基于共形鞅的变点检测：新的最优构造及现有方法的次优性
Change detection with conformal martingales: new optimal constructions, and suboptimality of existing methods
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中文总结 AI 辅助
研究分布自由的序贯变点检测，基于共形鞅提出极小极大最优的e-过程和e-检测器，将延迟从现有方法的Ω(T)和Ω(√ARL)改进为Θ(log T)和Θ(log ARL)。
中文摘要 AI 辅助
我们研究独立观测的分布自由序贯变点检测，其中变点前后的分布未知且不受限制。我们基于Vovk(2021)的共形检验鞅及相关的e-检测器，它们分别控制误报概率(PFA)和平均运行长度(ARL)。这些工作大多关注有效性，统计效率通常留给模拟验证。我们发展了一个全面的理论，研究共形p值在变点位于未知时间$T$的非可交换数据下的行为。我们利用该理论分析共形鞅方法在变点后的增长及由此产生的检测延迟，并证明标准现有方法在PFA和ARL控制上是次优的，可分别导致$\Omega(T)$和$\Omega(\sqrt{\text{ARL}})$的延迟。我们提出了不同的共形e-过程和e-检测器，它们被证明是极小极大最优的，延迟分别为$\Theta(\log T)$和$\Theta(\log \text{ARL})$，并且在模拟中具有更短的延迟。
英文摘要
We study distribution-free sequential changepoint detection for independent observations with unknown and unrestricted pre- and post-change laws. We build on the conformal test martingales and associated e-detectors of Vovk(2021), which control the probability of false alarm (PFA) and the average run length (ARL) respectively. The majority of these works focus on validity, with statistical efficiency usually left for simulations. We develop a comprehensive theory of how conformal p-values behave under non-exchangeable data with a changepoint at an unknown time $T$. We use this to analyze the post-change growth and resulting detection delay of conformal martingale methods, and prove that the standard existing methods are suboptimal for PFA and ARL control, and can lead to delays that are $Ω(T)$ and $Ω(\sqrt{\text{ARL}})$ respectively. We propose different conformal e-processes and e-detectors that are provably minimax optimal, with delays $Θ(\log T)$ and $Θ(\log \text{ARL})$ respectively, and have much shorter delays in simulations.
发表机构
- Wharton School, University of Pennsylvania（宾夕法尼亚大学沃顿商学院）
- Stanford University（斯坦福大学）
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