Robust changepoint detection in the variability of multivariate functional data

Abstract

We consider the problem of robustly detecting changepoints in the variability of a sequence of independent multivariate functions. We develop a novel changepoint procedure, called the functional Kruskal–Wallis for covariance changepoint procedure, based on rank statistics and multivariate functional data depth. The functional Kruskal–Wallis for covariance changepoint procedure allows the user to test for at most one changepoint or an epidemic period, or to estimate the number and locations of an unknown number of changepoints in the data. We show that when the ‘signal-to-noise’ ratio is bounded below, the changepoint estimates produced by the functional Kruskal–Wallis for covariance changepoint procedure attain the minimax localisation rate for detecting general changes in distribution in the univariate setting. We also provide the behaviour of the proposed test statistics for the at-most-one-change and epidemic settings under the null hypothesis and, as a simple consequence of our main result, these tests are consistent. In simulation, we show that our method is particularly robust when compared to similar changepoint methods. We present an application of the functional Kruskal–Wallis for covariance changepoint procedure to intraday asset returns and functional magnetic resonance imaging scans. As a by-product of our main result, we provide a concentration result for integrated functional depth functions, which may be of general interest.