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Identification of the change point: an overview

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Abstract

When a control chart addresses an out-of-control condition, a root-cause analysis should be started to identify and eliminate the special cause(s) of variation manifested in the process. The change point refers to the time when a special cause(s) takes place in the process and leads it to a departure from the in-control condition to an out-of-control condition. Identification of the change point is considered as an essential step for a root-cause analysis in both univariate and multivariate processes. If a change manifests in a normally distributed process mean, variance, or both, then the change point should be identified in the process mean, variance, or both, respectively. This paper attempts to comprehensively review the researches that considered the mean change point in different environment corresponding to univariate and multivariate normal processes.

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References

  1. Pyzdek T (2003) Quality engineering handbook, 2nd edn. Marcel Dekker, Basel

    Google Scholar 

  2. Shewhart WA (1931) Economic control of quality of manufactured product. ASQ, Milwaukee, 1980

    Google Scholar 

  3. Manual 7A (2002) Manual on the presentation of data and control chart analysis. Seventh edition

  4. American National Standards Institute (1996) Bl–B3, Guide for quality control charts, control chart method of analyzing data, control chart method for controlling quality during production

  5. Nelson LS (1984) The Shewhart control chart—test for special causes. J Qual Technol 16:237–239

    Google Scholar 

  6. Nelson LS (1985) Interpreting Shewhart X bar control charts. J Qual Technol 17:114–116

    Google Scholar 

  7. Roberts SW (1985) Properties of control chart zone test. Bell Syst Tech J 37:84–114

    Google Scholar 

  8. Duncan AJ (1986) Quality control and industrial statistics, 5th edn. Irwin, Illinois

    Google Scholar 

  9. Grant E. L. (1988) Leavenworth R.S. statistical quality control. 6th Ed., McGraw-Hill Book Company, New York, NY

  10. Page ES (1954) Continuous inspection schemes. Biometrika 41:100–114

    MathSciNet  MATH  Google Scholar 

  11. Lucas JM (1973) A modified V mask control scheme. Technometrics 15:833–847

    Google Scholar 

  12. Lucas JM, Crosier RB (1982) Fast initial response for CUSUM quality control schemes: give your CUSUM a head start. Technometrics 24:199–205

    Article  Google Scholar 

  13. Roberts SW (1959) Control chart tests based on geometric moving averages. Technometrics 1:239–250

    Article  Google Scholar 

  14. Lucas JM, Saccucci MS (1990) Exponentially weighted moving average control schemes: properties and enhancements. Technometrics 32:1–12

    Article  MathSciNet  Google Scholar 

  15. Montgomery DC (2005) Introduction to statistical quality control. Hoboken, N.J. Wiley

  16. Velasco T, Rowe MR (1993) Back propagation artificial neural networks for the analysis of quality control charts. Comput Ind Eng 25(1–4):397–400

    Article  Google Scholar 

  17. Hwarng HB, Hubele NF (1993) Back-propagation pattern recognizer for X bar control charts: methodology and performance. Comput Ind Eng 24(2):219–235

    Article  Google Scholar 

  18. Hwarng HB, Hubele NF (1993) X bar control chart patter identification through efficient off-line neural network training. IIE Trans 25(3):27–40

    Article  Google Scholar 

  19. Cheng C (1995) A multi-layer neural network model for detecting changes in the process mean. Comput Ind Eng 28(1):51–61

    Article  Google Scholar 

  20. Cheng CS (1997) A neural network approach for the analysis of control chart patterns. Int J Prod Res 35(3):667–697

    Article  MATH  Google Scholar 

  21. Chang SI, Aw CA (1996) A neural fuzzy control chart for detecting and classifying process mean shifts. Int J Prod Res 34(8):2265–2278

    Article  MATH  Google Scholar 

  22. Guh RS, Tannock TD (1997) A neural Network approach to characterize pattern parameters in process control charts. J Intell Manuf 10(5):449–462

    Article  Google Scholar 

  23. Noorossana R, Farrokhi M, Saghaei A (2003) Using neural networks to detect and classify out-of-control signals in autocorrelated processes. Qual Reliab Eng Int 19(6):493–504

