On changepoint detection in functional data using empirical energy distance

TL;DR

This paper introduces a new family of test statistics based on empirical energy distance for detecting changepoints in dependent, multivariate functional data, capable of identifying various types of distributional changes.

Contribution

It develops a novel, easily implementable statistical method for changepoint detection in complex functional data, including multiple changepoints and changes near data endpoints.

Findings

Effective detection of changepoints close to sample edges

Good performance in finite-sample simulations

Successful application to financial and temperature datasets

Abstract

We propose a novel family of test statistics to detect the presence of changepoints in a sequence of dependent, possibly multivariate, functional-valued observations. Our approach allows to test for a very general class of changepoints, including the "classical" case of changes in the mean, and even changes in the whole distribution. Our statistics are based on a generalisation of the empirical energy distance; we propose weighted functionals of the energy distance process, which are designed in order to enhance the ability to detect breaks occurring at sample endpoints. The limiting distribution of the maximally selected version of our statistics requires only the computation of the eigenvalues of the covariance function, thus being readily implementable in the most commonly employed packages, e.g. R. We show that, under the alternative, our statistics are able to detect changepoints occurring even very close to the beginning/end of the sample. In the presence of multiple changepoints, we propose a binary segmentation algorithm to estimate the number of breaks and the locations thereof. Simulations show that our procedures work very well in finite samples. We complement our theory with applications to financial and temperature data.