Bootstrap methods for epistemic fuzzy data

• Autorzy: Grzegorzewski Przemyslaw , Romaniuk Maciej

Identyfikatory

DOI

10.34768/amcs-2022-0021

• Języki publikacji: EN

Abstrakty

EN

Fuzzy numbers are often used for modeling imprecise perceptions of the real-valued observations. Such epistemic fuzzy data may cause problems in statistical reasoning and data analysis. We propose a universal nonparametric technique, called the epistemic bootstrap, which could be helpful when the existing methods do not work or do not give satisfactory results. Besides the simple epistemic bootstrap, we develop its several refinements that aim to reduce the variance in statistical inference. We also perform an extended simulation study to examine statistical properties of the approaches considered. The discussion of the results is supplemented by some hints for practical use.

Słowa kluczowe

EN

bootstrap method; fuzzy data; fuzzy numbers; hypotheses testing

PL

metoda bootstrap; dane rozmyte; testowanie hipotez

• Wydawca: Oficyna Wydawnicza Uniwersytetu Zielonogórskiego
• Czasopismo: International Journal of Applied Mathematics and Computer Science
• Rocznik: 2022
• Tom: Vol. 32, no. 2
• Strony: 285--297
• Opis fizyczny: Bibliogr. 48 poz., tab., wykr.

Twórcy

autor

Grzegorzewski Przemyslaw

• Faculty of Mathematics and Information Science, Warsaw University of Technology, Koszykowa 75, 00-662 Warsaw, Poland; Systems Research Institute, Polish Academy of Sciences, Newelska 6, 01-447 Warsaw, Poland

autor

Romaniuk Maciej

• Systems Research Institute, Polish Academy of Sciences, Newelska 6, 01-447 Warsaw, Poland; Warsaw School of Information Technology, Newelska 6, 01-447 Warsaw, Poland