    Article  Google Scholar 

  24. Hotelling H (1947) In: Eisenhart C, Hastay MW, Wallis WA (eds) Multivariate quality control—illustrated by the air testing of sample bombsights. Techniques of statistical analysis. McGraw-Hill, New York

    Google Scholar 

  25. Woodall WH, Ncube MM (1985) Multivariate CUSUM quality-control procedures. Technometrics 27(3):285–292

    Article  MathSciNet  MATH  Google Scholar 

  26. Healy JD (1987) A note on multivariate CUSUM procedures. Technometrics 29(4):409–412

    Article  Google Scholar 

  27. Crosier RB (1988) Multivariate generalization of cumulative sum quality control schemes. Technometrics 30(3):291–302

    Article  MathSciNet  MATH  Google Scholar 

  28. Pignatiello JJ, Runger GC (1990) Comparisons of multivariate CUSUM charts. J Qual Technol 22(3):173–186

    Google Scholar 

  29. Ngai HM, Zhang J (2001) Multivariate cumulative sum control charts based on projection pursuit. Stat Sin 11(3):747–766

    MathSciNet  MATH  Google Scholar 

  30. Chan LK, Zhang J (2001) Cumulative sum control charts for the covariance matrix. Stat Sin 11(3):767–790

    MathSciNet  MATH  Google Scholar 

  31. Qiu PH, Hawkins DM (2001) A rank-based multivariate CUSUM procedure. Technometrics 43(2):120–132

    Article  MathSciNet  Google Scholar 

  32. Qiu PH, Hawkins DM (2003) A nonparametric multivariate cumulative sum procedure for detecting shifts in all directions. J R Stat Soc Ser D-The Statistician 52(2):151–164

    Article  MathSciNet  Google Scholar 

  33. Runger GC, Testik MC (2004) Multivariate extensions to cumulative sum control charts. Qual Reliab Eng Int 20(6):587–606

    Article  Google Scholar 

  34. Lowry CA, Woodall WH, Champ CW, Rigdon SE (1992) A multivariate exponential weighted moving average control chart. Technometrics 34(1):46–53

    Article  MATH  Google Scholar 

  35. Rigdon SE (1995) An integral equation for the in-control average run length of a multivariate exponentially weighted moving average control chart. Journal of Statistical Computations and Simulation 52(4):351–365

    Article  MathSciNet  MATH  Google Scholar 

  36. Yumin L (1996) An improvement for MEWMA in multivariate process control computers and industrial engineering. 31(3–4): 779–781

  37. Runger GC, Prabhu SS (1996) A Markov chain model for the multivariate exponentially weighted moving average control chart. J Am Stat Assoc 91(436):1701–1706

    Article  MathSciNet  MATH  Google Scholar 

  38. Kramer HG, Schmid W (1997) EWMA charts for multivariate time series. Seq Anal 16(2):131–154

    Article  MathSciNet  MATH  Google Scholar 

  39. Prabhu SS, Runger GC (1997) Designing a multivariate EWMA control chart. J Qual Technol 29(1):8–15

    Google Scholar 

  40. Fasso A (1999) One-sided MEWMA control charts. Communications in statistics -theory and methods. 28(2): 381–401

  41. Borror CM, Montgomery DC, Runger GC (1990) Robustness of the EWMA control chart to non-normality. J Qual Technol 31(3):309–316

    Google Scholar 

  42. Runger GC, Keats JB, Montgomery DC, Scranton RD (1999) Improving the performance of a multivariate exponentially weighted moving average control chart. Qual Reliab Eng Int 15(3):161–166

    Article  Google Scholar 

  43. Tseng S, Chou R, Lee S (2002) A study on a multivariate EWMA controller. IIE Trans 34(6):541–549

    Google Scholar 

  44. Yeh AB, Lin DKJ, Zhou H, Venkataramani C (2003) A multivariate exponentially weighted moving average control chart for monitoring process variability. J Appl Stat 30(5):507–536

    Article  MathSciNet  MATH  Google Scholar 

  45. Testik MC, Runger GC, Borror CM (2003) Robustness properties of multivariate EWMA control charts. Qual Reliab Eng Int 19(1):31–38

