arXiv 2409.079482024-09math.STcs.ITmath.ITstat.TH

基于失配CUSUM的最快变化检测

Quickest Change Detection Using Mismatched CUSUM

University of Florida（佛罗里达大学）

Austin Cooper, Sean Meyn

备注：: Extended version of extended abstract for the Allerton Conference on Communication, Control, and Computing, September 2024

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AI总结：

本文针对最快变化检测问题，提出失配CUSUM形式的停止规则，基于隐马尔可夫模型推导线性函数类下的最优解，建立条件独立模型的渐近理论，数值验证了近似的准确性。

摘要：

最快变化检测关注从部分观测序列{Y_k:k≥0}中估计未知变化时间τ_a。本文考虑CUSUM形式的停止规则：X_{n+1}=max{0,X_n+F(Y_{n+1})}，τ_s=min{n≥0:X_n≥H}，其中函数F和阈值H为设计参数。观测值与变化时间通过隐马尔可夫模型联合建模，F从规定函数类Ψ中选取，以最小化加权准则E[(τ_s−τ_a)_++κ(τ_s−τ_a)_−]。当Ψ为线性函数类时，最优解F^*由凸规划刻画，其对偶可推广经典似然比构造，该结论基于κ→∞渐近 regime下的分析。本文证明隐马尔可夫模型存在最快变化检测文献中常用的渐近等价条件独立近似，进而为更广泛的条件独立模型类建立设计与渐近理论，使结论不依赖特定POMDP简化。结合更新理论与反射随机游走的大偏差理论，对每个F∈Ψ得到最优阈值与平均代价的渐近精确近似，误差随κ→∞消失。数值实验表明，所得近似在κ取中等值时仍保持准确。

英文摘要：

Quickest change detection concerns estimation of an unknown change time $τ_a$ from a sequence of partial observations $\{Y_k:k\ge 0\}$. We consider stopping rules of CUSUM form, \[ \mathcal{X}_{n+1} = \max\{0,\mathcal{X}_n+F(Y_{n+1})\}, \quad τ_s=\min\{n\ge 0:\mathcal{X}_n\ge \textrm{H}\}, \] where the function $F$ and threshold $\textrm{H}$ are design parameters. The observations and change time are modeled jointly through a hidden Markov model, and $ F$ is selected from a prescribed function class $\mathcal{G}$ to minimize the weighted criterion \[ \textsf{E}\bigl[ (τ_s-τ_a)_+ + κ(τ_s-τ_a)_- \bigr]. \] When $\mathcal{G}$ is a linear function class, the optimizer $F^*$ is characterized by a convex program, whose dual yields extensions of classical likelihood-ratio constructions. This conclusion is based on analysis that is asymptotic in the regime $κ\to\infty$. We show that the hidden Markov model admits an asymptotically equivalent conditionally independent approximation of the type commonly used in the quickest change detection literature. We then develop the design and asymptotic theory for a substantially broader class of conditionally independent models, so that the resulting conclusions are not tied to the particular POMDP reduction. Combining renewal theory and large deviations for reflected random walks, we obtain for each $F\in\mathcal{G}$ asymptotically accurate approximations of the optimal threshold and average cost, with error vanishing as $κ\to\infty$. It is found in numerical experiments that the resulting approximations are accurate for moderate values of $κ$.

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