ARTA過程的Xbar管制圖統計設計

The Statistical Design of the Xbar Control Chart for ARTA processes

林煥唐(Huan-Tang Lin)

指導教授 ： 陳慧芬

中原大學/工學院/工業工程研究所/碩士(2006年)

https://doi.org/10.6840/cycu200600655

摘要

本研究旨在探討 管制圖應用於相關性資料之設計。假設品質特性之測量值服從ARTA過程，且為一種任意邊際分配共變數穩定的時間數列。過去文獻對相關性資料管制圖設計的探討多著墨於常態性資料，有關相關性非常態資料之管制圖設計較為罕見。
本研究假設每個製程在製造後立即被檢測，因此在設計 管制圖時只考慮樣本數和管制上下限的距離兩個設計參數。本文構建一相關性非常態資料之管制圖設計最佳化模型，係在使用者製程穩定且固定管制圖偵測績效條件下，針對特定偏移資料尋求管制圖之設計參數，俾具有最佳的偵測能力。本文提出一個結合回溯近似法的模擬流程來求解，並進一步分析相關性非常態資料對設計參數的影響。
經模擬實驗結果顯示，樣本數的設計會隨著資料的正相關增強而增加；當資料之負相關較強時，樣本數並不會隨著相關性程度呈現單調性的變化。此外，非常態的邊際分配，exponential, lognormal和t在相同的p階自我相關性下需要比常態更大的樣本數。(我們只顯示p=1的實驗結果。)
此外我們在不同的相關性資料下比較ARMAST管制圖跟 管制圖。我們選擇AR(1)過程跟邊際分配為U(0,1)的ARTA(1)過程。結果指出 管制圖對於非常態的資料有較佳的穩健性。

關鍵字

管制圖 ； ARTA過程 ； 非常態性 ； 相關性資料 ； 回溯近似法

並列摘要

This thesis is to explore the control chart design for application to the corre-lated data. We assume that the quality characteristic X follows an ARTA(p) process, which is a covariance stationary process with an arbitrary marginal distribution. A considerable number of previous works have been dealing with the control chart de-sign for correlated normality data, little has been endeavored to the effect of the cor-related non-normality data.
In this thesis, we assume that each item is inspected and its quality characteristic is measured at the moment it is produced. Hence, we consider only two parameters: sample size m and control-limit factor k for the design chart. We develop an op-timization model for the control chart design with correlated non-normality data. The rationale for the model is to look for the optimal design parameters so as to maximize the out-of-control performance with specific mean shift, under a fixed performance of in-control process. We propose a procedure of simulation, incorporated with retro-spective approximation, to find the optimal design of control chart. In addition, we also examine the effects of correlated non-normality data on the design parameters.
From the results of simulation experiments, our findings reveal that the control chart design requires larger sample size as the data get higher positively correlated. For stronger negative correlation, the sample size does not change monotonically with the magnitude of correlation. Additionally, a larger sample is required for non-normal marginal distributions, such as exponential, lognormal, and t distributions, than for the normal marginal distribution with same lag-p autocorrelations. (Our empirical results only show p=1.)
In addition, we compare the chart to the ARMAST chart in different corre-lated data. The data processes we selected are AR(1) process and ARTA(1) with U(0,1) marginal distribution. The results indicate that the application of chart is more robust than ARMAST chart with non-normality marginal distribution.

並列關鍵字

retrospective approximation ； ARTA process ； control chart ； non-normality ； correlated data

參考文獻

Alwan, L.C. and H. V. Roberts (1988) “Time-Series Modeling for Statistical Process Control.” Journal of Business and Economic Statistics 6, 87-95.

Cario, M. C. and B. L. Nelson (1996) “Autoregressive to Anything: Time-series Input Processes for Simulation.” Operations Research Letters 19, 51-58.

Cario, M. C. and B. L. Nelson (1998) “Numerical Methods for Fitting and Simulating Autoregressive-to-anything Processes.” INFORMS Journal on Computing 10, 72-81.

Chen, H. and B.W. Schmeiser (2001) “Stochastic Root Finding via Retrospective Ap-proximation.” IIE Transactions 33, 259-275.

Harris, T. J. and W. H. Ross (1991) Statistical Process Control Procedures for Corre-lated Observations.” The Canadian Journal of Chemical Engineering 69, 48-57.