Distances and Inference for Covariance Functions

• Activity Number: 507
• Type: Invited
• Date/Time: Wednesday, August 7, 2013 : 10:30 AM to 12:20 PM
• Sponsor: Section on Statistical Learning and Data Mining
• Author(s): John Aston*+ and Davide Pigoli and Ian L. Dryden and Piercesare Secchi
• Companies: University of Warwick and Politecnico di Milano and University of Nottingham and Politecnico di Milano
• Keywords: Functional Data Analysis ; Shape Analysis ; Statistical Linguistics

Abstract

A framework is developed for inference concerning the covariance operator of a functional random process, where the covariance operator itself is an object of interest for the statistical analysis. Distances for estimating and then comparing positive definite covariance matrices are investigated for functional data. In particular, an infinite dimensional analogue of the Procrustes size and shape distance is developed. The convergence of the finite dimensional approximations to the infinite dimensional distance metrics is shown. To perform inference, a Fr\'{e}chet estimator for the average covariance function is introduced, and methods for extrapolation of the estimates are developed, a problem with known difficulties when using the standard Euclidean metric. The proposed techniques are applied to a philological study of cross-linguistic dependence where the use of covariance operators has been suggested as a way to incorporate quantitative phonetic information. It is shown that distances between languages derived from phonetic covariance functions can provide insight into relationships between the Romance languages.