t ( + ) h δ t + ) (12) 2 + (ω hs ) 2, where Θ 1 = (, k,ω,,δ); { C s,t (h s, h t ; Θ 2 ) = k 1 e ahδ t os[a(ω h s )] + k 2 os[a(ω h s )] + k 3 e ahδ t } e a da

9 Nonseparale Spae-Time Covariane Models 31 ( hδ t = k + ) 1 ( h δ t + ) 2 + (ω hs ) + k (ω h s ) + k 2 3 where Θ 2 = (, k 1, k 2, k 3,ω,,δ). h δ t +, (13) Example 4. Given the funtions C s (h s ; a,ω) = os[a(2ω h s )], a > 0, ω R, hδ C t (h t ;a,,δ) = e a2 t, 1 δ 2, >0, ( ) φ(a,) = 2 e a2, > 0, π sine the hypotheses of Theorem 1 are satisfied, two new lasses of nonseparale spae-time ovarianes are otained: C s,t (h s, h t ; Θ 1 ) = where Θ 1 = (, k,ω,,δ); C s,t (h s, h t ; Θ 2 ) = k os[a(2ω h s )] e a2 hδ t = 2k π = k + e h δ t ( ) 2 e a2 da π e a2 ( hδ t +) os[a(2ω h s )] da ω2 hs 2 h δ t { k 1 os[a(2ω h s )] e a2 hδ t + k 3 e a2 hδ t = k 1 + e where Θ 2 = (, k 1, k 2, k 3,ω,,δ). h δ t }( 2 π +, (14) ) e a2 da ω2 hs 2 h δ t + + k 2 e ω2 hs 2 + k 2 os[a(2ω h s )] + k 3 h δ t +, (15)

10 32 De Iao, Myers, and Posa Example 5. Given the funtions C s (h s ; a,,α) = e a2 hs α e hs a 2, 1 α 2, >0, hδ C t (h t ;a,,δ) = e a2 t φ(a,) = 2 e h t a 2, 1 δ 2, >0, π e a2, > 0, sine the hypotheses of Theorem 1 are satisfied, two new lasses of nonseparale spae-time ovarianes are otained: C s,t (h s, h t ; Θ 1 ) = = 2 π = k where Θ 1 = (,, k,α,,δ); C s,t (h s, h t ; Θ 2 ) = [ k 1 e ( a2 hs α + k 3 e ( a2 hδ t = k 1 ke (a2 hs α + hs a 2 ) e (a2 h δ t + h t a 2) 2 π e a2 da ke [a2 (+ hs α + hδ t + h s α + h s α + hδ t e 2 + hs a 2 ) e ( a2 h δ t + h t ] + h t a 2 ) 2 π e a2 da + hδ t e 2 )+ hs +h t a 2 ] da (+ hs α + hδ t )( h s +h t ), (16) a 2 ) + k 2 e ( a2 hs α (+ hs α + hδ t e 2 (+ hs α ) h s + k 3 where Θ 2 = (,, k 1, k 2, k 3,α,,δ). Remarks. + hδ t + hs a 2 ) )( h s +h t ) + k2 e 2 + h s α (+ hδ t )h t, (17) Even though the spae-time ovariane or variogram models (sine any of the aove ovariane models an e written in terms of variograms) look different from the spatial and temporal strutures whih are used in the

11 Nonseparale Spae-Time Covariane Models 33 Figure 1. Integrated produt sum variogram models (9) and (11) respetively in (A) and (B), with Θ 2 = (4000; 8; 2; 180; 220; 70; 1; 3; 1). integrals (4) (7), they retain the main features of the separate omponents in spae and time; the integrated produt sum variogram models orresponding to (9) and (11), derived from Examples 1 and 2, with Θ 2 = (4000; 8; 2; 180; 220; 70; 1; 3; 1), furnish a spae-time variogram surfae whih is onvex oth in h s and in h t, starting from spatial and temporal exponential variogram models (Fig. 1(A) and (B)); when the separate spatial and temporal strutures are Gaussian models, the spae-time variograms orresponding to (9) and (11), with Θ 2 = ( ; 8 2 ; 2; 180; 220; 70; 2; 3; 2), turn out to e onave oth in h s and in h t, espeially for small spatial and temporal lags (Fig. 2(A) and (B)); Figure 2. Integrated produt sum variogram models (9) and (11) respetively in (A) and (B), with Θ 2 = ( ;8 2 ; 2; 180; 220; 70; 2; 3; 2).

