Non-Fully Symmetric Space-Time Matérn Covariance Functions

Transcription

1 Non-Fully Symmetric Space-Time Matérn Covariance Functions Tonglin Zhang and Hao Zhang Purdue University Abstract The problem of nonseparable space-time covariance functions is essential and difficult in spatiotemporal data analysis. Considering that a fully symmetric space-time covariance function is inappropriate for many spatiotemporal processes, this article provides a way to construct a non-fully symmetric nonseparable space-time correlation function from any given marginal spatial Matérn and marginal temporal Matérn correlation functions. Using the relationship between a spatial Matérn correlation function and the characteristic function of a multivariate t-distribution, a modification of Bochner s representation is provided and a non-fully symmetric space-time Matérn model is obtained. KEY WORDS: Bochner s representation; Characteristic functions; Multivariate t-distributions; Non-fully symmetric space-time correlation functions; Space-time Matérn model. 1 Introduction For a second-order stationary spatiotemporal process Y (s, t), s R d and t R, its covariance function, C(h, u) = Cov(Y (s, t), Y (s + h, t + u)), h R d, u R, is essential to prediction and estimation. A space-time covariance function C is called fully symmetric (Gneiting, 2002) if C(h, u) = C( h, u) = C(h, u) = C( h, u), h R d, u R. (1) Department of Statistics, Purdue University, 250 North University Street, West Lafayette, IN , tlzhang@purdue.edu Department of Statistics, Purdue University, 250 North University Street, West Lafayette, IN , zhanghao@purdue.edu 1

2 In the purely spatial context, this property is also known as axial symmetry (Scaccia and Martin, 2005). Since it has been pointed out by Gneiting (2002) that (1) is often violated when environmental processes are considered dynamically, it is more appropriate to use non-fully symmetric space-time covariance functions in applications. A space-time covariance function is a spatial covariance function if time is ignored or a temporal covariance function if space is ignored. If time is not involved, a great interest is to consider the Matérn family of spatial correlation functions as M d (h ν, a) = (a h )ν 2 ν 1 Γ(ν) K ν(a h ) =M ν (a h ), h R d, a, ν > 0, (2) where K ν is a modified Bessel function of the second kind, a and ν are scale and smoothness parameters respectively, and M ν (z) = z ν K ν (z)/[2 ν 1 Γ(ν)], z R. The Matérn family is isotropic in space. It was proposed by Matérn (1986) and has received more attention since some theoretical work by Handcock and Stein (1993) and Stein (1999). A nice review and discussion on Matérn family is given by Guttorp and Gneiting (2006). The Matérn spatial correlation family has been used in many applications (Lee and Shaddick, 2010; North, Wang, and Genton, 2011). An isotropic correlation function is inappropriate in modeling a spatiotemporal process since the temporal axis and the spatial axis are of different scales and the Euclidean distance in the product space in not suitable. The construction of space-time correlation functions is an interesting but difficult problem. There have been good developments in recent years on the construction of space-time correlation functions. The simplest case is the separable one provided by the product of a spatial correlation function and a temporal correlation function, but it does not model the space-time interaction (Cressie and Huang, 1999; Stein, 2005) and also too restrictive for spacetime data analysis. A more detailed discussion of the shortcomings of separable models can be found in Kyriakidis and Journel (1999). Recently, much effort has been put on the construction of nonseparable space-time covariance functions. Many models have been proposed. Examples include the product-sum model (De Iaco, Myers, and Posa, 2001), the mixture models (Ma, 2002, 2003b), and anisotropic spatial component models (Mateu, Porcu, and Gregori, 2008; Porcu, Gregori, and Mateu, 2006; Porcu, Mateu, and Saura, 2008). In the construction of these models, many methods have been proposed. Examples include the spectral representation method (Cressie and Huang, 1999; Stein, 2005), the completely 2

3 monotone function method (Gneiting, 2002), the linear combination method (Ma, 2005), the convolution-based method (Rodrigues and Diggle, 2010), the stochastic different equation method (Kolovos, et. al, 2004), and the mixture representation method (Fonseca and Steel, 2011). There have been a number of studies on the properties of space-time correlation functions, including, for example, the test for separability (Brown, Karesen, and Tonellato, 2000; Fuentes, 2006; Li, Genton, Sherman, 2007; Mitchell, Genton, and Gumpertz, 2005), the evaluation of spatial or temporal margins (De Iaco, Posa, and Myers, 2013), and types of nonseparability (De Iaco and Poca, 2013). An important issue is the assessment of full symmetry. Although fully symmetry is a desirable assumption from a computational point of view, it may not be appropriate in applications (Shao and Li, 2009). Atmospheric, environmental, and geophysical processes are often under the influence of prevailing air or water flows, resulting in a lack of full symmetry (Gneiting, Genton, and Guttorp, 2007). Nonseparable and fully symmetric space-time covariance functions can be constructed by mixtures of separable covarianc functions (De Iaco, Myers, and Posa, 2002; Ma, 2003a). A geometric non-fully symmetric space-time covariance function can be formulated using a geometric transformation on a fully symmetric space-time covariance function (Stein, 2005). There is a great need of covariance functions which are non-fully symmetric in the spatiotemporal domain (Mateu, Porcu, and Gregori, 2008). In this work, we focus on the construction of a non-fully symmetric space-time (NFSST) Matérn correlation model that satisfies any given Matérn margins. Specifically, given any spatial Matérn correlation function M d (h ν 1, a 1 ) (called the spatial Matérn margin) and a temporal Matérn correlation function M 1 (u ν 2, a 2 ) (called the temporal Matérn margin), we construct an NFSST Matérn correlation function C(h, u) satisfying C(h, 0) = M d (h ν 1, a 1 ), C(0, u) = M 1 (u ν 2, a 2 ). (3) Although the model is non-fully symmetric, its spatial and temporal margins are both isotropic. We allow the two smoothness parameters and the two scale parameters to be arbitrary. To our best knowledge, our model is the first one with these properties. The remainder of this article is organized as follows. In Section 2, we provide an approach, modified from Bochner s representation for correlation functions. In Section 3, we employ the approach to construct an NFSST Matérn model. In Section 4, we apply our NFSST model to a 3

