Research Article Novel Construction of Copulas Based on ( α , β ) Transformation for Fuzzy Random Variables

The paper introduces a method for the construction of bivariate copulas with the usage of speciﬁc values of the parameters α and β ( ( α , β ) transformation) and the parameters κ and λ in their domain. The produced bivariate copulas are deﬁned in four subrectangles of the unit square. The bounds of the produced copulas are investigated, while a novel construction method for fuzzy copulas is introduced, with the usage of the produced copulas via ( α , β ) transformation in four subrectangles of the unit square. Following this construction procedure, the production of an inﬁnite number of copulas and fuzzy copulas could be possibly achieved. Some applications of the proposed methods are presented.


Introduction
Copulas are a significant member of aggregation functions on the unit interval [0,1]. e ability to construct aggregation functions with numerous processes and methods is of great importance, since it is essential for the researchers to not deviate from the real-life data. Sklar [1] presented the concept of copulas, by means of a mathematical tool that describes the stochastic dependence structure within random variables. ere are several procedures regarding the construction of copulas, based on given ones, in the literature, such as the construction of asymmetric multivariate copulas [2], which is connected with the product of copulas and the generalization of Archimedean copulas. Another construction of copulas, produced by the gluing of two or more copulas, is presented in [3]. In [4], three types of ordinal sums based on product copula are introduced as construction methods. A representation via the g-ordinal sums of copulas is introduced in [5]. In [6], a method for the construction of bivariate copulas by the modifications of given copulas on some subrectangles of the unit square is contained. Two different representations of 2-increasing aggregation functions, via the lower and the upper margins and a copula, are provided in [7]. e construction of copulas as a patchwork-like assembly of arbitrary copulas, with nonoverlapping rectangles as patches, is included in [8].
In [9], the set of copulas with the given horizontal section was studied and extended. e family of (α, β)-homogenous copulas was introduced in [10]. A general construction of copulas with given a horizontal and a vertical section is introduced in [11].One of the most important methods is the flipped and survival copulas [12]. ose are special cases of the (α, β)-transformation, which is a more general construction method [12].
On the other hand, regarding the real-life problems, researchers may handle data possibly imprecise. In order to deal with imprecise or vague information, fuzzy sets [13] are the most adequate tools for someone to establish. In [14], the fuzzy random variables are provided in order to represent the relationship between random experiments results and nonstatistical imprecise data.
us, the notion of fuzzy copulas is introduced in [15] to describe the stochastic dependence structure between two fuzzy random variables.
is paper shares two main purposes. e first is to provide a novel construction method for copulas, more general than the existing ones, based on the (α, β)-transformation. e second is to present a new fuzzy copula construction between two fuzzy random variables, via the construction mentioned before. e investigation of fuzzy copulas is of great importance, as in many circumstances, researchers need to fuse or aggregate probabilistic and fuzzy information [16][17][18]. In the present paper, we aim to provide a novel construction method of copulas, in order to produce a construction procedure of fuzzy copulas that no attempt has been made since the concept of fuzzy copulas was recently developed. e properties of the proposed copula and fuzzy copula construction methods are taken into consideration. Meanwhile, the bounds of the constructed copulas are presented. In conclusion, the proposed methods are illustrated via some numerical examples. e paper adopts the following structure: in Section 2, some necessary notions of copulas, fuzzy sets, and fuzzy random variables are presented. In Section 3, the novel construction of copulas and the bounds of the produced copulas are presented. In Section 4, the new construction method of fuzzy copulas is introduced. In Section 5, concluding remarks are mentioned.

Preliminaries
In this section, some basic definitions are provided, in order for the new construction method of copulas and fuzzy copulas to be introduced.

