Wang, Limin
(2008)
Karhunen-Loeve expansions and their applications.
PhD thesis, London School of Economics and Political Science.
Abstract
The Karhunen-Loeve Expansion (K-L expansion) is a bi-orthogonal stochastic process expansion. In the field of stochastic process, the Karhunen-Loeve expansion decomposes the process into a series of orthogonal functions with the random coefficients. The essential idea of the expansion is to solve the Fredholm integral equation, associated with the covariance kernel of the process, which defines a Reproducing Kernel Hilbert Space (RKHS). This either has an analytical solution or special numerical methods are needed. This thesis applies the Karhunen-Loeve expansion to some fields of statistics. The first two chapters review the theoretical background of the Karhunen-Loeve expansion and introduce the numerical methods, including the integral method and the expansion method, when the analytical solution to the expansion is unavailable. Chapter 3 applies the theory of the Karhunen-Loeve expansion to the field of the design experiment using a criteria called "maximum entropy sampling". Under such setting, a type of duality is set up between maximum entropy sampling and the D- optimal design of the classical optimal design. Chapter 4 uses the Karhunen-Loeve expansion to calculate the conditional mean and variance for a given set of observations, with application to prediction. Chapter 5 extends the theory of the Karhunen- Loeve expansion from the univariate setting to the multivariate setting: multivariate space, univariate time. Adaptations of numerical methods of Chapter 2 are also provided for the multivariate setting, with a full matrix development. Chapter 6 applies the numerical method developed in Chapter 5 to the emerging area of multivariate functional data analysis with a detailed example on a trivariate autoregressive process.
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