Approximating The Likelihood Ratio in Linear-Gaussian State-Space Models for Change Detection

Change-point detection methods are widely used in signal processing, primarily for detecting and locating changes in a considered model. An important family of algorithms for this problem relies on the likelihood ratio (LR) test. In state-space models (SSMs), the time series is modeled through a Markovian latent process. In this paper, we focus on the linear-Gaussian (LG) SSM, in which the LR-based methods require running a Kalman filter for every candidate change point. Since the number of candidates grows with the length of the time series, this strategy is inefficient in short time series and unfeasible for long ones. We propose an approximation to the LR which uses a constant number of filters, independently on the time-series length. The approximated LR relies on the Markovian property of the filter, which forgets errors at an exponential rate. We present theoretical results that justify the approximation, and we bound its error. We demonstrate its good performance in two numerical examples.

1. INTRODUCTION

The detection of changes in signals is a problem of broad interest, with applications in biological science [1],[2], climate science [3], speech processing [4], and finance [5], among many others. The problem has had various treatments, stemming from disciplines such as signal processing, industrial process monitoring, control systems, and statistics [6]-[9]. Change-point detection methods involve assessing how well one or several given statistical models represent an observed signal. They are generally used to detect and locate an abrupt change in the considered model [10] and for further post-processing purposes, e.g., segmentation of signals [11].