# I seek papers about 1) probabilistic IF - THEN rules; 2) fuzzy functions approximation

Shkondin (shkondin@post.krc.karelia.ru)
Sun, 1 Nov 1998 22:33:48 +0100 (MET)

Hello.

Can anybody help me to find papers concerning the following subjects:

1) Probabilistic IF - THEN rules.

It's known that random sets can be used as a basis for representing
fuzzy measures (possibility, probability). For random set S, its
distribution can be described by a fuzzy set with membership function
m_s(x)=Pr("x belong to S").
We may consider the rule base:

A_i --> B_i , i=1,...,N,

where A_i and B_i - random sets. (One of the feasible interpretations
of the rule A_i --> B_i is "if x belongs to A_i then y is B_i").
Let input A be a random set. Then one can produce the output B using
probability calculations.
I would also like to know about applications of this approach,
for example for representing conditional probability distribution
in bayesian networks.

2) Fuzzy rule base is not the universal fuzzy approximator.

Let G be a continuous mapping: G: X --> Y
where X, Y are a subsets of fuzzy sets.
Let G also be fuzzy monotonic, that is:
IF A_1 is in A_2 THEN G(A_1) is in G(A_2)
Then in the common case traditional fuzzy rule bases
A_i --> B_i, A_i belong to X, B_i belong to Y, i=1,...,N,
can not approximate G with sufficient accuracy.
I would also like to know about ways of resolving this problem.

Thanks.

My e-mail shkondin@post.krc.karelia.ru

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