Part 5: Spatial-Temporal Modeling

Transcription

1 Part 5: Spatial-Temporal Modeling 1 Introductory Model Formulation For a broad spectrum of applications within the realm of spatial statistics, commonly the data vary across both space and time giving rise to spatial-temporal data. There has been some debate about what we should call spatial-temporal data (see Roddick, Hornsby and Spiliopoulou, 2001) but for this course, we will include in this category data for which we need to account for both time- and space-dependencies. Due to the proliferation of data sets that are both spatially and temporally indexed, spatialtemporal modeling has received increased attention in the last few years. General model formulation. Consider the case of point-referenced data where time is discretized. We can look at space-time indexed data Y (s, t) in two ways: Writing Y (s, t) = Y s (t) as a spatially-varying time series model; and Writing Y (s, t) = Y t (s) as a temporally-varying spatial model. It is important to differentiate between the two types of models as one may be interested in the time-dependent patterns allowing for spatial dependent between time series (first model) or spacedependent patterns borrowing information from spatial processes that are close in time (second model). For exploratory analysis, we obtain the singular value decomposition (or EOF); assume T < n, (where T is the number of observations over time, and n the number of location sites), we can write Y = UDV = T l=1 d l U l V T l where U is a n n orthogonal matrix with columns U l = (u l (s 1 ),..., u l (s n )) and V a T T orthogonal matrix with columns V l = (v l (t 1 ),..., v l (t T ), D is a n T matrix of the form (, 0) T where is T T diagonal with diagonal entries D l. Assume d l s are arranged in decreasing order of their absolute values. U l Vl T is referred to as the l th empirical orthogonal function. We can write Note that Y (s i, t) = d l u l (s i )v l (t). Y Y T = T d 2 l U lul T. l=1 The first and second (maybe we need more than that) EOFs provide most of the information of the spatial structure. EOFs are a useful exploratory tool to learn about the spatial structure of the data, but for full inference we need a full spatiotemporal model. From a methodological point of view, the introduction of time into spatial modeling brings substantial complexity in the scope of the modeling, as we need to make decisions regarding spatial correlation, temporal correlation, and how space and time interact. There are some quick approaches to dealing with space and time dependencies: Spatial-temporal data analysis with methods for random fields in R d+1 ; Separate spatial analysis for each time point; or separate temporal analysis for each location. The first approach is not appropriate since time and space are not directly comparable. Space has not past, present, and future and the spatial coordinate units are not comparable to temporal units. 1

2 The last two approaches may be viewed as conditional methods as we analyze either the space given time or time given space. A two-stage variation of these two approaches is to combine the results from the conditional analysis in a second stage. Two-stage approaches have a series of drawbacks: If it is not possible to analyze the data spatially for a specific time point, then data collected at that time will not contribute in the second stage when the spatial analyses are combined. Data that are sparse in time or space may present difficulties in this regard. There are many ways to summarize the results from the first-stage analysis. To reasonably combine statistics into comprehensive measures requires information about the temporal correlation between statistics which is not available under this approach. Interpolation of observations in a continuous space-time process should take into account interaction between the spatial and temporal components and allow for predictions in time and space. Separate analysis in time (space) allows