4 meteorological data. In Section 5, we provide a discussion. 2 Bochner s Representation According to Bochner s representation (Bochner, 1955), a real continuous space-time function C on R d R is a (stationary) space-time covariance function iff C(h, u) = e iht s+iut F (ds, dt), (4) R d R for any s, h R d and t, u R, where F is the spectral measure satisfying F (A, B) = F ( A, B) for any Borel A R d and B R. We assume F is a finite measure in (4) since we need V(Y (s, t)) be finite. If C(h, u) is integrable, then F is absolutely continuous with respect to the Lebesgue measure on R d R and Bochner s representation (4) becomes C(h, u) = e iht s+iut f(s, t)dtds, h R d, u R, (5) R d R where the non-negative integrable function f(s, t) is called the spectral density and satisfies f(s, t) = f( s, t). If the variance of the process is one, then F is a probability measure in (4) and the corresponding f in (5) is a probability density function. Hence a correlation function can be viewed as a characteristic function of a random variable provided that the characteristic function is real. Bochner s representation is powerful in the construction of space-time correlation functions. It can be specified as a spatial correlation function if time is ignored or as a temporal correlation function if space is ignored. In the spatiotemporal context, a few nonseparable space-time covariance models can be constructed using Bochner s representation (Cressie and Huang, 1999; Stein, 2005). Bochner s representation has a few modifications. One of the modifications is used in the construction of the generalized Gauchy spatial correlation function (Devroye, 1990). This modification can be generalized to the spatiotemporal context for a space-time covariance function. Theorem 1 Let A(h, u) be a space-time covariance function. Let Z 1 and Z 2 be positive random variables with joint CDF G. Then C(h, u) = 0 0 A(hz 1, uz 2 )G(dz 1, dz 2 ) (6) 4

5 is a valid space-time covariance function. Theorem 2 Let Z 1 and Z 2 be independent positive random variables on (0, ) in Theorem 1. The necessary and sufficient conditions for C(h, u) to be separable in (6) is that A(h, u) be separable. The general way in our approach is to use a trivial A(h, u) to construct a non-trivial C(h, u). As the choice of Z 1 and Z 2 is generally flexible, many different families of space-time correlation functions may be obtained if different types of Z 1 and Z 2 are utilized. In the following section, we provide a way to construct a non-fully symmetric space-time Matérn model by using particular Z 1 and Z 2 with a Gaussian characteristic function A(h, u). 3 NFSST Matérn Model Our main idea is to provide special A(h, u), Z 1, and Z 2 in Theorem 2 such that C(h, u) is non-fully symmetric and satisfies (3). According to (6), if A(h, u) is non-fully symmetric, then C(h, u) is also non-fully symmetric. Therefore, we need to provide a non-fully symmetric A(h, u) in the construction. The way to choose A(h, u), Z 1, and Z 2 in motivated from the following. 3.1 Spatial Matérn Model The spatial Matérn correlation function (2) can be viewed as the characteristic function of a random variable on R d with a spectral density m d (x ν, a) = Γ(ν + d/2) π d 2 a d Γ(ν) 1 (1 + x 2 a 2 ) ν+ d 2, x R d, a, ν > 0. (7) In an alternative way, the spatial Matérn model can be defined by the following theorem. Theorem 3 Let u be a d-dimensional N(0, I d ) random variable and V be a univariate Γ(ν, 1/2) random variable. If u and V are independent, then m d (s ν, a) is the PDF and M d (h ν, a) is the characteristic function of x = au/ V. Corollary 1 If ν, a > 0, then for any h R d there is M d (h ν, a) = 1 2 ν Γ(ν) 0 v ν 1 e 1 2 ( a2 h 2 v +v) dv. (8) 5

6 Another expression of the modified Bessel function is derived by comparing (8) with (2): 0 v ν 1 e 1 2 ( z2 v +v) dv = 2 z ν K ν (z) = 2 ν Γ(ν)M ν (z), z R, ν > 0. (9) This expression is useful in the numerical purpose for our NFSST Matérn model in Section Space-Time Matérn Model If u 1 N(0, I d ) and U 2 N(0, 1) independently, then the characteristic function of (u 1, U 2 ) is E(e i(ht u 1 +uu 2 ) ) = e 1 2 ( h 2 +u 2). Let V 1 Γ(ν 1, 1/2) and V 2 Γ(ν 2, 1/2) independently, which are also independent of (u 1, U 2 ). Then, ( C(h, u) = E e 1 2 ( a 2 1 h 2 V 1 ) + a2 2 u2 ) V 2 is a valid space-time correlation function, which is separable and satisfies (1) and (3). If the dependence between u 1 and U 2 is imposed, then a nonseparable space-time Matérn model is obtained. Let u 1 N 0, I d r. 0 r T 1 U 2 Since the eigenvalues of the variance-covariance matrix are either 1 or 1 r 2, the distribution is valid if and only if r < 1. The characteristic function of u 1 and U 2 is A r (h, u) = e 1 2 ( h 2 +2ur T h+u 2). Definition 1 (NFSST Matérn Model). Let M r,a1,a 2,ν 1,ν 2 (h, u) = E ( e 1 2 ( a 2 1 h 2 V 1 ) + 2a 1 a 2 urt h + a2 2 u2 ) V1 V V 2 2, (10) where V 1 and V 2 are independent Γ(ν 1, 1/2) and Γ(ν 2, 1/2) random variables, respectively. If a 1, a 2, ν 1, ν 2 R + and r R d with r < 1, then M r,a1,a 2,ν 1,ν 2 (h, u) is called an NFSST Matérn correlation function. The class of space-time correlation functions M = {M r,a1,a 2,ν 1,ν 2 : a 1, a 2, ν 1, ν 2 R +, r R d, r < 1} is called the NFSST Matérn model. 6