    Article  Google Scholar 

  46. Testik MC, Borror CM (2004) Design strategies for the multivariate exponentially weighted moving average control chart. Qual Reliab Eng Int 20(6):571–577

    Article  Google Scholar 

  47. Chen GM, Cheng SW, Xie HS (2005) A new multivariate control chart for monitoring both location and dispersion. Communications in Statistics - Simulation and Computation 34(1):203–217

    Article  MathSciNet  MATH  Google Scholar 

  48. Zorriassatine F, Tannock JDT, O’Brien C (2003) Using novelty detection to identify abnormalities caused by mean shifts in bivariate processes. Comput Ind Eng 44(3):385–408

    Article  Google Scholar 

  49. Hwarng HB (2004) Detecting process mean shift in the presence of autocorrelation: a neural network based monitoring scheme. Int J Prod Res 42(3):573–595

    Article  MATH  Google Scholar 

  50. Guh RS (2007) On-line identification and quantification of mean shifts in bivariate processes using a neural network-based approach. Qual Reliab Eng Int 23(3):367–385

    Article  Google Scholar 

  51. Hwarng HB (2008) Toward identifying the source of mean shifts in multivariate SPC: a neural network approach. Int J Prod Res 46(20):5531–5559

    Article  MATH  Google Scholar 

  52. Guh RS, Shiue YR (2008) An effective application of decision tree learning for on-line detection of mean shift in multivariate control charts. Comput Ind Eng 55(2):475–493

    Article  Google Scholar 

  53. Shi J, Zhou S (2009) Quality control and improvement for multistage systems: a survey. IIE Trans 41(9):744–753

    Article  Google Scholar 

  54. Liu J (2010) Variation reduction for multistage manufacturing processes: a comparison survey of statistical-process-control vs. stream-of-variation methodologies. Quality and Reliability Engineering International 26(7):645–661

    Article  Google Scholar 

  55. Samuel TR, Pignatiello JJ Jr, Calvin JA (1998) Identifying the time of a step change with X bar control charts. Qual Eng 10(3):521–527

    Article  Google Scholar 

  56. Bassevile M, Nikiforov IV (1993) Detection of abrupt changes: theory and applications. Prentice Hall, New Jersey

    Google Scholar 

  57. Csorgo M, Horvath L (1997) Limit theorem in change-point analysis. Wiley, New York

    Google Scholar 

  58. Pignatiello JJ Jr, Simpson JR (2002) A magnitude-robust control chart for monitoring and estimating step change for normal process means. Qual Reliab Eng Int 18(6):429–441

    Article  Google Scholar 

  59. Pignatiello JJ Jr, Samuel TR (2001) Estimation of the change point of a normal process mean in SPC application. J Qual Technol 33(1):82–95

    Google Scholar 

  60. Noorossana R, Atashgar K, Saghaee A (2011) An integrated solution for monitoring process mean vector. Int J Adv Manuf Technol 56(5):755–765

    Article  Google Scholar 

  61. Perry MB, Pignatiello JJ Jr (2006) Estimation of the change point of a normal process mean with a linear trend distribution in SPC. Qual Technol Qual Manag 3(3):325–334

    MathSciNet  Google Scholar 

  62. Atashgar K, Noorossana R (2011) An integrating approach to root-cause analysis of a bivariate mean vector with a linear trend disturbance. Int J Adv Manuf Technol 52(1):407–420

    Article  Google Scholar 

  63. Perry MB, Pignatiello JJ Jr, Simpson JR (2007) Change point estimation for monotonically changing Poisson rate in SPC. Int J Prod Res 45(8):1791–1813

    Article  Google Scholar 

  64. Perry BM, Pignatiello JJ Jr, Simpson JR (2007) Estimating the change point of the process fraction non-conforming with monotonic change disturbance in SPC. Qual Reliab Eng Int 23(3):327–339

    Article  Google Scholar 

  65. Noorossana R, Shadman A (2009) Estimating the change point of a normal process mean with a monotonic change. Qual Reliab Eng Int 25(1):79–90

