Flexible resampling for fuzzy data

Grzegorzewski, Przemyslaw ; Hryniewicz, Olgierd ; Romaniuk, Maciej

International Journal of Applied Mathematics and Computer Science, Tome 30 (2020) no. 2, pp. 281-297.

Résumé

In this paper, a new methodology for simulating bootstrap samples of fuzzy numbers is proposed. Unlike the classical bootstrap, it allows enriching a resampling scheme with values from outside the initial sample. Although a secondary sample may contain results beyond members of the primary set, they are generated smartly so that the crucial characteristics of the original observations remain invariant. Two methods for generating bootstrap samples preserving the representation (i.e., the value and the ambiguity or the expected value and the width) of fuzzy numbers belonging to the primary sample are suggested and numerically examined with respect to other approaches and various statistical properties.

Détail

Keywords: bootstrap, fuzzy data, fuzzy number, fuzzy sample, imprecise data, resampling
Mots-clés : bootstrap, dane rozmyte, liczba rozmyta, dane niedokładne

Bibliographie

[1] Ban, A., Brândaş, A., Coroianu, L., Negruţiu, C. and Nica, O. (2011). Approximations of fuzzy numbers by trapezoidal fuzzy numbers preserving the ambiguity and value, Computers and Mathematics with Applications 61(5): 1379–1401.

[2] Ban, A., Coroianu, L. and Grzegorzewski, P. (2015). Fuzzy Numbers: Approximations, Ranking and Applications, Polish Academy of Sciences, Warsaw.

[3] Casals, M.R., Corral, N., Gil, M. Á., López, M.T., Lubiano, M.A., Montenegro, M., Naval, G. and Salas, A. (2013). Bertoluzza et al.’s metric as a basis for analyzing fuzzy data, METRON 71(3): 307–322.

[4] Casella, G. (2003). Introduction to the silver anniversary of the bootstrap, Statistical Science 18(2): 133–134.

[5] Chanas, S. (2001). On the interval approximation of a fuzzy number, Fuzzy Sets and Systems 122(2): 353–356.

[6] Colubi, A. (2009). Statistical inference about the means of fuzzy random variables: Applications to the analysis of fuzzy- and real-valued data, Fuzzy Sets and Systems 160(3): 344–356.

[7] 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.

[8] Davison, A.C., Hinkley, D.V. and Schechtman, E. (1986). Efficient bootstrap simulation, Biometrika 73(3): 555–566.

[9] De Angelis, D. and Young, G.A. (1992). Smoothing the bootstrap, International Statistical Review 60(1): 45–56.

[10] Delgado, M., Vila, M. and Voxman, W. (1998). On a canonical representation of a fuzzy number, Fuzzy Sets and Systems 93(1): 125–135.

[11] Denoeux, T., Masson, M.-H. and Hébert, P.-A. (2005). Nonparametric rank-based statistics and significance tests for fuzzy data, Fuzzy Sets and Systems 153(1): 1–28.

[12] Dubois, D. and Prade, H. (1987). The mean value of a fuzzy number, Fuzzy Sets and Systems 24(3): 279–300.

[13] Efron, B. (1979). Bootstrap methods: Another look at the jackknife, Annals of Statistics 7(1): 1–26.

[14] Gao, J.-Q., Fan, L.-Y., Li, L. and Xu, L.-Z. (2013). A practical application of kernel-based fuzzy discriminant analysis, International Journal of Applied Mathematics and Computer Science 23(4): 887–903, DOI: 10.2478/amcs-2013-0066.

[15] 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.

[16] 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.

[17] Graham, R., Hinkley, D.V., John, P.W.M. and Shi, S. (1990). Balanced design of bootstrap simulations, Journal of the Royal Statistical Society B 52(1): 185–202.

[18] Grzegorzewski, P. (2008). Trapezoidal approximations of fuzzy numbers preserving the expected interval—Algorithms and properties, Fuzzy Sets and Systems 159(11): 1354–1364.

[19] Grzegorzewski, P. (2018). The Kolmogorov–Smirnov goodness-of-fit test for interval-valued data, in E. Gil et al. (Eds), The Mathematics of the Uncertain: A Tribute to Pedro Gil, Springer International Publishing, Cham, pp. 615–627.

[20] Grzegorzewski, P. and Hryniewicz, O. (2002). Computing with words and life data, International Journal of AppliedMathematics and Computer Science 12(3): 337–345.

[21] 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.

[22] Hall, P., DiCiccio, T. and Romano, J. (1989). On smoothing and the bootstrap, Annals of Statistics 17(2): 692–704.

[23] Heilpern, S. (1992). The expected value of a fuzzy number, Fuzzy Sets and Systems 47(1): 81–86.

[24] Jimenez, M. and Rivas, J.A. (1998). Fuzzy number approximation, International Journal of Uncertainty, Fuzziness and Knowledge-based Systems 6(1): 68–78.

[25] 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.

[26] Lubiano, M.A., Salas, A., Carleos, C. and 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.

[27] 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.

[28] Pedrycz, W. (1994). Why triangular membership functions?, Fuzzy Sets and Systems 64(1): 21–30.

[29] Puri, M. and Ralescu, D.A. (1986). Fuzzy random variables, Journal of the Mathematical Analysis and Applications 114(2): 409–422.

[30] 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, Cham, pp. 127–147.

[31] Ramos-Guajardo, A. and Grzegorzewski, P. (2016). Distance-based linear discriminant analysis for interval-valued data, Information Sciences 372: 591–607.

[32] 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.

[33] 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 International Publishing, Cham, pp. 437–448.

[34] Romaniuk, M. and Hryniewicz, O. (2019a). Discrete and smoothed resampling methods for interval-valued fuzzy numbers, IEEE Transactions on Fuzzy Systems, DOI: 10.1109/TFUZZ.2019.2957253, (in press).

[35] Romaniuk, M. and Hryniewicz, O. (2019b). Interval-based, nonparametric approach for resampling of fuzzy numbers, Soft Computing 23(14): 5883–5903.

[36] Silverman, B.W. and Young, G.A. (1987). The bootstrap: To smooth or not to smooth?, Biometrika 74(3): 469–479.

[37] Sinova, B., Gil, M.A., Colubi, A. and Aelst, S.V. (2012). The median of a random fuzzy number. The 1-norm distance approach, Fuzzy Sets and Systems 200: 99–115.

[38] 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.