7 Lemma 1 Let d 2, h, r, and u be positive in (10). If r T h = 0, then M r,a1,a 2,ν 1,ν 2 (h, u) = M d (h ν 1, a 1 )M 1 (u ν 2, a 2 ); if r T h > 0, then M d (h ν 1, a 1 )M 1 (u ν 2, a 2 ) < M r,a1,a 2,ν 1,ν 2 (h, r); if r T h < 0, then M d (h ν 1, a 1 )M 1 (u ν 2, a 2 ) > M r,a1,a 2,ν 1,ν 2 (h, r). Let D r,a1,a 2,ν 1,ν 2 (h, u) = M r,a1,a 2,ν 1,ν 2 (h, u) M d (h ν 1, a 1 )M 1 (u ν 2, a 2 ) be the difference between the NFSST Matérn and the separable space-time Matérn correlation functions. For any d 2, there is D r,a1,a 2,ν 1,ν 2 (h, u)d r,a1,a 2,ν 1,ν 2 ( h, u) 0 (11) and the equality holds iff r T h = 0 or u = 0. If d = 1, then D r,a1,a 2,ν 1,ν 2 (h, u)d r,a1,a 2,ν 1,ν 2 ( h, u) 0 (12) and the equality holds iff at least one of u, h R and r ( 1, 1) is zero. Theorem 4 M r,a1,a 2,ν 1,ν 2 is fully symmetric or separable iff r = 0. An NFSST Matérn correlation function can be separable or nonseparable. If r 0, then M r,a1,a 2,ν 1,ν 2 (h, u) is nonseparable in the whole space but separable in the subspace {h R d : r T h = 0}. Its spatial margin is a spatial Matérn correlation function and its temporal margin is a temporal Matérn correlation function. The two margins can be arbitrary. Given the two margins, M r,a1,a 2,ν 1,ν 2 can be constructed by introducing additional parameter r. Let r = (r 1,, r d ). Then, r can also be expressed via a polar transformation on R d as r 1 = r cos θ 1, j 1 r j = r ( sin θ j ) cos(θ j ), j = 2,, d 1, k=1 d 1 r d = r sin θ k, k=1 (13) where θ 1,, θ d 2 [0, π] and θ d 1 [0, 2π]. One can also use ( r, θ 1,, θ d 1 ) to describe nonseparability. In practice, (13) is convenient in interpretation. 7

8 3.3 Algorithm As (10) cannot be used for the numerical purpose, we decide to provide a Taylor expansion for M r,a1,a 2,ν 1,ν 2 such that we can compute its numerical values. The basic way is to solve the Taylor expansion of the right side of (10) with positive h, r, and u as where and M r,a1,a 2,ν 1,ν 2 (h, u) = = = ( ( 1) n (a 1 a 2 ur T h) n E V n 2 1 V n 2 2 e 1 2 ( a 2 n! ( 1) n (a 1 a 2 ur T h) n n=0 n=0 n! b n,r,a1,a 2,ν 1,ν 2 (h, u) n=0 ( E V n 2 1 e a 2 1 h 2 2V 1 ) E 1 h 2 V 1 ) + a2 2 u2 ) V 2 ( V n 2 2 e a 2 2 u2 b n,r,a1,a 2,ν 1,ν 2 (h, u) = ( 1)n (a 1 a 2 ur T h) n 2 ν 1+ν 2Γ(ν1 )Γ(ν 2 )n! D ν 1 n 2 (a 1 h )D ν2 n 2 (a 2 u ) D α (z) = 0 2V 2 ) (14) v α 1 e 1 2 ( z2 v +v) dv, α R. (15) The difference between D α (z) and the integral in (9) is that we allow α 0 in the definition of D α, which is not contained in (9). Values of D α when α 0 cannot be directly obtained by Corollary 1. If α > 0, then D α (z) is well-defined for any z R. If α 0, then D α (z) is only well-defined for z 0. To have a well-defined Taylor expansion in (14), we need to have a way to compute D α (z) for any α 0 with z 0 and also a way to justify the convergence rate of the Taylor expansion. Theorem 5 If α > 0, then D α (z) = 2 α Γ(α)M α ( z ) for all z R. If α < 0 and z > 0, then D α ( z ) = 2 α Γ( α ) z 2α M α ( z ). If z > 0, then D 0 ( z ) = 4 z 2 [M 2 ( z ) M 1 ( z )]. Corollary 2 If α 0 and β > α then z 2β D α (z) is well-defined and continuous in all z R. Using Theorem 5 and Corollary 2, we derive a nice property of M ν ( z ) as M ν ( z ) 1 lim = 2 2ν Γ(1 ν), z 0 z 2ν Γ(ν + 1) implying that if ν (0, 1) then for sufficiently small z there is M ν (z) = 1 2 2ν Γ(1 ν) z 2ν + o( z 2ν ). (16) Γ(ν + 1) 8

9 Let h 0 = h/ h, u 0 = u/ u, and r 0 = r/ r. Then b n,r,a1,a 2,ν 1,ν 2 (h, u) = ( 1)n (u 0 r T 0 h 0 ) n 2 ν 1+ν 2Γ(ν1 )Γ(ν 2 )n! r n d n,ν1 (a 1 h )d n,ν2 (a 2 u ), (17) where d n,ν (z) = z n D ν n ( z ), z R. (18) 2 If neither 2ν 1 nor 2ν 2 is an integer, then it enough to consider d n,ν (z) = 2 ν n 2 Γ( ν n 2 ) z 2ν n M ν n 2 ( z ), z R, (19) in the computation of (17); otherwise, one also needs to consider d n, n 2 (z) = 4 z n 2 [M 1 ( z ) M 2 ( z )], z R. (20) It is enough to consider both (19) and (20) in the computation of M r,a1,a 2,ν 1,ν 2 (h, u), but (20) is not used if neither 2ν 1 nor 2ν 2 is an integer. For instance, if 0 < ν 1, ν 2 < 1/2 then, only using (19), we obtain d n,ν1 (a 1 h ) = 2 n 2 ν 1 Γ( n 2 ν 1)(a 1 h ) 2ν 1 M n 2 ν 1(a 1 h ), and d n,ν2 (a 2 u ) = 2 n 2 ν 1 Γ( n 2 ν 2)(a 2 u ) 2ν 2 M n 2 ν 2(a 2 u ) for any n 1, which yields M r,a1,a 2,ν 1,ν 2 (h, u) =M d (h ν 1, a 1 )M 1 (u ν 2, a 2 ) + n=1 ( 1) n (u 0 r T 0 h 0 ) n Γ( n 2 ν 1)Γ( n 2 ν 2) 2 2(ν 1+ν 2 ) n Γ(ν 1 )Γ(ν 2 )n! r n (a 1 h ) 2ν 1 (a 2 u ) 2ν 2 M n 2 ν 1(a 1 h )M n 2 ν 2(a 2 u ), 0 < ν 1, ν 2 < 1/2. (21) If ν 1 = ν 2 = 1/2 then, also using (20), we obtain d 1, 1 (a 1 h ) = 4 2 a 1 h [M 1(a 1 h ) M 2 (a 1 h )], and d 1, 1 (a 2 h ) = 4 2 a 2 u [M 1(a 2 u ) M 2 (a 2 u )], 9