predictions in space (time) only. Therefore, joint analysis of spatio-temporal data are preferable to separate analysis. In the joint analysis, we write the process as {Y (s, t) : s R(t) R d, t T }. The dependence of the domain R on time symbolizes the condition where the spatial domain changes over time. For simplicity, we assume R(t) = R. In space-time processes, there are two distances between points: the distance between points in space h ij = s i s j, and the distance between points in time k ij = t i t j. The spatio-temporal lag between Y (s, t) and Y (s + h, t + k) is [h, k]. An isotropic correlation function in space and time but anistropic is space-time is Corr(Y (s i, t i ), Y (s j, t j )) = R(θ s h ij 2 + θ t k ij 2 ) where θ s and θ t are the spatial and temporal parameters. We define stationarity and isotropy as follows: A spatio-temporal covariance function is (second-order) stationary in space and time if Cov(Y (s i, t i ), Y (s j, t j )) = C(s i s j, t i t j ). The covariance function is further isotropic in space if Cov(Y (s i, t i ), Y (s j, t j )) = C( s i s j, t i t j ). The functions C(h, 0) and C(0, k) are spatial and temporal covariance functions. For a spatiotemporal covariance function to be valid, it needs to be positive-definite; that is, for any set (s 1, t 1 ),..., (s k, t k ) and real numbers a 1,..., a k : 1.1 Covariance Functions k i=1 j=1 k a i a j C(s i s j, t i t j ) 0. Separable Covariance Functions. A separable spatio-temporal covariance function decomposes Cov(Y (s i, t i ), Y (s j, t j )) to separate spatial and temporal components. Valid separable covariance functions are: Cov(Y (s, t), Y (s + h, t + k)) = C s (h; θ s )C t (k; θ t ) 2

3 and Cov(Y (s, t), Y (s + h, t + k)) = C s (h; θ s ) + C t (k; θ t ) where C s (h; θ s ) and C t (k; θ t ) are spatial and temporal covariance functions. Separable covariance functions are commonly used since they are easy to interpret, to compute and to work with, and valid, provided that the components are valid covariance functions. Separability assumption implies that dependence attenuates in a multiplicative manner across space and time. The primary drawback of separable models is not to incorporate space-time interactions. For the multiplicative separable function, the spatial covariances at time lag k and lag u (assuming k u) have the same shape, they are proportion to each other. The temporal and spatial components represent dependencies in time given spatial location and space given time. The processes do not act upon each other. Non-Separable Covariance Functions. One important difficulty in non-separable covariance functions is their validity. Methods for constructing valid non-separable functions include the monotone function approach of Gneiting, 2002; the spectral method of Cressie and Huang 1999; the spectral method of Chen, Fuentes and Davis; the mixture approach of Ma, 2002; and the partial differential equation approach of Jones and Zhang, Monotone Function Approach. Some of the advantages of this method are that it does not require operations in the spectral domain and it constructs valid covariance functions from components whose validity is easy to check. Let [h, k] a lag vector in R d R and choose two functions φ(t) and ψ(t) for t 0 such that φ(t) is monotone and ψ(t) is positive with a monotone derivative (e.g. φ(t) = e ct, c > 0 and ψ(t) = at + 1, a > 0).A valid covariance function is then Cov(Y (s, t), Y (s + h, t + k)) = C(h, k) = σ 2 ψ( k 2 ) d/2 φ ( h 2 /ψ( k 2 ) ). For example, for φ(t) = e ctγ, c > 0 and ψ(t) = (at α +1) β, a > 0 with 0 < γ, α 1 and 0 β 1 and for d = 2, the covariance function is σ 2 ( C(h, k) = (a k 2α + 1) β exp c h 2γ ) (a k 2α + 1) βγ. For β = 0, the covariance function does not depend on the time lag. Multiplying this function with a temporal covariance function leads to a separable model when β = 0. For h = 0, the resulting temporal covariance function C t (k) multiplied with C t (k) C(h, k) provides a valid