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Simulation of Random LR Fuzzy Intervals

Maciej Romaniuk, Abbas Parchami, Przemysław Grzegorzewski

TL;DR

Starting from a simulation perspective on the piecewise linear LR fuzzy numbers with the interval-valued cores, their limiting behavior is considered and this leads to the numerically efficient algorithm for simulating a sample consisting of such fuzzy values.

Abstract

Random fuzzy variables join the modeling of the impreciseness (due to their ``fuzzy part'') and randomness. Statistical samples of such objects are widely used, and their direct, numerically effective generation is therefore necessary. Usually, these samples consist of triangular or trapezoidal fuzzy numbers. In this paper, we describe theoretical results and simulation algorithms for another family of fuzzy numbers -- LR fuzzy numbers with interval-valued cores. Starting from a simulation perspective on the piecewise linear LR fuzzy numbers with the interval-valued cores, their limiting behavior is then considered. This leads us to the numerically efficient algorithm for simulating a sample consisting of such fuzzy values.

Simulation of Random LR Fuzzy Intervals

TL;DR

Starting from a simulation perspective on the piecewise linear LR fuzzy numbers with the interval-valued cores, their limiting behavior is considered and this leads to the numerically efficient algorithm for simulating a sample consisting of such fuzzy values.

Abstract

Random fuzzy variables join the modeling of the impreciseness (due to their ``fuzzy part'') and randomness. Statistical samples of such objects are widely used, and their direct, numerically effective generation is therefore necessary. Usually, these samples consist of triangular or trapezoidal fuzzy numbers. In this paper, we describe theoretical results and simulation algorithms for another family of fuzzy numbers -- LR fuzzy numbers with interval-valued cores. Starting from a simulation perspective on the piecewise linear LR fuzzy numbers with the interval-valued cores, their limiting behavior is then considered. This leads us to the numerically efficient algorithm for simulating a sample consisting of such fuzzy values.
Paper Structure (6 sections, 1 theorem, 16 equations, 3 figures, 3 algorithms)

This paper contains 6 sections, 1 theorem, 16 equations, 3 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Simulation approach for piecewise LRFIs
  4. Limiting behavior
  5. Simulation of a fuzzy random sample
  6. Conclusion

Key Result

Theorem 4.1

Let $\tilde{X}_k = \left[f_O, f_{C^l}, f_{C^r}, f_{S^l},f_{S^r} \right]_k$ be a $k$-knot piecewise LRFI. Then for any fixed $x$, $\tilde{X}$ converges in probability to the random fuzzy number $\tilde{X}(x)$ as $k\to\infty$, where $\tilde{X}(x)$ has the following membership function

Figures (3)

  • Figure 1: The membership function of $\tilde{X}_k$ in Example \ref{['ExamA1']}.
  • Figure 2: Limiting behavior of $\tilde{X}_k = [N(0,1), U(0,1),U(0,1),U(0,2),\mathop{\mathrm{Exp}}\nolimits(1)]_k$ as a function of $k$.
  • Figure 3: Exemplary sample with $n=3$ elements generated for $\tilde{X} = [N(0,1), U(0,1),U(0,1),U(0,2),\mathop{\mathrm{Exp}}\nolimits(1)]$.

Theorems & Definitions (3)

  • Example 3.1
  • Theorem 4.1
  • proof