7.1 Fuzzy empirical correlation coefficient
For real data fuzziness can be present in different ways:
a. (xi, yi)* can be fuzzy two-dimensional vectors.
b. (xi*, yi*) are pairs of fuzzy numbers.
c. One coordinate is precise and the other is fuzzy.
In order to apply the extension principle for the generalization of the empirical correlation coefficient, first the fuzzy data have to be combined into a fuzzy vector of the 2n-dimensional Euclidean space 2n. For this combination the minimum t-norm is used.
In case (a) the fuzzy data consist of two-dimensional fuzzy vectors
with corresponding vector-characterizing functions ξi(., .), i = 1(1)n.
The combined fuzzy sample is a 2n-dimensional fuzzy vector whose vector-characterizing function ξ(., …, .) is given by its values
Applying the extension principle to the function
the characterizing function ψr*(·) of the generalized (fuzzy) empirical correlation coefficient r* is given by its values
The support of r* is a subset of the ...