Nonparametric Operator-Regularized Covariance Function Estimation for Functional Data

• Activity Number: 346 - Recent Advances in Nonparametric Statistical Methods
• Type: Contributed
• Date/Time: Tuesday, July 31, 2018 : 10:30 AM to 12:20 PM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Xiaoke Zhang* and Raymond Wong
• Companies: George Washington University and Texas A&M University
• Keywords: Functional data analysis; low-rank estimation; reproducing kernel Hilbert space; spectral regularization

Abstract

In functional data analysis (FDA), covariance function is fundamental both as a critical quantity for understanding elementary aspects of functional data and as an indispensable ingredient for many advanced FDA methods. A new class of nonparametric covariance function estimators is developed in terms of various spectral regularizations of an operator associated with a reproducing kernel Hilbert space. The covariance estimators are automatically positive semi-definite without any additional modification steps. An unconventional representer theorem is established to provide a finite dimensional representation for this class of covariance estimators. Trace-norm regularization is particularly studied to further achieve a low-rank representation, another desirable property which leads to dimension reduction, and is often needed in advanced FDA approaches. An efficient algorithm is developed based on the accelerated proximal gradient method. This resulted estimator is shown to enjoy an excellent rate of convergence under both fixed and random designs. The outstanding practical performance of the trace-norm-regularized estimator is demonstrated by the analysis of a traffic dataset.