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

Authors: Stylianos Giakoumakis; Basil Papadopoulos
Journal: Journal of Mathematics
 (2021)

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## Abstract

The paper introduces a method for the construction of bivariate copulas with the usage of specific values of the parametersαandβ (α,β transformation) and the parameters κandλ 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.


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## Body

## 1. Introduction

Copulas are a significant member of aggregation functions on the unit interval [0, 1]. The 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. There 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]. The 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. The 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]. Those 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. Thus, the notion of fuzzy copulas is introduced in [15] to describe the stochastic dependence structure between two fuzzy random variables.This paper shares two main purposes. The first is to provide a novel construction method for copulas, more general than the existing ones, based on theα,β-transformation. The second is to present a new fuzzy copula construction between two fuzzy random variables, via the construction mentioned before. The investigation of fuzzy copulas is of great importance, as in many circumstances, researchers need to fuse or aggregate probabilistic and fuzzy information [16–18]. 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. The 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.The paper adopts the following structure: in Section2, 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.


## 2. Preliminaries

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


### 2.1. Copulas, the Crisp Approach, Notions, and Definitions

Definition 1.
(see [1]). A 2-dimensional copula is a function with domain DomC=0,1×0,1 and range RanC=0,1 that is grounded and double-increasing, i.e., it satisfies the following conditions:(1)Cx,0=C0,y=0,Cx,1=x,C1,y=y,VCx1,x2×y1,y2≥0,where x,y,x1,x2,y1,y2∈0,1 and x1≤x2,y1≤y2.The definition of survival copula is provided as follows, according to [12].Definition 2.
(see [12]). The survival copula of a copula is defined as(2)C^x,y=x+y−1+C1−x,1−y.According to Nelsen [12], let XandY be two random variables with joint distribution function FXY, with margins FX and FY. Then, there exists a copula C, such that(3)FXYx,y=CFXx,FYy,∀x,y∈ℝ×ℝ.Also, ifC is a copula and FX and FY are cumulative distribution functions, then FX,Y is a joint distribution function, with margins FX and FY. If FX−1 and FY−1 are the inverses of FX and FY, respectively, then(4)Cu,v=FXYFX−1u,FY−1v,∀u,v∈0,1×0,1.As it was mentioned in Section1, a construction method of copulas is α,β-transformation that is defined [12] for parameters αandβ, that is, α,β∈0,12. The transformation of a copula C into the copula Cα,β is defined on the unit square by(5)Cα,βx,y=VCα1−x,x+α1−x×β1−y,y+β1−y.As a result, for specific values of the parametersα and β (in their domain), different copulas can be constructed. In case that α=0 and β=0, we obtain the copula:(6)C0,0x,y=Cx,y.In case thatα=0 and β=1, we obtain the copula:(7)C0,1x,y=x−Cx,1−y.In case thatα=1 and β=0, we obtain the copula:(8)C1,0x,y=y−C1−x,y.In case thatα=1 and β=1, we obtain the copula:(9)C1,1x,y=x+y−1+C1−x,1−y=C^x,y.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.