10 which yields M r,a1,a 2, 1 2, 1 2 (h, u) h ( + a =e u 1 a ) 2 8rT h 0 πa 1 a 2 u h [M 1(a 1 h ) M 2 (a 1 h )][M 1 (a 1 u ) M 2 (a 2 u )] + n=2 ( 1) n 2 n 2 (u 0 r T 0 h 0 ) n Γ( n 1 n 1 )Γ( 2 πn! 2 ) r n (a 1 h )(a 2 u )M n 1 2 (a 1 h )M n 1 (a 2 u ). 2 (22) As D 0 (z) with z = a 1 h or z = a 2 u may appear in (14) if either 2ν 1 or 2ν 2 is an integer, we need (20) in the computation. Theorem 6 The right size of (14) absolutely uniformly converges in h R d, u R, r R d exponentially fast if r 1 ϵ for any ϵ (0, 1). Theorem 5 provides a way to compute b n,r,a1,a 2,ν 1,ν 2 (h, u) for a given n. A numerical algorithm is obtained if (14) is employed. Corollary 2 concludes that b n,r,a1,a 2,ν 1,ν 2 (h, u) is continuous in all h and u for any n. Theorem 6 concludes that (14) is valid and its right side is absolutely continuous in h, u, and r if r < 1 ϵ for any given ϵ > 0. The expansion uniformly converges to a separable model as r 0. The convergence rate of (14) is exponential, which means that the expansion converges fast. Therefore, the algorithm based on (14) is efficient. 3.4 Correlation Functions with Ridges Space-time correlation functions may have a kind of discontinuity in correlations of certain linear combinations of the random field. For a nonnegative integer m, suppose that Y (s, t) is m times mean square differentiable spatiotemporal Gaussian random field in its second coordinate, and write Y m (s, t) as the mth mean square derivative. The covariance function of Y m is C m (h, u) = ( 1) m 2m C(h, u)/ u 2m. Let ρ m ϵ (h, u) = Cor(Y m (0, ϵ) Y m (0, 0), Y m (h, u + ϵ) Y m (h, u)) and let ρ m (t, s) be its limit as ϵ 0, assuming that the limit exists. Suppose d = 1 such that we can use C m (h, u) for h, u R to express C m. According to Proposition 1 of Stein (2005), if C m is a continuous function on R R and there exist 0 < α 1 < < α p < 2 and even functions C 1,, C p on R with C 1 (0) 0 such that p C m (h, u) = C m (h, 0) + C j (h) u α j + R h (u), (23) j=1 10

11 where R h (u) = O(u 2 ) as u 0 for any given h and R h ( ) has a bounded second derivative, then lim C 1 (h){ u + ϵ α 1 2 u α 1 + u ϵ α 1 ϵ 0 ρ m 2C 1 (0)ϵ α 1 ϵ (h, u) = 0, (24) sup u R where C ρ m 1 (h)/c 1 (0), u = 0, (h, u) = 0, u 0. Assume (23) holds for m = 0. Using (24) with m = 0 for ρ 0 (h, u), one can consider the mean squared error (MSE) of the ordinary kriging predictor of Y (0, ϵ) based on Y (0, 0), Y (h, 0), and Y (h, ϵ), which is 2{C 0 (0) C1(h)/C 2 1 (0)}ϵ α 1 +o(ϵ α 1 ). As the MSE of the ordinary kriging predictor of Y (0, ϵ) based on Y (0, 0) is 2C 1 (0)ϵ α 1 + o(ϵ α 1 ), the lack of continuity of ρ 0 (h, u) along the h axis leads to best linear unbiased predictors that depend on observations whose correlations are bounded away from 1 as ϵ 0, implying that correlation functions satisfying (23) have ridges along their axes. Stein (2005) points out that many separable correlation functions satisfy (23). All examples of Cressie and Huang (1999) satisfy (23). The sum of product form models with functions of just space and just time (De Iaco, Myers, and Posa, 2001) may share ridges along their axes. It would appear that nonseparable covariance functions proposed by Gneiting (2002) may not be smoother along their axes than at the origin. For a fully symmetric space-time correlation function C(h, u) on R R. If there is an 0 < α 1 < 2 such that 0 C 1 (h) = lim u 0 u α 1 [C(h, u) C(h, 0)] exists, then for sufficiently small u there is C( h, u ) = C(h, u ) = C(h, 0) + C 1 (h) u α 1 + O( u α 1 ), implying that C 1 (h) is an even function. Therefore, a fully symmetric space-time correlation probably satisfies (23). As our family is not fully symmetric, we consider a possible expansion related to (23). It is obtained if (16), (17), and (19) are combined. In particular, we consider the Taylor expansion for given h and r satisfying r T h 0. We conclude that for sufficiently small u with ν 2 (0, 1/2) there is M r,1,1,ν1,ν 2 (h, u) =M ν1 ( h ) sign(ur T h)c 1 ( h ) u 2ν 2 + C 2 ( h ) u 2ν 2 + o( u 2ν 2 ), where C 1 ( h ) > 0. Therefore, our model does not satisfy (23), implying that M r,a1,a 2,ν 1,ν 2 does not belong to the covariance family with a covariance ridge problem considered by Proposition 1 of Stein (2005). 11

12 (a): u=0 (b): u=0.1 h h h 1 h 1 (c): u=0.2 (d): u=0.5 h h h 1 h 1 Figure 1: Contour plots for M r,a1,a 2,ν 1,ν 2 (h, u) as functions of h = (h 1, h 2 ) for selected u when r = (0.5, 0), a 1 = a 2 = 1, and ν 1 = ν 2 = The Case When d = 2 We specify M r,a1,a 2,ν 1,ν 2 (h, u) in the case when d = 2 since it is important in practice. r = (r 1, r 2 ) and h = (h 1, h 2 ) when d = 2. If r and h are positive, then h 0 = (h 01, h 02 ) = (h 1 / h, h 2 / h ) and r 0 = (r 01, r 02 ) = (r 1 / r, r 2 / r ) are well defined. Equation (10) becomes ) )] M r,a1,a 2,ν 1,ν 2 (h, u) = E [(e a 1 a 2 u 0 u r h (r 01 h 01 +r 02 h 02 ) V1 V 2 (e 1 2 ( a 2 1 h 2 + a2 2 u2 ) V 1 V 2. (25) If u 0 > 0 then, given u and h, M r,a1,a 2,ν 1,ν 2 (h, u) is minimized at h 0 = r 0 and maximized at h 0 = r 0 ; otherwise, M r,a1,a 2,ν 1,ν 2 (h, u) is maximized at h 0 = r 0 and minimized at h 0 = r 0. If h 0 and u 0 are vertical, then M r,a1,a 2,ν 1,ν 2 (h, u) = M 2 (h ν 1, a 1 )M 1 (u ν 2, a 2 ). This is satisfied if r 01 h 01 + r 02 h 02 = 0. To study the performance of M in the spatial domain, we use (17) to numerically compute the values of the correlation function on the left side of (25) for selected u when r = (0.5, 0), a 1 = a 2 = 1, and ν 1 = ν 2 = 5 (Figure 1). It shows that the right side of (25) reduces to M 2 (h ν 1, a 1 ) if u = 0 (Figure 1(a)). The contour plots are more symmetric for small positive u values than those for large positive u values. The values of the correlation function strictly decreases in u or h increases along a certain direction. The speed of the decrease depends 12 Let

