Phase II Nonparametric Profile Monitoring and Decision Making on Process Quality via a Mixed Model

Document Type : Research Paper


1 Industrial Engineering Department, Islamic Azad University, South Campus, Tehran, Iran

2 Industrial Engineering Department, Iran University of Science and Technology, Tehran 16844, Iran


In many statistical process control applications, the quality of a process is characterized by a profile. A profile is a function in terms of one or more explanatory variables. In profile monitoring, one is interested to monitor the performance of a process or product using this functional relationship. Control charts for monitoring nonparametric profiles are useful when the relationship is too complex to be described parametrically. Most of the existing control charts in the literature are suitable for monitoring parametric profiles. This article focuses on nonparametric profile monitoring when within-profile autocorrelation is present. Our proposed phase II control chart considers mixed-effect model and uses the framework of a general smoothing spline analysis of variance (SS-ANOVA) along with Hoteling T2 control scheme. The proposed method is especially suitable for categorical data. Numerical results show that the proposed method is capable of detecting profile shifts and identifying the exact location of problematic segments.


Main Subjects

[1] Amiri, A., Jensen, W.A., Kazemzadeh, R.B., (2010)“ A Case Study on Monitoring Polynomial Profiles in the Automotive Industry.” Quality and Reliability Engineering International, 26, 2010, 509-520.
[2] Yeh, A.B., Huwang, L., Li, Y.M., (2009) “Profile Monitoring for a Binary Response.” IIE Transactions, 41,  pp. 931 – 941.
[3] Noorossana, R., Izadbakhsh, H., “ Profile Monitoring for Multinomial Responses” (In Farsi). International Journal of Industerial Engineering and Production Management, in press.
[4] Noorossana, R., Eyvazian, M., Amiri, A., Mahmoud, M.A., (2010) “ Statistical Monitoring of Multivariate Multiple Linear Regression Profiles in Phase I with Calibration Application.” Quality and Reliability Engineering International, 26,  pp. 291-303.
[5] Izadbakhsh, H., Noorossana, R., Saghaei, A., (2010) “Phase II Profile Monitoring for Ordinal Responses.” Advanced Manufacturing Technology, (Under Review), 2012.
[6] Ho, L.L., El, Said M., Kim, R.W., “Monitoring the Parameters of the Market Model by Linear Profile Procedures.” Economic Quality Control, 25, 2010, pp. 81-96.
[7] Zou, C., Ning, X., Tsung, F., (2010)“LASSO-Based Multivariate Linear Profile Monitoring.” Annals of Operations Research, 192, pp. 1-17.
[8] Woodall, W.H., Spitzner, D.J., Montgomery, D.C., Gupta, S., (2004) “Using Control Charts to Monitor Process and Product Quality Profiles.” Journal of Quality Technology, 36, 2004, pp. 309-320
[9] Stover, F. S., and Brill, R. V. (1998), “.Statistical Quality Control Applied to Ion”  Chromatography Calibrations, Journal of Chromatography, A, 804, 37-43.
10] Kang, L., and Albin, S. L. (2000), “ On-Line Monitoring When the Process Yields a Linear Profile.” Journal of Quality Technology, 32, 418-426.
[11] Mahmoud, M. A., and Woodall, W. H. (2004). “Phase I Monitoring of Linear Profiles with Calibration Application.” Technometrics 46. 380-391.
[12] Woodall, W.H. (2007), “.Current Research on Profile Monitoring.” Revista Producão,17, 420-425.
[13] Brill, R.V.,(2001) “A Case Study for Control Charting a Product Quality Measure that is a Continuous Function Over Time.” Presentation at the 45th Annual Fall Technical Conference, Toronto, Ontario,
[14] Walker, E., Wright, S.P., “Comparing Curves Using Additive Models.” J. Qual. Technol, Vol. 34, 2002, pp. 118–129.
[15]Jeong, M. K. ,  Lu, J. C. , Wang, N. (2006). “Wavelet-based SPC procedure for complicated functional data.” International Journal of Production Research, Volume 44, Issue 4 , 729 – 744.
Hotelling, H.H. (1947). “Multivariate Quality Control Illustrated by the Air Testing of Sample Bombsights,” Techniques of Statistical Analysis, 111-184.
[16] Williams, J.D., Woodall, W.H., Birch, J.B., (2007) “Statistical Monitoring of Nonlinear Product and Process Quality Profiles.” Quality and Reliability Engineering International, Vol. 23,  pp. 925–941.
[17]Jin, J. and Shi, J. (2001) “Automatic feature extraction of waveform signals for in-process diagnostic performance improvement.” Journal of Intelligent Manufacturing, 12, 140–145.
