Continuum Direction Vectors in High Dimensional Low Sample Size Data

Continuum Direction Vectors in High Dimensional Low Sample Size Data Public Deposited

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Lee, Myung Hee
- Affiliation: College of Arts and Sciences, Department of Statistics and Operations Research

Abstract

This dissertation consists of three parts regarding High Dimensional Low Sample Size (HDLSS) data analysis. Dimension reduction techniques in high dimensional space, based on a small number of direction vectors, will be the common theme. In the first part of the dissertation, Continuum Regression, originally proposed by Stone and Brooks (1990), will be understood as a family of methods for searching among direction vectors. Continuum Regression includes three popular methods- Ordinary Least Squares, Partial Least Squares, and Principal Component Regression - as special cases. The novel use of Continuum Regression in HDLSS settings will be illustrated by an application to microarray experiments. In the second part of the dissertation, we will extend the Continuum Regression idea to the challenging case of paired HDLSS data. The extended method, Continuum Canonical Correlation, is proposed, as a family of methods for searching direction vectors over two high dimensional spaces simultaneously. The last part of the dissertation studies the HDLSS asymptotic behavior of the maximum covariance direction vectors over two data spaces, i.e., the singular vectors of the sample cross-covariance matrix. We find some conditions under which consistency and strong inconsistency of the singular vectors in HDLSS is observed.

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