ABSTRACT

Data with more variables than the number of samples are emerging in
various fields, such as microarray experiments, medical images, and
signal processing. Data of this type motivate a new family of asymptotics, along with the development of new methodologies. This talk addresses some issues of the analysis and the asymptotic theory, relevant to High Dimensional Low Sample Size (HDLSS) data.

In the first part of the talk, Continuum Regression, originally proposed by Stone and Brooks (1990), is viewed as a family of direction searching methods. The novel use of Continuum Regression in HDLSS settings will be illustrated by an application to microarray experiments. In the second part, we extend the Continuum Regression idea to the challenging case of paired HDLSS data. The extended method, Continuum Canonical orrelation, is proposed as a family of methods for simultaneously searching for two direction vectors over both high dimensional spaces.

Lastly, the HDLSS asymptotic behavior of the maximum covariance direction will be analyzed. Asymptotic conditions under which the first pairs of the sample maximum covariance direction vectors are consistent and other conditions where they are strongly inconsistent are formulated in terms of the underlying covariance matrix. Also, the limiting distribution of the sample maximum covariance is derived.