Separable approximations of space-time covariance matrices

Marc G. Genton
2007 Environmetrics  
Statistical modeling of space-time data has often been based on separable covariance functions, that is, covariances that can be written as a product of a purely spatial covariance and a purely temporal covariance. The main reason is that the structure of separable covariances dramatically reduces the number of parameters in the covariance matrix and thus facilitates computational procedures for large space-time data sets. In this paper, we discuss separable approximations of nonseparable
more » ... time covariance matrices. Specifically, we describe the nearest Kronecker product approximation, in the Frobenius norm, of a space-time covariance matrix. The algorithm is simple to implement and the solution preserves properties of the space-time covariance matrix, such as symmetry, positive definiteness, and other structures. The separable approximation allows for fast kriging of large space-time data sets. We present several illustrative examples based on an application to data of Irish wind speeds, showing that only small differences in prediction error arise while computational savings for large data sets can be obtained. If the process Z itself is separable, that is Z(s, t) = Z S (s)Z T (t), where Z S is a purely spatial stochastic process with nonstationary covariance C S and Z T is a purely temporal stochastic process with nonstationary covariance C T , and Z S is independent of Z T , then the property (3) holds. Indeed, we 688 M. G. GENTON separability approximation error index as a function of β in the bottom-right panel of Figure 1 . The bold curve is for m = 4 and the other curves for larger matrices with m = 10, 20, and 50. The vertical dashed line is atβ = 0.681. Overall, we can see that the separability approximation error index is not larger than 2%, even in the most nonseparable settings described previously, thus indicating a Figure 1. Covariance matrix based on (9) for the Irish wind speed data estimated during the period 1961-1970. Top row: contour plots of the entries of (left) and of the entries of its nearest Kronecker product approximation (right). Bottom row: singular values of R( ) (left) and separability approximation error index as a function of β (right). The bold curve is for m = 4 and the other curves for m = 10, 20, and 50. The vertical dashed line is atβ = 0.681 We have proposed a general methodology for computing separable approximations of space-time covariance matrices and illustrated that it results in small differences in prediction error while providing computational savings for large space-time data sets. Although we do not claim that separable
doi:10.1002/env.854 fatcat:5hqvxpzw7jazhpnkuynn7sgrim