In my work I need to use the concept of fuzzy proximity relations. A
fuzzy proximity relation is basically a degree of closeness between
two variables. If the degree of a fuzzy proximity relation between
two variables A and B is 1 this means simply A=B. If the degree is 0
knowledge of A does not imply anything about the value of B, and vice
versa. An intermediary degree alpha means linguistically that the
value of A is close to B to a degree alpha.
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My question is as follows. Does anyone of you know any paper where
fuzzy relations (and in particular fuzzy proximity relations) are
treated from a formal frequentist point of view?.
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My view of the problem is that the notion of closeness must be seen as
a fuzzy set and it is then possible to treat the membership
distribution as a random set. For the extremal values of a fuzzy
proximity relation this makes sense. If for instance that the fuzzy
proximity relation between A and B is 1 and we know the value of A,
say, A = A0. B must then be one of the values with maximal (or maybe
even unity) membership degreee in the fuzzy set "close to A given
A = A0". If several values of B have the maximal degree of closeness, B
should be assigned one of them with uniform probability. On the other
hand if the fuzzy proximity relation between A and B is 0 one would
expect that for any value of A, B would be distributed according to
some probability distribution (or random set).
However, I get confused when I want to deal with intermediary degrees
of fuzzy proximity relations. For a special problem I can imagine
some formalism with probability distributions and expectancies and
variances. But I am sure that this must have been done generally
somewhere for fuzzy proximity relations as well as other fuzzy
relations.
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If anyone has any pointers to articles or is willing to share
knowledge here I would greatly appreciate the effort. My only source
is the (though excellent) Klir and Folger book which just gives a
review of fuzzy relations.
Tom Houlder
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