Flexible bootstrap based on the canonical representation of fuzzy numbers

Abstract

A new resampling approach for simulating bootstrapped samples of fuzzy numbers is proposed. The secondary samples consist of fuzzy numbers which preserve the canonical representation (i.e., the value and ambiguity) of fuzzy numbers belonging to the primary sample, although may differ from the initial ones. This way the resulting bootstrap distribution has a richer support than obtained with the conventional method. Numerical experiments concerning two statistical tests for the expected value of a fuzzy random variable illustrate the suggested VAbootstrap method and some of its properties.

Flexible Bootstrap for Fuzzy Data Based on the Canonical Representation

“…To avoid undesired repetitions which often appear in bootstrap Romaniuk and Hryniewicz [17][18][19] proposed a resampling method which enrich secondary samples with fuzzy observations imitating those from the primary sample but containing some incremental spreads on their 𝛼-cuts. Then Grzegorzewski et al [21,28] suggested another approach for generating bootstrap samples which may differ from the primary one but preserve the two-parameter canonical representation of each fuzzy observation, i.e., its value and ambiguity or the expected value and the width. We briefly discuss this method below, before introducing in Section 5 a new flexible bootstrap algorithm which preserves three-parameter canonical representation comprising the value, ambiguity and the fuzziness of a fuzzy number.…”

“…for each i = 1, … , n, we draw randomly one triple from the set (21) and denoting it by (Val * , Amb * , Fuzz * ), we follow the consecutive transformations as in Section 4.2 which lead us to Eq. ( 16), where besides the given value Val * and the ambiguity Amb * one can find two unknown constants c and s. However, let us recall that there exist a one-to-one correspondence between s and the fuzziness of a fuzzy number, provided we restrict our attention to trapezoidal fuzzy numbers only.…”

“…In the same manner as in Section 7.1, we conduct a power study for the C2-test. Because of the form of the hypotheses (21), the special approach has to be also applied. The number of rejections is estimated under increasing shift 𝜖 ∈ ℝ of realizations of the second initial fuzzy sample, i.e., Y, if both of the samples X, Y are generated using the same type of fuzzy numbers.…”

“…Flexible bootstrap was suggested for the first time in Ref. [21], where the algorithm for generating fuzzy numbers with fixed twoparameter canonical representation, i.e., the value and ambiguity, was considered. In this paper, we examine extensively this kind of the flexible bootstrap for triangular and trapezoidal fuzzy numbers.…”

خودارزیابی سیستم‌های تولید شبکه‌یی درمحیط‌های فازی شهودی (مطالعه‌موردی:شرکت سیم و کابل مغان)

“…=@ EV ( ÃI ) '13 |x]@= Q R= xO=iDU= =@ w CU= xOW h} QaD EV ÃI = (a 1 0 + a1 + 4a2 + a3 + a 3 0) =8 ( 15) [31]…”