12 34 De Iao, Myers, and Posa Figure 3. Integrated produt sum variogram models (13) and (15) respetively in (A) and (B), with Θ 2 = (8; 180; 220; 70; 10 4 ;3;1). models (13) and (15) present paraoli ehavior in h s for small spatial lags; they are also either onvex in h t (Fig. 3(A) and (B)), where Θ 2 = (8; 180; 220; 70; 10 4 ;3;1), if the temporal variogram is an exponential model, or onave in h t (Fig. 4(A) and (B)), with Θ 2 = (8; 180; 220; 70; 10 4 ;3;2), espeially for small lags, if the temporal variogram is a Gaussian model; model (17), with Θ 2 = ( ;8 2 ; 180; 220; 70; 1; 3; 1), an e used to desrie an almost omplete lak of spae-time orrelation, that is, it is lose to a pure nugget effet model in spae-time domain (Fig. 5). Figure 4. Integrated produt sum variogram models (13) and (15) respetively in (A) and (B), with Θ 2 = (8; 180; 220; 70; 10 4 ;3;2).

13 Nonseparale Spae-Time Covariane Models 35 Figure 5. Integrated produt sum variogram model (17) with Θ 2 = ( ;8 2 ; 180; 220; 70; 1; 3; 1). SOME PRACTICAL ASPECTS Given a spatial-temporal data set, it is neessary to know how to use the data to generate a model of the form (6) or (7), that is, how to hoose the funtion φ, the spatial and the temporal variograms, as well as the oeffiients using the data. In the following only the pratial aspets of using (7) are onsidered, sine users an easily extend the following tehnique in order to utilize (6). The first step is to take advantage of a asi property of the variogram, γ (0) = 0. Hene, from (7) it follows that γ s,t (h s, 0) = (k 2 + k 1 C t (0; a)) γ s (h s ; a)φ(a) da, (18) and γ s,t (0, h t ) = (k 3 + k 1 C s (0; a))γ t (h t ; a)φ(a) da. (19)

14 36 De Iao, Myers, and Posa Seondly, if it is assumed that 1. φ is a density funtion; 2. γ s (h s ; a) and γ t (h t ; a) are standardized variograms with sill values equal to 1, that is, C s (0; a) = 1 and C t (0; a) = 1; and 3. (k 2 + k 1 C t (0; a))c s (0; a) = (k 2 + k 1 ) and (k 3 + k 1 C s (0; a))c t (0; a) = (k 3 + k 1 ) are the sill values, respetively, of γ s,t (h s, 0) and γ s,t (0, h t ) (De Iao, Myers, and Posa, 2001); then the relations etween γ s,t (h s, 0) and γ s (h s ; a), γ s,t (0, h t ), and γ t (h t ; a) are stritly linked to the funtion φ. The aove assumptions are satisfied, for example, if the parameter a is related only to the ranges of the spatial and temporal variogram models, γ s (h s ; a) and γ t (h t ; a) (see examples in the previous setion). The funtion φ, whih appears in the spatial-temporal variogram model in (7), an e otained as follows: models for γ s ( ; a) and γ t ( ; a), dependent on a, an e hosen y looking at the ehavior of the sample spatial and temporal variograms (denoted y ˆγ s,t (r s, 0) and ˆγ s,t (0, r t ), respetively); for a disrete numer of values a 1,...,a m for the parameter a, one an otain multiple spatial and temporal theoretial urves γ s ( ; a j ),γ t ( ;a j ), j =1,...,m, whih provide different fits to the orresponding sample variograms ˆγ s,t (r s, 0) and ˆγ s,t (0, r t ). In pratie, the minimum and maximum values of the sequene (a i, i = 1,...,m) are hosen in suh a way that the orresponding theoretial variograms are not too far from the sample variograms; evaluate how well eah γ s ( ; a j ) and γ t ( ; a j ) fits the data. This measure of the goodness of fit an e used to define a likelihood of fit and hene a proaility density: w s j = w t j = 1 ( ˆγs,t(rs,0) (k2 + k1)γs(rs;a j ) (k 2 + k 1 )γ s (r s ;a j ) N s 1 ( ˆγs,t(0,rt) (k3 + k1)γt(rt;a j ) (k 3 + k 1 )γ t (r t ;a j ) N t ) 2, j = 1,...,m (20) ) 2, j = 1,...,m (21) where N s and N t are, respetively, the numer of spatial and temporal lags for the sample variograms;

