A Distribution Free Multivariate Control Chart Based on Run Test

doi:10.3969/j.issn.1007-7375.2020.02.016

Abstract

Abstract: A new distrbution free multivariate control chart based on run test is proposed. First, the shortest Hamilton path of the observations is determined by means of Kruskal algorithm, a new EWMA control chart with sliding window (HAMEWMA) is based on the number of runs of the above shortest Hamilton path. Monte Carlo simulation is used to study the control effect of HAMEWMA control chart under different conditions (dimensions, mean shift, controlled sample size and distributions of observations). Compared with some non-parametric multivariate control charts (DFEWMA, SREWMA, SSEWMA, RTC), HAMEWMA has a superior performance. The results show that: when mean shift is large, HAMEWMA has a better monitoring performance. When distributions are non-normal, HAMEWMA also performs well, even better than when the distribution is normal. Last but not the least, the HAMEWMA chart is more suitable for monitoring in high-dimensional situations

Key words: distribution-free, multivariate, non-parametric, run test, Hamilton path

CLC Number:

TP277

PEI Dezhao, LI Yanting. A Distribution Free Multivariate Control Chart Based on Run Test[J]. Industrial Engineering Journal, 2020, 23(2): 124-132,149.

References

[1] HOTELLING H. Multivariate quality control-illustrated by the air testing of sample bombsights[M]//EISENHART C, HASTAY M W, WALLIS W A. Techniques of statistical analysis. New York: McGraw Hill, 1947: 111-184.
[2] LOWRY C A, WOODALL W H, CHAMP C W, et al. A multivariate exponentially weighted moving average control chart[J]. Technometrics, 1992, 34(1): 46-53
[3] CROSIER R B. Multivariate generalizations of cumulative sum quality-control schemes[J]. Technometrics, 1988, 30(3): 291-303
[4] MAKIS V. Multivariate Bayesian control chart[J]. Operations Research, 2008, 56(2): 487-496
[5] QIU P. Introduction to statistical process control[M]. Boca Raton, FL: CRC Press, 2014.
[6] QIU P. Some perspectives on nonparametric statistical process control[J]. Journal of Quality Technology, 2018, 50(1): 49-65
[7] BORROR C M, MONTGOMERY D C, RUNGER G C. Robustness of the EWMA control chart to non-normality[J]. Journal of Quality Technology, 1999, 31(3): 309-316
[8] STOUMBOS Z G, SULLIVAN J H. Robustness to non-normality of the multivariate EWMA control chart[J]. Journal of Quality Technology, 2002, 34(3): 260-276
[9] TESTIK M C, RUNGER G C, BORROR C M. Robustness properties of multivariate EWMA control charts[J]. Quality and Reliability Engineering International, 2003, 19(1): 31-38
[10] QIU P, LI Z. Distribution-free monitoring of univariate processes[J]. Statistics and Probability Letters, 2011, 81(12): 1833-1840
[11] LIU R Y. Control charts for multivariate processes[J]. Publications of the American Statistical Association, 1995, 90(432): 1380-1387
[12] HAWKINS D M. MABOUDOU-TCHAO E M. Self-starting multivariate exponentially weighted moving average control charting[J]. Technometrics, 2007, 49(2): 199-209
[13] DENG H, RUNGER G, TUV E. System monitoring with real-Time contrasts[J]. Journal of Quality Technology, 2012, 44(1): 9-27
[14] LI J, ZHANG X, JESKE D R. Nonparametric multivariate cusum control charts for location and scale changes[J]. Journal of Nonparametric Statistics, 2013, 25(1): 1-20
[15] SUN R, TSUNG F. A Kernel-distance-based multivariate control chart using support vector methods[J]. International Journal of Production Research, 2003, 41(13): 2975-2989
[16] QIU P. Distribution-free multivariate process control based on log-linear modeling[J]. IIE Transactions, 2008, 40(7): 664-677
[17] ZOU C, TSUNG F. A multivariate sign EWMA control chart[J]. Technometrics, 2011, 53(1): 84-97
[18] SULLIVAN J H, JONES L A. A self-starting control chart for multivariate individual observations[J]. Technometrics, 2002, 44(1): 24-33
[19] ZOU C, JIANG W, TSUNG F. A lasso-based diagnostic framework for multivariate statistical process control[J]. Technometrics, 2011, 53(3): 297-309
[20] ZOU C, WANG Z, TSUNG F. A spatial Rank-based multivariate EWMA control chart[J]. Naval Research Logistics, 2012, 59(2): 91-110
[21] CHEN N, ZI X, ZOU C. A Distribution-free multivariate control chart[J]. Technometrics, 2015, 58(4): 448-459
[22] BICKEL P J. A distribution free version of the Smirnov two sample test in the p-variate case[J]. The Annals of Mathematical Statistics, 1969, 40(1): 1-23
[23] MUKHERJEE A, CHAKRABORTI S. A distribution-free control chart for the joint monitoring of location and scale[J]. Quality & Reliability Engineering International, 2012, 28(3): 335-352
[24] BOONE J M, CHAKRABORTI S. Two simple Shewhart-type multivariate nonparametric control charts[J]. Applied Stochastic Models in Business and Industry, 2012, 28(2): 130-140
[25] ZHOU M, ZI X, GENG W, et al. A distribution-free multivariate change-point model for statistical process control[J]. Communications in Statistics-Simulation and Computation, 2015, 44(8): 1975-1987
[26] LI Z, DAI Y, WANG Z. Multivariate change point control chart based on data depth for phase I analysis[J]. Communications in Statistics-Simulation and Computation, 2014, 43(6): 1490-1507
[27] CHOWDHURY S, MUKHERJEE A, CHAKRABORTI S. A new distribution-free control chart for joint monitoring of unknown location and scale parameters of continuous distributions[J]. Quality and Reliability Engineering International, 2014, 30(2): 191-204
[28] BISWAS M, MUKHOPADHYAY M, GHOSH A K. A distribution-free two-sample run test applicable to high-dimensional data[J]. Biometrika, 2014, 101(4): 913-926
[29] NELSON P R, STEPHENSON P L. Runs tests for group control charts[J]. Communications in Statistics, 1996, 25(11): 2739-2765
[30] NELSON L S. Supplementary runs tests for Np control charts[J]. Journal of Quality Technology, 1997, 29(2): 225-227
[31] INAGAKI J, HASEYAMA M, KITAJIMA H. A new genetic algorithm for routing the shortest route via several designated points[C]. IEEE International Symposium on Circuits & Systems, Sydney: Institute of Electrical and Electronics Engineers Inc., 2001.
[32] MOHEMMED A W, SAHOO N C, GEOK T K. Solving shortest path problem using particle swarm optimization[J]. Applied Soft Computing, 2008, 8(4): 1643-1653
[33] EUSUFF M, LANSEY K, PASHA F. Shuffled frog-leaping algorithm: a memetic meta-heuristic for discrete optimization[J]. Engineering Optimization, 2006, 38(2): 26
[34] MOOD A M. The distribution theory of runs[J]. Annals of Mathematical Statistics, 1940, 11(4): 367-392
[35] MICHAEL M. UCI machine learning repository[DB/OL]. (2008-11-19)[2019-04-01]. http://archive.ics.uci.edu/ml.