HMM conditional-likelihood based change detection with strict delay tolerance

Abstract

Hidden Markov models (HMMs) have been widely used for anomaly and change point detection due to their representation power and computational efficiency in capturing statistical dependencies in time series. However, often information is integrated over relatively long observation windows, with detections made when the observed sequence’s likelihood under the (null) HMM deviates significantly from its typical range. Three related limitations are: i) use of long windows entails large decision delay, which may e.g. fail to prevent machine failure/damage; ii) typical approaches do not narrowly identify an interval within which the change point occurred. Such information could be useful e.g. for process control, where one wants to know how long it takes for control inputs to induce desired change points; iii) The decision statistic is usually the likelihood of the data in the current window, without consideration of past observations. This is suboptimal – this likelihood should be conditioned on past observations to optimally account for statistical dependency in the time series. In this paper, we propose a framework for change point detection which overcomes all of these limitations: i) it applies a standard HMM Forward recursion, but used to evaluate the likelihood of an observation subsequence conditioned on the subsequence’s entire past. This approach is used to efficiently evaluate the conditional likelihoods of all intervals of fixed length (hence with fixed delay, d), until a change point is first detected. Here d is a design parameter whose proper value (needed to have a quick response/mitigate damage) may be known for a given application domain; ii) the algorithm narrowly estimates the interval within which a detected change point lies; iii) we propose a novel performance criterion well-matched to low-delay, narrowly localized change point detection – the true detection interval rate (TDIR) – and also evaluate the false positive rate (FPR) and the bias and variance of the estimated change point, all as a function of d. The proposed method is shown to outperform a CUSUM algorithm, symbolic time series analysis (STSA) methods, and a standard HMM method (evaluating the unconditioned likelihood) for instability onset in combustion systems and fatigue failure initiation in a material.