Bibliografia

• [1] Ban, A., Coroianu, L. and Grzegorzewski, P. (2015). Fuzzy Numbers: Approximations, Ranking and Applications, Polish Academy of Sciences, Warsaw.
• [2] Casella, G. (2003). Introduction to the silver anniversary of the bootstrap, Statistical Science 18(2): 133–134.
• [3] Colubi, A., Fernández-García, C. and Gil, M. (2002). Simulation of random fuzzy variables: An empirical approach to statistical/probabilistic studies with fuzzy experimental data, IEEE Transactions on Fuzzy Systems 10(3): 384–390.
• [4] Couso, I. and Dubois, D. (2014). Statistical reasoning with set-valued information: Ontic vs. epistemic views, International Journal of Approximate Reasoning 55(7): 1502–1518.
• [5] Couso, I. and Sánchez, L. (2011). Inner and outer fuzzy approximations of confidence intervals, Fuzzy Sets and Systems 184(1): 68–83.
• [6] Davison, A.C. and Hinkley, D.V. (1997). Bootstrap Methods and Their Application, Cambridge University Press, Cambridge.
• [7] De Angelis, D. and Young, G.A. (1992). Smoothing the bootstrap, International Statistical Review 60(1): 45–56.
• [8] Denoeux, T. (2011). Maximum likelihood estimation from fuzzy data using the EM algorithm, Fuzzy Sets and Systems 183(1): 72–91.
• [9] Efron, B. (1979). Bootstrap methods: Another look at the jackknife, Annals of Statistics 7(1): 1–26.
• [10] Ferson, S., Kreinovich, V., Hajagos, J., Oberkampf, W. and Ginzburg, L. (2007). Experimental uncertainty estimation and statistics for data having interval uncertainty, Technical Report SAND2007-0939, Applied Biomathematics, New York.
• [11] Gil, M., Montenegro, M., González-Rodríguez, G., Colubi, A. and Casals, M. (2006). Bootstrap approach to the multi-sample test of means with imprecise data, Computational Statistics and Data Analysis 51(1): 148–162.
• [12] Giné, E. and Zinn, J. (1990). Bootstrapping general empirical measures, Annals of Probability 18(2): 851–869.
• [13] González-Rodríguez, G., Montenegro, M., Colubi, A. and Gil, M. (2006). Bootstrap techniques and fuzzy random variables: Synergy in hypothesis testing with fuzzy data, Fuzzy Sets and Systems 157(19): 2608–2613.
• [14] Grzegorzewski, P. (2000). Testing statistical hypotheses with vague data, Fuzzy Sets and Systems 112(3): 501–510.
• [15] Grzegorzewski, P. (2001). Fuzzy tests—defuzzification and randomization, Fuzzy Sets and Systems 118(3): 437–446.
• [16] Grzegorzewski, P. and Goławska, J. (2021). In search of a precise estimator based on imprecise data, Joint Proceedings of the IFSA-EUSFLAT-AGOP 2021 Conferences, Bratislava, Slovakia, pp. 530–537.
• [17] Grzegorzewski, P. and Hryniewicz, O. (2002). Computing with words and life data, International Journal of Applied Mathematics and Computer Science 12(3): 337–345.
• [18] Grzegorzewski, P., Hryniewicz, O. and Romaniuk, M. (2019). Flexible bootstrap based on the canonical representation of fuzzy numbers, Proceedings of EUSFLAT 2019, Prague, Czech Republic, pp. 490–497.
• [19] Grzegorzewski, P., Hryniewicz, O. and Romaniuk, M. (2020a). Flexible bootstrap for fuzzy data based on the canonical representation, International Journal of Computational Intelligence Systems 13(1): 1650–1662.
• [20] Grzegorzewski, P., Hryniewicz, O. and Romaniuk, M. (2020b). Flexible resampling for fuzzy data, International Journal of Applied Mathematics and Computer Science 30(2): 281–297, DOI: 10.34768/amcs-2020-0022.
• [21] Grzegorzewski, P. and Romaniuk, M. (2021). Epistemic bootstrap for fuzzy data, Joint Proceedings of the IFSAEUSFLAT-AGOP 2021 Conferences, Bratislava, Slovakia, pp. 538–545.
• [22] Hall, P., DiCiccio, T. and Romano, J. (1989). On smoothing and the bootstrap, Annals of Statistics 17(2): 692–704.
• [23] Hukuhara, M. (1967). Integration des applications measurables dont la valeur est un compact convexe, Funkcialaj Ekvacioj 10: 205–223.
• [24] Kołacz, A. and Grzegorzewski, P. (2019). Asymptotic algorithm for computing the sample variance of interval data, Iranian Journal of Fuzzy Systems 16(4): 83–96.
• [25] Kroese, D.P., Taimre, T. and Botev, Z.I. (2011). Handbook of Monte Carlo Methods, Wiley, Hoboken.
• [26] Kruse, R. (1982). The strong law of large numbers for fuzzy random variables, Information Sciences 28(3): 233–241.
• [27] Kwakernaak, H. (1978). Fuzzy random variables, Part I: Definitions and theorems, Information Sciences 15(1): 1–15.
• [28] Lubiano, M.A., Montenegro, M., Sinova, B., de la Rosa de Sáa, S. and Gil, M.A. (2016). Hypothesis testing for means in connection with fuzzy rating scale-based data: Algorithms and applications, European Journal of Operational Research 251(3): 918–929.
• [29] Lubiano, M.A., Salas, A., Carleos, C., de la Rosa de Sáa, S. and Gil, M.A. (2017). Hypothesis testing-based comparative analysis between rating scales for intrinsically imprecise data, International Journal of Approximate Reasoning 88: 128–147.
• [30] 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: 31–49.
• [31] Nguyen, H., Kreinovich, V., Wu, B. and Xiang, G. (2012). Computing Statistics under Interval and Fuzzy Uncertainty, Springer, Berlin/Heidelberg.
• [32] Parchami, A. (2018). EM algorithm for maximum likelihood estimation by non-precise information, https://cran.r-project.org/package=EM.Fuzzy.
• [33] Pedrycz, W. (1994). Why triangular membership functions?, Fuzzy Sets and Systems 64(1): 21–30.
• [34] Piegat, A. (2005). A new definition of the fuzzy set, International Journal of Applied Mathematics and Computer Science 15(1): 125–140.
• [35] Ramos-Guajardo, A., Blanco-Fernández, A. and González-Rodríguez, G. (2019). Applying statistical methods with imprecise data to quality control in cheese manufacturing, in P. Grzegorzewski et al. (Eds), Soft Modeling in Industrial Manufacturing, Springer, Berlin/Heidelberg, pp. 127–147.
• [36] Ramos-Guajardo, A. and Grzegorzewski, P. (2016). Distance-based linear discriminant analysis for interval-valued data, Information Sciences 372: 591–607.
• [37] Ramos-Guajardo, A. and Lubiano, M. (2012). k-Sample tests for equality of variances of random fuzzy sets, Computational Statistics and Data Analysis 56(4): 956–966.
• [38] Romaniuk, M. (2019). On some applications of simulations in estimation of maintenance costs and in statistical tests for fuzzy settings, in A. Steland et al. (Eds), Stochastic Models, Statistics and Their Applications, Springer, Cham, pp. 437–448.
• [39] Romaniuk, M. and Hryniewicz, O. (2019). Interval-based, nonparametric approach for resampling of fuzzy numbers, Soft Computing 23: 5883–5903.
• [40] Romaniuk, M. and Hryniewicz, O. (2021). Discrete and smoothed resampling methods for interval-valued fuzzy numbers, IEEE Transactions on Fuzzy Systems 29(3): 599–611.
• [41] Sevinc, B., Cetintav, B., Esemen, M. and Gurler, S. (2019). RSSampling: A pioneering package for ranked set sampling, The R Journal 11(1): 401–415.
• [42] Shao, J. and Tu, D. (1995). The Jackknife and Bootstrap, Springer, New York.
• [43] Silverman, B.W. and Young, G.A. (1987). The bootstrap: To smooth or not to smooth?, Biometrika 74(3): 469–479.
• [44] Suresh, H. and Guttag, J.V. (2021). A framework for understanding sources of harm throughout the machine learning life cycle, Equity and Access in Algorithms, Mechanisms, and Optimization (EAAMO ’21), New York, USA.
• [45] Vavasis, S.A. (1991). Nonlinear Optimization: Complexity Issues, Oxford University Press, New York.
• [46] Wang, D. and Hryniewicz, O. (2015). A fuzzy nonparametric Shewhart chart based on the bootstrap approach, International Journal of Applied Mathematics and Computer Science 25(2): 389–401, DOI: 10.1515/amcs-2015-0030.
• [47] Wolfe, D.A. (2004). Ranked set sampling: An approach to more efficient data collection, Statistical Science 19(4): 636–643.
• [48] Zadeh, L.A. (1973). Outline of a new approach to the analysis of complex systems and decision processes, IEEE Transactions on Systems, Man and Cybernetics SMC-3(1): 28–44.