Minimizing Change-Point Estimation Error

TL;DR

This paper develops and analyzes algorithms for estimating change-points across multiple sequences, focusing on minimizing estimation error rather than detection, with theoretical bounds and practical extensions for non-constant intensities.

Contribution

It introduces a scan-CUSUM algorithm that achieves optimal bounds for change-point estimation error and extends it to non-constant intensity functions with asymptotic zero-error performance.

Findings

The scan-CUSUM algorithm attains the no error probability upper bound.

The proposed method approaches the lower bound for expected estimation distance.

Extension to non-constant intensities achieves asymptotically zero error.

Abstract

In this paper we consider change-points in multiple sequences with the objective of minimizing the estimation error of a sequence by making use of information from other sequences. This is in contrast to recent interest on change-points in multiple sequences where the focus is on detection of common change-points. We start with the canonical case of a single sequence with constant change-point intensities. We consider two measures of a change-point algorithm. The first is the probability of estimating the change-point with no error. The second is the expected distance between the true and estimated change-points. We provide a theoretical upper bound for the no error probability, and a lower bound for the expected distance, that must be satisfied by all algorithms. We propose a scan-CUSUM algorithm that achieves the no error upper bound and come close to the distance lower bound. We next consider the case of non-constant intensities and establish sharp conditions under which estimation error can go to zero. We propose an extension of the scan-CUSUM algorithm for a non-constant intensity function, and show that it achieves asymptotically zero error at the boundary of the zero-error regime. We illustrate an application of the scan-CUSUM algorithm on multiple sequences sharing an unknown, non-constant intensity function. We estimate the intensity function from the change-point profile likelihoods of all sequences and apply scan-CUSUM on the estimated intensity function.