自相關過程的Ｘbar管制圖設計─模式與無模式兩種

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自相關過程的Ｘbar管制圖設計─模式與無模式兩種

Designs of Xbar Control Charts for Autocorrelated Processes─Model-Based and Model-Free

鄭郁諺 , Ph.D　　Advisor：陳慧芬　　

英文

Xbar管制圖；階段二；階段一；非常態性；無模式之設計；估計之管制上下限；模式為基之設計；無分配之設計；平均連串長度；自相關過程； Phase II ； Xbar chart ； nonnormality ； Phase I ； estimated control limits ； model-free design ； distribution-free design ； average run length ； model-based design ； Autocorrelated processes

Abstract │ Reference (74) │ Altmetrics

Abstract 〈TOP〉

本論文研究針對自相關過程，探討具有對稱管制上下限的Xbar管制圖之設計問題。Xbar管制圖是在1931年由蕭華特博士提出，是偵測製程平均數偏移一個有用的工具。Xbar管制圖的設計問題在於決定樣本平均數Xbar的樣本數和管制上下限的距離因素。

傳統Xbar管制圖的假設是樣本平均數服從獨立且常態分配。然而，在實務應用上，品質特性測量值卻可能具有自我相關性和非常態性。再者，當樣本平均數的標準差未知但給定一組管制內的資料時，則樣本平均數的標準差可以藉由這組管制內的資料估計，但是具有隨機性。自我相關性，非常態性，和參數估計會影響特定管制圖的績效指標(如：平均連串長度)下之樣本數和管制上下限的距離因素。

本論文研究包含四個子問題。子問題一探討非常態性，自我相關性，和參數估計對Xbar管制圖的績效指標之影響。子問題二到子問題四考慮Xbar管制圖的三個設計問題，分別在三個假設之下：(1)資料過程已知(以模式為基之設計表示)，(2)已知資料的自我共變數，但邊際分配未知(以無分配之設計表示)，(3)資料過程完全未知(以無模式之設計表示)。

針對子問題一，我們得到的結果有：(1) 非常態性對於平均連串長度和連串長度的標準差具有非單調性的影響。(2) 當管制內Xbar資料的個數增加時，平均連串長度和連串長度的標準差將遞減且收斂到已知樣本平均數的標準差的情形。(3) 當樣本平均數的間隔一自我相關性增加時，管制內連串長度等於2的機率遞減且管制內連串長度分配的尾部機率值增加。因此，管制內平均連串長度和連串長度的標準差增加。(4) 針對自我相關性和參數估計的影響，當樣本平均數的間隔一自我相關性之絕對值遞增，管制內Xbar資料的個數遞減，或兩者同時發生時，則連串長度分配的尾部機率值增加。除此之外，我們也探討樣本平均數Xbar的樣本變異數之變異數的漸近結果並針對此結果對管制內Xbar資料的個數提出建議。(5) 針對非常態性和參數估計的影響，當偏移量是小量或無偏移時，平均連串長度在具有非常態性，參數估計的組合和具有常態性，參數已知的組合之差距會隨著管制內Xbar資料的個數遞增而遞增。當偏移量是大量時，則此差距會隨著管制內Xbar資料的個數遞增而遞減。(6) 自我相關性，非常態性和參數估計對於平均連串長度和連串長度的標準差具有非單調性的影響。

針對子問題二，我們提出一演算法，在限定管制內平均連串長度為一定值之下，用以決定使得管制外平均連串長度最小化之樣本數及管制上下限的距離因素之參數組合。我們藉由模擬實驗來比較：(1) EWMAST(針對穩態資料過程之指數加權移動平均)管制圖，(2) SCC(特殊原因)管制圖，(3) ARMAST(針對穩態資料過程之自我相關移動平均)管制圖，(4) 我們提出的Xbar管制圖設計。當品質特性測量值服從ARMA(1, 1)(一階自我迴歸和一階移動平均)過程時，ARMAST管制圖勝過我們提出的Xbar管制圖設計，EWMAST管制圖，和SCC管制圖，原因是ARMAST管制圖是針對ARMA(1, 1)過程而設計。當品質特性測量值具有自我相關性和非常態性時，我們提出的Xbar管制圖設計優於ARMAST管制圖。

