Model-Free Change Point Detection for Mixing Processes

Abstract

This paper considers the change point detection problem under dependent samples. In particular, we provide performance guarantees for the MMD-CUSUM test under exponentially $\alpha$ , $\beta$ , and fast $\phi$ -mixing processes, which significantly expands its utility beyond the i.i.d. and Markovian cases used in previous studies. We obtain lower bounds for average-run-length ( $ {\mathtt {ARL}}$ ) and upper bounds for average-detection-delay ( $ {\mathtt {ADD}}$ ) in terms of the threshold parameter. We show that the MMD-CUSUM test enjoys the same level of performance as the i.i.d. case under fast $\phi$ -mixing processes. The MMD-CUSUM test also achieves strong performance under exponentially $\alpha$ / $\beta$ -mixing processes, which are significantly more relaxed than existing results. The MMD-CUSUM test statistic adapts to different settings without modifications, rendering it a completely data-driven, dependence-agnostic change point detection scheme. Numerical simulations are provided at the end to evaluate our findings.