Q: Fuzzy singleton?

jg.campbell (jg.campbell@ulst.ac.uk)
Fri, 9 Jul 1999 08:10:10 +0200 (MET DST)

In [Wang, 1994] the following statement appears on p. 22 in a definition
of 'fuzzifier':

"The fuzzifier performs a mapping from a crisp point x = (x1, x2, ...
xn)T (T -- transpose), [member of U,] into a fuzzy set A' in U. There
are (at least) two possible choices of this mapping:

- Singleton fuzzifier: ... [definition of A' as having single point of
support at x, memb(x) = 1, memb(x')=0 for all x'!=x]

- Nonsingleton fuzzifier: ... "

Then the statement is made: "It seems that only the singleton fuzzifier
has been used."

Later, p. 25, eqn. 2.46, a class of fuzzy systems is defined in which it
is stated that a singleton fuzzifier is used, _along with Gaussian
membership functions_.

Questions: (a) Surely it is untrue to say "...only the singleton
fuzzifier has been used"; (b) If the fuzzifier is singleton, how can the
statement about Gaussian membership functions have any relevance?

Clearly I am missing something obvious -- and since I have implemented
fuzzy systems and written about them, I am almost afraid to ask such a
basic question!

The same question arises from my reading of (Wang and Mendel, 1992).

Wang, L.-X, Adaptive Fuzzy Systems and Control, Prentice Hall, 1994

Wang, L.-X., and J.M. Mendel
Fuzzy basis functions, Universal Approximation, and Orthogonal
Least Squares Learning
IEEE Trans on Neural Networks, Vol. 3, No. 5, 1992

TIA,

Jon Campbell

--
Jonathan G Campbell Univ. Ulster Magee College Derry BT48 7JL N. Ireland
+44 1504 375367 JG.Campbell@ulst.ac.uk  http://www.infm.ulst.ac.uk/~jgc/

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