spatio-temporal covariance function; it is separable for β = 0 and non-separable otherwise. Since the separable and non-separable models are nested, a statistical test for H 0 : β = 0 can be carried out. The test statistic may be based on restricted log-likelihood ratio. Spectral Method. Assuming that C is continuous and its spectral distribution function possesses a spectral density f(w; τ) 0, according to Bochner s Theorem C(h; u) = exp(ih w + iuτ)f(w; τ) dwdτ. Therefore, the spectral density f(w; τ) is the Fourier transform of the spatial-temporal covariance function 1 f(w; τ) = (2π) d+1 exp( iw t x iτt)c(x, t) dxdt. 3

4 For example, the Matérn spatial spectral density is given by f(w) = (φ 2 + w 2 ) ν d/2. Chen, Fuentes and Davis, 2006 proposed the following spatial-temporal spectral density that has a separable model as a particular case. The spectral density f of Y changes with space and time to explain how the spatial temporal dependency varies on the domain of interest. Locally (in a neighborhood of s i = (x i, t i ), to allow lack of stationarity), they propose the following parametric model for f, f si (w, τ) = γ i (α 2 i β 2 i + β 2 i w 2 + α 2 i τ 2 + ɛ w 2 τ 2 ) ν i under stationarity, the parameters of the model do not change with location (e.g. it is simply α, rather than α i ). It is assumed that γ i, α i and β i are positive, ν i > (d + 1)/2 and ɛ [0, 1]. The parameters αi 1 explains the rate of decay of the spatial correlation. For the temporal correlation, the rate of decay is explained by the parameter βi 1, whereas γ i is a scale parameter. When ɛ = 1, the previous equation can be written as f si (w, τ) = γ i (α 2 i β 2 i + β 2 i w 2 + α 2 i τ 2 + w 2 τ 2 ) ν i = γ i (α 2 i + w 2 ) ν i (β 2 i + τ 2 ) ν i. Therefore the corresponding spatial-temporal covariance is separable, both spatial component and temporal component are Matérn type covariances. On the other hand, when ɛ = 0, f si (w, τ) = γ i (α 2 i β 2 i + β 2 i w 2 + α 2 i τ 2 ) ν i. The corresponding spatial-temporal covariance is a 3-d Matérn type covariance with an extra parameter, which can be considered a conversion factor between the units in the space and time domains. That is, when ɛ = 0, C(x, t) = σ 2 2 ν 1 Γ(ν) ( (x, ρt ) r ) ( ) (x, ρt ) K ν r where r is the range parameter, σ 2 is the sill, ν > 0 measures the smoothness of Y, ρ which is new, is a scale factor to take into account the change of units between the spatial and temporal domains. Therefore, this is a d + 1 Matérn type covariance, but it takes into account the change of units between space domain and temporal domain. In summary, the new class spectral density is nonseparable for 0 ɛ < 1, and separable for ɛ = 1. Therefore, the parameter ɛ plays a role for separability. It controls the interaction between the spatial component and the temporal component. When ɛ equals 0 and 1, there are exact forms for the corresponding spatial-temporal covariances. Otherwise, the corresponding spatial-temporal covariance has to be computed numerically. Cressie and Huang, 1999 propose a generic approach to developing parametric models for spatialtemporal processes. The method relies heavily on spectral representations for the theoretical spacetime covariance structure, and generalizes the results of Matern for pure spatial processes. In essence, Matérn constructs a number of parametric families for spatial processes by direct inversion of spectral densities. Cressie and Huang show that the same ideas can be used to construct families of spatial-temporal covariances. First, Cressie and Huang represent the stationary spatial-temporal covariance C(h, u) C(h, u) = exp(ih T w + iuτ)g(w, τ) dwdτ 4

