近年來，許多高科技公司應用指數加權移動平均(EWMA, exponentially weighted moving average)與累和(CUSUM, cumulative sum)管制圖來偵測製程平均之微量變動，但此類管制圖係假設觀察值彼此獨立。然而隨著電腦科技的日新月異及生產過程在自動化技術不斷改進下，蒐集製程資料的間隔時間已大幅縮短，使得樣本間存在著高度自我相關性。因此，若將此類資料以傳統的管制圖進行監測極易產生誤判，而導致不必要的成本浪費。近年來已有多位學者提出配適原始資料的時間數列模型，若模型配適正確，且模型的殘差彼此獨立，即可利用殘差管制圖監控製程的品質。
過去多位研究學者藉由不同統計模型的選取準則協助廠商選擇正確的時間數列模型，這些準則依其不同統計模型，大致可分為具有漸進有效性(如AIC和AICC)與一致性(如BIC與SIC)之準則兩大類，另有學者提出結合前述兩類優點之WIC準則。基於上述模型選取準則各有其適用性，因此本研究著重於樣本數不同之情形下，如何利用適當的準則選取正確的統計模型。研究之成果可作為業界在處理自我相關製程資料與監控時的重要參考。

The exponentially weighted moving average control chart (EWMA) and the cumulative sum control chart (CUSUM) for detecting the small sustained process change have recently been used by high-tech companies. When the process data are correlated, one should first select a suitable model to fit the data and then use the EWMA or CUSUM control charts to monitor the change of residual. The effectiveness of using residual control charts of EWMA and CUSUM depend crucially on the appropriateness of the model selected. In this paper, several order-selection criteria including AICC (bias-corrected Akaike’s information criterion), BIC (Akaike’s Bayesian modification of AIC), and WIC (weighted average information criterion, proposed by Wu and Sepulveda 1998) are used to select the model order when the process data follows a time series model like ARMA (Auto-regressive and moving average). The performances of these criteria are further demonstrated by simulation and numerical examples. The results show that residual control charts using WIC are more appropriate than other criteria such as AIC, AICC, BIC and SIC when the sample size is between 30 and 100.

【1】 Adams, B. M. and Tseng, L. T., “Robustness of Forecast-Based Monitoring Schemes,” Journal of Quality Technology, 30, 328-329 (1998).
【2】 Akaike, H., “A New Look at the Statistical Model Identification,” IEEE Transactions on Automatic Control, AC-19, 716-723 (1974).
【3】 Akaike, H., “A Bayesian Analysis of The Minimum AIC Procedure,” Annals of the Institute of Statistical Mathematics, 30, 9-14 (1978).
【4】 Alwan, A. J. and Alwan, L. C., “Monitoring Autocorrelated Processes Using Multivariate Quality Control Charts,” Proceedings of the Decision Sciences Institute Annual Meeting, 3, 2106-2108 (1994).
【5】 Alwan, L.C. and Roberts, H. V., “Time-Series Modeling for Statistical Process Control,” Journal of Business and Economic Statistics, 6, 87-95 (1988).
【6】 Bengtsson, T. and Cavanaugh, J. E., “An improved Akaike information criterion for state-space model selection,” Computational Statistics & Dara Analysis, 50, 2635-2654 (2006).
【7】 Box, G. E. P., Jenkins, G. M. and Reinsel, G. C., Time Series Analysis, Forecasting and Control, 3rd Ed., Prentice-Hall, London (1994).
【8】 Box, G. E. P. and Pierce, D. A., “Distribution of Residual Autocorrelations in Autoregressive-Integrated Moving Average Time Series Models,” Journal of American Statistical Association, 65, 1509-1526 (1970).
【9】 Crowder, S. V., “A Simple Method for Studying Run-Length Distributions of Exponentially Weighted Moving Average Charts,” Technometrics, 29, 401-407 (1987a).
【10】 Crowder, S. V., “Average Run Lengths of Exponentially Weighted Moving Average Control Charts,” Journal of Quality Technology, 19, 161-164 (1987b).
【11】 Crowder, S. V., “Design of Exponentially Weighted Moving Average Schemes,” Journal of Quality Technology, 21, 155-162 (1989).
【12】 Gan, F. F., “An Optimal Design of CUSUM Quality Control Charts,” Journal of Quality Technology, 23, 279-286 (1991).
【13】 Goel, A. L. and Wu, S. M., “Determination of A.R.L. and a Contour Nomogram for CUSUM Charts to Control Normal Mean,” Technometrics, 13, 221-230 (1971).
【14】 Goldsmith, P. L. and Whitefield, H., “Average Run Lengths in Cumulative Chart Quality Control Schemes,” Technometrics, 3, 11-20 (1961).
【15】 Harris, T. J. and Ross, W. H., “Statistical Process Control Procedure for Correlated Observations,” Canadian Journal of Chemical Engineering, 69, 48-57 (1991).