    Article  Google Scholar 

  66. Atashgar K, Noorossan R (2010) Identifying change point in a bivariate normal process mean vector with monotonic changes. Int J Ind Syst Eng Prod Manag 21(1):1–13

    Google Scholar 

  67. Woodall WH (2000) Controversies and contradictions in statistical process control. J Qual Technol 32:341–350

    Google Scholar 

  68. Sullivan JH (2002) Detection of multiple change point from clustering individual observations. J Qual Technol 34(4):371–383

    Google Scholar 

  69. Box GEP, Cox DR (1964) An analysis of transformations. J R Stat Soc B26:211–243

    MathSciNet  Google Scholar 

  70. Siegmund D (1986) Boundary crossing probabilities and statistical applications. Ann Stat 14(2):361–404

    Article  MathSciNet  MATH  Google Scholar 

  71. Meeker WQ, Escobar LA (1995) Teaching about approximate confidence region based on maximum likelihood estimation. Am Stat 49:48–53

    Google Scholar 

  72. Page ES (1955) Control charts with warning lines. Biometrika 42:213–254

    MathSciNet  Google Scholar 

  73. Page ES (1955) A test for a change in a parameter occurring at an unknown point. Biometrika 42:523–527

    MathSciNet  MATH  Google Scholar 

  74. Page ES (1957) On problems in which a change in a parameter occurs at an unknown point. Biometrika 44:248–252

    MATH  Google Scholar 

  75. Page ES (1961) Cumulative sum control charts. Technometrics 3:1–9

    Article  MathSciNet  Google Scholar 

  76. Srivastava MS (1994) Comparison of CUSUM and EWMA procedures for detecting a shift in the mean or an increase in the variance. J Appl Stat 1(4):445–468

    MATH  Google Scholar 

  77. Hinkley DV (1970) Inferences about the change-point in a sequence of random variables. Biometrika 57:1–17

    Article  MathSciNet  MATH  Google Scholar 

  78. Hinkley DV, Hinkley EA (1970) Inference about the change-point in a sequence of binomial variables. Biometrika 57(3):477–488

    Article  MathSciNet  MATH  Google Scholar 

  79. Pettitt AN (1979) A simple cumulative sum type statistic for the change-point problem with zero–one observations. Biometrika 67(1):79–84

    Article  MathSciNet  Google Scholar 

  80. Quandt R (1958) The estimation of the parameters of a linear regression system obeying two separating regimes. J Am Stat Assoc 53:873–880

    Article  MathSciNet  MATH  Google Scholar 

  81. Quandt R (1960) Test of hypothesis that a linear regression system obeys two separate requires. J Am Stat Assoc 5(3):324–330

    Article  MathSciNet  Google Scholar 

  82. Quandt R (1972) A new approach to estimating switching regressions. J Am Stat Assoc 67:3116–310

    Article  Google Scholar 

  83. Robertson T, Wright FT, Dykstra RL (1988) Order restricted statistical inference. Wiley, New York

    MATH  Google Scholar 

  84. Ayer M, Brunk HD, Ewing GM, Reid WT, Silverman E (1955) An empirical distribution function for sampling with incomplete information. Ann Math Stat 26:641–647

    Article  MathSciNet  MATH  Google Scholar 

  85. Best MJ, Chakravarti N (1990) Active set algorithms for monotonic regression: a unifying framework. Math Program 47:425–439

    Article  MathSciNet  MATH  Google Scholar 

  86. Grotzinger SJ, Witzgall C (1984) Projections onto order simplexes. Appl Math Optim 12:247–270

    Article  MathSciNet  MATH  Google Scholar 

  87. Hinkley DV (1971) Inferences about the change-point from cumulative sum test. Biometrika 58:1–17

    MathSciNet  Google Scholar 

  88. Nishina K (1992) A comparison of control charts from the viewpoint of change-point estimator. Qual Reliab Eng Int 8(6):537–541

    Article  Google Scholar 

  89. Hawkins DM, Qiu P, Kang CW (2003) The change point model for statistical process control. J Qual Technol 35(4):355–366