### 2.2. Fuzzy Sets, Notions, and Definitions

LetX be a universal set. Each function A˜:X⟶0,1 is called a fuzzy set of X, where μA˜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. The α-cuts of a fuzzy set A˜ are defined by A˜α=x:μA˜x≥α,∀α∈0,1, where A˜0, which is called support, is the closure in the topology of X of the union of all the α-cuts [19], i.e., A˜0=∪α∈0,1A˜α¯=x:μA˜x>0¯. Now, a fuzzy set A˜ of ℝ is called a fuzzy number if1
∃x∈ℝ:μA˜x=1, which means that A˜ is normal2
∀k∈0,1 and x1,x2∈ℝ; we have(10)μA˜1−kx1+kx2≥minμA˜x1,μA˜x2,which means thatA˜ is convex fuzzy set3
∀α∈0,1,A˜α is a nonempty compact interval in ℝ, which means that A˜ has compact supportThe interval of theα-cuts is denoted by A˜α=A˜αL,A˜αR, where A˜αL=infx:x∈A˜α and A˜αR=supx:x∈A˜α. We denote the set of all fuzzy numbers by ℱℝ. In [20], based on [21], some of the operations of α-cuts were presented as follows.LetA˜,B˜∈ℱℝ and A˜α=A˜αL,A˜αR,B˜α=B˜αL,B˜αR be their α-cuts, respectively. Then, ∀α∈0,1, and the fuzzy addition and the scalar multiplication were defined as follows:(11)A˜⊕B˜α=A˜αL+B˜αL,A˜αR+B˜αR,λ⊗A˜α=λ,λA˜αL,A˜αR=λA˜αL,λA˜αR,respectively, where the scalar λ∈ℝ+ was identified as the interval λ,λ.Definition 3.
(see [22]). Let A˜ be a fuzzy number A˜∈ℱℝ and x∈ℝ. Then, the index Cr:ℱℝ×ℝ⟶0,1 is defined by(12)CrA˜≤x=12supy≤xμA˜y+1−supy>xμA˜y.
This gives the credibility degree, that is,A˜ is less than or equal to x.Remark 1.
(see [22]). Let A˜∈ℱℝ and x∈ℝ, then1
(13)CrA˜≤x=1,if and only if(14)A˜0R≤x.2
For any fixedA˜, CrA˜≤x is a nondecreasing function with respect to x, i.e.,(15)∀x1<x2,CrA˜≤x1≤CrA˜≤x2.3
Cr is self-dual, i.e.,(16)CrA˜>x=1−CrA˜≤x.Definition 4.
(see [23]). Let A˜∈ℱℝ and α∈0,1, then the α-pessimistic value of A˜ is given by A˜α=infx∈A˜0:CrA˜≤x≥α. Furthermore, A˜α is a nondecreasing function of α∈0,1.Remark 2.
(see [24]). For a given A˜∈ℱℝ, let A˜α be defined ∀α∈0,1, by(17)A˜α=A˜2αL,α∈0,12,A˜21−αR,α∈12,1.
Then, theα-cuts of A˜ are given by(18)A˜α=A˜α/2,A˜1−α/2.Lemma 1.
(see [15]). Let A˜,B˜∈ℱℝ and let λ∈ℝ. Then,(19)A˜⊕B˜α=A˜α+B˜α,λ⊗A˜α=λA˜α,λ≥0,λA˜1−α,λ≤0.Proof.
The proof of Lemma1 can be found in [15].Definition 5.
(see [24]). Let A˜,B˜∈ℱℝ. ∀α∈0,1. If A˜α≤B˜α, then A˜≺¯B˜, and if A˜α=B˜α, then A˜=B˜.Example 1.
Letx∈ℝ and A˜ be a nonsymmetric triangular fuzzy number with membership function given by(20)μA˜x=0,−∞<x≤5,x−53,5≤x≤8,1,x=8,9−x,8≤x≤9,0,9≤x<+∞.
Then, the credibility degree thatA˜ is less than or equal to x is given by(21)CrA˜≤x=0,−∞<x≤5,x−56,5≤x≤8,12,x=8,x−72,8≤x≤9,1,9≤x<+∞.
As a result, we obtain theα-pessimistic values of A˜ by(22)A˜α=6α+5,0<α≤12,2α+7,12≤α≤1.


### 2.3. Fuzzy Random Variables

The 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 the extraction of the data of the experiments is affected by randomness. Thus, 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⟶ℱℝ is called a fuzzy random variable.The 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˜αL:Ω⟶ℝ and X˜αR:Ω⟶ℝ are random variables, then X˜:Ω⟶ℱℝ is a fuzzy random variable.Remark 3.
(see [28]). In the next relationships, summarize the data of the two dimensional variable X˜αL,X˜αR, in the one-dimensional variable X˜α.(23)X˜α=X˜2αL,0<α≤12,X˜21−αR,12≤α≤1,X˜α=X˜α/2,X˜1−α/2,α∈0,1.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.


