Providing comprehensive control chart for monitoring of linear and nonlinear profiles using functional data analysis.

Subject Areas : Statistics

M. Bahri ¹ , A. Hadi-Vencheh ²

1 - Department of Industrial Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran
2 - Department of Mathematics, Isfahan (Khorasgan) Branch, Islamic Azad University, Isfahan, Iran

Received: 2018-02-27 Accepted : 2019-07-06 Published : 2020-01-21

Keywords: پروفایل مانیتورینگ, نمودارهای کنترلی, کنترل فرآیند آماری, آنالیز داده‌های تابعی, فاز II,

Abstract :

Considering profiles as functional variables, two control charts are proposed for their monitoring in phase II. Due to its conformity with the nature of real-world profiles, applying functional model leads to proposed control charts obtained through functional data analysis techniques with desired features. These include simplicity in calculation and possibility of using them for different profiles such as linear and non-linear (even in the presence of complex within-profile Autocorrelation forms). These features distinguish the functional model from the regression models common in profile monitoring. Simulated computer simulations show that, in different states, the proposed control charts have a lower average run length than other methods, which indicates the desired performance of the proposed functional approach. Morevere, in some non-linear cases with complex autocorrelation, other methods completely fail, and only the proposed control charts are able to detect the occurring deviation, and even in these cases, the average run length of these control charts is highly desirable.

References:

[1]          Qiu, P., C. Zou, and Z. Wang, Nonparametric profile monitoring by mixed effects modeling. Technometrics, 2010. 52(3): p. 265-277.

[2]          Woodall, W.H., Current research on profile monitoring. Producao, 2007. 17(3): p. 420-425.

[3]          Mestek, O., J. Pavlík, and M. Suchánek, Multivariate control charts: Control charts for calibration curves. Fresenius' Journal of Analytical Chemistry, 1994. 350(6): p. 344-351.

[4]          Stover, F.S. and R.V. Brill, Statistical quality control applied to ion chromatography calibrations. Journal of Chromatography A, 1998. 804(1–2): p. 37-43.

[5]          Woodall, W.H., et al., Using control charts to monitor process and product quality profiles. Journal of Quality Technology, 2004. 36(3): p. 309-320.

[6]          Noorossana, R., A. Saghaei, and A. Amiri, Statistical analysis of profile monitoring. Vol. 865. 2012: John Wiley & Sons.

[7]          Kang, L. and S.L. Albin, On-line monitoring when the process yields a linear profile. Journal of Quality Technology, 2000. 32(4): p. 418-426.

[8]          Kim, K., M.A. Mahmoud, and W.H. Woodall, On the monitoring of linear profiles. Journal of Quality Technology, 2003. 35(3): p. 317-328.

[9]          Gupta, S., D.C. Montgomery, and W.H. Woodall, Performance evaluation of two methods for online monitoring of linear calibration profiles. International Journal of Production Research, 2006. 44(10): p. 1927-1942.

[10]       Neter, J., W. Wasserman, and M.H. Kutner, Applied Linear Statistical Models, 3rd edition., in Richard D. Irwin, Inc., Boston, MA.1990.

[11]       Chang, T.-C. and F.-F. Gan, Monitoring linearity of measurement gauges. Journal of Statistical Computation and Simulation, 2006. 76(10): p. 889-911.

[12]       Zhu, J. and D.K.J. Lin, Monitoring the slopes of linear profiles. Quality Engineering, 2010. 22(1): p. 1-12.

[13]       Noorossana, R., et al. Monitoring Process Performance Using Linear Profiles. in Proceedings of the 3rd International Industrial Engineering Conference. 2004. Tehran, Iran.

[14]       Healy, J.D., A Note on Multivariate CUSUM procedure. Technometrics, 1987. 29: p. 409-412.

[15]       Zou, C., F. Tsung, and Z. Wang, Monitoring general linear profiles using multivariate exponentially weighted moving average schemes. Technometrics, 2007. 49(4): p. 395-408.

[16]       Mahmoud, M.A., Phase I analysis of multiple linear regression profiles. Communications in Statistics: Simulation and Computation, 2008. 37(10): p. 2106-2130.

[17]       Jensen, W.A., J.B. Birch, and W.H. Woodall, Monitoring correlation within linear profiles using mixed models. Journal of Quality Technology, 2008. 40(2): p. 167-183.

[18]       Noorossana, R., M. Eyvazian, and A. Vaghefi, Phase II monitoring of multivariate simple linear profiles. Computers and Industrial Engineering, 2010. 58(4): p. 563-570.

[19]       Kazemzadeh, R.B., R. Noorossana, and A. Amiri, Phase I monitoring of polynomial profiles. Communications in Statistics - Theory and Methods, 2008. 37(10): p. 1671-1686.

[20]       Williams, J.D., W.H. Woodall, and J.B. Birch, Statistical monitoring of nonlinear product and process quality profiles. Quality and Reliability Engineering International, 2007. 23(8): p. 925-941.

