Adaptive, Robust Functional and Image Regression in Functional Mixed Models

• Activity Number: 386
• Type: Invited
• Date/Time: Tuesday, August 2, 2011 : 2:00 PM to 3:50 PM
• Sponsor: General Methodology
• Author(s): Hongxiao Zhu and Philip J. Brown and Jeffrey S. Morris*+
• Companies: Statistical and Applied Mathematical Sciences Institute and University of Kent at Canterbury and The University of Texas MD Anderson Cancer Center
• Keywords: Functional Data Analysis ; Functional Mixed Models ; Robust Methods ; Outlier Detection ; Image Analysis ; Bayesian Methods

Abstract

New methods have been developed in recent years to analyze functional and image data, many of which involve extensions of linear regression such as functional regression and functional mixed models. Existing methods, however, tend to be sensitive to outliers, as no analogs to robust linear regression have been developed for the functional setting. Here, we discuss a unified Bayesian method for robust functional regression, whereby a functional response of unspecified form is regressed on a set of linear predictors. The method is developed within the general functional mixed model framework, which can simultaneously model multiple factors and accommodate between-function correlation induced by the experimental design. We demonstrate outstanding robustness properties, doing an excellent job estimating functional regression coefficients even in the presence of Cauchy errors and random effects, and yet not trading off much efficiency when the true likelihood is Gaussian. We also observed remarkable adaptive smoothing properties in our estimates of the fixed and random effect functions, which arise from an interaction of the robust likelihood and adaptive sparsity priors.