【16】 Hunter, J. S., “The Exponentially Weighted Moving Average,” Journal of Quality Technology, 18, 203-210 (1986).
【17】 Hurvich, C. M. and Tsai, C. L., “Regression and Time Series Model Selection in Small Samples,” Biometrika, 76, 297-307 (1989).
【18】 Lee, H. L., So, K. C. and Tang, C. S., “The Value of Information Sharing in a Two-Level Supply Chains,” Management Science, 46, 626-643 (2000).
【19】 Ljung, G. M. and Box, G. E. P., “On a Measure of Lack of Fit in Time Series Models,” Biometrika, 65, 297-303 (1978).
【20】 Lu, C. W. and Reynolds, M. R., “EWMA Control Charts for Monitoring the Mean of Autocorrelated Processes,” Journal of Quality Technology, 31, 166-188 (1999).
【21】 Lu, C. W. and Reynolds, M. R., “CUSUM Charts for Monitoring An Autocorrelated Processes,” Journal of Quality Technology, 33, 316-334(2001).
【22】 Lucas, J. M. and Crosier, R. B., “Fast Initial Response for CUSUM Quality-Control Schemes: Give Your CUSUM a Head Start,” Technometrics, 24, 199-205 (1982).
【23】 Lucas, J. M. and Saccucci, M. S., “Exponentially Weighted Moving Average Control Schemes: Properties and Enhancements,” Technometrics, 32, 1~12 (1990).
【24】 Montgomery, D. C., Introduction to Statistical Quality Control, 5th Ed., Wiley, New York (2005).
【25】 Montgomery, D. C. and Mastrangelo, C. M., “Some Statistical Process Control Methods for Autocorrelated Data,” Journal of Quality Technology 23, 179-193 (1991).
【26】 Page, E. S., “Cumulative Sum Charts,” Technometrics, 3, 1-9 (1961).
【27】 Roberts, S. W., “Control Chart Tests Based on Geometric Moving Averages,” Technometrics, 1, 239-250 (1959).
【28】 Robinson, P. B. and Ho, T. Y., “Average Run Lengths of Geometric Moving Average Charts by Numerical Methods,” Technometrics, 20, 85-93 (1978).
【29】 Runger, G. C., Willemain, T. R. and Prabhu, S., “Average Run Lengths for CUSUM Control Charts Applied to Residuals,” Communication in Statistics-Theory and Methods, 24, 273-282 (1995).
【30】 Ryan, T. P., Discussion in Montgomery, D. C. and C. M. Mastrangelo, “Some Statistical Process Control Methods for Autocorrelated Data,” Journal of Quality Technology, 23, 200-203 (1991).
【31】 Schwartz, G., “Estimating The Dimension of A Model,” Annals of Statistics, 6, 461-464 (1978).
【32】 Tseng, S. and Adams, B. M., “Monitoring autocorrelated Processes with an Exponentially Weighted Moving Average Forcast,” Journal of Statistical Computation and Simulation, 50, 187-195 (1994).
【33】 VanBrackle III, L. N. and Reynolds Jr., M. R., “EWMA and CUSUM Control Charts in the Presence of Correlation,” Communications in Statistics: Simulation and Computation, 26, 979-1008 (1997).
【34】 Wardell, D. G., Moskowitz, H. and Palnte, R. D., “Run-Length Distributions of Special-Cause Control Charts for Correlated Processes,” Technometric, 36, 3-17 (1994).
【35】 Wei, William W. S., Time Series Analysis, Univariate and Multivariate Methods, 2nd Ed., Addison-Wesley, California (1994).
【36】 Wu, T. J. and Sepulveda, A., “The Weighted Average Information Criterion for Order Selection in Time Series and Regression Models,” Statistics & Probability Letters, 39, 1-10 (1998).
【37】 Wu, T. J., Pan, J. N., Chen, S. F. and Li, P. T., “An Application of Information Criteria to the Detection of Process Change,” Proceedings of the Asia Pacific Management Conference, 191-203 (2002).
【38】 Yashchin, E., “Performance of CUSUM Control Schemes for Serially Correlated Observations,” Technometrics, 35, 37-52 (1993).
【39】 Zhang, H. M. and Wang, P., “A New Way to Estimate Orders in Time Series,” Journal of Time Series Analysis, 15, 545-559 (1994).
【40】 潘浙楠、林明毅，在製程平均微量變動下管制圖正確選用之探討與研究，品質學報，第六卷，1-28頁，(1999)。
【41】 潘浙楠、陳必達，自我相關環保管制圖的比較研究－以台北地區空氣污染資料為例，中國統計學報，第42卷，31-62頁，(2004)。
【42】 林茂文，時間數列分析與預測－增訂版，華泰書局，(1992)。