13 (a): r=(0,0) (b): r=(0.1,0) h h h 1 h 1 (c): r=(0.2,0) (d): r=(0.5,0) h h h 1 h 1 Figure 2: Contour plots for M r,a1,a 2,ν 1,ν 2 (h, u) as functions of h = (h 1, h 2 ) for selected r when u = 0.5, a 1 = a 2 = 1, and ν 1 = ν 2 = 5. on the direction of r, which is maximized at the positive direction but minimized at the negative direction of r. To compare, we also use (17) to study the performance of M in the spatial domain when r varies. We numerically compute the values of the correlation function on the left side of (25) for selected r when u = 0.5, a 1 = a 2 = 1, and ν 1 = ν 2 = 5 (Figure 2). It shows that curves in Figure 1(a) and Figure 2(a) are parallel. Both of them are isotropic. Values of curves in Figure 2(a) equals values of curves in Figure 1(a) multiplied by M 5 (0.5) = 815. Curves in the rest plots of Figure 2 are not isotropic. The magnitude of anisotropy increases as r increases. Curves in Figure 1(d) and Figure 2(d) are identical. They provide the extremely anisotropic case in those displayed in Figure 2. Note that both r and h can be expressed by their norms and their angles. We also use the polar coordinates to express the Taylor expansion of M r,a1,a 2,ν 1,ν 2 (h, u). The result based on the polar transformation is easy to interpret. Let the angle of r be ω and the angle of h be η, ω, η ( π, π]. Then, r = (r 1, r 2 ) = ( r cos ω, r sin ω), h = (h 1, h 2 ) = ( h cos η, h sin η), and r T h = r 1 h 1 + r 2 h 2 = r h cos(ω η), which yields cos(ω η) = (r 1 h 1 + r 2 h 2 )/( r h ) = 13

14 (a): η=0 (b): η=π 8 h h u u (c): η=π 4 (d): η=π 2 h h u u Figure 3: Contour plots for M r,a1,a 2,ν 1,ν 2 (h, u) as functions of h and u for selected η when r = (0.5, 0), a 1 = a 2 = 1, and ν 1 = ν 2 = 5, where η is the angle between h and the horizontal axis. r 01 h 01 + r 02 h 02 and b n,r,a1,a 2,ν 1,ν 2 (h, u) = [ sign(u) cos(ω η)]n 2 ν 1+ν 2Γ(ν1 r n d n,ν1 (a 1 h )d n,ν2 (a 2 u ). (26) )Γ(ν 2 )n! If u and h are positive, then M r,a1,a 2,ν 1,ν 2 (h, u) < M 2 (h ν 1, a 1 )M 1 (u ν 2, a 2 ) if ω η < π/2 and M r,a1,a 2,ν 1,ν 2 (h, u) > M 2 (h ν 1, a 1 )M 1 (u ν 2, a 2 ) if ω η > π/2. Using the polar expression of b n,r,a1,a 2,ν 1,ν 2 (h, u) in (26), we can also interpret Figure 1 according to a rotation of r: if r is rotated by an orthogonal transformation then the corresponding contour plot of M r,a1,a 2,ν 1,ν 2 (h, u) is also rotated with the same angle of the transformation. To study the performance of M in the spatiotemporal domain, we use (26) to numerically compute the values of M r,a1,a 2,ν 1,ν 2 (h, u) (Figure 3). We still use r = (0.5, 0), a 1 = a 2 = 1, and ν 1 = ν 2 = 5 in the computation such that ω = 0 and cos(ω η) = cos η in (26). It shows that the value of the correlation function is not symmetric about zero in u when η π/2. If η = π/2, then cos(η) = 0, implying that M r,a1,a 2,ν 1,ν 2 (h, u) is separable. If η (π/2, π], then cos(η) = cos(π η), implying that corresponding plots for η [π/2, π] can be derived by simply reflecting the plots for η [0, π/2]. Note that M r,a1,a 2,ν 1,ν 2 (h, u) is only separable in the sub-space {h R 2 : η = π/2}. The lower-right panel in Figure 3 is just a special case of Lemma 1. 14

15 (a): η=0 (b): η=π 8 r r u u (c): η=π 4 (d): η=π 2 r r u u Figure 4: Contour plots for M r,a1,a 2,ν 1,ν 2 (h, u) as functions of r and u for selected η when h = (0.5, 0), a 1 = a 2 = 1, and ν 1 = ν 2 = 5, where η is the angle between h and the horizontal axis. In the end, we evaluate the performance of M in the temporal domain. We let r change but h be fixed at h = (0.5, 0) (Figure 4). It shows that values of M r,a1,a 2,ν 1,ν 2 (h, u) decreases in r if η < π/2 and u is positive. Values of M r,a1,a 2,ν 1,ν 2 (h, u) increases in r if η < π/2 and u is negative. Values of M r,a1,a 2,ν 1,ν 2 (h, u) does not vary in r if η = π/2. This is expected based on (14). A nice property is that M r,a1,a 2,ν 1,ν 2 (h, u) is symmetric about r in the spatial domain. In particular, for any distinct h 1, h 2 R 2 satisfying h 1 = h 2, if h 1 and h 2 are symmetric about r, then M r,a1,a 2,ν 1,ν 2 (h 1, u) = M r,a1,a 2,ν 1,ν 2 (h 2, u); otherwise, M r,a1,a 2,ν 1,ν 2 (h 1, u) > M r,a1,a 2,ν 1,ν 2 (h 2, u) if the angle between h 1 and r is less than the angle between h 2 and r. In practice, r provides the strongest and weakest dependence direction between space and time. For positive u, the dependence between space and time is the strongest if the direction of the space change is the opposite of r and it is the weakest if the direction is the same of r. This provides a nice way to interpret our family in applications. 15

