基于曼−惠特尼U统计量的在线变点检测

    Online Change-Point Detection Based on Mann-Whitney U Statistic

    • 摘要: 变点检测旨在发现时间序列中统计特性发生突变的时刻,广泛应用于信号处理的各个领域。滑动窗曼−惠特尼U统计量(Mann-Whitney U Statistic,MWUS)是一种基于秩统计量的非参数变点检测方法,对脉冲噪声干扰具有较强的鲁棒性。针对其忽略滑动窗引入的时间相关且仅依赖边缘分布设定阈值而导致在线虚警率难控的问题,本文提出了基于低通滤波的MWUS在线检测方法(Low-Pass Filtering MWUS,LPFMWUS)。首先,推导了无变点零假设下MWUS序列的功率谱密度及理想情况下(无噪且只存在突变点时)能量谱密度的闭式解,揭示了噪声的带通特性与变点的低通特性,据此引入低通滤波。然后,证明了零假设下MWUS序列及其经有限冲激响应滤波后序列的遍历性,进而构建滤波后统计量的多元联合分布模型及动态阈值设定方法。最后仿真表明,本文所提出的动态阈值设定方法可以精确控制在线虚警率;LPFMWUS通过提高信噪比可以提升检测性能,定位精度优于代表性的累积和(cumulative sum,CUSUM)方法(参数方法)与非参数无界变点局部检测(Non-Parametric UNbounded Changepoint Local,NUNCL)方法。

       

      Abstract: Change-point detection (CPD) aims to identify abrupt changes in the statistical properties of a time series, and has been widely applied across various signal processing domains. The sliding-window Mann–Whitney U statistic (MWUS) , a non-parametric rank-based CPD method, exhibits strong robustness against impulsive noise interference. However, MWUS neglects the temporal correlation introduced by the sliding window and determines thresholds solely based on marginal distributions, leading to uncontrollable false alarm rates. To address this, we propose a low-pass filtering MWUS (LPFMWUS) method. We derive closed-form expressions for the power spectral density of the MWUS sequence under the null hypothesis (no change point) and for the energy spectral density under the ideal scenario (noise-free with only a change point) . This analysis reveals that the noise exhibits band-pass characteristics, while the change point exhibits low-pass characteristic, motivating the application of a low-pass filter. Leveraging the proven ergodic properties of the MWUS sequence and its finite impulse response filtered version under the null hypothesis, we construct a multivariate joint distribution model for the filtered statistics and develop a dynamic threshold setting method. Simulations demonstrate that: (1) the proposed dynamic threshold setting method can precisely control the online false alarm rate; (2) LPFMWUS improves detection performance by enhancing the signal-to-noise ratio, and its localization accuracy outperforms that of representative methods, including the parametric cumulative sum (CUSUM) method and the Non-Parametric UNbounded Changepoint Local (NUNCL) method.

       

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