5 where C(h, u) is a stationary spatial-temporal covariance function in which h represents a d- dimensional spatial vector and u is a scalar time component. The function g(w, τ) may be written as a scalar Fourier transform in τ g(w, τ) = 1 e iuτ h(w, u) du 2π where h(w, u) is its inverse. Therefore, C(h, u) = The next step is to write e iht w h(w, u) dw. (1) h(w, u) = k(w)ρ(w, u) (2) where k(w) is the spectral density of a pure spatial process and ρ(w, u) for each w is a valid temporal autocorrelation function in u. Cressie and Huang remark that any smooth space-time covariance function can be written in the form (1) and (2), and they also impose the conditions: (C1) For each w, ρ(w, ) is a continuous temporal autocorrelation function, ρ(w, u) du < ; (C2) k(w) dw < and k(w) > 0. Under those conditions, the generic formula for C(h, u) becomes C(h, u) = e iht w k(w)ρ(w, u) dw. (3) When ρ(w, u) is independent of w, (3) reduces again to a separable model. Cressie et al. developed seven special cases of (3). For example, ρ(w, u) = exp ( w 2 u 2 ) exp( δu 2 ), δ > 0 4 k(w) = exp ( c 0 w 2 ), c 0 > 0 4 which lead to C(h, u) ) 1 (u 2 exp ( h 2 + c 0 ) d/2 u 2 exp( δu 2 ) (4) + c 0 The condition δ > 0 is needed to ensure the condition (C1) is satisfied at w = 0, but the limit of (4) as δ 0 is also a valid spatial-temporal covariance function, leading to the three parameter family σ 2 C(h, u) = (a 2 u 2 exp ( b2 h 2 ) + 1) d/2 a 2 u Mixture Approach. If Y S (s) and Y T (t) are two purely spatial and temporal processes with covariance functions C S (h; θ s ) and C T (k; θ t ), respectively, then Y (s, t) = Y S (s)y T (t) has the separable product covariance function C(h, k) = C S (h; θ s )C T (k; θ t ), if Y S (s) and Y T (t) are uncorrelated. Space-time interactions can be incorporated by mixing product covariance or correlation functions. Ma (2002) considers the probability mass functions π ij = P ((U, V ) = (i, j)) where (U, V ) is a bivariate random vector with support on the non-negative integers. Conditional on (U, V ) = (i, j), the spatio-temporal process has correlation function R(h, k U = i, V = j) = (R S (h)) i (R T (k)) j, 5

6 then the correlation function of the unconditional process is the non-separable model R(h, k) = (R S (h)) i (R T (k)) j π ij. i=1 j=1 Ma, 2002 terms this model a positive power mixture; if R(u) is a correlation model in R d then (R(u)) i is also a valid correlation model in R d for any positive integer i. The method of power does not require a bivariate, discrete mass function. A non-separable model can be constructed for the univariate case also R(h, k) = RS(h)R i T i (k)π i. i=1 If U is a random variable with support of positive integers, and P (U = i) = π i, then its probability generating function is G(w) = w i π i, 0 w 1. i=1 Therefore, the correlation function above takes the role of w i in G(w). The procedure will be to obtain the probability generating function and replace w with R S (h)r T (k). Note: In the covariance functions above, we have used C S (h) and C T (k) stationary functions though this is not necessary the case. The development of the mixture models applies to nonstationary functions also. 2 The Spatio-Temporal Semivariogram To estimate the parameters of the covariance function, similar methods to spatial correlation estimation may be applied here. The semivariogram for a stationary spatio-temporal process is γ(h, k) = 1 V[Y (s, t) Y (s + h, t + k)] = 2 = V[Y (s, t)] Cov(Y (s, t), Y (s + h, t + k)] = C(0 d, 0) C(h, k). The corresponding empirical spatio-temporal semivariogram estimator is ˆγ(h, k) = 1 N(h, k) (i,j) N(h,k) (Y (s i, t i ) Y (s j, t j )) 2. The set N(h, k) consists of the points that are within spatial distance h and time lag k of each other; N(h, k) is the number of distinct pairs in that set. This empirical estimator for the joint spatiotemporal semivariogram is different from a conditional estimator of the spatial semivariogram at time t which would be used in a two-stage method: ˆγ t (h) = 1 N t (h) (i,j) N t (h) (Y (s i, t) Y (s j, t)) 2. A weighted least squares fit of the joint spatio-temporal empirical semivariogram to a model γ(h, k; θ) estimates θ by minimizing m s m t j=1 l=1 N(h j, k l ) 2γ(h j, k l ; θ) (ˆγ(h j, k l ) γ(h j, k l ; θ)) 2 6

