A Constant-per-Iteration Likelihood Ratio Test for Online Changepoint Detection for Exponential Family Models

Kes Ward; Gaetano Romano; Idris Eckley; Paul Fearnhead

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arXiv · 2302.04743

A Constant-per-Iteration Likelihood Ratio Test for Online Changepoint Detection for Exponential Family Models

Kes Ward·Gaetano Romano·Idris Eckley·Paul Fearnhead

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Abstract

Online changepoint detection algorithms that are based on likelihood-ratio tests have been shown to have excellent statistical properties. However, a simple online implementation is computationally infeasible as, at time $T$, it involves considering $O(T)$ possible locations for the change. Recently, the FOCuS algorithm has been introduced for detecting changes in mean in Gaussian data that decreases the per-iteration cost to $O(\log T)$. This is possible by using pruning ideas, which reduce the set of changepoint locations that need to be considered at time $T$ to approximately $\log T$. We show that if one wishes to perform the likelihood ratio test for a different one-parameter exponential family model, then exactly the same pruning rule can be used, and again one need only consider approximately $\log T$ locations at iteration $T$. Furthermore, we show how we can adaptively perform the maximisation step of the algorithm so that we need only maximise the test statistic over a small subset of these possible locations. Empirical results show that the resulting online algorithm, which can detect changes under a wide range of models, has a constant-per-iteration cost on average.

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Kes Ward, Gaetano Romano, Idris Eckley, Paul Fearnhead. 2023-02-09. A Constant-per-Iteration Likelihood Ratio Test for Online Changepoint Detection for Exponential Family Models. https://arxiv.org/abs/2302.04743

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