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Some topics concerning the singular value decomposition and generalized singular value decomposition
Description
This dissertation involves three problems that are all related by the use of the singular value decomposition (SVD) or generalized singular value decomposition (GSVD). The specific problems are (i) derivation of a generalized singular value expansion (GSVE), (ii) analysis of the properties of the chi-squared method for regularization parameter selection in the case of nonnormal data and (iii) formulation of a partial canonical correlation concept for continuous time stochastic processes. The finite dimensional SVD has an infinite dimensional generalization to compact operators. However, the form of the finite dimensional GSVD developed in, e.g., Van Loan does not extend directly to infinite dimensions as a result of a key step in the proof that is specific to the matrix case. Thus, the first problem of interest is to find an infinite dimensional version of the GSVD. One such GSVE for compact operators on separable Hilbert spaces is developed. The second problem concerns regularization parameter estimation. The chi-squared method for nonnormal data is considered. A form of the optimized regularization criterion that pertains to measured data or signals with nonnormal noise is derived. Large sample theory for phi-mixing processes is used to derive a central limit theorem for the chi-squared criterion that holds under certain conditions. Departures from normality are seen to manifest in the need for a possibly different scale factor in normalization rather than what would be used under the assumption of normality. The consequences of our large sample work are illustrated by empirical experiments. For the third problem, a new approach is examined for studying the relationships between a collection of functional random variables. The idea is based on the work of Sunder that provides mappings to connect the elements of algebraic and orthogonal direct sums of subspaces in a Hilbert space. When combined with a key isometry associated with a particular Hilbert space indexed stochastic process, this leads to a useful formulation for situations that involve the study of several second order processes. In particular, using our approach with two processes provides an independent derivation of the functional canonical correlation analysis (CCA) results of Eubank and Hsing. For more than two processes, a rigorous derivation of the functional partial canonical correlation analysis (PCCA) concept that applies to both finite and infinite dimensional settings is obtained.
ContributorsHuang, Qing (Author) / Eubank, Randall (Thesis advisor) / Renaut, Rosemary (Thesis advisor) / Cochran, Douglas (Committee member) / Gelb, Anne (Committee member) / Young, Dennis (Committee member) / Arizona State University (Publisher)
Created2012
Description
The concentration factor edge detection method was developed to compute the locations and values of jump discontinuities in a piecewise-analytic function from its first few Fourier series coecients. The method approximates the singular support of a piecewise smooth function using an altered Fourier conjugate partial sum. The accuracy and characteristic features of the resulting jump function approximation depends on these lters, known as concentration factors. Recent research showed that that these concentration factors could be designed using aexible iterative framework, improving upon the overall accuracy and robustness of the method, especially in the case where some Fourier data are untrustworthy or altogether missing. Hypothesis testing methods were used to determine how well the original concentration factor method could locate edges using noisy Fourier data. This thesis combines the iterative design aspect of concentration factor design and hypothesis testing by presenting a new algorithm that incorporates multiple concentration factors into one statistical test, which proves more ective at determining jump discontinuities than the previous HT methods. This thesis also examines how the quantity and location of Fourier data act the accuracy of HT methods. Numerical examples are provided.
ContributorsLubold, Shane Michael (Author) / Gelb, Anne (Thesis director) / Cochran, Doug (Committee member) / Viswanathan, Aditya (Committee member) / Economics Program in CLAS (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2016-05