In this paper, we consider a nonparametric Shewhart chart for fuzzy data. We utilize the fuzzy data without transforming them into a real-valued scalar (a representative value). Usually fuzzy data (described by fuzzy random variables) do not have a distributional model available, and also the size of the fuzzy sample data is small. Based on the bootstrap methodology, we design a nonparametric Shewhart control chart in the space of fuzzy random variables equipped with some L2 metric, in which a novel approach for generating the control limits is proposed. The control limits are determined by the necessity index of strict dominance combined with the bootstrap quantile of the test statistic. An in-control bootstrap ARL of the proposed chart is also considered.
@article{bwmeta1.element.bwnjournal-article-amcv25i2p389bwm,
author = {Dabuxilatu Wang and Olgierd Hryniewicz},
title = {A fuzzy nonparametric Shewhart chart based on the bootstrap approach},
journal = {International Journal of Applied Mathematics and Computer Science},
volume = {25},
year = {2015},
pages = {389-401},
zbl = {1322.93066},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-amcv25i2p389bwm}
}
Dabuxilatu Wang; Olgierd Hryniewicz. A fuzzy nonparametric Shewhart chart based on the bootstrap approach. International Journal of Applied Mathematics and Computer Science, Tome 25 (2015) pp. 389-401. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-amcv25i2p389bwm/
[000] Cen, Y. (1996). Fuzzy quality and analysis on fuzzy probability, Fuzzy Sets and Systems 83(2): 283-290.
[001] Cheng, C.-B. (2005). Fuzzy process control: Construction of control charts with fuzzy numbers, Fuzzy Sets and Systems 154(2): 287-303.
[002] Couso, I., Dubois, D., Montes, S. and Sanchez, L. (2007). On various definitions of the variance of a fuzzy random variable, Proceedings of the 5th International Symposium on Imprecise Probabilities and Their Applications, Prague, Czech Republic, www.sipta.org/isipta07/proceedings/ 056.html.
[003] Dubois, D. and Prade, H. (1983). Ranking fuzzy numbers in the setting of possibility theory, Information Sciences 30(3): 183-224. | Zbl 0569.94031
[004] Efron, B. and Tibshirani, R. (1993). An Introduction to the Bootstrap, Chapman-Hall, New York, NY. | Zbl 0835.62038
[005] Faraz, A. and Shapiro, A. (2010). An application of fuzzy random variables to control charts, Fuzzy Sets and Systems 161(20): 2684-2694. | Zbl 1206.93062
[006] Feng, Y., Hu, L. and Shu, H. (2001). The variance and covariance of fuzzy random variables and their applications, Fuzzy Sets and Systems 120(3): 487-497. | Zbl 0984.60029
[007] Féron, R. (1976). Ensembles aléatoires flous, Comptes Rendus de l'Academie des Sciences Serie A 282: 903-906. | Zbl 0327.60004
[008] Gil, M., López-Diaz, M. and Ralescu, D.A. (2006). Overview on the development of fuzzy random variables, Fuzzy Sets and Systems 157(19): 2546-2557. | Zbl 1108.60006
[009] Grzegorzewski, P. and Hryniewicz, O. (2000). Soft methods in statistical quality control, Control and Cybernetics 29(1): 119-140. | Zbl 1030.90019
[010] Gülbay, M. and Kahraman, C. (2006). Development of fuzzy process control charts and fuzzy unnatural pattern analysis, Computational Statistics and Data Analysis 51(1): 433-451. | Zbl 1157.62560
[011] Gülbay, M. and Kahraman, C. (2007). An alternative approach to fuzzy control charts: Direct fuzzy approach, Information Sciences 177(6): 1463-1480. | Zbl 1120.93332
[012] Höppner, J. (1994). Statistische Prozeßkontrolle mit FuzzyDaten, Ph.D. thesis, University of Ulm, Ulm.
[013] Höppner, J. and Wolff, H. (1995). The design of a Shewhart control chart for fuzzy data, Technical report, University of Ulm, Würzburg.
