Abstract

Let $\mathbf{X}=(X_1,X_2,\ldots,X_n)$ be a time series, that is a sequence of random variable indexed by the time $ t=1,2,\ldots,n $. We assume the existence of a segmentation $\tau=(\tau_1,\tau_2,\ldots,\tau_n)$ such that $X_i$ is a family of independent identically distributed (i.i.d) random variable for i $\in (\tau_k,\tau_k+1],~and~k=0,\ldots,K$ where by convention $\tau_o$ and $\tau_{K+1}=N$. In the literature, it exist two main kinds of change points detections: The change points on-line and the change points off-line. In this work, we consider only the change point analysis (off-line), when number of change points is unknown. The result obtained is based on Filtered Derivative method where we use a second step based on False Discovery Rate. We compare numerically this new method with the Filtered Derivative with p-Value.

Highlights

  • Change-point detection is an important problem in many applications, and it has been well-studied for a long time, see e.g. the textbooks (Basseville & Nikirov, 1993; Brodsky & Darkhovsky, 1993; Csorgo & Horvath, 1997), or (Huskova & Meintanis, 2006b; Gombay & Serban, 2009) for an updated overview

  • We propose to replace the family of single hypothesis tests of Step 2 in Filtered Derivative with p-value (FDpV) method by the use of the False Discovery Rate

  • We propose a new method derived from the FDpV one: We replace Step 2 of FDpV by False Discovery Rate method (FDR) and we call this method FDqV

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Summary

Introduction

Change-point detection is an important problem in many applications, and it has been well-studied for a long time, see e.g. the textbooks (Basseville & Nikirov, 1993; Brodsky & Darkhovsky, 1993; Csorgo & Horvath, 1997), or (Huskova & Meintanis, 2006b; Gombay & Serban, 2009) for an updated overview. We only consider the a posteriori problem In this century, the state to the art method was the Penalized Least Square Criterion (PLS): When the number of change point is known, PLS minimizes a contrast function (Bai & Perron, 1998; Lavielle & Moulines, 2000). Cumulative sum can be iteratively computed and leads to algorithms with both time and memory complexity of order O(n) Among these methods, the Filtered Derivative has been introduced by Benveniste and Basseville (1984) and Basseville and Nikirov (1993). We propose to replace the family of single hypothesis tests of Step 2 in FDpV method by the use of the False Discovery Rate.

Description of the Problem
Change Point Analysis: A Toy Model
Comparison Criterions
Penalized Least Square Method
A New Method for Change Point Analysis
Simulations Based on one Realization
How to Choose the Extra-Parameters?
Errors of Type I and Type II at Step 1
Choice of the Window A
Error of Type I and Type II at Step 2

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