16 Figure 5: Locations of Weather Stations in Shandong Province, China in July Application 4.1 Shandong Temperature Data We applied our M to the Chinese daily temperature data in this section. The data were provided by the Climatic Data Center, National Meterological Information Center, China Meteorological Administration. They contained the average daily temperature, the lowest daily temperature, and the highest daily temperature at 756 weather stations from 1951 to 2007 in all of the nine climatic zones in China. In order to avoid the consideration of climate zones and seasonal patterns, we decided to focus on our analysis for data within a single month and a single province. We extracted daily highest temperature data in July 2007 in Shandong province. Shandong is located in Northern China, extended from o to o latitude north and o to o longitude east. Its area is about thousand square kilometers. After extraction, the data set contained 20 stations in the province from the first day to the last day in July 2007 (Figure 5). By excluding one missing value, it had 619 observations of daily highest temperature values. The impact of altitude was removed by regressing the daily highest temperature across sites on their monthly averages. 16

17 Let Y (s, t) be the daily highest temperature relative to its site average. A geostatistical model Y (s, t) = µ + Z(s, t) + ϵ(s, t) (27) was basically considered to analyze the spatiotemporal variations of Y (s, t), where ϵ(s, t) was a white noise process and Z(s, t) was a mean zero stationary Gaussian process. function of Z(s, t) was modeled by M as The covariance E[Z(s, t)z(s + h, t + u)] = τ 2 M r,a1,a 2,ν 1,ν 2 (h, u), (28) where 1/a 2 (given by km) and 1/a 2 (given by day) were considered as two correlation lengths for space and time, respectively. There were seven parameters on the right side of (28). Together the variance parameter σ 2 in the white noise process, our statistical model contained eight parameters for the spatiotemporal variation of Y (s, t). We used η = σ 2 /(τ 2 + σ 2 ) to represent the nugget effect parameter and θ = (ν 1, 1/a 1, ν 2, 1/a 2, r 1, r 2 ) to represent the correlation parameters, where r 1 represented the correlation between longitude and time and r 2 represented the correlation between latitude and time. The nugget effect disappeared if η = 0. We attempted to estimate η and θ by the profile likelihood approach. In particular, let Y (s i, t i ), i = 1,, 619 be the ith observation in the data set. Then, where Cov[Y (s i, t i ), Y (s j, t j )] = (σ 2 + τ 2 )ρ ij = κ 2 ρ ij, ρ ij = ρ ij (η, θ) = Corr[Y (s i, t i ), Y (s j, t j )] = (1 η)m r,a1,a 2,ν 1,ν 2 (s j s i, t j t i ), i j. Let R = R(η, κ 2 ) = (ρ ij ) i,j=1,,619 be the correlation matrix of the observations. The loglikelihood function of the data was l(µ, κ 2, η, θ) = log 2π 1 2 log κ2 1 1 log det(r) 2 2κ (y 2 1µ) R 1 (y 1µ), (29) where y = (Y (s 1, t 1 ),, Y (s 619, t 619 )) and 1 is the vector with all of its elements equal to one. Given η and θ, the conditional MLE of µ was and the conditional MLE of κ 2 was ˆµ(η, θ) = (1 R 1 1) 1 1 R 1 y (30) ˆκ 2 (η, θ) = [1 R 1 1 yr 1 1(1 R 1 1) 1 1 R 1 y]. (31) 17

18 (a): Parallel to r^ (b): Vertical to r^ Sample Correlations Sample Correlations Location Change Location Change Figure 6: Values of Ĉ(s i s j, 1) when they are greater than corresponding to the direction parallel to ˆr (left) and the counterclockwise direction vertical to ˆr (right). Putting (30) and (31), we obtained the profile loglikelihood function as l P (η, θ) = 619 2π (1 + log ) log det(r) 2 2 log(y My), (32) where M = R 1 R 1 1(1 R 1 1)1 R 1. We computed the profile MLEs of η and θ by maximizing l P (η, θ). by ˆη and ˆθ. Let them be denoted Then, they were also the MLEs of η and θ respectively, and the profile MLEs of µ and κ 2, also their MLEs, were ˆµ = ˆµ(ˆη, ˆκ 2 ) and ˆκ 2 = ˆκ(ˆη, ˆθ) (Patefield, 1977). We carried out a Newton-Raphson algorithm to compute ˆη and ˆθ. Their values were ˆη = , ˆθ = (ˆν 1, 1/â 1, ˆν 2, 1/â 2, ˆr 1, ˆr 2 ) = (0.2674, 1005, , , , 679). After that, we had ˆµ = and ˆκ 2 = Since the corresponding estimates of correlation parameters in M r,a1,a 2,ν 1,ν 2 were ˆr = ( , 679) 0, we concluded that the estimate of Matérn correlation model was nonseparable. To confirm, we investigated the properties of sample correlations between stations when time change was one day. In particular, we computed Ĉ(s i s j, 1) = Ĉorr[Y (s i, t), Y (s j, t + 1)] = 30 t=1 [Y (s i, t) Ȳi][Y (s j, t + 1) Ȳj] { 30 t=1 [Y (s i, t) Ȳi] 2 } 1 2 { 30 t=1 [Y (s j, t + 1) Ȳj] 2 }