## 2.1. Copulas, the Crisp Approach, Notions, and Definitions

Definition 1.
(see [1]). A 2-dimensional copula is a function with domain DomC=0,1×0,1 and range RanC=0,1 that is grounded and double-increasing, i.e., it satisfies the following conditions:(1)Cx,0=C0,y=0,Cx,1=x,C1,y=y,VCx1,x2×y1,y2≥0,where x,y,x1,x2,y1,y2∈0,1 and x1≤x2,y1≤y2.The definition of survival copula is provided as follows, according to [12].Definition 2.
(see [12]). The survival copula of a copula is defined as(2)C^x,y=x+y−1+C1−x,1−y.According to Nelsen [12], let XandY be two random variables with joint distribution function FXY, with margins FX and FY. Then, there exists a copula C, such that(3)FXYx,y=CFXx,FYy,∀x,y∈ℝ×ℝ.Also, ifC is a copula and FX and FY are cumulative distribution functions, then FX,Y is a joint distribution function, with margins FX and FY. If FX−1 and FY−1 are the inverses of FX and FY, respectively, then(4)Cu,v=FXYFX−1u,FY−1v,∀u,v∈0,1×0,1.As it was mentioned in Section1, a construction method of copulas is α,β-transformation that is defined [12] for parameters αandβ, that is, α,β∈0,12. The transformation of a copula C into the copula Cα,β is defined on the unit square by(5)Cα,βx,y=VCα1−x,x+α1−x×β1−y,y+β1−y.As a result, for specific values of the parametersα and β (in their domain), different copulas can be constructed. In case that α=0 and β=0, we obtain the copula:(6)C0,0x,y=Cx,y.In case thatα=0 and β=1, we obtain the copula:(7)C0,1x,y=x−Cx,1−y.In case thatα=1 and β=0, we obtain the copula:(8)C1,0x,y=y−C1−x,y.In case thatα=1 and β=1, we obtain the copula:(9)C1,1x,y=x+y−1+C1−x,1−y=C^x,y.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.


## 2.2. Fuzzy Sets, Notions, and Definitions

LetX be a universal set. Each function A˜:X⟶0,1 is called a fuzzy set of X, where μA˜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. The α-cuts of a fuzzy set A˜ are defined by A˜α=x:μA˜x≥α,∀α∈0,1, where A˜0, which is called support, is the closure in the topology of X of the union of all the α-cuts [19], i.e., A˜0=∪α∈0,1A˜α¯=x:μA˜x>0¯. Now, a fuzzy set A˜ of ℝ is called a fuzzy number if1
∃x∈ℝ:μA˜x=1, which means that A˜ is normal2
∀k∈0,1 and x1,x2∈ℝ; we have(10)μA˜1−kx1+kx2≥minμA˜x1,μA˜x2,which means thatA˜ is convex fuzzy set3
∀α∈0,1,A˜α is a nonempty compact interval in ℝ, which means that A˜ has compact supportThe interval of theα-cuts is denoted by A˜α=A˜αL,A˜αR, where A˜αL=infx:x∈A˜α and A˜αR=supx:x∈A˜α. We denote the set of all fuzzy numbers by ℱℝ. In [20], based on [21], some of the operations of α-cuts were presented as follows.LetA˜,B˜∈ℱℝ and A˜α=A˜αL,A˜αR,B˜α=B˜αL,B˜αR be their α-cuts, respectively. Then, ∀α∈0,1, and the fuzzy addition and the scalar multiplication were defined as follows:(11)A˜⊕B˜α=A˜αL+B˜αL,A˜αR+B˜αR,λ⊗A˜α=λ,λA˜αL,A˜αR=λA˜αL,λA˜αR,respectively, where the scalar λ∈ℝ+ was identified as the interval λ,λ.Definition 3.
(see [22]). Let A˜ be a fuzzy number A˜∈ℱℝ and x∈ℝ. Then, the index Cr:ℱℝ×ℝ⟶0,1 is defined by(12)CrA˜≤x=12supy≤xμA˜y+1−supy>xμA˜y.
This gives the credibility degree, that is,A˜ is less than or equal to x.Remark 1.
(see [22]). Let A˜∈ℱℝ and x∈ℝ, then1
(13)CrA˜≤x=1,if and only if(14)A˜0R≤x.2
For any fixedA˜, CrA˜≤x is a nondecreasing function with respect to x, i.e.,(15)∀x1<x2,CrA˜≤x1≤CrA˜≤x2.3
Cr is self-dual, i.e.,(16)CrA˜>x=1−CrA˜≤x.Definition 4.
(see [23]). Let A˜∈ℱℝ and α∈0,1, then the α-pessimistic value of A˜ is given by A˜α=infx∈A˜0:CrA˜≤x≥α. Furthermore, A˜α is a nondecreasing function of α∈0,1.Remark 2.
(see [24]). For a given A˜∈ℱℝ, let A˜α be defined ∀α∈0,1, by(17)A˜α=A˜2αL,α∈0,12,A˜21−αR,α∈12,1.
Then, theα-cuts of A˜ are given by(18)A˜α=A˜α/2,A˜1−α/2.Lemma 1.
(see [15]). Let A˜,B˜∈ℱℝ and let λ∈ℝ. Then,(19)A˜⊕B˜α=A˜α+B˜α,λ⊗A˜α=λA˜α,λ≥0,λA˜1−α,λ≤0.Proof.
The proof of Lemma1 can be found in [15].Definition 5.
(see [24]). Let A˜,B˜∈ℱℝ. ∀α∈0,1. If A˜α≤B˜α, then A˜≺¯B˜, and if A˜α=B˜α, then A˜=B˜.Example 1.
Letx∈ℝ and A˜ be a nonsymmetric triangular fuzzy number with membership function given by(20)μA˜x=0,−∞<x≤5,x−53,5≤x≤8,1,x=8,9−x,8≤x≤9,0,9≤x<+∞.
Then, the credibility degree thatA˜ is less than or equal to x is given by(21)CrA˜≤x=0,−∞<x≤5,x−56,5≤x≤8,12,x=8,x−72,8≤x≤9,1,9≤x<+∞.
As a result, we obtain theα-pessimistic values of A˜ by(22)A˜α=6α+5,0<α≤12,2α+7,12≤α≤1.


