Abstract
The paper introduces a method for the construction of bivariate copulas with the usage of specific values of the parameters ( transformation) and the parameters 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 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.
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 -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 -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 the -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 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.
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 and range that is grounded and double-increasing, i.e., it satisfies the following conditions:where and .
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
According to Nelsen [12], let be two random variables with joint distribution function , with margins and . Then, there exists a copula , such that
Also, if is a copula and and are cumulative distribution functions, then is a joint distribution function, with margins and . If and are the inverses of and , respectively, then
As it was mentioned in Section 1, a construction method of copulas is -transformation that is defined [12] for parameters , that is, . The transformation of a copula into the copula is defined on the unit square by
As a result, for specific values of the parameters and (in their domain), different copulas can be constructed. In case that and , we obtain the copula:
In case that and , we obtain the copula:
In case that and , we obtain the copula:
In case that and , 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.
2.2. Fuzzy Sets, Notions, and Definitions
Let be a universal set. Each function is called a fuzzy set of , where ’s interpretation is the membership degree of in the fuzzy set . Crisp (classical) sets are special cases of fuzzy sets, with , or . The -cuts of a fuzzy set are defined by , where , which is called support, is the closure in the topology of of the union of all the -cuts [19], i.e., . Now, a fuzzy set of is called a fuzzy number if1, which means that is normal2 and ; we have which means that is convex fuzzy set3 is a nonempty compact interval in , which means that has compact support
The interval of the -cuts is denoted by , where and . We denote the set of all fuzzy numbers by . In [20], based on [21], some of the operations of -cuts were presented as follows.
Let and be their -cuts, respectively. Then, , and the fuzzy addition and the scalar multiplication were defined as follows:respectively, where the scalar was identified as the interval .
Definition 3. (see [22]). Let be a fuzzy number and . Then, the index is defined byThis gives the credibility degree, that is, is less than or equal to .
Remark 1. (see [22]). Let and , then1 if and only if2For any fixed , is a nondecreasing function with respect to x, i.e.,3 is self-dual, i.e.,
Definition 4. (see [23]). Let and , then the -pessimistic value of is given by . Furthermore, is a nondecreasing function of .
Remark 2. (see [24]). For a given , let be defined , byThen, the -cuts of are given by
Lemma 1. (see [15]). Let and let . Then,
Proof. The proof of Lemma 1 can be found in [15].
Definition 5. (see [24]). Let . . If , then , and if , then .
Example 1. Let and be a nonsymmetric triangular fuzzy number with membership function given byThen, the credibility degree that is less than or equal to is given byAs a result, we obtain the -pessimistic values of by
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 be the set of all possible outcomes of a random experiment, let be the -algebra of the subsets of , and let be the probability measure on the measurable space . Now, suppose that the probability space describes the random experiment. If , is a random variable on , then 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 be a probability space. If , the two mappings and are random variables, then is a fuzzy random variable.
Remark 3. (see [28]). In the next relationships, summarize the data of the two dimensional variable , in the one-dimensional variable .
Definition 7. (see [15]). If and are independent , then and are independent fuzzy random variables. If and are identically distributed , then and 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 the -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 and , with the adequate adjustments. The construction is achieved through the following theorem.
Theorem 1. Let and be fixed in [0, 1] and be any copula. Then, the function , defined byis also a copula.
Proof. The proof that the function is well defined and the boundary conditions of Definition 1 are satisfied is straightforward. In order to prove that is double-increasing, the proof that is double-increasing in each one of the four rectangles of the domain is needed. For the first rectangle, we have and , and with the usage of the first branch of , we obtainsince and is double-increasing as a copula. Hence, is double-increasing in the rectangle . For the rectangle , we have and , and with the usage of the second branch of , we obtainsince and is double-increasing as a copula. Hence, is double-increasing in the rectangle . For the rectangle , we have and , and with the usage of the third branch of , we obtainsince and is double-increasing as a copula. Hence, is double-increasing in the rectangle . For the rectangle , we have and , and with the usage of the fourth branch of , we obtainsince and is double-increasing as a copula. Hence, is double-increasing in the rectangle . Finally, is double-increasing in the rectangle . As a result, is a copula, and the proof is completed.
