Model-Free Change Point Detection for Mixing Processes

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.