## 2.3. Fuzzy Random Variables

The 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 the extraction of the data of the experiments is affected by randomness. Thus, 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⟶ℱℝ is called a fuzzy random variable.The 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˜αL:Ω⟶ℝ and X˜αR:Ω⟶ℝ are random variables, then X˜:Ω⟶ℱℝ is a fuzzy random variable.Remark 3.
(see [28]). In the next relationships, summarize the data of the two dimensional variable X˜αL,X˜αR, in the one-dimensional variable X˜α.(23)X˜α=X˜2αL,0<α≤12,X˜21−αR,12≤α≤1,X˜α=X˜α/2,X˜1−α/2,α∈0,1.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.


## 3. The Novel Construction Method of Copulas

In this section, a novel construction method of copulas is provided, via theα,β-transformation, in four different subrectangles of the unit square. This 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. The construction is achieved through the following theorem.Theorem 1.
Letκ and λ be fixed in [0, 1] and C:0,1×0,1⟶0,1 be any copula. Then, the function ℭ, defined by(24)ℭx,y=Cxκ,yλκλ,x,y∈0,κ×0,λ,x−Cxκ,1−y1−λκ1−λ,x,y∈0,κ×λ,1,y−C1−x1−κ,yλ1−κλ,x,y∈κ,1×0,λ,x+y−1+C1−x1−κ,1−y1−λ1−κ1−λ,x,y∈κ,1×λ,1,is also a copula.Proof.
The proof that the functionℭ 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≤x1≤x2≤κ and 0≤y1≤y2≤λ, and with the usage of the first branch of ℭ, we obtain(25)Vℭx1,x2×y1,y2=κλVCx1κ,x2κ×y1λ,y2λ≥0since κ,λ∈0,1 and C is double-increasing as a copula. Hence, ℭ is double-increasing in the rectangle 0,κ×0,λ. For the rectangle 0,κ×λ,1, we have 0≤x1≤x2≤κ and λ≤y1≤y2≤1, and with the usage of the second branch of ℭ, we obtain(26)Vℭx1,x2×y1,y2=κ1−λVCx1κ,x2κ×1−y21−λ,1−y11−λ≥0since κ,1−λ∈0,1 and C is double-increasing as a copula. Hence, ℭ is double-increasing in the rectangle 0,κ×λ,1. For the rectangle κ,1×0,λ, we have κ≤x1≤x2≤1 and 0≤y1≤y2≤λ, and with the usage of the third branch of ℭ, we obtain(27)Vℭx1,x2×y1,y2=1−κλVC1−x21−κ,1−x11−κ×y1λ,y2λ≥0since 1−κ,λ∈0,1 and C is double-increasing as a copula. Hence, ℭ is double-increasing in the rectangle κ,1×0,λ. For the rectangle κ,1×λ,1, we have κ≤x1≤x2≤1 and λ≤y1≤y2≤1, and with the usage of the fourth branch of ℭ, we obtain(28)Vℭx1,x2×y1,y2=1−κ1−λVC1−x21−κ,1−x11−κ×1−y21−λ,1−y11−λ≥0since 1−κ,λ∈0,1 and C is double-increasing as a copula. Hence, ℭ is double-increasing in the rectangle κ,1×0,λ. Finally, ℭ is double-increasing in the rectangle 0,12. As a result, ℭ is a copula, and the proof is completed.Remark 4.
LetC be the product copula Π in Theorem 1. Then, ∀x,y∈0,1,ℭx,y=Πx,y.The flowchart in Figure1 illustrates the novel copula construction process.Figure 1
Flowchart of copula construction process.The following example illustrates the construction of a copula, with the usage of Theorem1.Example 2.
Letκ=2/3, λ=3/4, and C1x,y=xy/x+y−xy, which is a member of the Ali-Mikhail-Haq family of copulas [12]. According to Theorem 1, the following copula is constructed:(29)ℭ1x,y=6xy9x+8y−12xy,x,y∈0,23×0,34,12x2y+6x−6xy−9x212xy+8−8y−9x,x,y∈0,23×34,1,12xy2−8y2+6y−6xy−8y−9x+9+12xy,x,y∈23,1×0,34,9x2+8y2−12x2y−12xy2+30xy−15x−14y+69x+8y−5−12xy,x,y∈23,1×34,1.The produced copula of Example1, is presented in Figure 2.Figure 2
