Flexible Bootstrap for Fuzzy Data Based on the Canonical Representation

Abstract

Several new resampling methods for generating bootstrap samples of fuzzy numbers are proposed. To avoid undesired repetitions in the secondary samples we do not draw randomly directly observations from the primary samples but construct them allowing for some modifications in their membership functions, however only such which do not disturb the canonical representation of the initial fuzzy numbers. We consider both two-parameter and three-parameter canonical representations, as well as the triangular and trapezoidal outputs in the secondary samples. Numerical experiments concerning some statistical tests based on fuzzy samples show that the suggested methods may appear helpful in statistical reasoning with imprecise data.

Resampling Fuzzy Numbers with Statistical Applications: FuzzyResampling Package

“…Because of the previously mentioned shortcomings of Efron's approach, several modifications were proposed in the case of real-valued data. New resampling methods for fuzzy data were introduced recently in Grzegorzewski et al (2019Grzegorzewski et al ( , 2020b; Romaniuk (2019); . All of them are implemented in FuzzyResampling package.…”

“…The second group of methods contains the following flexible bootstrap algorithms: VA-method (Grzegorzewski et al, 2019(Grzegorzewski et al, , 2020a, EW-method (Grzegorzewski et al, 2020a), VAF-method (Grzegorzewski et al, 2020b) and VAA-method (Grzegorzewski and Romaniuk, 2022b). Contrary to d-method or w-method, flexible approaches can be applied to primary samples of any fuzzy numbers, but the generated outputs consist of TPFNs.…”

“…In particular, to obtain a single TPFN four values c, s, l, r are generated independently using random distributions described in Table 2, where N(mu,sigma) stands for the normal distribution with the mean mu and standard deviation sigma, U (0,a) denotes the uniform distribution on the interval (0, a), Exp (lambda) stands for the exponential distribution with the parameter lambda, etc. More details concerning various types of simulated fuzzy numbers can be found in, e.g., Grzegorzewski et al (2020b); Grzegorzewski and Romaniuk (2022a). If we generate a sample of n fuzzy numbers using any of the considered functions we receive the output as a matrix with n rows and four columns corresponding to values a 1 , a 2 , a 3 , a 4 defined in (1) (abbreviated further as "the ends") if the parameter increases is set to FALSE, otherwise producing values l, c − s, c + s, r described in (2) (abbreviated further as "the increments").…”

“…For the second group, we are interested in the preservation of some basic characteristics of the initial values, like their canonical representation (Delgado et al, 1998). Both approaches seem to be fruitful in many aspects of the statistical inference (see Grzegorzewski et al (2019Grzegorzewski et al ( , 2020b; Grzegorzewski and Romaniuk (2022b); Romaniuk (2019); ) and may be helpful in improving the results of some real-life data analyses.…”

Bootstrap Based Diagnostics for Survival Regression Model with Interval and Right-Censored Data

“…The theoretical aspect of the single bootstrap was studied by Singh (1981) and, Abramovitch and Singh (1985). More recent works are by Zhuang, Xu, and Pang (2021) who proposed an improved two-stage method combined with fractional-random-weight bootstrap to analyze interval failure data, and Grzegorzewski, Hryniewicz, and Romaniuk (2020) who proposed a new methodology for simulating bootstrap samples of fuzzy numbers. Some earlier works on influence diagnostics for survival models are by authors such as Pettitt and Daud (1989), Escobar and Meeker Jr (1992), Weissfeld and Schneider (1990), Lesaffre and Verbeke (1998) and Ortega, Cancho, and Bolfarine (2006).…”