[014] Hryniewicz, O. (2006). Possibilistic decisions and fuzzy statistical tests, Fuzzy Sets and Systems 157(19): 2665-2673. | Zbl 1099.62008
[015] Kanagawa, A., Tamaki, F. and Ohta, H. (1993). Control charts for process average and variability based on linguistic data, International Journal of Production Research 2(4): 913-922. | Zbl 0769.62076
[016] Körner, R. (1997). On the variance of fuzzy random variables, Fuzzy Sets and Systems 92(1): 83-93. | Zbl 0936.60017
[017] Körner, R. (2000). An asymptotic α-test for the expectation of random fuzzy variables, Journal of Statistical Planning and Inference 83(2): 331-346. | Zbl 0976.62013
[018] Kruse, R. and Meyer, K. (1987). Statistics with Vague Data, D. Riedel, Dordrecht. | Zbl 0663.62010
[019] Kwakernaak, H. (1978). Fuzzy random variables, Part I: Definitions and theorems, Information Sciences 15(1): 1-15. | Zbl 0438.60004
[020] Kwakernaak, H. (1979). Fuzzy random variables, Part II: Algorithms and examples for the discrete case, Information Sciences 17(3): 253-278. | Zbl 0438.60005
[021] Liu, R. and Tang, J. (1996). Control charts for dependent and independent measurements based on bootstrap methods, Journal of American Statistical Association 91(436): 1694-1700. | Zbl 0881.62104
[022] Montenegro, M., Colubi, A., Casals, M. and Gil, M. (2004). Asymptotic and bootstrap techniques for testing the expected value of a fuzzy random variable, Metrika 59(1): 31-49. | Zbl 1052.62048
[023] Näther, W. (2000). On random fuzzy variables of second order and their application to linear statistical inference with fuzzy data, Metrika 51(3): 201-221. | Zbl 1093.62557
[024] Näther, W. (2006). Regression with fuzzy random data, Computational Statistics and Data Analysis 51(1): 235-252. X̅ control chart, Jour | Zbl 1157.62463
[025] Nelson, L. (1985). Interpreting Shewhart X̅ control chart, Journal of Quality Technology 17(2): 114-116.
[026] Puri, M. and Ralescu, D. (1986). Fuzzy random variables, Journal of Mathematical Analysis and Applications 114(2): 409-422. | Zbl 0592.60004
[027] Raz, T. and Wang, J. (1990). Probabilistic and membership approaches in the construction of control charts for linguistic data, Production Planning & Control 1(3): 147-157.
[028] Senturk, S. and Erginel, N. (2009). Development of fuzzy x̅̃ - R̃ and x̅̃ - S̃ control charts using α-cuts, Information Sciences 179(10): 1542-1551.
[029] Shu, M.-H. and Wu, H.-C. (2011). Fuzzy X̅ and r control charts: Fuzzy dominance approach, Computers & Industrial Engineering 61(3): 676-685.
[030] Taleb, H. (2009). Control charts applications for multivariate attribute processes, Computers & Industrial Engineering 56(1): 399-410.
[031] Taleb, H. and Limam, M. (2002). On fuzzy and probabilistic control charts, International Journal of Production Research 40(12): 2849-2863.
[032] Wang, J. and Raz, T. (1990). On the construction of control charts using linguistic variables, International Journal of Production Research 28: 477-487.
[033] Wetherill, B. and Brown, D. (1991). Statistical Process Control, Chapman and Hall, London. | Zbl 0724.62099
[034] Woodall, W.H. Tsui, K.-L. and Tucker, G.R. (1997). A review of statistical and fuzzy quality control charts based on categorical data, in H.-H. Lenz and P.-Th. Wilrich (Eds.), Frontiers in Statistical Quality Control 5, Physica-Verlag, Heidelberg, pp. 83-89. | Zbl 0900.62538
[035] Zadeh, L. (1965). Fuzzy sets, Information and Control 8(3): 338-353. | Zbl 0139.24606
[036] Zadeh, L. (1975). The concept of a linguistic variable and its application to approximate reasoning, Parts 1 and 2, Information Sciences 8(3): 199-249, 8(4): 301-357. | Zbl 0397.68071