7 where m s and m t are the number of spatial and temporal lag classes. A fit of the conditional spatial semi-variogram at time t minimizes m s j=1 N t (h j ) 2γ(h j, t; θ) (ˆγ(h j, t) γ(h j, t; θ)) 2. 3 Spatio-Temporal Point Processes A spatio-temporal point process is a spatio-temporal random field with a random spatial index R and a temporal index T. According to the type of the nature of the temporal component we distinguish three types of processes: Earthquake process: Events are unique to spatial locations and time points, only one event can occur at a particular location and time. This type of process may be applied not only to earthquakes but also to burglary patterns for example. Explosion process: The idea of an explosion process is the generation of the spatial point process at a time t which itself is a realization in a stochastic process. The realization of an explosion process consists of locations s i1,..., s ini R at time t i T. Temporal events occur with intensity γ(t) and produce point patterns with intensity λ t (s). An example of such a spatiotemporal process is the distribution of acorns around an oak tree. The time at which the acorns fall each year can be considered a temporal random process. The distribution of the acorns is a point process with some intensity, possibly spatially-varying. Note that for this prcess, the time points at which the point patterns are observed are the realization of a stochastic process; otherwise, if the observation times are chosen by the experimenter, the spatio-temporal pattern is referred to as a point pattern sampled in time. Birth-Death process: This process is useful to model objects that are placed at random location by birth at time t b and exist at that location for a random time t l. Cressie, 1993 refers to such a process as a space-time survival point process. At time t, an event is recorded at location s if a birth occurred at s at time t b < t and the object has a lifetime of t b + t l > t. Rathbum and Cressie, 1994 formulate the spatio-temporal distribution of longleaf pines through a birth-death process. The realization of a birth-death process observed at fixed time points can be indistinguishable from temporally sampled point pattern. Events observed at location s at time t i but not at time t i+1 could be due to the death of a spatially stationary object or due to the displacement of a non-stationary object between the two time points. Intensity Measures. We extend the first and second order intensity measures of a spatial point process to the spatio-temporal scenario. For this, we define N(ds, dt) the number of events in an infinitesimal cylinder with base ds and height dt (Dorai-Raj, 2001). Contrast this to the definition under spatial processes, where we consider N(ds) with ds the infinitesimal disk (ball) of area (volume) ds. The spatio-temporal intensity of the process Y (s, t) is then defined as the average events per unit volume as the cylinder is shrunk around the point (s, t): λ(s, t) = lim ds, dt 0 E(N(ds, dt)). ds dt To consider the marginal spatial intensity, we integrate out the time λ(s) = λ(s, v) dv and obtain the marginal temporal intensity by λ(t) = T R 7 λ(u, t) du.

8 The conditional spatial intensity at time t is defined as λ(s t) = E(N(ds, t)) lim ds 0 ds and the conditional temporal intensity at location s as λ(t s) = lim dt 0 E(N(s, dt)). dt Second-order intensities can also be extended to the spatio-temporal case; let A i = ds i dt i be an infinitesimal cylinder containing the point (s i, t i ). The second-order spatio-temporal intensity is defined as λ 2 (s i, s j ; t i, t j ) = lim A i, A j 0 E(N(A i )N(A j )). A i A j Stationarity and Complete Randomness. We define Y (s, t) is first-order stationary in space if λ(t s) = λ (t); Y (s, t) is first-order stationary in time if λ(s t) = λ (s); Y (s, t) is first-order stationary in space and time if λ(s, t) does not depend on s and t. Second order stationarity in space and time requires that λ(s, t) does not depend on s and t, and