Produced copula ofC1 with κ=2/3 and λ=3/4 (left axis represents y, right axis represents x, and vertical axis represents C1x,y).Fréchet [29] and Hoeffding [30] introduced the Fréchet–Hoeffding [12] bounds of copulas for any x,y∈0,1, maxx+y−1,0≤Cx,y≤minx,y. In order to present the bounds of the produced copulas ℭ, we provide the following theorem.Theorem 2.
Letκ and λ be fixed in [0, 1] and Candℭ be two copulas of Theorem 1. Then, the functions ℭ∼ and ℭ∼ with Domℭ∼=Domℭ∼=0,12 and Ranℭ∼=Ranℭ∼=0,1, given by(30)ℭ∼x,y=max0,λx+κy−κλ,x,y∈0,κ×0,λ,x−min1−λx,κ1−y,x,y∈0,κ×λ,1,y−min1−κy,λ1−x,x,y∈κ,1×0,λ,x+y−1+max0,1−λ1−x++1−κ1−y−1−κ1−λ,x,y∈κ,1×λ,1,ℭ∼x,y=minλx,κy,x,y∈0,κ×0,λ,x−max0,1−λx++κ1−y−κ1−λ,x,y∈0,κ×λ,1,y−max0,λ1−x++1−κy−1−κλ,x,y∈κ,1×0,λ,x+y−1+min1−λ1−x1−κ1−y,x,y∈κ,1×λ,1,respectively, are copulas, and ℭ∼x,y≤ℭx,y≤ℭ∼x,y, for every copula, ∀x,y∈0,1.Proof.
The proof thatℭ∼ and ℭ∼ have Domℭ∼=Domℭ∼=0,12 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 ∀x1,x2∈0,κ and ∀y1,y2∈0,λ, with x1≤x2 and y1≤y2:(31)κλVMx1κ,x2κ×y1λ,y2λ≥0since κ, λ∈0,1, and M is double-increasing as a copula. Hence, ℭ∼ is double-increasing in 0,κ×0,λ. Next, ∀x1,x2∈0,κ and ∀y1,y2∈λ,1, with 0≤x1/κ≤x2/κ≤1 and λ≤y1≤y2≤1⇒0≤1−y2/1−λ≤1−y1/1−λ≤1:(32)κ1−λVWx1κ,x2κ×1−y21−λ,1−y11−λ≥0since κ,1−λ∈0,1, and W is double-increasing as a copula. Hence, ℭ∼ is double-increasing in 0,κ×λ,1. Next, ∀x1,x2∈κ,1 and ∀y1,y2∈0,λ, with κ≤x1≤x2≤1⟹0≤1−x2/1−κ≤1−x1/1−κ≤1 and 0≤y1/λ≤y2/λ≤1 :(33)1−κλVW1−x21−κ,1−x11−κ×y1λ,y2λ≥0since 1−κ,λ∈0,1, and W is double-increasing as a copula. Hence, ℭ∼ is double-increasing in κ,1×0,λ. Next, ∀x1,x2∈κ,1 and ∀y1,y2∈λ,1, with κ≤x1≤x2≤1⇒0≤1−x2/1−κ≤1−x1/1−κ≤1 and λ≤y1≤y2≤1⇒0≤1−y2/1−λ≤1−y1/1−λ≤1(34)1−κ1−λVM1−x21−κ,1−x11−κ×1−y21−λ,1−y11−λ≥0since 1−κ,1−λ∈0,1 and M is double-increasing as a copula. Hence, ℭ∼ is double-increasing in κ,1×λ,1. As a result, ℭ∼ is a copula. The 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(35)max0,x+y−1≤Cx,y≤minx,y⇔κλmax0,xκ+yλ−1≤κλCxκ,yλ≤κλminxκ,yλ⇔max0,λx+κy−κλ≤ℭx,y≤minλx,κy.
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.The present construction method of copulas that can be achieved through Theorem1, and based on the fact that ℭ∼≠ℭ∼, of Theorem 2, can lead us in the result that there exists an infinite number of copulas that can be constructed via this transformation. The 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.Example 3.
Letκ=7/8, λ=2/2, and copulas ℭ∼ and ℭ∼ of Theorem 2. Then, the produced copulas for those specific values of κ,λ are given by(36)ℭ∼x,y=max0,2x2+7y8−7216,x,y∈0,78×0,22,x−min2−2x2,71−y8,x,y∈0,78×22,1,y−miny8,21−x2,x,y∈78,1×0,22,x+y−1++max0,16−72−82−2x−2y16,x,y∈78,1×22,1,ℭ∼x,y=min2x2,7y8,x,y∈0,78×0,22,x−max0,2−2x2−7y8+7216,x,y∈0,78×22,1,y−max0,328−2x2+y8,x,y∈78,1×0,22,x+y−1++min2−21−x2,1−y8,x,y∈78,1×22,1.The plots ofℭ∼ and ℭ∼ are illustrated in Figures 3 and 4, respectively.Figure 3
ℭ∼ with κ=7/8 and λ=2/2 (left axis represents y, right axis represents x, and vertical axis represents ℭ∼x,y).Figure 4
ℭ∼ with κ=7/8 and λ=2/2 (left axis represents y, right axis represents x, and vertical axis represents ℭ∼x,y).


