1. 1 Literature Review on Profile Monitoring Jyh-Jen Horng Shiau Institute of Statistics National Chiao Tung University (交通大學統計所 洪志真 ) Sept. 25, 2009 NCTS Industrial Statistics Research Group Seminar

2. 2 Outline Introduction Linear Profile Monitoring Multiple and Polynomial Regression Binary Responses Multivariate Linear Profiles Autocorrelated Profiles Nonlinear Profile Monitoring Mixed Models Use of Wavelets and Other Nonparametric Methods

3. 3 Introduction

4. 4 Profile Monitoring Briefly summarizing, Woodall et al. (2004) highlighted the following important issues when monitoring profiles: 1. the usefulness of carefully distinguishing between Phase I and Phase II applications, 2. the decision regarding whether or not to include some between profile variation in common cause variation, 3. the use of methods capable of detecting any types of shifts in the shape of the profile, and 4. the use of the simplest adequate profile model.

5. 5 Literature Review A literature survey and areas for future research on profile monitoring can be found in Woodall et al. (2004) and Woodall (2007). Woodall, W.H, Spitzner, D.J., Montgomery, D.C., and Gupta, S. (2004). “Using Control Charts to Monitor Process and Product Quality Profiles.” J. Qual. Technol., vol. 36, pp. 309-320. Woodall, W.H. (2007). “Current Research on Profile Monitoring.” Revista Producao, 17, 420-425.

6. 6 Two Phases for Linear Profile Monitoring Simple Linear Profile Monitoring Linear Profiles with Correlation Binary Response

7. 7 Phase I Monitoring for Linear Profile The purpose of phase I analysis is to evaluate the stability of a process and to estimate process parameters. Phase I monitoring of simple linear profiles: Stover, F. S., and Brill, R. V. (1998). “Statistical quality control applied to ionchromatography calibrations”. Journal of Chromatography A, 804(1– 2),37–43. Mestek, O., Pavlik, J., and Suchánek, M. (1994). “Multivariate Control Charts: Control Charts for Calibration Curves.” Fresenius J. Anal. Chem., 350, 344-351. Kang, L., and Albin, S.L. (2000). “On-Line Monitoring When the Process Yields a Linear Profile.“ J. Qual. Technol., 32, 418-426.

8. 8 Phase I Monitoring for Linear Profile Kim, K., Mahmoud, M.A., Woodall, W.H. (2003). “On the Monitoring of Linear Profiles.” J. Qual. Technol., 35, 317-328. Mahmoud, M.A., and Woodall, W.H. (2004). “Phase I Monitoring of Linear Profiles with Calibration Application.” Technometrics, 46, 380-391. Mahmoud, M.A., Parker, P.A., Woodall, W. H., and Hawkins, D. M. (2007). “A Change Point Method for Linear Profile Data.” Qual Reliab Eng Int., 23, 247-268.

9. 9 In phase II analysis, we are interested in detecting shifts in the process parameters as soon as possible. Phase II monitoring of simple linear profiles: Kang and Albin (2000), Kim et al. (2003), Noorossana, R., Amiri, A., Vaghefi, A., and Roghanian, E. (2004). “Monitoring Quality Characteristics Using Linear Profile”. Proc. 3rd Conf. International Industrial Engineering, Tehran, Iran Gupta, S., Montgomery, D.C., Woodall, W.H. (2006). “Performance Evaluation of Two Methods for Online Monitoring of Linear Calibration Profiles.“ Int. J. Prod. Res., 44, 1927-1942 Phase II Monitoring for Linear Profile

10. 10 Phase II Monitoring for Linear Profile Zou, C., Zhang, Y., and Wang, Z. (2006). “Control Chart Based on Change-Point Model for Monitoring Linear Profiles”. IIE Transactions, 38, 1093-1103. Zou, C., Zhou, C.,Wang, X. and Tsung, F. (2007). ” A self-starting control chart for linear profiles”. Journal of Quality Technology, 39, 364-375. Noorossana R, Amiri A (2007). “Enhancement of linear profiles monitoring in phase II”. AmirKabir J Sci Tech,18, 19–27 in Farsi Saghaei, A., Mehrjoo, M. and Amiri, A. (2009). “A CUSUM-based method for monitoring simple linear profiles”. The International Journal of Advanced Manufacturing Technology.

11. 11 Linear Profile Monitoring Mahmoud and Woodall (2004) phase I monitoring with calibration applications the use of a univariate control chart to monitor error standard deviation in conjunction with a global F-test to monitor the regression coefficients in Phase I.