Remark 4. Let be the product copula in Theorem 1. Then, .
The flowchart in Figure 1 illustrates the novel copula construction process.
The following example illustrates the construction of a copula, with the usage of Theorem 1.
Example 2. Let , , and , which is a member of the Ali-Mikhail-Haq family of copulas [12]. According to Theorem 1, the following copula is constructed:
The produced copula of Example 1, is presented in Figure 2.
Fréchet [29] and Hoeffding [30] introduced the Fréchet–Hoeffding [12] bounds of copulas for any , . In order to present the bounds of the produced copulas , we provide the following theorem.
Theorem 2. Let and be fixed in [0, 1] and be two copulas of Theorem 1. Then, the functions and with and , given byrespectively, are copulas, and , for every copula, .
Proof. The proof that and have and , and that satisfy the boundary conditions of copulas is straightforward. In addition, for the proof that and are double-increasing, we have to examine this in each one of the four rectangles of their domain. For the function , we have that and , with and :since , , and is double-increasing as a copula. Hence, is double-increasing in . Next, and , with and :since , and is double-increasing as a copula. Hence, is double-increasing in . Next, and , with and :since , and is double-increasing as a copula. Hence, is double-increasing in . Next, and , with and since and is double-increasing as a copula. Hence, is double-increasing in . As a result, is a copula. The proof that is a copula can be considered in the same manner. For the case of the first rectangle, we have and , and with the usage of the first branch of copula and the Fréchet–Hoeffding [12] bounds of copulas, we obtainFor the cases of the other three rectangles, with the same approach as the case of the first rectangle, we obtain the desirable equations. Hence, and are copulas, and the bounds of every copula , and as a result, the proof is completed.
The present construction method of copulas that can be achieved through Theorem 1, and based on the fact that , 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 , we are using for each of the four branches only the copula C.
Example 3. Let , , and copulas and of Theorem 2. Then, the produced copulas for those specific values of are given by
The plots of and are illustrated in Figures 3 and 4, respectively.
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 and at , with the following α-cuts:
As a result, 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 , the following conditions hold:where , with and .
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. Let and be fixed in , be any fuzzy copula and be produced by any copula with the usage of Theorem 1. Then, at , defined byis the fuzzy copula of the fuzzy random variables and .
Proof. For the first branch of , we have that, for and ,As a result, . The proof that can be considered in the same manner. The next step is to examine the second condition that holds in the special case of and . In this case, becomes , and the proof is straightforward. In the last step, we have that and , where and andSince , , we have thatAs a result, the proof for the first branch is completed. Next, for the case of the second branch, of , we have that and for ,Hence, we obtainFor the examination of the third condition, we have that and , where and andThis holds if we consider that since , then . Hence, the proof of the second branch is completed. The cases of the third and the fourth branches of can be examined in a similar way as the first and the second branches of . Hence, the proof that is a fuzzy copula is completed.
The 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 and be random variables that have joint distribution function, given byAs a result, and . Let and be two nonsymmetric triangular fuzzy numbers and let and . Hence, we have that and . Therefore, we obtain the -pessimistic values of and , respectively, byBased on the fact that , has uniform distribution and has Weibull distribution. Hence,Also, we have the -pessimistic values of and given as follows:Hence, we obtain the next -pessimistic values:Now, , and we get the inverse functions of and byAlso, we have thatThus, we obtain the left and right parts of the -cuts of the fuzzy copula bywhere . Now, based on Theorem 3, we have that, for the first branch of ,where, and . Next, for the second branch of , we have that and ,whereFor the third branch of , we have that and ,whereFinally, for the case of the fourth branch of , we have that and ,where
In Example 3 of [15], the production of a fuzzy copula was established. In the illustrative Example 4 of the present paper, the generalization of this result was presented with the usage of 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 the 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 -dimensional copulas . 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.
Data Availability
No data were used to support the findings of the study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.