## 4. The Novel Construction Method of Copulas for Fuzzy Random Variables

The 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:(37)C˜x,yα=FF˜X˜−1xα/2,F˜Y˜−1yα/2,FF˜X˜−1x1−α/2,F˜Y˜−1y1−α/2.As a result,C˜ is a joint fuzzy distribution function [15]. The following proposition examines the properties of the fuzzy copula.Proposition 1.
(see [15]). For a fuzzy copula C˜x,y, the following conditions hold:(38)C˜x,0=C˜0,y=0˜,C˜x,1=x˜,C˜1,y=y˜,C˜x2,y2⊕C˜x1,y1≻C˜x1,y2⊕C˜x2,y1,where x,y,x1,x2,y1,y2∈0,1, with x1<x2 and y1<y2.The 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. This is achieved through the following theorem.Theorem 3.
Letκ and λ be fixed in 0,1, C˜:0,1×0,1⟶0,1 be any fuzzy copula and ℭ:0,1×0,1⟶0,1 be produced by any copula C:0,1×0,1⟶0,1 with the usage of Theorem 1. Then, ℭ˜x,y at x,y∈0,1×0,1, defined by(39)ℭ˜x,y=C˜xκ,yλ⊗κλ,x,y∈0,κ×0,λ,x˜⊕C˜xκ,1−y1−λ⊗κλ−1,x,y∈0,κ×λ,1,y˜⊕C˜1−x1−κ,yλ⊗κ−1λ,x,y∈κ,1×0,λ,x˜⊕y˜⊕−1˜⊕C˜1−x1−κ,1−y1−λ⊗⊗1−κ1−λ,x,y∈κ,1×λ,1,is the fuzzy copula of the fuzzy random variables X˜ and Y˜.Proof.
For the first branch ofℭ˜, we have that, for x=0 and ∀y∈0,λ,(40)ℭ˜0,yα=C˜0κ,yλκλα=FF˜X˜−10α/2,F˜Y˜−1yλα/2κλ,FF˜X˜−101−α/2,F˜Y˜−1yλ1−α/2κλ=FF˜X˜1−α/20−1,F˜Y˜−1yλα/2κλ,FF˜X˜α/20−1,F˜Y˜−1yλ1−α/2κλ=F−∞,F˜Y˜−1yλα/2κλ,F−∞,F˜Y˜−1yλ1−α/2κλ=0,0=0˜α,∀α∈0,1.
As a result,ℭ˜0,y=0˜. The proof that ℭ˜x,0=0˜ can be considered in the same manner. The 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 ∀x1,x2∈0,κ and ∀y1,y2∈0,λ, where x1<x2 and y1<y2 and(41)C˜x2,y2⊕C˜x1,y1≻C˜x1,y2⊕C˜x2,y1⇔.
Sinceκ,λ≥0, ∀α∈0,1, we have that(42)κλ⊗C˜x2,y2⊕C˜x1,y1α>κλ⊗C˜x1,y2⊕C˜x2,y1α⇔ℭ˜x2,y2⊕ℭ˜x1,y1≻ℭ˜x1,y2⊕ℭ˜x2,y1.
As a result, the proof for the first branch is completed. Next, for the case of the second branch, ofℭ˜, we have that ∀x∈0,κ and for y=1,(43)x˜⊕C˜xκ,01−λ⊗κλ−1α==x˜α/2−FF˜X˜−1xκ1−α/2,F˜Y˜−101−α/2κ1−λ,x˜1−α/2−FF˜X˜−1xκα/2,F˜Y˜−10α/2κ1−λ=x˜α/2−FF˜X˜−1xκ1−α/2,F˜Y˜α/20−1κ1−λ,x˜1−α/2−FF˜X˜−1xκα/2,F˜Y˜1−α/20−1κ1−λ=x˜α/2−FF˜X˜−1xκ1−α/2,−∞κ1−λ,x˜1−α/2−FF˜X˜−1xκα/2,−∞κ1−λ.
