| Summary: | 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.
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