12. 12 Linear Profile Monitoring Mahmoud et al. (2007) (Phase I) Zou et al. (2006) (Phase II) their proposed methods based on likelihood ratio statistics based on the change-point model for monitoring the linear profiles particularly when the process parameters are all unknown.

13. 13 Linear Profile Monitoring Zou, Zhou, Wang, and Tsung (2007) proposed a self starting Phase II control chart for monitoring linear profiles when the process parameters are not known or cannot be reasonably estimated due to lack of large Phase I samples.

14. 14 Linear Profile Monitoring MCUSUM procedure for phase II monitoring of simple linear profiles. Noorossana et al. (2004) proposed MCUSUM procedure in combination with R chart Noorossana and Amiri (2007) proposed MCUSUM procedure in combination with a χ2 control chart

15. 15 Linear Profile Monitoring A standard assumption in the monitoring of simple linear regression profiles is that the errors are independent and identically distributed, usually with an assumed normal distribution. Noorossana et al. (2004) studied the effect of non- normality of the error terms, using t-distributions as alternatives, on the performance of the EWMA/R method of Kang and Albin (2000). Noorossana, R., Vaghefi, S.A., and Amiri, A. (2004). “Effect of Non-normality on the Monitoring of Linear Profiles”. Proceedings of the 2nd International Industrial Engineering Conference, Riyadh, Saudi Arabia.

16. 16 Linear Profile Monitoring Autocorrelated errors frequently occur within profile monitoring data because successive measurements are often made close in space and/or time. Noorossana and Soleimani (2007) studied the effect of ignoring autocorrelation of the errors within profiles. Soleimani et al. (2009) investigated linear profiles with within profile autocorrelation. Transform data first to eliminate autocorrelation, then apply methods in Kang and Albin (2000) and Kim et al (2003) Noorossana et al. (2008) investigated the effect of the first order autoregressive autocorrelation between linear profiles over time. They proposed three methods based on a time series approach to eliminate the autocorrelation effect.

17. 17 Noorossana, R. and Soleimani, P. (2007) “Effect of Within Profile Autocorrelation on the Performance of Linear Profiles. To appear in the Proceedings of the 5 th International Industrial Engineering Conference, Tehran, Iran (in Farsi), Noorossana, R., Amiri, A., and Soleimani, P. (2008). “On the Monitoring of Autocorrelated Linear Profiles.” Commun. Statist-Theory Meth., 37, 425.442 (Between) Soleimani, P., Noorossana, R., and Amiri, A., (2009) “Simple Linear Profiles Monitoring in the Presence of Within Profile Autocorrelation” Computers & Industrial Engineering, 57, 1015-1021 Linear Profile Monitoring

18. 18 Multiple and Polynomial Regression Profile Monitoring Zou, Wang, and Tsung (2007) proposed a multivariate EWMA (MEWMA) control chart based on likelihood ratio statistics for monitoring phase II general linear profiles that included both a polynomial regression and a multiple linear regression relationship. They considered diagnostic methods after an out-of-control signal, including a change- point approach.

19. 19 Multiple and Polynomial Regression Profile Monitoring Noorossana, Amiri, and Soleimani (2008) and Kazemzadeh, Noorossana, and Amiri (2007, 2009) considered linear and polynomial profiles over time and modeled autocorrelation between profiles as a first order autoregressive process, respectively. It is assumed that due to time collapse between profiles, the corresponding observations in two successive profiles are autocorrelated. Based on a time series approach.

20. 20 Multiple and Polynomial Regression Profile Monitoring Zou C, Tsung F, and Wang Z (2007). “Monitoring general linear profiles using multivariate exponentially weighted moving average schemes”. Technometrics, 49, 395–408 Kazemzadeh, R. B., Noorossana, R., and Amiri, A. (2007). “Statistical monitoring of autocorrelated polynomial profiles”. In Proceedings of the 9th Islamic Countries Conference on Statistical Sciences. [Phase II ] Kazemzadeh, R. B., Noorossana, R., and Amiri, A. (2008a). Phase I monitoring of polynomial profiles. Communications in Statistics-Theory and Methods, 37, 1671–1686. Kazemzadeh, R. B., Noorossana, R., & Amiri, A. (2008b). “Monitoring polynomial profiles in quality control applications”. The International Journal of Advanced Manufacturing Technology. [Phase II ] Kazemzadeh, R. B., Noorossana, R., and Amiri, A. (2009), “Phase II monitoring of autocorrelated polynomial profiles in AR(1) processes”. To appear in The International Journal of Science and Technology. Scientia Iranica.