針對子問題三，我們提出兩個無分配方法(以方法1和2表示)，在限定管制內平均連串長度為一定值之下，用以決定使得管制外平均連串長度最小化之樣本數及管制上下限的距離因素之參數組合。方法1和2之基本概念分別為假設樣本平均數服從具獨立性之常態分配和AR(1)(一階自我迴歸)過程。為了符合中央極限定理，我們以調整樣本數至少30來修正方法1和2。修正後的方法2優於修正後的方法1。我們針對R&W(Runger和Willemain)管制圖，DFTC(無分配之表格化累積和)管制圖，和我們提出的修正後方法2做比較。R&W管制圖和DFTC管制圖在自我相關性輕微到中度和偏移量輕微或較大時表現較好。而在高度自我相關性和偏移量中度或較大時，我們提出的修正後方法2表現較好。

針對子問題四，我們針對完全未知之自我相關過程，提出一個無模式Xbar管制圖設計，並且是在給定一組管制內品質特性測量值資料之條件下。樣本平均數標準差是由無重疊分批平均數法估計。我們提出演算法在限定管制內平均連串長度為一定值之下，用以決定使得管制外平均連串長度最小化之三個參數：批量數、樣本數、管制上下限的距離因素。在此三個參數的搜尋過程中，我們計算平均連串長度在於樣本平均數和分批平均數服從具獨立性的常態分配之假設下。為了符合中央極限定理，我們也修正批量數和樣本數至少30。我們藉由模擬實驗來評估此演算法，而測試例子包含了AR(1)過程，ARTA(1)(AutoRegressive To Anything of order 1)過程且邊際分配分別為t分配和指數分配。實驗結果顯示我們的無模式演算法有不錯效果而且在自我相關性輕微或自我相關性較大但偏移量小量或輕微時表現較好。而當資料服從ARTA(1)過程且邊際分配為指數分配時，我們的無模式Xbar管制圖設計表現較差。

Parallel Abstract 〈TOP〉

This thesis considers the design problem of the Xbar chart with symmetric control limits for autocorrelated data processes. The Xbar chart, proposed by W.A. Shewhart in 1931, is a useful tool to detect a shift in the process mean.  The design problem of the Xbar chart is to determine the sample size n of Xbar and the control-limit factor k (number of standard deviations away from the center line).

A conventional assumption of the Xbar chart is that the sample means are independent and normally distributed. Nevertheless, in practice the quality characteristic measurements may be autocorrelated and nonnormally distributed.  Furthermore, when the standard deviation of Xbar is unknown but in-control observations are given, the standard deviation of Xbar needs to be estimated and it is random.  The autocorrelation, nonnormality, and parameter estimation affect the values of the design parameters n and k for meeting specified control chart performances, e.g., average run length.

This thesis consists of four subproblems.  Subproblem 1 considers the effects of the nonnormality, autocorrelation, and parameter estimation on the Xbar chart's performance.  Subproblems 2 to 4 consider the design problem of the Xbar chart assuming that the data process is known (denoted as model-based design), has known autocovariance structure but unknown marginal distribution (denoted as distribution-free design), and is unknown (denoted as model-free design), respectively.

For Subproblem 1, we show that (1) The nonnormality has nonmonotonic effects on the mean and standard deviation of the run length. (2) When the number m of in-control Xbar observations increases, the mean and standard deviation of run length decrease and converge to the corresponding values of m=infinity. (3) As the lag-1 autocorrelation phi of sample means increases, P{N_0=2} (where N_0 is the in-control run length) decreases and the tail probabilities increase.  Therefore, the ARL (Average Run Length) and standard deviation of run length increase. (4) For the simultaneous effect of autocorrelation and estimation, the distribution of the run length has a heavier tail as |phi| increases, m decreases, or |phi| increases and m decreases simultaneously. We also study the asymptotic result of Var(S_{Xbar}^2) to recommend the m.  (5) For the simultaneous effect of nonnormality and estimation, for small shift magnitude or shift magnitude = 0., the difference of  the ARL between the nonnormal and estimation case with the normal and known variance case increases as m increases.  For a large shift magnitude (>=2), the difference of the ARL between the nonnormal and estimation case with the normal and known variance case decreases as m increases.  (6) For the simultaneous effect of autocorrelation, nonnormality, and estimation: we conclude that the three-dimensional effect is nonmonotic on the mean and standard deviation of run length.