19 Figure 7: China Monsoon in Summer for all i, j = 1,, 20, where we used Ȳi = 30 t=1 Y (s i, t)/30 and Ȳj = 30 i=1 Y (s i, t + 1)/30. We obtained 400 sample correlation values. We had 326 of those were positive and 74 of those were negative. For sample correlations with absolute values greater than 0.2, we had 203 of those were positive and 17 of those negative. For sample correlations with absolute values greater than, we had 132 of those were positive and 7 of those were negative. For sample correlations with absolute values greater than, we had 68 of those were positive and 1 of those was negative. All of 23 sample correlations with absolute values greater than 0.5 were positive. To justify whether a non-fully symmetric space-time covariance model was more appropriate than a full symmetric space-time covariance model, we collected the values of Ĉ(s i s j, 1) for those greater than. We plotted these values along the direction parallel to ˆr (Figrue 6(a)) and along the counterclockwise direction vertical to ˆr (Figrue 6(b), i.e., the angle of ˆr plus π/2). We found that most of large Ĉ(s i s j, 1) values were at the oppsite direction of ˆr, indicating that the sample correlations were not symmetric along ˆr. Along the direction vertical to ˆr, we found that large Ĉ(s i s j, 1) values were almost symmetric. Therefore, we concluded that our model was more appropriate than a fully symmetric model in the analysis of the data set. 4.2 Irish Wind Speed Data We also applied our M to the Irish wind speed data set, which is available in the R package gstat. The data set contained the daily average wind speeds from 1961 to 1978 at 12 stations in Ireland. As recommended by Gneiting (2002), we removed Rosslare in our analysis. The whole period 19

20 Figure 8: A typical North Atlantic low-pressure area moving across Ireland (obtained from Wikipedia) of the data has been previously analyzed by many authors (Fonseca and Steel, 2011; Gneiting, 2002; Stein, 2005, e.g.). In order to understand the non-fully symmetry, we focused on the wind speed in July The final data contained 11 stations in 31 days. The size of the data was n = = 341. Following Haslett and Raftery (1989), we used the square root transformation of the wind speed because this transformation made the data nearly Gaussian. After that, we obtained the response values by removing the station averages. We computed the MLEs of η and θ by exactly the same method that we displayed in the previous data. We had η = and θ = (ν 1, 1/a 1, ν 2, 1/a 2, r 1, r 2 ) = (0.6353, 645.2, , , , 963). The estimated model was nonseparable as r = 0. 5 Discussion The article proposes an approach to constructing a non-fully symmetric space-time Mate rn correlation function based on an arbitrary spatial Mate rn margin and an arbitrary temporal Mate rn margin. The basic idea is to use the spectral densities of the two margins to construct the spectral density of a space-time correlation model. A seperable model is derived if the dependence between the two margins is ignored, while a nonseparable model is derived if the dependence is taken into account. Since a closed form formula generally do not exist, it is important to provide a Taylor expansion for the model such that numerical values can be computed. Although the space-time Mate rn model is focused on, the proposed approach can be used to 20

21 construct other models. Theoretically, for any given A(h, u) in Theorem 1, infinite families of Z 1 and Z 2 can be selected. Therefore, the proposed approach can be used to construct infinite many space-time correlation models, where one of those is the space-time Matérn model that has been discussed in the article. The nonseparable space-time Matérn model contains two smoothness parameters, two scale parameters, and a vector correlation parameter. The vector correlation parameter provides the direction for dependence between space and time. The space-time dependence is symmetric if the change of spatial locations belongs to the orthogonal space of the correlation parameter; otherwise, non-full symmetry appears. The significant difference between our proposed model and many previous models is the nonfull symmetry. The proposed approach directly provides a non-fully symmetric spectral density in the space-time domain. One does not need to consider a way to make a space-time correlation function non-fully symmetric via a certain transformation. As our model contains the separable Matérn correlation model as a special case, the justification of separability of spatiotemporal data in geostatistics can be possibly defined as a hypothesis testing problem, and this is an interesting and important research topic in the future. A Proofs Proof of Theorem 1: Let F be the spectral distribution of A(h, u). Without the loss of generality, assume F is a probability measure, which is generated by a random vector x 1 R d and X 2 R. If (x 1, X 2 ) is independent of (Z 1, Z 2 ), then ) A(hZ 1, uz 2 ) = E (e i(z 1h T x 1 +Z 2 ux 2 ) Z 1, Z 2 and C(h, u) = 0 [ =E E =E 0 [ E e i(z 1h T x 1 +z 2 ux 2 ) ( e i(ht x 1 Z 1 +ux 2 Z 2 ) ( e i(ht x 1 Z 1 +ux 2 Z 2 ) ). ] G(dz 1, dz 2 ) ) ] Z 1, Z 2 Therefore, C(h, u) is a characteristic function of x 1 Z 1 and X 2 Z 2, implying that C(h, u) is a valid space-time correlation function. If F is not a probability measure, then C(h, u) is a covariance function but not a correlation function. 21

22 Proof of Theorem 2: The sufficiency is directly implied by the expression of C(h, u). For the necessity, if C(h, u) is separable then, using the uniqueness of the inverse Fourier transformation, we conclude that Z 1 x 1 and Z 2 X 2 are independent. Since (x 1, X 2 ), Z 1, and Z 2 are independent, (Z 1 x 1, Z 1 ) is independent of Z 2 and (Z 2 x 2, Z 2 ) is independent of Z 1. Let z 1 and z 2 be real values such that P (Z 1 dz 1 ) and P (Z 2 dz 2 ) are positive if dz 1 and dz 2, the Lebesgue measures of dz 1 and dz 2 respectively, are positive. For any B 1 B(R d ) and B 2 B(R), P (x 1 B 1, X 2 B 2 ) = lim dz 1 0, dz 2 0 P (Z 1 x 1 z 1 B 1, Z 2 X 2 z 2 B 2 Z 1 dz 1, Z 2 dz 2 ) P (Z 1 x 1 z 1 B 1, Z 2 X 2 z 2 B 2, Z 1 dz 1, Z 2 dz 2 ) = lim dz 1 0, dz 2 0 P (Z 1 dz 1, Z 2 dz 2 ) = lim dz 1 0, dz 2 0 P (Z 1 x 1 z 1 B 1, Z 1 dz 1 )P (Z 2 X 2 z 2 B 2, Z 2 dz 2 ) P (Z 1 dz 1 )P (Z 2 dz 2 ) = lim P (Z 1 x 1 z 1 B 1 Z 1 dz 1 )P (Z 2 X 2 z 2 B 2 Z 2 dz 2 ) dz 1 0, dz 2 0 =P (x 1 B 1 )P (X 2 B 2 ). Therefore, x 1 and X 2 are independent, implying that A(h, u) is separable. Proof of Theorem 3: The joint PDF of (u, V ) is f 1 (u, v) = v ν 1 2 ν+ d 2 π d 2 Γ(ν) e 1 2 ( u 2 +v). The inverse transformation of (x, V ) = (au/ V, V ) is (u, v) = ( V x/a, V ). The corresponding Jacabian determinant is (V/a) d 2. The joint PDF of (x, V ) is f 2 (x, v) = v ν+ d ν+ d 2 a d 2 π d 2 Γ(ν) e 1 x 2 (1+ 2 a 2 )v. Integrationg out v in f 2 (x, v), we obtain the marginal PDF of x, which is m d (x ν, a), implying that the characteristic function of x is M d (h ν, a). Proof of Corollary 1: Straightforwardly, there is ( ) [ ( M d (h ν, a) = E e iht x = E E e iht (au/ )] ( ) V ) V = E e a2 h 2 2V, which yields (8). Proof of Lemma 1: The conclusion is implied by comparing ( M r,a1,a 2,ν 1,ν 2 (h, u) = E e a T 1a 2ur h V1 V 2 e 1 2 ( a2 1 h 2 + a2 2 u2 ) V 1 V 2 ) 22