21. 21 Multivariate Linear Profiles In other practical applications, two or more profiles are required in order to model the quality of a product or process effectively. Several response variables explained by several explanatory or independent variables -- using multivariate regression Saghaei et al. (2008) proposed the use of a control chart scheme in phase II monitoring of multivariate linear profiles. Saghaei, A., Noorossana, R., Eyvazian, M., and Vaghefi, A. (2008), “Statistical Monitoring of Multivariate Linear Profiles”, Proceedings of the IEEE IEEM, Singapore.

22. 22 Profile Monitoring - Binary Data Yeh et al. (2009) studied how the profile functional relationship between the response and predictor variables can be monitored when the response variable of interest is binary using logistic regression (model the link function by multiple regression ). Phase I profile monitoring Yeh, A. B., Huwang, L., and Li, Y.-M. (2009). “Profile monitoring for a binary response”. IIE Transactions, 41, 931 – 941.

23. 23 Nonlinear Profile Monitoring There have also been several examples in which quality of a product or process is characterized by nonlinear relationships between two or more variables. Brill (2001) presented an example based on a chemical property referred to as UMPH. Walker and Wright (2002) used density profiles of engineered wood boards to illustrate their approach. Williams et al. (2007) fit a nonlinear regression to model the vertical density profiles (VDP) data. They also presented the estimated dose-response curve of a manufactured drug.

24. 24 Nonlinear Profile Monitoring Williams et al. (2007) extended the use of control charts based on three different formulations of the T 2 statistics to monitor the coefficients resulting from a nonlinear regression model fit to profile data for Phase I application.

25. 25 Nonlinear Profile Monitoring Ding et al. (2006) used several types of data reduction methods for nonlinear profile data, in particular principal component analysis (PCA) and independent component analysis (ICA). They also studied methods for clustering analysis for profile data.

26. 26 Nonlinear Profile Monitoring Moguerza et al. (2007) used support vector machines (SVM) to monitor the fitted curves themselves instead of monitoring the parameters of models fitting the curves. Chen and Nembhard (2007) proposed a high- dimensional control chart approach to profile monitoring based on the adaptive Neyman test statistic for the coefficients of discrete Fourier transforms of profiles.

27. 27 Nonlinear Profile Monitoring Walker, E., Wright, S. P. (2002). “Comparing curves using additive models”. J. Qual. Technol., 34,118–129. Ding, Y., Zeng, L., and Zhou, S. (2006). Phase I analysis for monitoring nonlinear profiles in manufacturing processes. Journal of Quality Technology, 38, 199–216. Moguerza, J. M., Munoz, A., and Psarakis, S. (2007). “Monitoring Nonlinear Profiles using Support Vector Machines”. Lecture Notes in Computer Science, 4756, p. 574- 583. The publisher is Springer-Verlag. Williams, J.D., Birch, J.B., Woodall, W.H. and Ferry, N.M. (2007). Statistical monitoring of heteroscedastic dose- response profiles from high-throughput screening. Preprint, Journal of Agricultural, Biological and Environmental Statistics, 12, 216–235.

28. 28 Nonlinear Profile Monitoring Chen, S. and Nembhard, H. B. (2007). “A High- Dimensional Control Chart for Profile Monitoring”. Resubmitted to Technometrics. Williams, J. D., Woodall, W. H., and Birch, J. B. (2007). “Statistical Monitoring of Nonlinear Product and Process Quality Profiles.” Qual Reliab Eng Int., 23, 925-941, Moguerza, J. M., Munoz, A., & Psarakis, S. (2007). Monitoring nonlinear profiles using support vector machines. Lecture Notes in Computer Science, 4789, Jensen, W.A., Birch, J.B., Woodall, W.H. (2008) “Monitoring Correlation within Linear Profiles Using Mixed Models”. Journal of Quality Technology, 40, 167-183. Vaghefi, A., Tajbakhsh, S. D., & Noorossana, R. (2009). Phase II monitoring of nonlinear profiles. Communications in Statistics: Theory and Methods, 38, 1834-1851.

29. 29 Mixed Model Mosesova et al. (2007) proposed a mixed model approach with an example from the automotive industry. Jensen et al. (2008) proposed linear mixed models (LMM) to monitor the linear profiles in an attempt to account for the autocorrelation within a linear profile, especially when the data are unbalanced or when there are missing data. (Phase I monitoring) Jensen and Birch (2009) proposed use of nonlinear mixed model (NMM) to monitor the nonlinear profiles in order to account for the correlation structure. These authors discussed the issue of aligning the curves prior to analysis. This step is needed in some applications.