Hence, we obtain(44)x˜⊕C˜xκ,01−λ⊗κλ−1α=x˜α/2,x˜1−α/2=x˜α,∀α∈0,1⇒ℭ˜x,0=x˜.
For the examination of the third condition, we have that∀x1,x2∈0,κ and ∀y1,y2∈λ,1, where x1<x2 and y1<y2 and(45)x2˜α−C˜x2κ,1−y21−λκ1−λ1−α+x1˜α−C˜x1κ,1−y11−λκ1−λ1−α>x2˜α−C˜x2κ,1−y11−λκ1−λ1−α+x1˜α−C˜x1κ,1−y21−λκ1−λ1−α⇔C˜x2κ,1−y21−λκ1−λα+C˜x1κ,1−y11−λκ1−λα<C˜x2κ,1−y11−λκ1−λα+C˜x1κ,1−y21−λκ1−λα.
This holds if we consider that sinceλ≤y1<y2≤1, then 0≤1−y2/1−λ<1−y1/1−λ≤1. Hence, the proof of the second branch is completed. The 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.The flowchart of Figure5 illustrates the novel fuzzy copula construction process.Figure 5
Flowchart of 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.
LetX and Y be random variables that have joint distribution function, given by(46)FXYx,y=x+11−e−yx−1e−y+2,x,y∈−1,1×0,∞.
As a result,X∼U−1,1 and Y∼Weibull1,1. Let Θ˜=2;1,4T and Γ˜=8;5,9T be two nonsymmetric triangular fuzzy numbers and let X˜=Θ˜+X and Y˜=Γ˜+Y. Hence, we have that X˜=X+2;1,4T and Y˜=Y+8;5,9T. Therefore, we obtain the α-pessimistic values of X˜ and Y˜, respectively, by(47)X˜α=X+2α+1,0<α≤12,X+4α,12≤α≤1,Y˜α=Y+6α+5,0<α≤12,Y+2α+7,12≤α≤1.
Based on the fact that∀α∈0,1, X˜α has uniform distribution and Y˜α has Weibull distribution. Hence,(48)X˜α∼U2α,4α−1,0<α≤12,U4α−1,α+1,12≤α≤1,Y˜α∼Weibull6α+5,1,1,0<α≤12,Weibull2α+7,1,1,12≤α≤1.
Also, we have theα-pessimistic values of Θ˜ and Γ˜ given as follows:(49)Θ˜α=2α+1,0<α≤12,4α,12≤α≤1,Γ˜α=6α+5,0<α≤12,2α+7,12≤α≤1.
Hence, we obtain the nextα-pessimistic values:(50)F˜Xxα=FX˜1−αx=ℙX˜1−α≤x=ℙX≤x−Θ˜1−α⇔F˜Xxα=x−Θ˜1−α+12,x∈Θ˜1−α−1,Θ˜1−α+1,F˜Yyα=FY˜1−αy=ℙY˜1−α≤y=ℙY≤y−Γ˜1−α⇔F˜Yyα=1−eΓ˜1−α−y,y∈Γ˜1−α,∞.
Now,∀α∈0,1, and we get the inverse functions of F˜Xxα and F˜Yyα by(51)F˜Xuα−1=2u+Θ˜1−α−1,F˜Yvα−1=Γ˜1−α−ln1−v.