15 Nonseparale Spae-Time Covariane Models 37 y plotting a j versus w s j and w t j, j = 1,...,m, one an easily define the measure φ; finally, a spae-time variogram model an e otained y solving the integral (7). It is evident that the aove model still has an unknown parameter k 1, whih an e estimated in either of two ways: y minimizing W (k 1 ), the weighted least-squares value (Cressie, 1993), given y N s N t ( ) ˆγs,t (r s, r t ) γ s,t (r s, r t ; k 1 ) 2 W (k 1 ) = L(r s, r t ), γ s,t (r s, r t ; k 1 ) s where L(r s, r t ) is the ardinality of the set t L(r s, r t ) ={(s+h s,t+h t ) A,(s,t) A:h s Tol(r s ) and h t Tol(r t )}, Tol(r s ),Tol(r t ) are, respetively, speified tolerane regions around r s and r t, N s and N t are, respetively, the numer of spatial vetor lags and the numer of temporal lags, while ˆγ s,t is the sample spae-time variogram; y omputing the sill value of γ s,t from the sample spae-time variogram ˆγ s,t and solving the linear system of the following equations: k 2 + k 1 = estimated sill value of γ s,t (r s, 0), k 3 + k 1 = estimated sill value of γ s,t (0, r t ), k 1 + k 2 + k 3 = estimated sill value of γ s,t (r s, r t ). The method desried aove for generating the measure µ provides a very useful result for approximation and optimization. Theorem 2. Let φ(a) e a proaility density funtion and let γ s (r s ; a) and γ t (r t ; a) e standardized variograms. The spatial and temporal variograms, defined, respetively, in (18) and (19), satisfy the following inequalities: ( ) [γ s,t (r s, 0) ˆγ s,t (r s, 0)] 2 E a [γ s (r s ; a) ˆγ s,t (r s, 0)] 2, N s N s ( ) [γ s,t (0, r t ) ˆγ s,t (0, r t )] 2 E a [γ t (r t ; a) ˆγ s,t (0, r t )] 2, N t N t

16 38 De Iao, Myers, and Posa where r s and r t have een defined in Setion 2, N s and N t are, respetively, the numer of spatial vetor lags, and the numer of temporal lags, and ˆγ s,t (r s, 0) and ˆγ s,t (0, r t ) are the sample spatial and temporal variograms. Proof: Sine φ is a density funtion defining γ s,t (r s, 0) = γ s,t (0,r t ) = 0 0 γ s (r s, a)φ(a) da, γ t (r t,a)φ(a) da, it follows that γ s,t (r s, 0) = E a (γ s (r s, a)), γ s,t (0, r t ) = E a (γ t (r t, a)). Likewise, ar a (γ s (r s, a)) = E a (γ s (r s, a) γ s,t (r s, 0)) 2 = E a (γ s (r s, a) ˆγ s,t (r s, 0)) 2 (γ s,t (r s, 0) ˆγ s,t (r s, 0)) 2, ar a (γ t (r t, a)) = E a (γ t (r t, a) γ s,t (0, r t )) 2 = E a (γ t (r t, a) ˆγ s,t (0, r t )) 2 (γ s,t (0, r t ) ˆγ s,t (0, r t )) 2. Sine the variane is nonnegative, the following inequalities hold for any lag r s and r t : (γ s,t (r s, 0) ˆγ s,t (r s, 0)) 2 E a (γ s (r s, a) ˆγ s,t (r s, 0)) 2, (γ s,t (0, r t ) ˆγ s,t (0, r t )) 2 E a (γ t (r t, a) ˆγ s,t (0, r t )) 2, hene, the theorem follows. This result ensures that the spatial and temporal variogram models, γ s,t (r s, 0) and γ s,t (0, r t ), otained y the aove proedure, provide a etter average fitting to the spatial and temporal sample variograms than the fit otained y just using γ s (r s, ) and γ t (r t, ).

17 Nonseparale Spae-Time Covariane Models 39 AN APPLICATION TO AN AIR POLLUTION STUDY The methods desried aove have een applied to hourly average onentrations of NO 2 (µg/m 3 ) measured during August 1997 in 18 survey stations in Milan distrit. After removing the seasonal omponent y the standard tehnique of moving averages (Brokwell and Davis, 1987), residuals, availale for all stations, were used for the strutural analysis. The steps for generating the spae-time variogram model are listed elow. By examining the shapes of the sample spatial and temporal variograms, ˆγ s,t (, 0) and ˆγ s,t (0, ), the following exponential models have een hosen for γ s (, a) and γ t (, a): hs a γ s (h s, a) = 1 e 4414, (22) γ t (h t, a) = 1 e a h t (23) The sill values have een set to 400 and 250, respetively, for the spatial and temporal strutures, that is, k 2 + k 1 = 400 and k 3 + k 1 = 250. For a = 1, 400γ s (h s, a) and 250γ t (h t, a) are onsidered to e a good fit to the estimated spatial and temporal variograms (Fig. 6(A) and (B)). After hoosing 10 values of a, from a 1 = 0.25 to a 10 = 2.75, as many spatial and temporal theoretial urves, from (22) and (23), have een otained. Figure 6. Sample variograms, models for different values of the parameter a, and integrated models for time (A) and spae (B).