Copulas, the Crisp Approach, Notions, and Definitions
Definition 1 (see [1] e definition of survival copula is provided as follows, according to [12]. Definition 2 (see [12]). e survival copula of a copula is defined as According to Nelsen [12], let X and Y be two random variables with joint distribution function F XY , with margins F X and F Y . en, there exists a copula C, such that Also, if C is a copula and F X and F Y are cumulative distribution functions, then F X,Y is a joint distribution function, with margins F X and F Y . If F − 1 X and F − 1 Y are the inverses of F X and F Y , respectively, then As it was mentioned in Section 1, a construction method of copulas is (α, β)-transformation that is defined [12] for parameters α and β, that is, (α, β) ∈ [0, 1] 2 . e transformation of a copula C into the copula C α,β is defined on the unit square by As a result, for specific values of the parameters α and β (in their domain), different copulas can be constructed. In case that α � 0 and β � 0, we obtain the copula: In case that α � 0 and β � 1, we obtain the copula: In case that α � 1 and β � 0, we obtain the copula: In case that α � 1 and β � 1, we obtain the copula: In the first case, we get the original copula; in the second and third cases, we get the flipped copulas; and in the last case, we get the survival copula.

Fuzzy Sets, Notions, and Definitions.
Let X be a universal set. Each function A: X ⟶ [0, 1] is called a fuzzy set of X, where μ A (x)'s interpretation is the membership degree of x in the fuzzy set A. Crisp (classical) sets are special cases of fuzzy sets, with A(x) � 0, or A(x) � 1. e α-cuts of a fuzzy set A are defined by , which is called support, is the closure in the topology of X of the union of all the α-cuts [19], i.e., (10) which means that A is convex fuzzy set 3 ∀α ∈ [0, 1], A [α] is a nonempty compact interval in R, which means that A has compact support e interval of the α-cuts is denoted by . We denote the set of all fuzzy numbers by F(R). In [20], based on [21], some of the operations of α-cuts were presented as follows.
Definition 3 (see [22]). Let A be a fuzzy number (A ∈ F(R)) and x ∈ R. en, the index Cr: is gives the credibility degree, that is, A is less than or equal to x.
Remark 1 (see [22]). Let A ∈ F(R) and x ∈ R, then 1 Cr if and only if 2 For any fixed A, Cr A ≤ x is a nondecreasing function with respect to x, i.e., 3 Cr is self-dual, i.e., Definition 4 (see [23]). Let A ∈ F(R) and α ∈ (0, 1], then the α-pessimistic value of A is given by Remark 2 (see [24]). For a given A ∈ F(R), let A α be defined ∀α ∈ (0, 1], by en, the α-cuts of A are given by Lemma 1 (see [15]). Let A, B ∈ F(R) and let λ ∈ R. en, Proof. e proof of Lemma 1 can be found in [15].
Example 1. Let x ∈ R and A be a nonsymmetric triangular fuzzy number with membership function given by en, the credibility degree that A is less than or equal to x is given by As a result, we obtain the α-pessimistic values of A by

Fuzzy Random
Variables. e concept of fuzzy random variables is one of the most adequate tools to handling the results of random experiments, expressed in nonexact terms. In order to integrate randomness and vagueness, random fuzzy sets and random fuzzy numbers [25], are introduced. In most real-life problems, the nature of the data of the experiments is affected by fuzziness, and the procedure of Journal of Mathematics the extraction of the data of the experiments is affected by randomness. us, the definition of fuzzy random variables to be considered in this paper was given in [24], as in the following definition. Definition 6. (see [24]). Let Ω be the set of all possible outcomes of a random experiment, let A be the σ-algebra of the subsets of Ω, and let P be the probability measure on the measurable space (Ω, A). Now, suppose that the probability space (Ω, A, P) describes the random experiment. If ∀α ∈ [0, 1], X α : Ω ⟶ R is a random variable on (Ω, A, P), then X: Ω ⟶ F(R) is called a fuzzy random variable. e notion of fuzzy random variables was introduced in [14,26]. In [27], the notion of fuzzy random variables was formalized with the following approach: let (Ω, A, P) be a probability space. If ∀α ∈ [0, 1], the two mappings X L α : Ω ⟶ R and X R α : Ω ⟶ R are random variables, then X: Ω ⟶ F(R) is a fuzzy random variable.

Remark 3.
(see [28]). In the next relationships, summarize the data of the two dimensional variable (X Definition 7 (see [15]). If X α and Y α are independent ∀α ∈ [0, 1], then X and Y are independent fuzzy random variables. If X α and Y α are identically distributed ∀α ∈ [0, 1], then X and Y are identically distributed fuzzy random variables.

