The RKHS Formulation of Stochastic Data Analysis Problems

Abstract

Traditional multivariate statistics focuses on the analysis of data that are vectors of finite length. However, modern data collection methods are now frequently returning observations that should be viewed as the result of digitized recording or sampling from stochastic processes. Data sets of this type occur in the context of functional data analysis (where the observation process has a one-dimensional index set), image analysis and spatial statistics, for example. Goals for analyzing such high-dimensional data include the detection of structure and dimensionality reduction. In this talk, we consider general canonical correlation analysis as well as a dimension- reduction problem in nonparametric regression. A novelty of our approaches is the use of reproducing kernel Hilbert space as a device for representing the variable space. We will demonstrate that this enables a seamless extension from the multivariate setting to the functional setting, and motivates simple and efficient computational algorithms.