Model-Free Change Point Detection for Mixing Processes
Hao Chen, Abhishek Gupta, Yin Sun, Ness Shroff
TL;DR
This work extends model-free change point detection to dependent data by analyzing the MMD-CUSUM test under α, β, and φ-mixing processes. It provides nonparametric guarantees: ARL grows at least as exp(b^{γ/(γ+1)}) under exponential α/β-mixing and at least exp(b) under fast φ-mixing, while ADD remains O(b) across all three mixing notions, with dependence captured by the offset Δ and the finite bias C(r,h). The results demonstrate that MMD-CUSUM can achieve i.i.d.-like performance without requiring Markovian or HMM structure, making it robust and directly applicable to a broad class of dependent time series. Numerical simulations on a linear dynamical system corroborate the theory, showing exponential ARL growth with the threshold and linear ADD growth with b, and the discussion notes practical considerations, unbiased estimators, and potential extensions to spectral kernel methods. Overall, the paper establishes robust, data-driven change point detection guarantees for dependent data, significantly broadening the applicability of kernel-based quickest change detection.
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 $α$, $β$, and fast $φ$-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 (ARL) and upper bounds for average-detection-delay (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 $φ$-mixing processes. The MMD-CUSUM test also achieves strong performance under exponentially $α$/$β$-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.