23 with ( M d (h ν 1, a 1 )M 1 (h ν 2, a 2 ) = E e 1 2 ( a 2 1 h 2 V 1 ) + a2 2 u2 ) V 2, since it is drawn by looking at the sign of r T h. Proof of Theorem 4: The necessary and sufficient conditions for M r,a1,a 2,ν 1,ν 2 to be separable is implied by Theorem 2. The sufficiency of full symmetry is concluded using the definition of M r,a1,a 2,ν 1,ν 2. If d 2 then the necessity of full symmetry is implied by (11). If d = 1, then the necessity of full symmetry is implied by (12). Proof of Theorem 5: The conclusion for α > 0 can be directly implied by (9). If α < 0 and z 0, using variable transformation w = z 2 /v in the integral expression of D α (z) there is D α (z) = z 2α w α 1 e 1 2 ( z2 w +w) dw = z 2α w α 1 e 1 2 ( z2 w +w) dw = z 2α D α ( z ), 0 0 implying the conclusion for α < 0. If α = 0, then D 0 (z) = 2 [ ve v z 2 2 de z2 2 ( v ) ] 2v = z e 1 2 ( z2 v +v) dv implying the conclusion for α = = 2 z 2 D 2(z) 1 z 2 D 1(z), Proof of Corollary 2: If α < 0 then z 2β D α (z) = 2 α Γ( α ) z 2(β+α) M α ( z ) implying that it is continuous in z R. Write 1 z 2β D 0 (z) = z 2β v 1 e 1 2 ( z2 v +v) dv + z 2β v 1 e 1 2 ( z2 v +v) dv. (33) 0 The second term on the right side of (33) is continuous in z R. For any β > γ > 0, the first term is dominated by z 2β v γ 1 e 1 2 ( z2 v +v) dz = z 2β D γ (z), 0 which is continuous in all z R. Letting γ 0 and using the Dominated Convergence Theorem, we conclude that the first term on the right side of (33) is also continuous in all z R. Proof of Theorem 6: If n > 2(ν 1 n 2 ), then d n,ν1 (a 1 h ) = 2 n 2 ν 1 Γ( n 2 ν 1)(a 1 h ) 2ν 1 M n 2 ν 1(a 1 h ) 1 and d n,ν2 (a 2 u ) = 2 n 2 ν 2 Γ( n 2 ν 2)(a 2 u ) 2ν 2 M n 2 ν 2(a 2 u ), implying that b n,r,a1,a 2,ν 1,ν 2 (h, u) = ( 1)n (u 0 r T 0 h 0 ) n (a 1 h ) 2ν 1 (a 2 u ) 2ν 2 2 2(ν 1+ν 2) c n,ν1,ν Γ(ν 1 )Γ(ν 2 ) 2 (a 1 h, a 2 u, r ), 23

24 where Note that and c n,ν1,ν 2 (z 1, z 2, r) = 2n r n Γ( n ν 2 1)Γ( n ν 2 2) M n 2 n! ν 1(z 1 )M n 2 ν 2(z 2 ), z 1, z 2, r > 0. b n,r,a1,a 2,ν 1,ν 2 (h, u) (a 1 h ) 2ν 1 (a 2 u ) 2ν 2 2 2(ν 1+ν 2) Γ(ν 1 )Γ(ν 2 ) c n,ν 1,ν 2 (a 1 h, a 2 u, r ) c n,ν1,ν 2 (z 1, z 2, r) c n,ν1,ν 2 (r) = 2n r n Γ( n ν 2 1)Γ( n ν 2 2), n > 2(ν 1 ν 2 ). n! It is enough to show the uniform convergence of n=[2(ν 1 n 2 )]+1] c n,ν 1,ν 2 (r) in r [0, 1 ϵ] for any 0 < ϵ < 1. Using Stirling s approximation that Γ(z) 2πe z z z 1 2 (1 + o(1)) for sufficiently large z, there is c n+1,ν1,ν lim 2 (r) n c n,ν1,ν 2 (r) = lim ν 2 1 )Γ( n+1 ν 2 2 ) (n + 1)Γ( n ν 2 1)Γ( n ν 2 2) n 2rΓ( n+1 = lim n 2r ( n+1 ν 2 1 ) n 2 ν 1 ( n+1 n + 1 ν 2 2 ) n 2 ν 2 e (n+1 ν 1 ν 2 ) ( n ν 2 1) n 1 2 ν 1 ( n ν 2 2) n 1 2 ν 2 e (n ν 1 ν 2 ) 2r( n+1 ν 2 1 ) 1 2 ( n+1 ν 2 2 ) 1 2 = lim n (n + 1)e =r. ( n 2ν 1 ) n 1 2 ν 1 ( n 2ν 2 ) n 1 2 ν 2 Therefore, n=[2(ν 1 ν 2 )]+1 d n,ν 1,ν 2 (r) uniformly converges in r [0, 1 ϵ] for any 0 < ϵ < 1, implying that the right size of (14) absolutely uniformly converges in h R d, u R, r R d with r 1 ϵ for any 0 < ϵ < 1. References Bochner, S. (1955). Harmonic Analysis and the Theory of Probability, Berkeley and Los Angeles: University of California Press. Brown, P., Karesen, K., Tonellato, G.O.R.S. (2000). Blur-generated nonseparable space-time models. Journal of Royal Statistical Society Series B, 62, Cressie, N. and Huang, H.C. (1999). Classes of nonseparable, spatio-temporal stationary covariance functions. Journal of the American Statistical Association, 94, De Iaco, S., Myers, D., and Posa, D. (2001). Space-time analysis using a general product-sum model. Statistics and Probability Letters, 52,

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