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Matérn Cross-Covariance Functions for Multivariate Random Fields

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Matérn Cross-Covariance Functions for Multivariate Random Fields Matérn Cross-Covariance Functions for Multivariate Random Fields Tilmann GNEITING, William KLEIBER, and Martin SCHLATHER We introduce a flexible parametric family of matrix-valued covariance functions for multivariate spatial random fields, where each con- stituent component is a Matérn process. The model parameters are interpretable in terms of process variance, smoothness, correlation length, and colocated correlation coefficients, which can be positive or negative. Both the marginal and the cross-covariance functions are of the Matérn type. In a data example on error fields for numerical predictions of surface pressure and temperature over the North American Pacific Northwest, we compare the bivariate Matérn model to the traditional linear model of coregionalization. KEY WORDS: Co-kriging; Convolution; Cross-correlation; Gaussian spatial random field; Matérn class; Maximum likelihood; Multi- variate geostatistics; Numerical weather prediction; Positive definite. 1. INTRODUCTION which depends on the spatial separation vector, h ∈ Rd, only. The covariance function is isotropic if C(h ) = C(h ) when- Spatial stochastic process models are of ever-increasing im- 1 2 ever h =h , where ·is the Euclidean norm. Clearly, portance in a wealth of applications, ranging from the geo- 1 2 sciences and atmospheric sciences to environmental monitor- stationarity along with isotropy are not always realistic as- ing, economics, and other areas. The goals are as varied as sumptions; however, these models are the basic building blocks the fields in which spatial models are applied. The modeler of more sophisticated, anisotropic, and nonstationary models. might want to gain scientific insight in the processes studied, in A class of isotropic covariance functions that has received a which case the interpretability of the parameters is of great im- great deal of attention recently is the Matérn family (Matérn portance. Environmental risk analysis may require spatial sam- 1986; Guttorp and Gneiting 2006), which specifies the covari- 2 2 pling, which depends on the use of a model that allows for fast ance function as σ M(h|ν,a) where σ > 0 is the marginal simulation. Shared amongst very many applications is the need variance and for spatial interpolation to sites where no observations are avail- 1−ν 2 ν able. Whatever the goal of the study, it is frequently essential M(h|ν,a) = (ah) Kν(ah) (1) (ν) that several spatial variables be modeled simultaneously. The critical step then is to identify a suitable spatial dependence is the spatial correlation at distance h.HereKν is a mod- structure, not only within each variable, but between variables ified Bessel function of the second kind and a > 0 is a spatial as well. Here we focus on geostatistical formulations that rely scale parameter, whose inverse, 1/a, is sometimes referred to as on the fitting of covariance and cross-covariance structures for a correlation length. The smoothness parameter ν>0 defines Gaussian random fields on a Euclidean space, Rd. In this con- the Hausdorff dimension and the differentiability of the sample text, spatial interpolation is often referred to as co-kriging. Ex- 1 paths. If ν equals an integer plus 2 , the Matérn function reduces cellent expositions of geostatistical approaches to the model- to the product of an exponential function and a polynomial, in ing of spatial data include the monographs by Cressie (1993), that Goovaerts (1997), Chilès and Delfiner (1999), Stein (1999), n 1 (n + k)! n − Wackernagel (2003), and Banerjee, Carlin, and Gelfand (2004). M hn + , a = exp(−ah) (2ah)n k Gaussian Markov random fields provide an alternative that is 2 (2n)! k k=0 particularly suited to lattice data (Besag 1974; Mardia 1988; Rue and Held 2005; Sain and Cressie 2007). for n = 0, 1,... by Equation 8.468 of Gradshteyn and Ryzhik Suppose, for the moment, that we have a single spatial vari- (2000). This nests the popular exponential model that arises as | 1 = − able, and our univariate data arise from a spatial random field M(h 2 , a) exp( a h ). {Y(s) : s ∈ Rd}, which we assume to be Gaussian and second- For a positive integer k, the sample paths of a Gaussian order stationary with mean zero. Hence, the process admits a Matérn field are k times differentiable if and only if ν> covariance function, k (Handcock and Stein 1993; Banerjee and Gelfand 2003). C(h) = E(Y(s + h)Y(s)), A complementary description of the smoothness of a Matérn field is via the Hausdorff or fractal dimension of a sample path in Rd, which equals the maximum of d and d + 1 − ν (Adler Tilmann Gneiting is Professor, Institut für Angewandte Mathematik, Univer- 1981; Goff and Jordan 1988). For a differentiable field with sität Heidelberg, Germany. William Kleiber is Graduate Student, Department of Statistics, University of Washington (E-mail: [email