that λ 2 (s, s + h; t, t + k) = λ s (h, k). A spatio-temporal process is then completely spatio-temporally random (CSTR) process if it is a Poisson process in both time and space; that is, N(A, T ) P oisson(λ A T ). For this, λ(s, t) = λ and λ 2 (s, s + h; t, t + k) = λ 2. The CSTR is commonly used as a benchmark against which spatio-temporal patterns are tested. Space-time clustering is the most frequent alternative to CSTR. For spatio-temporal processes, a space-time clustering is one for which among events that are close in time, there are events that are closer in space than would be expected by chance (Cressie, 1993). Discussed in Cressie (1993) who references Fiksel, 1984 is a test for space-time clustering for an earthquake process. Let {(s i, t i ) : i = 1, 2,..} denote the locations and times of earthquake events ordered so that t 1 < t 2 <... Assume that the position of the (n + 1)th event depends on the earlier events (s 1, t 1 ),..., (s n, t n ). Let ds n+1 be an infinitesimal region located at s n+1. Define the transition density P (N(ds n+1 ) = 1 s 1,..., s n ) φ n (s n+1 s 1,..., s n ) = lim ν(ds n+1 ) 0 ν(ds n+1 assuming the limit exists. If φ n is known, then P (s n+1 A s 1,..., s n ) = A φ n (u s 1,..., s n )ν(du). For example, for an earthquake point process, Fiksel considers the parametric form φ n (s n+1 s 1,..., s n ) = λ2 2πn and conditional on n, the likelihood is which is maximized to obtain the MLE for λ. n exp( λ s n+1 s i ) i=1 l n (λ) = φ n 1 (s n s 1,..., s n 1 )...φ 1 (s 2 s 1 ) 8

9 4 Spatial-Temporal Modeling Denote Y (s, t) measurement at location s and time t. Y (s, t) = µ(s, t) + ɛ(s, t) where µ is the mean structure and ɛ are the residuals. We can express the mean function as µ(s, t) = x(s, t)β(s, t) where x are known covariates, and β are spatio-temporally varying coefficients. The error term ɛ can be rewritten as ɛ(s, t) = w(s, t) + e(s, t) where e is a Gaussian white noise process and w is a mean-zero spatiotemporal process. Then, this would be a hierarchical model with a conditionally independent first stage given µ and w. The small-scale spatial variations are captured in w(s, t). Common alternatives to defining this process are: model 1: w(s, t) = α(t) + w(s) model 2: w(s, t) = α s (t) model 3: w(s, t) = w t (s) The first one provides an additive form in temporal and spatial effects. The second provides temporal evolution at each spatial location. The third provides a spatial structure at each time. One other common approach in spatiotemporal estimation is Bayesian hierarchical modeling where µ(s, t) is a random spatiotemporal process. Gelfand et al proposed modeling the mean function as µ(s, t) = X(s, t)β(s, t) where X(s, t) are a set of covariates. They further discussed different possible models for β(s, t) under certain assumptions: 1. No interaction between time and space (β(s, t) = β(s) + β(t)); 2. Spatial dependence attenuated across time (β(s, t) (0, Σ), Σ [β(s,t)] = Σ [β(s)] Σ [β(t)] ); and 3. Spatially varying coefficient processes nested within time (β(s, t) = β (t) (s)). Gelfand, Banerjee and Gamerman 2005 introduced dynamic spatiotemporal models, Bowman, 2005 used a spatial autoregressive model and Gössl et al a semiparametric model for µ(s, t). Other relevant methods within the Bayesian hierarchical model framework are introduced by Huang and Cressie, 1996; Wikle and Cressie, 1999; Mardia et al., 1998; Tonellato, 1997; Sanso and Guenni,2000; Stroud et al.,2001; Banerjee, Gamerman and Gelfand, 2003; Xu and Wikle, Another common approach to spatiotemporal estimation is extending spatial smoothing (Nychka, 2000) to incorporate temporal dependence (Luo et al, 1998; Clark et al., 2006; Kamman and Wang, 2003). In spatiotemporal smoothing, the focus is on modeling the mean function µ(s, t). Luo et al, 1998 suggested using a tensor product of spatial and temporal effects for modeling µ(s, t) resulting in f(s, t) = g 1 (t) + g 2 (s) + g 1 (t)g 2 (s). Within the same line of models, Kamman and Wang, 2003 proposed a geoadditive model. They use an additive decomposition of the spatial and temporal components, µ(s, t) = β 0 + f(s) + g(t). The functions f and g are estimated using a semiparametric model formulation. We will see two of these examples ( Clark et al., 2006 and Luo et al, 1998) as part of this lecture and one of the assigned papers is Kamman and Wang, 2003, which will allow us to compare three non-parametric smoothing models. The Bayesian models will be deferred to the last part of this course when we will discuss spatial (temporal) Bayesian modelling. 9