For Subproblem 2, we propose an algorithm to compute values of the sample size n and control-limit factor k that minimize the out-of-control ARL while keeping the in-control ARL at a specified value. Simulation experiments are run to compare the proposed Xbar chart design with the EWMAST (Exponentially Weighted Moving Average STationary), SCC (Special Cause Control), and ARMAST (Auto-Regressive Moving-Average STationary) charts.  When the data process is ARMA(1, 1) (AutoRegressive of order 1 and Moving Average of order 1) process, the ARMAST chart outperforms the proposed Xbar chart, EWMA, and SCC because the ARMAST chart is designed based on the ARMA(1, 1) process.  When the data process is autocorrelated with nonnormal distribution, the proposed Xbar chart often performs better than the ARMAST chart.

For Subproblem 3, we propose two distribution-free methods (denoted as Methods 1 and 2) to compute the sample size n and control-limit factor k that minimize the out-of-control ARL while keeping the in-control ARL at a specified value.  Methods 1 and 2 are based on the assumptions that sample means are independently and normally distributed and that sample means follow an AR(1) (AutoRegressive of order 1) process, respectively.  We further modify Methods 1 and 2 by setting the sample size to be at least 30 to meet the central limit theorem.  The modified Method 2 outperforms the modified Method 1.  We compare the modified Method 2 with R&W (Runger and Willemain) and DFTC (Distribution-Free Tabular CuSum) charts. The R&W and DFTC charts perform better when the autocorrelation is small to moderate and the shift is small or large.  Our modified Method 2 performs better when the autocorrelation is high and the shift is moderate to large.

For Subproblem 4, we propose a model-free Xbar chart design for unknown autocorrelated data processes, when M observations of in-control quality characteristic measurements are given.  The standard deviation of Xbar is estimated by the NBM (nonoverlapping batch means) method.  The Xbar chart design parameters n and k and the NBM batch size omega are determined to minimize the out-of-control ARL while keeping the in-control ARL at a specified value.  During the search of n, k, and omega, the ARL values are computed assuming that the sample means and batch means are independently and normally distributed.  The computed optimal values of n and omega are further modified to be at least 30 for meeting the central limit theorem.  Simulation experiments are run to evaluate the performance of the proposed model-free Xbar chart using AR(1) processes and ARTA(1) (AutoRegressive To Anything of order 1) processes with t and exponential marginal distributions as testing examples. The experimental results show that our model-free method performs well in general; it performs better when the autocorrelation is small or when the autocorrelation is large but the shift is small or moderate.  The proposed model-free Xbar chart performs worse when the data process is ARTA(1) with an exponential marginal distribution.

Reference ( 74 ) 〈TOP〉

[20] Choobineh, F., Ballard, J. L. (1987). Control-Limits of QC Charts for Skewed Distributions Using Weighted-Variance. IEEE Transactions on Reliability 36}, 473-477.
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[23] Darling, D. A. and A. J. F. Siegert. (1953). The First passage problem for a continuous Markov process. Annals of Mathematical Statistics 24}, 624-639.
連結：
[24] Del Castillo, E. (1996). Evaluation of Run Length Distributions for Xbar Charts with Unknown Variance. Journal of Quality Technology 28}, 116-122.
連結：
[29] Goldsman, D., M. S. Meketon, and L. W. Schruben. (1990). Properties of standardized time series weighted area variance estimators. Management Science 36, 602-612.
連結：
[44] Lucas, J.M. and M.S. Saccucci (1990). Exponentially weighted moving average control schemes: Properties and enhancements. Technometrics 32, 1-12.
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