Scaling-invariant description of dependence between fuzzy variables: Towards a fuzzy version of copulas

To get a general description of dependence between n fuzzy variables x
₁
,..., X
_n
, we can use the membership function μ(χ
₁
,..., x
_n
) that describes, for each possible tuple of values (x
₁
,..., x
_n
) to which extent this tuple is possible. There are, however, many different ways to elicit these degrees. Different elicitations lead, in general, to different numerical values of these degrees - although, ideally, tuples which have a higher degree of possibility in one scale should have a higher degree in other scales as well. It is therefore desirable to come up with a description of the dependence between fuzzy variables that does not depend on the corresponding procedure and, thus, has the same form in different scales. In this paper, by using an analogy with the notion of copulas in statistics, we come up with such a scaling-invariant description. Our main idea is to use marginal membership functions μ
_i
(x
_i
) = max μ(x
_i
,..., x
_i-1
, x
_i
, x
_i+1
,..., x
_n
) and then describe the relationship between the fuzzy variables x
₁
,..., x
_n
by selecting, for some i from 1 to n, a function r
_i
(x
₁
,..., x
_n
) for which, for all the tuples (x
₁
,..., x
_n
), we have μ(x
₁
,..., x
_n
)= μ
_i
(r
_i
(x
₁
,..., x
_n
)).