Positive definite matrix-valued functions and matrix variogram modeling.

Abstract

In many applications in the physical and earth sciences there are multiple variables of interest which are correlated. In these cases, the spatial random function becomes vector-valued, in which spatial correlation and component (inter-variable) correlation come out simultaneously. We denote by Z(x) = (z₁(x), …, z(m)(x)ᵀ the vector-valued random function. Similarly the covariance and variogram structure of Z(x) play a central role in any prediction scheme. But the covariance function and variogram of Z(x) are no longer scalar functions. They are matrix-valued functions when m > 1 and have a positive (negative) definiteness property in a generalized sense. Any prediction technique for vector-valued spatial functions relies heavily on this property. Therefore, characterizing and modeling the matrix-valued covariance or variogram structure of Z(x) is extremely important in spatial statistics and become more difficult than in scalar cases. For instance, (a) there is a lack of standard models for the covariance function and variogram (23); (b) there is no efficient graphic aid for fitting models since the covariance function and variogram are matrix-valued functions; (c) there are many parameters need to be estimated. Even the basic analytic properties of matrix-valued positive definite functions are not clear. In this dissertation, we generalize the concept of (scalar) positive definite functions to matrix-valued functions which are related to correlations and variograms of vector-valued random functions, to analytically study matrix-valued (conditionally) positive definite functions beyond basic definitions, to create matrix-valued variogram models, to provide techniques for systematic variogram modeling.