Break Point Detection for Functional Covariance

TL;DR

This paper introduces a new statistical method based on CUSUM for detecting change points in the covariance structure of zero-mean functional data, with theoretical analysis and practical validation.

Contribution

It proposes a novel change point detection procedure for functional covariance that accounts for unknown parameters and provides asymptotic properties.

Findings

Method effectively detects change points in simulated data.

Application reveals covariance changes in rat brain signals post-stroke.

Theoretical insights into parameter influence on test performance.

Abstract

Many experiments record sequential trajectories where each trajectory consists of oscillations and fluctuations around zero. Such trajectories can be viewed as zero-mean functional data. When there are structural breaks (on the sequence of trajectories) in higher order moments, it is not always easy to spot these by mere visual inspection. Motivated by this challenging problem in brain signal analysis, we propose a detection and testing procedure to find the change point in functional covariance. The detection procedure is based on the cumulative sum statistics (CUSUM). The classical testing procedure for functional data depends on a null distribution which depends on infinitely many unknown parameters, though in practice only a finite number of these can be included for the hypothesis test of the existence of change point. This paper provides some theoretical insights on the influence of the number of parameters. Meanwhile, the asymptotic properties of the estimated change point are developed. The effectiveness of the proposed method is numerically validated in simulation studies and an application to investigate changes in rat brain signals following an experimentally-induced stroke.