FRand: MATLAB Toolbox for Fuzzy Random Number Simulation

Introduction

Random variable generation from a probabilistic point of view is based on the ideas, postulates, theorems of A.N. Kolmogorov ^[8] which have been widely applied to different sciences, including the theory of random processes theory (see Grimmet & Stirzaker ^[4] and Ross ^[13]). Even more, commercial simulation software packages including ARENA, ProModel, FlexSIM, MATLAB EventSIM etc. are successful commercial applications of random variable theory and methods.

Most of probabilistic methods are based on the cumulative probability function (which is monotonic non-decreasing and injective) and the idea behind is to use uniform random numbers to generate probabilistic variables. Further information about probabilistic random variable theory and methods can be found in Devroye ^[5], Law & Kelton^[9], Monahan^[10], Grimmet & Stirzaker^[4], Wilks ^[15] and Mood et al^[11].

Analogous concepts for fuzzy numbers have been proposed by Varón-Gaviria et al. ^[14] and Pulido-López et al. ^[12]who used the well known α-cuts representation of a fuzzy set and the cumulative membership function ψ_A(x) of A (see Figueroa-García & López-Bello^[2], ^[3] and Figueroa-García^[1]). This way, it is possible to efficiently generate random variables using uniform random numbers and fuzzy sets easy to implement in general purpose code/software like MATLAB.

This paper focuses on the implementation of two fuzzy random variable generation methods into a toolbox over MATLAB general purpose software. Some graphic analysis and statistical test are performed to verify independence and randomness of the obtained samples.

This paper is divided into six sections: Section 1 introduces the main problem. Section 2 presents a theoretical background; two methods for fuzzy random variable generation are introduced in Section 3; Section 4 presents the developed MATLAB toolbox; some experiments are presented in Section 5, and finally Section 6 shows some concluding remarks.

Theoretical background

First, we refer to well known definitions for fuzzy sets/numbers (see Klir & Folger^[6], Klir & Yuan^[7]) and the two implemented methods: α-cuts and cumulative membership function.

Basics on fuzzy sets

A fuzzy set is denoted by emphasized capital letters A with a membership function over a universal set measures the membership of a value regarding the concept/word/label is the class of all crisp sets, is the class of all fuzzy sets, is the class of all real-valued fuzzy sets, and is the class of all fuzzy numbers. Thus, a fuzzy set A is a set of ordered pairs of an element x and its membership degree,

A fuzzy number is defined as follows:

Definition 1 Let be a fuzzy subset of the reals. Then is a Fuzzy Number(FN) iff there exists a closed interval with a membership function such that:

Where is monotonic non-decreasing, continuous from the right, is monotonic non-increasing, continuous from the left,

The set of elements with membership degree larger than α are called the α-cut i.e. ^αA

and its α-level namely A_α is the set of elements with membership degree equal to α, i.e.

The support of , supp(A) is the set of all elements x ∈ X with nonzero membership:

and the core of is the set of elements with maximum membership degree:

A trapezoidal fuzzy number is shown in Figure 1.

Figure 1: Fuzzy number

Implementation of fuzzy random generation methods

There are essentially two kinds of fuzzy numbers: Singleton-core i.e. and interval-corei.e. core The α-cuts method is adequate for single-core fuzzy numbers (triangular, Gaussian, exponential, etc), and the cumulative membership function approach is adequate for interval-core sets (trapezoidal, bell shaped, etc.). Both procedures are shown next:

α-cuts procedure

This method was proposed by Varón-Gaviria et al. ^{[ ₁₄ ]}for single-core fuzzy sets. The main idea is to use the core c of to compute the partial areas namely Λ₁,Λ₂ around c, as follows:

so the normalized areas are defined as follows:

Procedure 1 show the implemented routine.

Cumulative membership procedure

For interval-core fuzzy numbers, we implement the proposal of Pulido-López et al. ^{[ ₁₂ ]}based on Eqs. (11) and (14), as shown in Procedure 2:

MATLAB Toolbox implementation

The implementation of both methods within MATLAB is divided into two parts: The code for all the implemented membership functions (useful for implementing fuzzy random variables as embedded code in other MATLAB routines/toolboxes) and the code for the GUI which are described as follows. Fig. 2 shows a screenshot of the MATLAB toolbox.

Figure 2: MATLAB Toolbox

Experimentation and statistical analysis of generated data

To test the toolbox, we generated n = 1000 realizations of the following fuzzy random variables:

• >> FT (44, 60, 85, 1000)

• >> FTz (10, 15, 20, 22, 1000)

• >> FG (50,0.3, 1000)

• >> FQe (200,0.8, 1000)

• >> FGT (12, 0.6, 10, 18, 14, 0.8, 1000)

Then its mean, variance, runs test, turning points test, GAP (gap test) with the Kolmogorov Smirnovtest, Chi-square test, and autocorrelation (min:↓andmax:↑) are computed to test if the obtained sequence is an independent. The results are shown in Table I.

Table I: Independence test on simulated variables(p-values)

Runs and turning points tests show statistical significance of the hypothesis of independence on samples at a 95% confidence level (see Table I). The autocorrelation test shows a Pearson test on autocorrelations for the 10 first lags where autocorrelation. ↓ is the lowest observed autocorrelation and autocorrelation. ↑ is the largest observed autocorrelation (at a 95% confidence level).

The evidence shows that the generated samples are independent since the hypothesis of randomness is significant at a 95% confidence level (p-value ¿0.05), and there is no evidence autocorrelated samples since the hypothesis of existence of autocorrelation is not significant at a 95% confidence level (p-value ¡0.05). With the two gaps test, in the first case the Kolmogorov-Smirnov test is evaluated with a critical value Dα = 0,043 for each membership function, all the values are less than Dα, therefore the 1000 numbers are randomly ordered, and finally the Chi-Square test, presented a value χ² = 44,98 with 95 % confidence level and for . All the values obtained allow us to infer that the hypothesis that data are randomly distributed is significant for every membership function.

Figures 3-7 shows the exact location of the simulated values and their membership degree α.

Figure 3: Simulated triangular random variables

Figure 4: Simulated trapezoidal random variables

Figure 5: Simulated quasi-exponential random variables

Figure 6: Simulated Gaussian random variables

Figure 7: Simulated double-exponential random variables

Note that Figs. 3-7 are good representations of each fuzzy number since the obtained shapes correspond to each membership function.

Concluding Remarks

The the oretical results of Varón-Gaviria et al. ^{[ ₁₄ ]} and Pulido-López et al. ^{[ ₁₂ ]}were successfully implemented as MATLAB functions. All of them are available to download for free, and the are compatible with MATLAB 9.2 R2017a and later versions.

A theoretical background has been provided to explain how to use the implemented MATLAB’s functions and toolbox. All functions are easy to understand, use, and are also complaint with MATLAB specifications for coding and compatibility.

Finally, some experiments are presented and tested using statistical tests on randomness and autocorrelation. The obtained results show generated samples to be independent and random, so the evidence supports the idea that the toolbox provides random samples.