Approximating the ARL of Changes in the Mean of a Seasonal Time Series Model with Exponential White Noise Running on a CUSUM Control Chart

Q3 Mathematics

WSEAS Transactions on Systems and Control Pub Date : 2023-11-15 DOI:10.37394/23203.2023.18.39

W. Peerajit

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Abstract

Control charts comprise an excellent statistical process control tool for monitoring industrial processes. Especially, the CUSUM control chart is very sensitive to small-to-moderate process parameter changes. The proposed approach utilizes the numerical integral equation (NIE) method to approximate the average run length (ARL) of changes in the mean of a seasonal time series model with underlying exponential white noise running on a CUSUM control chart. This was achieved by solving a system of linear equations and integration through partitioning and summation using the area under the curve of a function obtained by applying the Gauss-Legendre quadrature. A numerical study was conducted to compare the capabilities of the ARL derivations obtained using the NIE method and explicit formulas to detect changes in the mean of a long-memory model with exponential white noise running on a CUSUM control chart. The results reveal that the performances of both were comparable in terms of the accuracy percentage, which was greater than 95%, meaning that the ARL values were highly consistent. Thus, the NIE method can be used to validate ARL results for this situation.

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在 CUSUM 控制图上近似运行指数白噪声季节时间序列模型均值变化的 ARL

控制图是监控工业过程的绝佳统计过程控制工具。特别是，CUSUM 控制图对中小规模的过程参数变化非常敏感。所提出的方法利用数值积分方程 (NIE) 方法来近似计算在 CUSUM 控制图上运行的带有潜在指数白噪声的季节性时间序列模型平均值变化的平均运行长度 (ARL)。其方法是求解线性方程组，并利用通过应用高斯-回归正交获得的函数曲线下面积，通过分割和求和进行积分。通过数值研究，比较了使用 NIE 方法和显式公式获得的 ARL 推导在检测 CUSUM 控制图上运行的指数白噪声长记忆模型均值变化方面的能力。结果表明，二者在准确率方面表现相当，都大于 95%，这意味着 ARL 值高度一致。因此，NIE 方法可用于验证这种情况下的 ARL 结果。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

WSEAS Transactions on Systems and Control Mathematics-Control and Optimization

CiteScore

1.80

自引率

0.00%

发文量

期刊介绍： WSEAS Transactions on Systems and Control publishes original research papers relating to systems theory and automatic control. We aim to bring important work to a wide international audience and therefore only publish papers of exceptional scientific value that advance our understanding of these particular areas. The research presented must transcend the limits of case studies, while both experimental and theoretical studies are accepted. It is a multi-disciplinary journal and therefore its content mirrors the diverse interests and approaches of scholars involved with systems theory, dynamical systems, linear and non-linear control, intelligent control, robotics and related areas. We also welcome scholarly contributions from officials with government agencies, international agencies, and non-governmental organizations.