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 <inline-formula><tex-math notation="LaTeX">$\alpha$</tex-math></inline-formula>, <inline-formula><tex-math notation="LaTeX">$\beta$</tex-math></inline-formula>, and fast <inline-formula><tex-math notation="LaTeX">$\phi$</tex-math></inline-formula>-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 (<inline-formula><tex-math notation="LaTeX">$ {\mathtt {ARL}}$</tex-math></inline-formula>) and upper bounds for average-detection-delay (<inline-formula><tex-math notation="LaTeX">$ {\mathtt {ADD}}$</tex-math></inline-formula>) in terms of the threshold parameter. We show that the MMD-CUSUM test enjoys the same level of performance as the i.i.d. case under fast <inline-formula><tex-math notation="LaTeX">$\phi$</tex-math></inline-formula>-mixing processes. The MMD-CUSUM test also achieves strong performance under exponentially <inline-formula><tex-math notation="LaTeX">$\alpha$</tex-math></inline-formula>/<inline-formula><tex-math notation="LaTeX">$\beta$</tex-math></inline-formula>-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.