Hierarchical Models for Assessing Variability among Functions

• Activity Number: 89
• Type: Contributed
• Date/Time: Monday, August 9, 2004 : 8:30 AM to 10:20 AM
• Sponsor: Section on Bayesian Statistical Science
• Author(s): Sam Behseta*+ and Garrick L. Wallstrom and Robert E. Kass
• Companies: California State University, Bakersfield and University of Pittsburgh and Carnegie Mellon University
• Keywords: functional data analysis ; Bayesian statistics ; variance estimation ; neuronal data analysis

Abstract

We present a method for summarizing functional variation based on fits, with taking account of the estimation process. We show, for example, that the proportion of variance associated with the first principal component of a sample covariance matrix computed from estimated functions will be biased upward. Alternatively, we introduce three Bayesian methods of accounting for estimation variation, all based on hierarchical models using free-knots splines. Our approach extends the Bayesian Adaptive Regression Spline method of DiMatteo, Genovese and Kass (2001). We rely upon the notion of a hierarchical Gaussian Process model which uses the approximate normality of the estimated function values. We apply our Hierarchical Gaussian Process (HGP) model to neuronal data obtained from the primary motor cortex area of the brain. We demonstrate a considerably lower estimation of the proportion of variability attributed to the first principal component.