This thesis concerns the theory of positive-definite completions and its mutually beneficial connections to the statistics of function-valued or continuously-indexed random processes, better known as functional data analysis. In particular, it dwells upon ...
We consider the problem of positive-semidefinite continuation: extending a partially specified covariance kernel from a subdomain Omega of a rectangular domain I x I to a covariance kernel on the entire domain I x I. For a broad class of domains Omega call ...
Let X={Xu}u is an element of U be a real-valued Gaussian process indexed by a set U. We show that X can be viewed as a graphical model with an uncountably infinite graph, where each Xu is a vertex. This graph is characterized by the reproducing property of ...
We study the positive-definite completion problem for kernels on a variety of domains and prove results concerning the existence, uniqueness, and characterization of solutions. In particular, we study a special solution called the canonical completion whic ...
We develop a generalisation of Mercer’s theorem to operator-valued kernels in infinite dimensional Hilbert spaces. We then apply our result to prove a Karhunen-Loève theorem, valid for mean-square continuous random functions valued in a separable Hilbert s ...
