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研究生:王琪玲
研究生(外文):Chi-Ling Wang
論文名稱:I(d)過程的統計管制圖
論文名稱(外文):Statistical Control Charts of I(d) processes
指導教授:郭美惠郭美惠引用關係
指導教授(外文):Mei-Hui Guo
學位類別:碩士
校院名稱:國立中山大學
系所名稱:應用數學系研究所
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2002
畢業學年度:90
語文別:英文
論文頁數:77
中文關鍵詞:EWMA管制圖長相關過程EWRMS管制圖長記憶過程
外文關鍵詞:ARFIMA processlong-memory processEWRMSI(d) processEWMA
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本文主要在建立監控長相關過程 ( I(d) process ) 的管制圖,包含EWMA及EWRMS管制圖。論文中推導出與超幾何函數相關的EWMA準確(exact)漸近管制線。對於EWRMS則推導出Box及Pearson的逼近管制界線。並且利用蒙地卡羅模擬方法探討此管制圖的偵錯能力,如其ARL值。此管制圖可廣泛應用在具有長相關特性過程的監控,例如:河流年最低水位、空氣污染、臭氧含量監控等。


Long range dependent processes occur in many fields, it is important to monitor these processes to early detect their shifts. This paper considers the problem of detecting changes in an I(d) process or an ARFIMA(p,d,q) process by statistical control charts. The control limits of EWMA and EWRMS control charts of I(d) processes are established and analytic forms are derived. The average run lengths of these control charts are estimated by Monte Carlo simulations. In addition, we illustrate the performance of the control charts by empirical examples of I(d) processes and ARFIMA(1,d,1) processes.


Contents
1 Introduction 1
2 Preliminaries 3
2.1 The ARFIMA(p; d; q) Process . . . . . . . . . . . . . . . . 3
2.2 Generalized Hypergeometric Function . . . . . . . . . . . . 5
2.3 Control Charts . . . . . . . . . . . . . . . . . . . . . . . . 6
3 Control Charts for I(d) Processes 9
3.1 EWMA Control Chart
(The Exponentially Weighted Moving Average Control Chart) 9
3.2 EWRMS Control Chart
(Expontentially Weighted Root Mean Square) . . . . . . . 12
3.3 Control Charts for ARFIMA(p; d; q) Processes . . . . . . . 17
4 Simulation Results 18
4.1 Detecting Mean Shift . . . . . . . . . . . . . . . . . . . . . 18
4.2 Detecting Variance Change . . . . . . . . . . . . . . . . . . 19
4.3 Pre-I(d)ing Procedure . . . . . . . . . . . . . . . . . . . . 20
5 Examples 22
5.1 The Nile River's Annual Lowest Water Level . . . . . . . 22
5.2 Monitoring Air Pollution . . . . . . . . . . . . . . . . . . . 23
5.3 Monitoring Ozone . . . . . . . . . . . . . . . . . . . . . . . 24
6 Conclusions and Further Study 27
A Appendix A 30
B Appendix B 34
B.1 Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
B.2 Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56


References[1] Montogmery, D.C. (1997) Introduction to statistical quality control. John Wiley &Sons, Inc.[2] Imhof, J.P.(1961) Computing the distribution of quadratic forms in normal variables.Biometrika 148, pp.419-426.[3] Pearson, E.S. Note on an approximation to the distribution of non-central ˜ 2 .Biometrika 146, pp.364.[4] Anderson, T.W. (1971) The statistical analysis of time series. John Wiley & Sons,Inc.[5] Box, G.E.P. (1953) Some theorems on quadratic forms applied in the study of analysisof variance problems. (I) E ect of inequallity ofvariance in the one-way classi cation.Annals of Mathematical statistic 125, pp.290-302.[6] Macgregor, J.F. & Harris, T.J. (1993) The exponentially weighted moving variance.Journal of Quality Technology 25, pp.106-117.[7] Seaborn, J.B. (1991) Hypergeometric functions and their applications. Springer-Verlag New York, Inc.[8] Brockwell, P.J. & Davis, R.A. (1991) Time Series: Theory and Methods. Springer-Verlag New York, Inc.[9] Wei, William W.S. (1994) Time series analysis. Addison-Wesley, Inc.[10] Lu, C.W. & Reynolds, M.R., JR. (1993) Control charts for monitoring the mean andvariance of autocorrelated processes. Journal of Quality Technology 31, pp.259-274.[11] Lu, C.W. & Reynolds, M.R., JR. (1993) EWMA control charts for monitoring themean of autocorrelated processes. Journal of Quality Technology 31, pp.166-188.

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