    Google Scholar 

  90. Holbert D (1982) A Bayesian analysis of a switching linear model. J Econ 19:77–87

    Google Scholar 

  91. Hartigan JA (1990) Partition model. Communication in Statistics- Theory and Methods 19:2745–2756

    Article  MathSciNet  Google Scholar 

  92. Barry D, Hartigan JA (1993) A Bayesian analysis for change point problem. J Am Stat Assoc 88:309–319

    MathSciNet  MATH  Google Scholar 

  93. Crowley EM (1997) Product partition models for normal means. J Am Stat Assoc 92:192–198

    Article  MATH  Google Scholar 

  94. Quintana FA, Iglesias PL (2003) Bayesian clustering and product partition models. J R Stat Soc Ser B (Stat Methodol) 65:557–574

    Article  MathSciNet  MATH  Google Scholar 

  95. Quintana FA, Iglesias PL, Bolfarine H (2005) Bayesian identification of outliers and change-points in measurement error models. Adv Complex Syst 8:433–449

    Article  MathSciNet  MATH  Google Scholar 

  96. Loschi RH, Cruz FRB (2005) Extension to the product partition model: computing the probability of a change. Comput Stat Data Anal 48:255–268

    Article  MathSciNet  MATH  Google Scholar 

  97. Fearnhead P (2006) Exact and efficient Bayesian inference for multiple changepoint problems. Stat Comput 16:203–213

    Article  MathSciNet  Google Scholar 

  98. Fearnhead P, Liu Z (2007) Online inference for multiple changepoint problems. J R Stat Soc Ser B (Stat Methodol) 69:589–605

    Article  MathSciNet  Google Scholar 

  99. Rosangela H, Loschi JGP, Frederico RBC (2010) Multiple change-point analysis for linear regression models. Chil J Stat 1(2):93–112

    MathSciNet  MATH  Google Scholar 

  100. Turner CDJR, Sullivan JH, Batson RG, Woodall WH (2001) A program for retrospective change-point analysis of individual observations. J Qual Technol 33(2):242–243

    Google Scholar 

  101. McGee VE, Carleton WT (1970) Piecewise regression. J Am Stat Assoc 65:1109–1124

    Article  Google Scholar 

  102. Hawkins DM (1976) Point estimation of the parameters of piecewise regression model. Appl Stat 25:51–57

    Article  MathSciNet  Google Scholar 

  103. Dempster AP, Laird NM, Rubin DB (1977) Maximum likelihood from incomplete data via EM algorithm. J R Stat Soc Ser B (Stat Methodol) 39(1):1–38

    MathSciNet  MATH  Google Scholar 

  104. Vostrikova LJ (1981) Detecting disorder in multidimensional random process. Sov Math Doklady 24:55–59

    MATH  Google Scholar 

  105. Habibi R (2011) Change point detection using bootstrap methods. Advanced Modeling and Optimizing 13(3):341–347

    MathSciNet  Google Scholar 

  106. Ghazanfari M, Alaeddini A, Akhavan Niaki ST, Aryanezhad MB (2008) A clustering approach to identify the time of a step change in Shewhart control charts. Qual Reliab Eng Int 24(7):765–778

    Article  Google Scholar 

  107. Joseph L (1989) The multi-path change-point. PhD Thesis. Department of Mathematics and Statistics. McGill University, Montreal.

  108. Joseph L, Wolfson DB (1992) Estimation in multi-path change-point problems. Communications in Statistics - Theory and Methods 21:897–913

    Article  MATH  Google Scholar 

  109. Joseph L, Wolfson DB (1993) Maximum likelihood estimation in the multi-path problem. Ann Inst Stat Math 45(3):511–530

    Article  MathSciNet  MATH  Google Scholar 

  110. Joseph L, Wolfson DB, due Berger R, Lyle RM (1996) Change-point analysis of a randomized trial on the effects of calcium supplementation on blood pressure. In: Berry DA, Stangi DK (eds) Bayesian biostatistics. Marcel Dekker, New York

    Google Scholar 

  111. Joseph L, Woifson DB (1996) Estimation in the multi-path change-point problem for correlated data. Can J Stat 24(1):37–53

