Detecting change-points in a discrete distribution via model selection

TL;DR

This paper introduces a model selection-based method for detecting multiple change-points in the distribution of categorical variables, providing adaptive, computationally efficient procedures with proven theoretical guarantees.

Contribution

It proposes a novel two-stage, adaptive change-point detection method with linear computational complexity and theoretical performance guarantees.

Findings

First estimator satisfies an oracle inequality

Method is adaptive in the minimax sense

Computational complexity is linear in data size

Abstract

This paper is concerned with the detection of multiple change-points in the joint distribution of independent categorical variables. The procedures introduced rely on model selection and are based on a penalized least-squares criterion. Their performance is assessed from a nonasymptotic point of view. Using a special collection of models, a preliminary estimator is built. According to an existing model selection theorem, it satisfies an oracle-type inequality. Moreover, thanks to an approximation result demonstrated in this paper, it is also proved to be adaptive in the minimax sense. In order to eliminate some irrelevant change-points selected by that first estimator, a two-stage procedure is proposed, that also enjoys some adaptivity property. Besides, the first estimator can be computed with a complexity only linear in the size of the data. A heuristic method allows to implement the second procedure quite satisfactorily with the same computational complexity.