Characterizations Of Residual Impliactions Derived From Copulas

In probability and statistics,copulas play important roles theoretically as well as to address a wide range of problems in various application areas.It takes each marginal distribution as the independent variable and the joint distribution as the dependent variable,which can reflect the relationship between the marginal distribution and the joint distribution in the form of a function.The notion of a quasi-copula is a generalization of that of a copula.It was introduced in order to characterize the univariate distribution functions.Since copulas and triangular norms are special kinds of conjunctors,many scholars apply the research methods of triangular norms to quasi-copulas.In fuzzy logic systems,the left-continuous triangular norms can derive fuzzy implications,which is called residual implications.Conversely,the fuzzy implications that satisfies certain conditions can also derive left continuous triangular norms.Therefore,the left-continuous triangular norms and the fuzzy implications that certain conditions have a mutually uniquely deterministic relationship.However,Kawaguchi et al.pointed out that the associativity property and the commutativity property of t-norms are not unnecessarily restrictive conditions.Demirli et al.presented a characterization of residual implications derived from left-continuous semi-copulas.Moreover,Durate et al.characterized residual implications derived from quasi-copulas,and presented a characterization of residual implications derived from copulas as an open question.This paper starts from the open question and tries to give an answer to this question.Due to the promotion of practical applications,the study of discrete copulas has also attracted attentions of many scholars,and some properties of copulas have also been extended to discrete cases.Therefore,this paper also discusses the properties of discrete residual implications derived from discrete copulas.This paper focuses on the public problem of how to characterize the residual implications derived from copulas.And we use the conclusions to give several methods of constructing copulas and proper quasi-copulas.First,we start with the discrete quasicopula with a relatively simple structure,and proposed the characterization of residual implications derived from discrete quasi-copulas.There is a one-to-one correspondence between the set of all discrete copulas and the set of all permutation matrices.Then we proposed discrete residual implications derived from discrete copulas and proper quasicopulas,respectively.Secondly,we through a counterexample to show that there is a flaw in the characterization of the residual implication derived from the quasi-copula which was presented by Durante et al..Moreover,we revise it by modifying the variation of variables.Thirdly,we characterize the residual implications derived from copulas.Thus,we answer the above open question which was raised by Durate et al..Finally,the construction of copulas is one of the concerns of statistics.Based on the characterizations of fuzzy implications,we propose several new methods of constructing copulas and proper quasi-copulas.