30. 30 Mixed Model Mosesova, S. A., Chipman, H. A., Mackay, R. J.,; Steiner, S. H. (2007). Profile Monitoring Using Mixed-Effects Models. Submitted for publication. Jensen, W. A., Birch, J. B., and Woodall, W. H. (2008). Monitoring correlation within linear profiles using mixed models. Journal of Quality Technology, 40, 167–183. Jensen, W. A., and Birch, J. B. (2009). Profile monitoring via nonlinear mixed model. Journal of Quality Technology, 41, 18–34.

31. 31 Use of Wavelets Wavelets continue to be a popular way to represent profile data when simple models cannot adequately represent the shape of the profile. Jin and Shi (2001), Reis and Saraiva (2006), Zhou, Sun and Shi (2007), Jeong, Lu and Wang (2006), and Chicken and Pignatiello (2007) proposed wavelet-based methods.

32. 32 Use of Wavelets Wavelet methods are usually recommended when the shape of the profiles is too complicated for linear and nonlinear models to work well. As pointed out by Woodall et al. (2004), however, monitoring only a subset of the most significant wavelet coefficients based on in-control data can be dangerous in the sense that some out-of-control function changes will not be detectable.

33. 33 Use of Wavelets Jin, J., & Shi, J. (1999). Feature-preserving data compression of stamping tonnage information using wavelets. Technometrics, 41, 327–339. Jin, J. and Shi, J. (2001). Automatic feature extraction of waveform signals for in-process diagnostic performance improvement. J. Intell. Manuf., 12, 140-145. Reis, M. S. and Saraiva, P. M. (2006). Multiscale Statistical Process Control of Paper Surface Profiles. Quality Technology and Quantitative Management, 3, 263-282. Zhou, S. Y., Sun, B. C. and Shi, J. J. (2006). An SPC Monitoring System for Cycle-based Waveform Signals using Haar Transform. IEEE Transactions on Automation Science and Engineering, 3, 60- 72. Jeong, M. K., Lu, J. C. and Wang, N. (2006). Wavelet-based SPC Procedure for Complicated Functional Data. International Journal of Production Research, 44, 729-744. Chicken, E. and Pignatiello, J. J. JR., and Simpson, J. (2009). Statistical Process Monitoring of Nonlinear Profiles Using Wavelets. Journal of Quality Technology, 41, 198-212.

34. 34 B-Spline B-spline plays the role to define the shape of a profile Bernadette and Jonathan (2005) review three approaches for analyzing the results of experimental design when the response is a curve. 1.parametric nonlinear pre-processing, 2.pointwise functional regression; 3. B-spline transformation pre-processing. Chang and Yadama (2008) further combine wavelet transformation and B-spline for nonlinear profile monitoring.

35. 35 B-Spline Chang and Chou (2009) Studying the effectiveness of wavelet transformation in nonlinear profiles monitoring. Wavelet filtering is applied to separate signals containing shape information from noise of a nonlinear profile. B-spline approximation is applied to the filtered profile to reduce dimensionality. Examining the performance of B-spline fitting with and without wavelet filtering.

36. 36 B-Spline Bernadette, G. and Jonathan, N. (2005). Analysing the Results of a Designed Experiment when the Response is a Curve: Methodology and Application in Metal Injection Moulding. Quality and Reliability Engineering International, 21, 509-520. Chang, S.I. and Yadama, S. (2008). Statistical Process Control for Monitoring Nonlinear Profiles Using Wavelet Filtering and B-Spline Approximation. International Journal of Production Research. Chang, S. and Chou, S.-H. (2009). A Study of Using Wavelet Transformation and B-Spline Approximation for Nonlinear Profiles Monitoring. Proceedings of the 2009 Industrial Engineering Research Conference.

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62. 62 Phase I Monitoring of Nonlinear Profiles James D. Williams, William H. Woodall and Jeffrey B. Birch, 2003 Method 1 (sample covariance matrix) does not take into account the sequential sampling structure of the data: The overall probability of detecting a shift in the mean vector will decrease (See Sullivan and Woodall, 1996) Should not be used Method 2 (successive differences) accounts for the sequential sampling scheme, and gives a more robust estimate of the covariance matrix In the VDP example, both Methods 1 and 2 gave same result because No apparent shift in the mean vector There were only about two outliers Method 3 (intra-profile pooling) should be used when there is no profile- to-profile common cause variability Comparison of the three methods: Method 1 assumes all variability is due to common cause Method 3 assumes that no variability is due to common cause Method 2 is somewhere in the middle

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