The Novel Construction Method of Copulas
In this section, a novel construction method of copulas is provided, via the (α, β)-transformation, in four different subrectangles of the unit square. is becomes feasible by the jointing process of the four cases produced in Section 2.1 for specific values of the parameters α and β, with the adequate adjustments. e construction is achieved through the following theorem.
is also a copula.
Proof. e proof that the function C is well defined and the boundary conditions of Definition 1 are satisfied is straightforward. In order to prove that C is double-increasing, the proof that C is double-increasing in each one of the four rectangles of the domain is needed. For the first rectangle, we have 0 ≤ x 1 ≤ x 2 ≤ κ and 0 ≤ y 1 ≤ y 2 ≤ λ, and with the usage of the first branch of C, we obtain since κ, λ ∈ [0, 1] and C is double-increasing as a copula.
λ ≤ y 1 ≤ y 2 ≤ 1, and with the usage of the second branch of C, we obtain and with the usage of the third branch of C, we obtain  e flowchart in Figure 1 illustrates the novel copula construction process. e following example illustrates the construction of a copula, with the usage of eorem 1.
Obtain the copula ζ. via the copula C.
Stop Figure 1: Flowchart of copula construction process.
Proof. e proof that C ∼ and C ∼ have DomC ∼ � DomC ∼ � [0, 1] 2 and RanC ∼ � RanC ∼ � [0, 1], and that satisfy the boundary conditions of copulas is straightforward. In addition, for the proof that C ∼ and C ∼ are double-increasing, we have to examine this in each one of the four rectangles of their domain. For the function C ∼ , we have that ∀x 1 , x 2 ∈ [0, κ] and ∀y 1 , y 2 ∈ [0, λ], with x 1 ≤ x 2 and y 1 ≤ y 2 :  κ)) ≤ 1 and 0 ≤ (y 1 /λ) ≤ (y 2 /λ) ≤ 1 : . As a result, C ∼ is a copula. e proof that C ∼ is a copula can be considered in the same manner. For the case of the first rectangle, we have 0 ≤ x ≤ κ and 0 ≤ y ≤ λ, and with the usage of the first branch of copula C and the Fréchet-Hoeffding [12] bounds of copulas, we obtain For the cases of the other three rectangles, with the same approach as the case of the first rectangle, we obtain the desirable equations. Hence, C ∼ and C ∼ are copulas, and the bounds of every copula C, and as a result, the proof is completed. □ e present construction method of copulas that can be achieved through eorem 1, and based on the fact that C ∼ ≠ C ∼ , of eorem 2, can lead us in the result that there exists an infinite number of copulas that can be constructed via this transformation. e disadvantage of this method is that, for the construction of copula C, we are using for each of the four branches only the copula C.
/2, and copulas C ∼ and C ∼ of eorem 2. en, the produced copulas for those specific values of κ, λ are given by , , e plots of C ∼ and C ∼ are illustrated in Figures 3 and 4, respectively.

The Novel Construction Method of Copulas for Fuzzy Random Variables
e extension of copulas was achieved in [15], through the notion of fuzzy copula functions of two fuzzy random variables X and Y at (x, y) ∈ [0, 1] × [0, 1], with the following α-cuts: As a result, C is a joint fuzzy distribution function [15]. e following proposition examines the properties of the fuzzy copula.

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Journal of Mathematics e proof of this proposition can be found in [15]. Next, inspired by the fuzzy copula for fuzzy random variables, we propose a novel method for the construction of fuzzy copulas. is is achieved through the following theorem.