    Article  MATH  Google Scholar 

  112. Joseph L, Wolfson DB (1997) Analysis of panel data with change-points. Stat Sin 7(3):687–703

    MATH  Google Scholar 

  113. Asgharian M, Wolfson DB (2001) Covariates in multipath change-point problems: modelling and consistency of the MLE. Can J Stat 29(4):515–528

    Article  MathSciNet  MATH  Google Scholar 

  114. Nedumaran G, Pignatiello JJ Jr, Calvin JA (1998) Estimation of the time of a step-change with χ 2 control chart. Qual Eng 13(2):765–778

    Google Scholar 

  115. Noorossana R, Arbabzadeh N, Saghaei A (2008) Painabar K. Development of procedure of detection change point in multi environment. 6th International Conference on Industrial Engineering, Iran-Tehran. (Written in Persian language)

  116. Arbabzadeh N (2008) Estimating of the change point of a multivariate normal process. Thesis of degree of Master of science in industrial engineering. Iran University of Science and Technology. Industrial Engineering Faculty

  117. Zamba KD, Hawkins DM (2006) A multivariate change-point model for statistical process control. Technometrics 48(4):539–549

    Article  MathSciNet  Google Scholar 

  118. Hawkins DM, Qiu P (2003) The changepoint model for statistical process control. J Qual Technol 35(4):355–366

    Google Scholar 

  119. Li F, Runger GC, EUGENE TUV (2006) Supervised learning for change point detection. Int J Prod Res 44(14):2853–2868

    Article  MATH  Google Scholar 

  120. Ahmadzadeh F, Noorossana R (2008) Identifying the time of a step change with MEWMA control charts by artificial neural network. The 8th International Industrial Engineering and Engineering Management Conference, Singapore.

  121. Ahmadzadeh F (2009) Change point detection with multivariate control charts by artificial neural network. Int J Adv Manuf Technol. doi:10.1007/s00170-009-2193-6

  122. Blazek LW, Novic B, Scott MD (1987) Displaying multivariate data using polyplots. J Qual Technol 19(2):69–74

    Google Scholar 

  123. Jackson JE (1991) A user’s guide to principle components. Wiley, New York

    Book  Google Scholar 

  124. Fuchs C, Benjamin Y (1994) Multivariate profile charts for statistical process control. Technometrics 36(2):182–195

    Article  MATH  Google Scholar 

  125. Subramanyam N, Houshmand AA (1995) Simultaneous representation of multivariate and corresponding univariate x charts using a line graph. Qual Eng 7(4):681–692

    Article  Google Scholar 

  126. Mason Robert L, Tracy Nola D, Young JC (1995) Decomposition of T 2 for multivariate control chart interpretation. J Qual Technol 27(2):109–119

    Google Scholar 

  127. Mason Robert L, Tracy Nola D, Young John C (1997) A practical approach for interpreting multivariate T 2 control chart signals. J Qual Technol 29(4):396–406

    Google Scholar 

  128. Atienza OO, Ching LT, Wah BA (1998) Simultaneous monitoring of univariate and multivariate SPC information using boxplots. Int J Qual Sci 3(2):194–204

    Article  Google Scholar 

  129. Maravelakis PE, Bersimis S, Panaretos J, Psarakis S (2002) Identifying the out of control variable in a multivariate control chart. Commun Stat 31(12):2391–2408

    Article  MathSciNet  MATH  Google Scholar 

  130. Niaki STA, Abbasi B (2005) Fault diagnosis in multivariate control charts using artificial neural network. Qual Reliab Eng Int 21(8):825–840

    Article  Google Scholar 

  131. Aparisi F, Avendano G, Sanz J (2006) Techniques to interpret T 2 control chart signals. IIE Transaction 38(8):647–657

    Article  Google Scholar 

  132. Atashgar K, Noorossana R (2011) Diagnosing the source(s) of a monotonic change in the process mean vector. Int J Adv Manuf Technol. doi:10.1007/s00170-011-3654-2

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Atashgar, K. Identification of the change point: an overview. Int J Adv Manuf Technol 64, 1663–1683 (2013). https://doi.org/10.1007/s00170-012-4131-2

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