Abstract: Traditional control charts like the Shewhart control chart only utilize the current sample information, while the exponentially weighted moving average (EWMA) control chart combines current and historical data through a smoothing constant for improved shift detection ability. However, the performance of these parametric control charts relies heavily on the assumption that the process data follows a specific probability distribution, typically the normal distribution. When this parametric assumption is violated, the control charts can suffer from low detection power and frequent false alarms. This paper will introduce a new nonparametric adaptive exponentially weighted moving average (AEWMA) control chart based on the Wilcoxon signed-rank (SR) statistic to monitor process median shifts when the underlying data distribution is unknown or non-normal. The proposed AEWMA SR control chart leverages the robust properties of nonparametric statistics while inheriting the overall superior shift detection capabilities of adaptive schemes. The smoothing constant of the proposed adaptive exponentially weighted updating schemeis adjustable based on the magnitude of the monitoring statistic through a discrete error transmission function. This allows the AEWMA SR control chart to automatically emphasize recent or past observations to optimally detect different levels of shifts. To further enhance its detection rapidity, the authors study the properties of the AEWMA SR control chart under a variable sampling interval (VSI) strategy. Two sampling intervals are utilized: a shorter interval when the statistic falls in a warning zone around the center line to quickly detect any potential shifts, and a longer interval in the safety zone to reduce sampling costs. The exact run-length performance measures including the average run length (ARL) and the average time to signal (ATS) are derived using the Markov chain approach. An optimization procedure is developed to determine the optimal set of chart parameters (smoothing constants, error transmission function coefficients, control limits, and sampling intervals) that minimizes the out-of-control ARL over a range of shifts while constraining the in-control ARL. Extensive comparisons are made between various fixed and variable sampling interval configurations of AEWMA SR and EWMA SR control charts. The results demonstrate the superiority of the proposed VSI AEWMA SR control chart in providing robust and balanced shift detection performance across different magnitudes of shifts. Unlike individual EWMA control charts tuned for specific shifts, the AEWMA adaptation allows general sensitivity to a range of shifts. Furthermore, the VSI feature leads to substantially faster signaling times (lower out-of-control ATS values) compared to fixed-sampling charts. The paper also presents approaches to recursively calculate the probability mass functions of the run length under both in-control and out-of-control conditions. A case study on vibration acceleration monitoring data is provided, highlighting the rapidity of the VSI AEWMA SR control chart in detecting a shift compared to its fixed-sampling counterpart. In summary, this research introduces an effective nonparametric AEWMA control charting technique that offers robust median monitoring performance when data distributions are unknown, combines the advantages of adaptation and variable sampling intervals, and provides a comprehensive optimization and evaluation framework. The VSI AEWMA SR control chart is a valuable scheme for various applications where parametric assumptions may not hold and flexible, efficient shift detection is critical.

Key words: nonparametric adaptive EWMA control chart, variable sampling intervals, average run length, average time to signal

摘要： 本文基于Wilcoxon符号秩(Signed Rank,SR)检验统计量,提出了一种非参数自适应指数加权移动平均(Adaptive Exponentially Weighted Moving Average,AEWMA)控制图。所提出的AEWMA SR控制图结合了非参数统计量的稳健受控性能以及自适应控制图良好的整体偏移检测特性。同时,为了提高固定采样间隔下的非参数AEWMA SR静态控制图对异常偏移的检测效率,本文进一步研究了可变采样间隔(Variable Sampling Intervals,VSI)下的非参数AEWMA SR动态控制图设计问题。使用了Markov链方法计算控制图的精确平均运行链长(Average Run Length, ARL)和平均报警时间(Average Time to Signal, ATS)等性能指标。通过仿真分析比较了VSI AEWMA SR控制图、FSI AEWMA SR控制图和VSI EWMA SR控制图的统计性能。结果表明,所提出的VSI AEWMA SR控制图能兼顾对于不同大小偏移的敏感性,且变采样间隔的动态策略能显著提高控制图的检测效率。

关键词: 非参数AEWMA控制图, 变采样间隔, 平均运行链长, 平均报警时间