Also, we have that(52)F˜X−1uα=2u+Θ˜α−1,F˜Y−1vα=Γ˜α−ln1−v.
Thus, we obtain the left and right parts of theα-cuts of the fuzzy copula by(53)C˜αLu,v=FXYF˜X−1uα/2,F˜Y−1vα/2=2u+Θ˜α/21−e−Γ˜α/21−v2u+Θ˜α/2−2e−Γ˜α/21−v+2,C˜αRu,v=FXYF˜X−1u1−α/2,F˜Y−1v1−α/2=2u+Θ˜1−α/21−e−Γ˜1−α/21−v2u+Θ˜1−α/2−2e−Γ˜1−α/21−v+2,where C˜u,vα=C˜αLu,v,C˜αRu,v. Now, based on Theorem 3, we have that, for the first branch of ℭ˜,(54)ℭ˜u,vα=ℭ˜αLu,v,ℭ˜αRu,v,where(55)ℭ˜αLu,v=C˜αLuκ,vλκλ=2u/κ+Θ˜α/21−e−Γ˜α/21−v/λ2u/κ+Θ˜α/2−2e−Γ˜α/21−v/λ+2κλ,ℭ˜αRu,v=C˜αRuκ,vλκλ=2u/κ+Θ˜1−α/21−e−Γ˜α/21−v/λ2u/κ+Θ˜1−α/2−2e−Γ˜1−α/21−v/λ+2κλ,∀u∈0,κ, and v∈0,λ. Next, for the second branch of ℭ˜, we have that ∀u∈0,κ and v∈λ,1,(56)ℭ˜u,vα=ℭ˜αLu,v,ℭ˜αRu,v,where(57)ℭ˜αLu,v=u˜α/2+2u/κ+Θ˜α/21−e−Γ˜α/21−1−v/1−λ2u/κ+Θ˜α/2−2e−Γ˜α/21−1−v/1−λ+2κλ−1,ℭ˜αRu,v=u˜1−α/2+2u/κ+Θ˜1−α/21−e−Γ˜1−α/21−1−v/1−λ2u/κ+Θ˜1−α/2−2e−Γ˜1−α/21−1−v/1−λ+2κλ−1.
For the third branch ofℭ˜, we have that ∀u∈κ,1 and v∈0,λ,(58)ℭ˜u,vα=ℭ˜αLu,v,ℭ˜αRu,v,where(59)ℭ˜αLu,v=v˜α/2+21−u/1−κ+Θ˜α/21−e−Γ˜α/21−v/λ21−u/1−κ+Θ˜α/2−2e−Γ˜α/21−v/λ+2κ−1λ,ℭ˜αRu,v=v˜1−α/2+21−u/1−κ+Θ˜1−α/21−e−Γ˜1−α/21−v/λ21−u/1−κ+Θ˜1−α/2−2e−Γ˜1−α/21−v/λ+2κ−1λ.
Finally, for the case of the fourth branch ofℭ˜, we have that ∀u∈κ,1 and v∈λ,1,(60)ℭ˜u,vα=ℭ˜αLu,v,ℭ˜αRu,v,where(61)ℭ˜αLu,v=u˜α/2+v˜α/2−1+21−u/1−κ+Θ˜α/21−e−Γ˜α/21−1−v/1−λ21−u/1−κ+Θ˜α/2−2e−Γ˜α/21−1−v/1−λ+21−κ1−λ,ℭ˜αRu,v=u˜1−α/2+v˜1−α/2−1++21−u/1−κ+Θ˜1−α/21−e−Γ˜1−α/21−1−v/1−λ21−u/1−κ+Θ˜1−α/2−2e−Γ˜1−α/21−1−v/1−λ+21−κ1−λ.In Example3 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.


## 5. Conclusions

We provided a method for the construction of copulas in four subrectangles of the unit square, via theα,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. These topics are the basis for our future investigations.



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