I've got several indications that my question about fuzzy proximity
relations was not specific and badly formulated. Well, I try again...
Fuzzy proximity relations are described in the book of Klir and Folger
as an extension of mathematical relations where for instance the
members of a similarity class have fuzzy membership grades. Since
these extensions are done in terms of fuzzy variables I wondered if
they can be given a frequentistic interpretation.
Klir and Folger give the example of a fuzzy proximity relation between
three cities, A, B and C. A is ``very near'' to B to a degree 0.8 and
B is ``very near'' C to a degree 0.7. This is a nontransitive
relation as A is possibly (but not necessarily) ``very near'' to C to
a degree 0.5.
Intuitively I understand this well. However, I would like to assign a
frequentistic meaning of the numbers 0.5, 0.8 and 0.9 _analogously_to_
the_way_one_can_give_a_frequentistic_interpretation_of_a_fuzzy_set
by_first_viewing_the_membership_grades_as_possibility_measures.
What follows is how I reason. (This may well be wrong.... ;)
Let A and B be standard variables taking values on a set X. If we are
given the value of A together with the equivalence relation A = B this
can be used to infer the value of B (that is, if A = A0 -> B = A0) If
we are given the value of A but no relation between A and B, there is
no evidence of the value taken by B.
I would suppose that the same notion of inference could be extracted
from fuzzy relations. If the value of A is given together with a
similarity relation "A SR B, alpha" this should provide some evidence
of the value of B. (SR is the similarity relation, and alpha is the
degree of similarity. It is easier to consider similarity relations
as these are transitive in contrast to proximity relations.)
Following Klir and Folger relations can be expressed by membership
matrices. This should be equivalent to define a fuzzy set for each
state in X in where all states of X is assigned a membership grade.
For instance the following membership matrice for a reflexive
asymmetric relation
A B
A 1.0 0.7
B 0.4 1.0
could be seen as a fuzzy set defined on A where A has membership grade
1.0 and B membership grade 0.7 together with a fuzzy set defined on B
where A has membership grade 0.4 and B membership grade 1.0.
It should now be possible to interpret these fuzzy sets as fuzzy
possibility distributions with a following frequentistic
interpretation. Said another way, I would assume that the value of A
together with the degree of similarity should define a possibility
distribution on X representing an evidence of the value of B. For
different degrees of alpha the possibility distribution should hence
say that the higher the degree alpha, B is expected to take values
that are "more and more similar" to the value taken by A.
[Does this means that a fuzzy relation is basically just a fuzzy set
together with a variable which strengthens the linguistic meaning
described by the set?? (There was a name for such a variable, but I
don't remember...)]
My viewpoint is that there is an aspect of inference in similarity
relations and that the degree of this inference is given by the degree
of similarity (The degree could for instance be related to an
expectancy value). This aspect is not explicitly in the formalism of
mathematical relations, but the double nature of fuzzy sets seems to
imply that it should be possible to interpret fuzzy relations this
way.
*
The application where I want to apply such an interpretation of
similarity relation is basically as follows.
Let A and B be RANDOM variables taking values on a set X. Let B = B0
and consider the joint probability distribution P(A, B) and the
conditional probability distribution, P_A(A| B=B0). If we are given a
fuzzy similarity relation between A and B of degree 0 the variables
would be uncorrelatedm P(A, B) = P(A) * P(B). If we on the other hand
are given the equivalence relation A = B (a fuzzy similarity relation
of degree 1.0), the variables are perfectly correlated P(A=B0 | B=B0)
e this has clarified my problem.
Suggestions, comments, flames or article suggestions are welcome...
Tom Houlder
hou= 1.
For the intermediary degrees of alpha, the correlation is given in a
fuzzy way, "the value of A is probably close to B". I would suppose
it was possible to specify this correlation solely on the basis of a
fuzzy set (in this case a ``similar to'' or ``close to'' set) and a
degree of similarity. However, I do not know how to do it
properly..... ;)
I hope this has clarified my problem.
Suggestions, comments, flames or article suggestions are welcome...
Tom Houlder
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