Multiradial matrix covariance functions: characterization and applications (2014)
All results presented here concern to radial (isotropic) and multiradial (danisotropic) matrix-valued covariance functions. We specify some important properties of matrix-valued covariance functions associated to Multivariate Gaussian fields in a Euclidean space Rd. In particular, we focus (a) on the radially symmetric case and, the more general case, (b) on multiradial obtained through isotropy between components of the lag vector. We call the later set of functions the class of multiradial matrix-valued covariance functions or the class of d-anisotropic matrix-valued covariance functions, this case includes, as special case, space-time and fully symmetric correlation functions. The classes of radial and multiradial matrix-valued covariance functions are characterized as the scale mixture of a uniquely determined matrix-valued measure d(·), with d(b) − d(a) positive definite matrix for any 0 ≤ a ≤ b, with a, b ∈ Rn +, for some n ∈ Z+. We call the matrix function d(·) the m-Schoenberg measure. Such result is the analogue of Schoenberg (1938) theorem for the class of univariate stationary-isotropic covariance functions. We introduce the multivariate versions of radial and multiradial Mont´ee and Descente operators which were introduced by Matheron in the univariate-radial case, calling these matrix operators the m-Mont´ee and m-Descente, m ≥ 2, and prove that these operators change the smoothness of the mapped functions and they are dimensional walks, i.e. each one of these operators map a matrix-valued covariance function valid in Rd in another matrix-valued covariance function valid in Euclidean space of higher o lower dimension. Also, we characterize the associated m-Schoenberg measures of the new covariance matrix functions, it is set up the necessary conditions for the m-Mont´ee and m-Descente are well define and obtain examples where these operators as dimension walks are not well defined. Analogue of the Turning Bands operator established by Matheron for univariate covariance functions, we show the existence of projection operators that map a matrix-valued covariance function ϕ being positive definite on some Euclidean space Rd in another function, say ̺, being radial and positive definite on a Euclidean space of higher dimension, result which opens a future line of research in simulation of multivariate random fields. At the end, we present ascending dimensional walks, based on scale mixtures of Beta distribution function, that map a radial or multiradial matrix-valued covariance function valid in Rd into a radial or multiradial matrix-valued covariance function valid in a space of higher dimension.