Journal of Mathematics 9
is the fuzzy copula of the fuzzy random variables X and Y.
Proof. For the first branch of C, we have that, for x � 0 and ∀y ∈ [0, λ], As a result, C(0, y) � 0. e proof that C(x, 0) � 0 can be considered in the same manner. e next step is to examine the second condition that holds in the special case of κ � 1 and λ � 1. In this case, C becomes C, and the proof is straightforward. In the last step, we have that ∀x 1 , x 2 ∈ [0, κ] and ∀y 1 , y 2 ∈ [0, λ], where x 1 < x 2 and y 1 < y 2 and C x 2 , y 2 ⊕C x 1 , y 1 ≻C x 1 , y 2 ⊕C x 2 , y 1 ⇔. (41) Since κ, λ ≥ 0, ∀α ∈ (0, 1], we have that As a result, the proof for the first branch is completed. Next, for the case of the second branch, of C, we have that ∀x ∈ [0, κ] and for y � 1, Hence, we obtain For the examination of the third condition, we have that ∀x 1 , x 2 ∈ [0, κ] and ∀y 1 , y 2 ∈ [λ, 1], where x 1 < x 2 and y 1 < y 2 and is holds if we consider that since λ ≤ y 1 < y 2 ≤ 1, then 0 ≤ ((1 − y 2 )/(1 − λ)) < ((1 − y 1 )/(1 − λ)) ≤ 1. Hence, the proof of the second branch is completed. e cases of the third and the fourth branches of C can be examined in a similar way as the first and the second branches of C. Hence, the proof that C(x, y) is a fuzzy copula is completed. □ e flowchart of Figure 5 illustrates the novel fuzzy copula construction process. Now, in order to illustrate the novel construction of fuzzy copula, the following example is provided, based on an example that may be found in [15].
Example 4. Let X and Y be random variables that have joint distribution function, given by As a result, X ∼ U[− 1, 1] and Y ∼ Weibull(1, 1). Let Θ � (2; 1, 4) T and Γ � (8; 5, 9) T be two nonsymmetric triangular fuzzy numbers and let X � Θ + X and Y � Γ + Y. Hence, we have that X � (X + 2; 1, 4) T and Y � (Y + 8; 5, 9) T . erefore, we obtain the α-pessimistic values of X and Y, respectively, by Based on the fact that ∀α ∈ (0, 1], X α has uniform distribution and Y α has Weibull distribution. Hence, Weibull(2α + 7, 1, 1), Also, we have the α-pessimistic values of Θ and Γ given as follows: Hence, we obtain the next α-pessimistic values: Now, ∀α ∈ (0, 1], and we get the inverse functions of (F X (x)) α and (F Y (y)) α by Also, we have that us, we obtain the left and right parts of the α-cuts of the fuzzy copula by , v)]. Now, based on eorem 3, we have that, for the first branch of C, Start Choose the fuzzy copula C.
Obtain the fuzzy copula ζ. via the fuzzy copula C. Figure 5: Flowchart of fuzzy copula construction process. 12 Journal of Mathematics

Stop
where For the third branch of C, we have that ∀u ∈ [κ, 1] and v ∈ [0, λ], Finally, for the case of the fourth branch of C, we have that ∀u ∈ [κ, 1] and v ∈ [λ, 1], where C L α (u, v) � u α/2 + v α/2 − 1 + Journal of Mathematics 13 In Example 3 of [15], the production of a fuzzy copula C was established. In the illustrative Example 4 of the present paper, the generalization of this result was presented with the usage of (α, β) transformation, in four different subrectangles of the unit square.

Conclusions
We provided a method for the construction of copulas in four subrectangles of the unit square, via the (α, β) transformation, and we introduced the upper and lower bounds of the produced copulas of this method. Also, we developed a method to construct fuzzy copulas for fuzzy random variables. With the usage of those methods, we can conclude that the construction of an infinite number of copulas and fuzzy copulas can be achieved. Each one of the produced copulas and fuzzy copulas could be applied based on their adequacy in real-life problems. It may also be of interest considering these construction methods in the case of n-dimensional copulas n > 2. On the other hand, the case in which these construction methods could possibly be defined in more than four subrectangles of the unit square may be examined. In addition, the extension of those methods for Intuitionistic fuzzy sets [16,18] and Pythagorean fuzzy sets [31] could be possibly achieved, in order to develop aggregation operators for multiple attribute decision making algorithms. ese topics are the basis for our future investigations.

Data Availability
No data were used to support the findings of the study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.