18 40 De Iao, Myers, and Posa Figure 7. Spatial and temporal likelihoods of fit. (20) and (21) have een omputed for the 10 values of a and the plot of a j versus w s j and w t j, j = 1,...,10, is presented in Figure 7. Sine the family of funtions ka 2 e 2.7a has een used for the fit, the following density funtion φ has een derived: φ(a) = 9.84a 2 e 2.7a. By using (18) and (19), the following spatial and temporal variogram models have een otained: γ s,t (h s, 0) = γ s,t (0, h t ) = 0 ( = [ 400 (1 e a hs ( h s 4414 [ ( e a h t 8.22 )] [9.84a 2 e 2.7a ] da ) 3 ) ( ) = ( h t ) ; )] [9.84a 2 e 2.7a ] da The sample variograms of the residuals, the exponential models for different values of a, and the integrated models are shown in Figure 6(A) and (B); moreover, the sample temporal variogram of the original data is presented (Fig. 6(A)).

19 Nonseparale Spae-Time Covariane Models 41 Figure 8. Sample spae-time variogram surfae (A) and integrated spae-time variogram model (B) of residuals. Finally, the sill value of γ s,t, evaluated from the sample spae-time variogram ˆγ s,t (Fig. 8(A)), has een set to 470 (k 1 + k 2 + k 3 = 470). Hene, k 1 = 180, k 2 = 220, k 3 = 70 and the spae-time variogram model (Fig. 8(B)) has een otained y solving integral (7): γ s,t (h s, h t ) = ( ( h s h s h t 8.22 ) 3 70 ( ) h t 8.2 ) 3 CONCLUSIONS Estimating and modeling the orrelation of a spae-time proess is a relevant issue. In this paper, eginning with the produt and the produt sum ovariane models, nonintegrale spae-time ovariane models have een generated. These parametri families annot e otained, in general, from Cressie Huang representation. To use these models, the user still must fit a model to the data. Possile hoies an e determined after omputing the sample spatial-temporal variogram and inspeting its main features (suh as ehavior near the origin for small spatial and temporal lags, the sill values along spatial and temporal diretions). It is evident, for example, that one an hoose etween the integrated produt and the integrated produt sum just y looking at the spatial and temporal sill values, sine only the produt sum model an e used when the sill values are different. Several examples reprodue the most ommon spae-time variogram ehavior. Other

20 42 De Iao, Myers, and Posa pratial aspets, linked with the prolems of fitting the ovariane or variogram model to the data availale, were disussed and a ase study has een presented. REFERENCES Brokwell, P. J., and Davis, R. A., 1987, Time series: Theory and methods: Springer, New York, 577 p. Chilès, J., and Delfiner, P., 1999, Geostatistis: Wiley, New York, 687 p. Cressie, N., 1993, Statistis for spatial data: Wiley, New York, 900 p. Cressie, N., and Huang, H., 1999, Classes of nonseparale, spatio-temporal stationary ovariane funtions: J. Am. Stat. Asso., v. 94, no. 448, p De Cesare, L., Myers, D., and Posa, D., 1997, Spatial-temporal modeling of SO 2 in Milan Distrit, in Baafi, E. Y., and Shofield, N. A., eds., Geostatistis Wollongong 96, ol. 2: Kluwer Aademi, The Netherlands, p De Cesare, L., Myers, D., and Posa, D., 2001, Estimating and modeling spae-time orrelation strutures: Stat. Proa. Lett., v. 51, no. 1, p De Iao, S., Myers, D., and Posa, D., 2001, Spae-time analysis using a general produt sum model: Stat. Proa. Lett., v. 52, no. 1, p Gradshteyn, I. S., and Ryzhik I. M., 2000, Tales of integrals, series and produts: Aademi Press, New York, 1163 p. Kyriakidis, P. C., and Journel, A. G., 1999, Geostatistial spae-time models: A Rev. Math. Geol., v. 31, no. 6, p Matern, B., 1980, Spatial variation, leture notes in statistis, 2nd edn.: Springer, New York, ol. 36, 151 p. First edition pulished in Meddelanden fran Statens Skogsforskningsinstitute Swed., Band 49, no. 5, Myers, D. E., and Journel, A., 1990, ariograms with Zonal Anisotropies and Non-Invertile Kriging Systems: Math. Geol., v. 22, no. 7, p Posa, D., 1993, A simple desription of spatial-temporal proesses: Comput. Stat. Data Anal., v. 15, no. 4, p Rodriguez-Iture, I., and Meija, J. M., 1974, The design of rainfall networks in time and spae: Water Res. Res., v. 10, no. 4, p Rouhani, S., and Hall, T. J., 1989, Spae-time kriging of groundwater data, in Armstrong, M., ed., Geostatisti, ol. 2: Kluwer Aademi, The Netherlands, p Rouhani, S., and Myers, D., 1990, Prolems in spae-time kriging of geohydrologial data: Math. Geol., v. 22, no. 5, p