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 (FKWC) changepoint procedure, based on rank statistics and multivariate functional data depth. The FKWC changepoint procedure allows the user to test for at most one changepoint (AMOC) or an epidemic period, or to estimate the number and locations of an unknown amount of changepoints in the data. We show that when the ``signal-to-noise'' ratio is bounded below, the changepoint estimates produced by the FKWC procedure attain the minimax localization rate for detecting general changes in distribution in the univariate setting (Theorem 1). We also provide the behavior of the proposed test statistics for the AMOC and epidemic setting under the null hypothesis (Theorem 2) and, as a simple consequence of our main result, these tests are consistent (Corollary 1). In simulation, we show that our method is particularly robust when compared to similar changepoint methods. We present an application of the FKWC procedure to intraday asset returns and f-MRI scans. As a by-product of Theorem 1, we provide a concentration result for integrated functional depth functions (Lemma 2), which may be of general interest.