protected]). smoothness parameter ν>1, the Hausdorff dimension of a Martin Schlather is Professor, Institut für Mathematische Stochastik, Univer- sample path equals its topological dimension, d. Generally, the sität Göttingen, Germany. We are grateful to Jeff Baars, Clifford F. Mass, larger ν, the smoother the process. Adrian E. Raftery, the associate editor, and the referees for comments and discussions and/or providing data. This research was supported by the Alfried Krupp von Bohlen und Halbach Foundation, the National Science Foundation under awards ATM-0724721, DMS-0706745, and the VIGRE program, and by © 2010 American Statistical Association the Joint Ensemble Forecasting System (JEFS) under subcontract S06-47225 Journal of the American Statistical Association from the University Corporation for Atmospheric Research (UCAR). The re- September 2010, Vol. 105, No. 491, Theory and Methods search of Martin Schlather was supported by DFG grant FOR 916. DOI: 10.1198/jasa.2010.tm09420 1167 1168 Journal of the American Statistical Association, September 2010 Moving to a multivariate spatial process, let Y(s) = (Y1(s), on the scale and cross-covariance smoothness parameters. Sec- d ...,Yp(s)) , so that at each location, s ∈ R , there are p con- tion 3 turns to a data study on error fields for numerical pre- stituent components. We assume that the process is multivari- dictions of surface temperature and pressure over the Pacific ate Gaussian and second-order stationary with mean vector zero Northwest. For pressure, error fields are known to be smooth, and matrix-valued covariance function while for temperature they are rough, with a strongly negative ⎛ ⎞ colocated correlation coefficient, resulting in a situation that is C (h) ··· C (h) 11 1p difficult to accommodate with extant models. The paper closes ⎜ . ⎟ C(h) = ⎝ . .. ⎠ , (2) in Section 4 with a discussion and hints at extensions. C (h) ··· C (h) p1 pp 2. THE MULTIVARIATE MATÉRN MODEL where each C (h) = E(Y (s + h)Y (s)) is a univariate covari- ii i i As noted, we discuss conditions on the multivariate Matérn ance function, while Cij(h) = E(Yi(s + h)Yj(s)) is the cross- ≤ = ≤ model that result in a valid specification for the cross-covariance covariance function between process components 1 i j p. functions in (4), ranging from a parsimonious model in a lower It is difficult to specify nontrivial, valid parametric models for dimensional parameter space to a full bivariate model with the cross-covariance functions, because of the notorious require- most flexibility. ment of nonnegative definiteness. Specifically, if is the co- variance matrix of the random vector (Y(s1) ,...,Y(sn) ) , then 2.1 Parsimonious Multivariate Matérn Model for any vector a ∈ Rnp, the fitted covariance model must sat- We first consider a parsimonious multivariate Matérn model isfy aa ≥ 0. The prevalent way of guaranteeing this is to with marginal and cross-covariance functions that are of the use the linear model of coregionalization (LMC), where each form (3) and (4), with natural restrictions on the parameter component is represented as a linear combination of latent, space. independent univariate spatial processes (Goulard and Voltz 1992; Wackernagel 2003). For the LMC, the smoothness of Theorem 1. Let d ≥ 1 and p ≥ 2. Suppose that any component defaults to that of the roughest latent process, 1 νij = (νi + νj) for 1 ≤ i = j ≤ p, and thus the standard approach does not admit individually dis- 2 tinct smoothness properties, unless structural zeros are imposed and suppose that there is a common scale parameter, in that on the latent process coefficients. More generally, the lack of there exists an a > 0 such that flexibility of the LMC and related models for multivariate spa- a1 =···=ap = a and aij = a for 1 ≤ i = j ≤ p. tial random fields has been noted by various authors, including Cressie (1993, p. 141), Goovaerts (1997, p. 123), and recently Then the multivariate Matérn model provides a valid second- Rd Jun, Knutti, and Nychka (2008, p. 945) who study the spatial order structure in if d 1/2 d 1/2 structure of climate model biases individually, using Matérn (νi + ) (νj + ) ρ = β 2 2 ij ij 1/2 1/2 covariances, rather than jointly, because “we are not aware of (νi) (νj) flexible cross-covariance models that would be suitable.” 1 ( (νi + νj)) These challenges motivate the constructions hereinafter, × 2 for 1 ≤ i = j ≤ p, 1 + + d which preserve the Matérn covariance function marginally, (2 (νi νj) 2 ) while allowing nontrivial, flexible cross-covariance structures. where the matrix (β )p with diagonal elements β = 1for Specifically, in our multivariate Matérn model, each marginal ij i,j=1 ii i = 1,...,p and off-diagonal elements β for 1 ≤ i = j ≤ p is covariance function, ij symmetric and nonnegative definite. = 2 | = Cii(h) σi M(h νi, ai) for i 1,...,p, (3) The requirement that the components share the same scale is of the Matérn type with variance parameter σ 2 > 0, smooth- parameter, a, may seem overly restrictive at first.
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