[18]Williams, J. D., Woodall, W. H., and Birch, J. B. (2003) “Phase I Monitoring of Nonlinear Profiles.” paper presented at the 2003 Quality and Productivity Research Conference, Yorktown Heights, New York.
[19]Ding, Y., Zeng, L., Zhou, S. (2006)  “Phase I Analysis for Monitoring Nonlinear Profiles in Manufacturing Processes.” Journal of Quality Technology, 38, pp. 199-216.
[20] Jensen, W. A.; Birch, J. B.; and Woodall, W. H. (2006).“Profile Monitoring via Linear Mixed Models.” Technical Report 06-2, Department of Statistics, Virginia Polytechnic Institute & State University.
[21] Peynabar, K., and Jin, J. (2011). “Characterization of non-linear profiles variations using mixed-effect models and wavelet”s.  IIE Transactions, Volume 43, Issue 4 , 275 – 290.
[22] Jensen, W.A. and Birch, J.B. (2009) “ Profile monitoring via nonlinear mixed models.” Journal of Quality Technology, 41, 18–34.
[23] 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.
[24] Laird, N. M., andWare, J. H. (1982), “Random EffectsModels for Longitudinal Data.” Biometrics, 38, 963–974. [266]
[25] Diggle, P. J., Heagerty, P., Liang, K. Y. and Zeger, S. L. (2002). “Analysis of Longitudinal Data.” (2nd ed). Oxford University Press, Oxford, U.K.
[26] Abramovich, F. and Angelini, C. (2006), “Testing in Mixed-Effects FANOVA Models.” Journal of Statis-tical Planning and Inference, 136, 4326-4348.
[27] Gu, C. (2013) “Smoothing Spline ANOVA Models.” New York: Springer.
[28] Antoniadis, A. and Sapatinas, T. (2004). “Estimation and Inference in FunctionalMixed-effects Models.” Technical Report, Department of Mathematics and Statistics,University of Cyprus, Nicosia       
[29] Morris, J. S. and Carroll, R. (2006). “Wavelet-based Functional Mixed Models.” Journal of the Royal Statistical Society: Series B, 68(2), 179-199.
[30] Morris, J. S., Arroyo, C., Coull, B. A., Ryan, L. M. and Gortmaker, S. L. (2006).“Using Wavelet-Based Functional Mixed Models to characterize Population Heterogeneity in Accelerometer Profiles: A Case Study.” Journal of the American StatisticalAssociation, 101(476), 1352-1364.
[31]Qiu, P., Zou, C., & Wang, Z. (2010). “Nonparametric profile monitoring by mixed effects modeling.” (withdiscussions). Technometrics, 52, 265–277.
[32]Qiu, P., & Zou, C. (2010). “Control chart for monitoring nonparametric profiles with arbitrary design.” Statistica Sinica, 1655–1682
[33] Shi, M., Weiss R. E., and Taylor, J. M. G. (1996), “An Analysis of Paediatric CD4 Counts for Acquired Immune Deficiency Syndrome Using Flexible Random Curves.” AppliedStatistics, 45, 151-163.
[34] Rice, J. A. and Wu, C. O., (March 2001), “Nonparametric Mixed Effects Models for Unequally Sampled Noisy Curves.” Biometrics, 57, 253-259.
[36]Abdel-Salam, A.-S., and Birch, J. B. (2009), “Profile-Monitoring Analysis With Fixed and Random Effects Using Nonparametric and Semiparametric Methods.” contributed paper presented at the Joint Statistical Meetings, August 2009, Washington, DC. [286,287]
[37] Wahba, G. (1990) “Spline Models for Observational Data.” Philadelphia: Society for Industrial and Applied Mathematics.
[38] Chen, Z. (1993) Fitting multivariate regression Journal of the Royal Statistical Society. Series B (Methodological),Vol. 55, No. 2 , pp. 473-491
Demidenko, E. (2004), “ Mixed Models: Theory and Applications”. New York: Wiley.
[39]Gu, C., Bates, D. M., Chen, Z. and Wahba, G. (1989) “The computation of GCV functions through Householdertridiagonalization with application to the fitting of interaction spline models.” SIAM J. Matrix Anal. Applic.,10, 457–480
[40] Wahba, G. (1978) “ Improper priors, spline smoothing and the problem of guarding against model errors in regression.” J. R. Statist. Soc. B, 40, 364–372.
[41] Aronszajn, N. (1950) “Theory of reproducing kernels.” Trans. Am. Math. Soc., 68, 337–404.
 [42] Kimeldorf, G. S. and Wahba, G. (1971). “Some results on Tchebycheffian spline functions.” J. Math. Anal. Appl. 33 82-94.
[43] Harville, D. (1976). “Extension of the Gauss-Markov theorem to include the estimation of random effects.” Ann. Statist. 4 384-395.
[44] Hawkins, D. M., and Olwell, D. H. (1998), “Cumulative Sum Charts and Charting for Quality Improvement.” New York: Springer-Verlag.
[45] Zou, C., Tsung, F. and Wang, Z. (2008) “Monitoring profiles based on non-parametric regression methods.”  Technometrics, 50, 512–526.