Multiple Testing of Local Extrema for Detection of Structural Breaks in Piecewise Linear Models

TL;DR

This paper introduces a novel method for detecting change points in piecewise linear models by analyzing local extrema, controlling false discoveries, and offering computational efficiency, validated through theoretical and numerical studies.

Contribution

The paper presents a new change point detection method based on local extrema analysis, with proven FDR control, power consistency, and low computational complexity.

Findings

Ensures asymptotic FDR control and power consistency.

Requires only a single test for all candidate extrema.

Maintains FDR control and power in non-asymptotic cases.

Abstract

In this paper, we propose a new generic method for detecting the number and locations of structural breaks or change points in piecewise linear models under stationary Gaussian noise. Our method transforms the change point detection problem into identifying local extrema (local maxima and local minima) through kernel smoothing and differentiation of the data sequence. By computing p-values for all local extrema based on peak height distributions of smooth Gaussian processes, we utilize the Benjamini-Hochberg procedure to identify significant local extrema as the detected change points. Our method can distinguish between two types of change points: continuous breaks (Type I) and jumps (Type II). We study three scenarios of piecewise linear signals, namely pure Type I, pure Type II and a mixture of Type I and Type II change points. The results demonstrate that our proposed method ensures asymptotic control of the False Discover Rate (FDR) and power consistency, as sequence length, slope changes, and jump size increase. Furthermore, compared to traditional change point detection methods based on recursive segmentation, our approach only requires a single test for all candidate local extrema, thereby achieving the smallest computational complexity proportionate to the data sequence length. Additionally, numerical studies illustrate that our method maintains FDR control and power consistency, even in non-asymptotic cases when the size of slope changes or jumps is not large. We have implemented our method in the R package "dSTEM" (available from https://cran.r-project.org/web/packages/dSTEM).