A Distribution-Free Multivariate Control Chart

Monitoring multivariate quality variables or data streams remains an important and challenging problem in statistical process control (SPC). Although the multivariate SPC has been extensively studied in the literature, designing distribution-free control schemes are still challenging and yet to be addressed well. This paper develops a new nonparametric methodology for monitoring location parameters when only a small reference dataset is available. The key idea is to construct a series of conditionally distribution-free test statistics in the sense that their distributions are free of the underlying distribution given the empirical distribution functions. The conditional probability that the charting statistic exceeds the control limit at present given that there is no alarm before the current time point can be guaranteed to attain a specified false alarm rate. The success of the proposed method lies in the use of data-dependent control limits, which are determined based on the observations on-line rather than decided before monitoring. Our theoretical and numerical studies show that the proposed control chart is able to deliver satisfactory in-control run-length performance for any distributions with any dimension. It is also very efficient in detecting multivariate process shifts when the process distribution is heavy-tailed or skewed. Supplementary materials for this article are available online.

在统计过程控制（Statistical Process Control, SPC）领域，多变量质量变量与数据流的监控始终是一项兼具重要学术与应用价值的挑战性课题。尽管现有文献已对多变量统计过程控制（multivariate SPC）展开了广泛研究，但设计无分布控制方案仍存在显著挑战，尚未得到妥善解决。针对仅可获取少量参考数据集（reference dataset）的场景，本文提出了一种全新的非参数方法（nonparametric methodology），用于位置参数的监控。其核心思路为构建一系列条件无分布检验统计量：在给定经验分布函数（empirical distribution functions）的前提下，该类统计量的分布不受未知总体分布的约束。可保证在当前时刻之前未发出任何告警的条件下，当前控制图统计量超出控制限（control limit）的条件概率达到预设的虚警率（false alarm rate）。所提方法的成功之处在于采用了数据依赖型控制限：该控制限并非在监控启动前预先设定，而是基于在线观测数据实时确定。理论分析与数值实验结果表明，所提出的控制图可对任意维度、任意分布的过程实现令人满意的受控运行长度性能。当过程分布具有厚尾或偏态特征时，该方法在检测多变量过程偏移方面同样具备极高的效率。本文的补充材料可在线获取。