Partially Functional Linear Regression in High Dimensions

• Activity Number: 288
• Type: Invited
• Date/Time: Tuesday, August 11, 2015 : 8:30 AM to 10:20 AM
• Sponsor: Section on Statistics in Imaging
• Author(s): Fang Yao* and Kaijie Xue and Dehan Kong and Hao Helen Zhang
• Companies: University of Toronto and University of Toronto and The University of North Carolina at Chapel Hill and The University of Arizona
• Keywords: Brain image ; Functional data ; Functional linear regression ; Model selection ; Principal components ; Regularization

Abstract

In modern experiments, functional and non-functional data are often encountered simultaneously, where the observations are sampled from random processes together with a potentially large number of scalar covariates. The complex nature of such a scheme makes it difficult to apply existing methods for model selection and estimation. We propose a new class of partially functional linear models to characterize the regression between a scalar response and those covariates, including both functional and scalar types. The new approach provides a unified and flexible framework to simultaneously take into account multiple functional and ultra-high dimensional scalar predictors, identify important features and improve interpretability of the estimates. The underlying processes of the functional predictors are considered genuinely infinite-dimensional, and one of our contributions is to characterize the impact of regularization on the resulting estimates. We establish consistency and oracle properties under mild conditions, illustrate the performance of the proposed method with simulation studies, and then apply to the motivating functional magnetic resonance imaging brain image data.