** Marc G. Genton**

Department of Econometrics, University of Geneva

Department of Statistics, Texas A&M University

**Thursday, January 17, 2008**

**Mesa Laboratory, Chapman Room**

**Lecture 12:00am**

### Kronecker Product Approximation of Covariance and Other Matrices

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 talk, we first discuss separable approximations of nonseparable space-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. We discuss additional constraints on the pattern
of the approximation such as sparsity and Toeplitz structures. We describe
extensions of the methodology to rectangular matrices with application to
RCM data and present some R functions.