Break point detection for functional covariance

Abstract

Many neuroscience 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 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 fully functional testing procedure relies on a null distribution which depends on infinitely many unknown parameters, though in practice only a finite number of these parameters 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.

Change point analysis of covariance functions: A weighted cumulative sum approach

“…Repeating the proof of (31), one can establish (32) and (33), we only need to replace (37) with (39). Proceeding along the lines of ( 38)…”

“…Most closely related to the present paper are Sharipov and Wendler (2020), who develop change point detection methods for the covariance operator based on norms of a suitably constructed functional cumulative sum (CUSUM) process under general weak dependence conditions. The recent preprint Jiao et al (2020) considers change point detection as well as estimation using similar norm-based techniques and conditions to Sharipov and Wendler (2020), with applications to functional data objects derived from rat brain studies. A notable feature of each of these procedures is that they are based on norms of the standard CUSUM process.…”

Variation pattern classification of functional data

“…, where C(0) 𝑔,𝓁 (⋅) is the covariance operator of the 𝓁th quasi-stationary segment of group 𝑔, and 𝕀 (𝓁) 𝑔𝑘 = 1 if 𝑘 lies in the 𝓁th segment of group 𝑔 and 𝕀 (𝓁) 𝑔𝑘 = 0 otherwise. Here we applied the break point detection method developed in Jiao, Frostig & Ombao (2022). Binary segmentation was applied to determine all pronounced break points.…”

“…Step 1. For each group (𝑔 = 0, 1), apply break point detection method (e.g., Jiao, Frostig & Ombao (2022)) to find the structural break points in the covariance operator, and estimate the covariance operator for each local stationary subsequence.…”

Enhanced Change-Point Detection in Functional Means

“…In the change-point analysis of functional data, one major limitation of dimension reduction is that, when the selected basis functions are not aligned with the jump of mean function, the projection-based detector fails to detect the change points. To solve this problem, an alternative fully functional approach is employed in Horváth et al (2014), Aue et al (2018) and Jiao et al (2022), which does not rely on any dimension reduction. In the fully functional detection procedure, the null distribution involves infinitely many unknown parameters and requires additional truncation step, however.…”