[21]       Ding, Y., L. Zeng, and S. Zhou, Phase I analysis for monitoring nonlinear profiles in manufacturing processes. Journal of Quality Technology, 2006. 38(3): p. 199-216.

[22]       Williams, J., et al., Statistical monitoring of heteroscedastic dose-response profiles from high-throughput screening. Journal of agricultural, biological, and environmental statistics, 2007. 12(2): p. 216-235.

[23]       Young, T.M., P.M. Winistorfer, and S. Wang, Multivariate Control Charts of MDF and OSB Vertical Density Profile Attributes. Forest Products Journal, 1999. 49(5): p. 79-86.

[24]       Walker, E. and S.P. Wright, Comparing curves using additive models. Journal of Quality Technology, 2002. 34(1): p. 118-129.

[25]       Vaghefi, A., S.D. Tajbakhsh, and R. Noorossana, Phase II monitoring of nonlinear profiles. Communications in Statistics - Theory and Methods, 2009. 38(11): p. 1834-1851.

[26]       Zhang, H. and S. Albin, Detecting outliers in complex profiles using a χ2 control chart method. IIE Transactions (Institute of Industrial Engineers), 2009. 41(4): p. 335-345.

[27]       Chicken, E., J.J. Pignatiello Jr, and J.R. Simpson, Statistical process monitoring of nonlinear profiles using wavelets. Journal of Quality Technology, 2009. 41(2): p. 198-212.

[28]       Zou, C., P. Qiu, and D. Hawkins, Nonparametric control chart for monitoring profiles using change point formulation and adaptive smoothing. Statistica Sinica, 2009. 19(3): p. 1337.

[29]       Soleimani, P., R. Noorossana, and A. Amiri, Simple linear profiles monitoring in the presence of within profile autocorrelation. Computers & Industrial Engineering, 2009. 57(3): p. 1015-1021.

[30]       Noorossana, R., A. Amiri, and P. Soleimani, On the monitoring of autocorrelated linear profiles. Communications in Statistics - Theory and Methods, 2008. 37(3): p. 425-442.

[31]       Kazemzadeh, R.B., R. Noorossana, and A. Amiri, Phase II monitoring of autocorrelated polynomial profiles in AR(1) processes. Scientia Iranica, 2010. 17(1 E): p. 12-24.

[32]       Jensen, W.A. and J.B. Birch, Profile monitoring via nonlinear mixed models. Journal of Quality Technology, 2009. 41(1): p. 18-34.

[33]       Zou, C., Y. Zhang, and Z. Wang, A control chart based on a change-point model for monitoring linear profiles. IIE Transactions (Institute of Industrial Engineers), 2006. 38(12): p. 1093-1103.

[34]       Mahmoud, M.A., et al., A change point method for linear profile data. Quality and Reliability Engineering International, 2007. 23(2): p. 247-268.

[35]       Li, Z. and Z. Wang, An exponentially weighted moving average scheme with variable sampling intervals for monitoring linear profiles. Computers and Industrial Engineering, 2010. 59(4): p. 630-637.

[36]       Bahri, M., Hadi-Vencheh, A. Designing Statistical Test for Mean of Random Profiles. Industrial Engineering and Management Systems, 2016, 15(4): p. 432-445.‏

[37]       Ramsay, J.O., When the data are functions. Psychomrtrica, 1982. 47: p. 379-396.

[38]       Silverman, B. and J. Ramsay, Functional Data Analysis2005: Springer.

[39]       Horváth, L. and P. Kokoszka, Inference for functional data with applications. Vol. 200. 2012: Springer.

[40]       Scheer, C., Bedingte Konvergenz stochastischer Prozesse, 2003, University of Trier: Trier, Germany.

[41]       Ramsay, J.O. and B.W. Silverman, Applied functional data analysis: methods and case studies. Vol. 77. 2002: Springer.

[42]       Fan, J. and S.-K. Lin, Test of significance when data are curves. Journal of the American Statistical Association, 1998. 93(443): p. 1007-1021.

[43]       Ferraty, F. and P. Vieu, Nonparametric functional data analysis: theory and practice2006: Springer.

[44]       Montgomery, D.C., Introduction to Statistical Quality Control2008: Wiley.

[45]       MacGregor, J. and T. Harris, The exponentially weighted moving variance. Journal of Quality Technology, 1993. 25(2).

[46]       Mahmoud, M.A. and W.H. Woodall, Phase I analysis of linear profiles with calibration applications. Technometrics, 2004. 46(4): p. 380-391.

[47]       Mahmoud, M.A., The Monitoring of linear profiles and the inertial properties of control charts, 2004, Virginia Polytechnic Institute and State University.

[48]       Ok, E.A., Probability Theory with Economic Applications2006: book draft.

[49]       Box, G.E.P., Some Theorems on Quadratic Forms Applied in the Study of Analysis of Variance Problems, I. Effect of Inequality of Variance in the One-Way Classification. Annals of Mathematical Statistics, 1954. 25(2): p. 13