10 5 Homework For the next lecture, you will all read three papers: 1. Kamman, E.E., Wand, M.P.(2003), Geoadditive Models, Applied Statistics, 52(1), p Bowman, F.D., (2007), Spatiotemporal Models for Region of Interest Analyses of Functional Neuroimaging Data, Journal of the American Statistical Association, 478, Sanso B., Guenni L.(2000), A Nonstationary Multisite Model for Rainfall,J. of American Statistical Association, 452, Stein, M. (2005), SpaceTime Covariance Functions, Journal of the American Statistical Association, Vol. 100, No. 469, References [1] Chen, L., Fuentes, M., and Davis, J. (2006). Spatial-temporal Statistics for Ecological data, Chapter in Modern Statistical Computation, to appear. Editor Jim Clark and Alan Gelfand. Wiley, New York. [2] Clarke, E.D., Speirs,D.C., Heath, M.R.,Wood, S.N., Gurney, W.S.C.,Holmes, S.J.(2006) Calibrating remotely sensed chlorophyll-a data by using penalized regression splines, Appl. Statistics, 55(3), p [3] Cressie, N., Huang, H.C. (1999), Classes of nonseparable spato-temporal stationary covariance functions, JASA, 94, [4] Gelfand, A.E., Banerjee, S., Gamerman, D.(2005), Spatial process modelling for univariate and multivariate dynamic spatial data, Econometrics, 16, [5] Gelfand, A.E., Kim H.J., Sirmans, C.F., Banerjee, S., (2003), Spatial Modeling With Spatially Varying Coefficient Processes, Journal of the American Statistical Association, 462, [6] Gneiting, T. (2002), Nonseparable, stationary covariance functions for space-time data, JASA, 97, [7] Gössl, C., Auer, D.P., Fahrmeir, L. (2001), Bayesian Spatiotemporal Inference in Functional Magnetic Resonance Imaging, Biometrics, 57, p [8] Huang, H.C., Cressie, N. (1996), Spatio-temporal prediction of snow water equivalent using the Kalman filter, Computational Statistics and Data Analysis, 22, [9] Jones, R.H., Zhoang, Y. (1997), Models for continuous stationary space-time processes, In Gregoire, T.G, Brillinger, D.R., Diggle, P.J., Russek-Cohen, E., Warren, W.G., Wolfinger, R.D. (eds), Modeling Londitudinal and Spatially Correlated Data, Springer Verlang, NY, [10] Kamman, E.E., Wand, M.P.(2003), Geoadditive Models, Applied Statistics, 52(1), p1-18. [11] Luo, Z., Wahba G., Johnson, D.(1998), Spatial-Temporal Analysis of Temperature Using Smoothing Spline ANOVA, Journal of Climate, 11,

11 [12] Ma, C. (2002), Spatio-temporal covariance functions generated by mixtures, Mathematical Geology, 34, [13] Nychka, D.W.(2000), Spatial-Process Estimates as Smoothers, in Smoothing and Regression. [14] Sanso B., Guenni L.(2000), A Nonstationary Multisite Model for Rainfall,J. of American Statistical Association, 452, 1089 [15] Schabenberger, O., Gotway, C.A. (2005), Statistical Methods for Spatial Data Analysis, Chapman & Hall, CRC. [16] Waller L.A., Carlin B.P., Xia H., Gelfand A.E., (1997), Hierarchical Spatio-Temporal Mapping of Disease Rates, Journal of the American Statistical Association, 438, [17] Wikle C.K., Cressie, N.A.C.(1999), A Dimension-Reduced Approach to Space-Time Kalman Fltering, Biometrika, 86(4), [18] Wikle, C.K., Milliff R.F., Nychka, D., Berliner, L.M., Spatiotemporal Hierachical Bayesian Modeling, 2001, Journal of the American Statistical Association, 96, 454, [19] Xu, K., Wikle, C.K., Fox, N.I., (2005), A Kernel-based Spatio-temporal Dynamical Model for nowcasting Weather Radar Reflectivities, Journal of the American Statistical Association, 472,