# Re: Q: Fuzzy singleton?

jg.campbell (jg.campbell@ulst.ac.uk)
Fri, 16 Jul 1999 19:58:10 +0200 (MET DST)

In article <7m2htn\$f4q\$1@nnrp1.deja.com>,
jg.campbell <jg.campbell@ulst.ac.uk> wrote:
> 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

Yes, see below!

> -- 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
>

[When one gets no reply to a posted question, or no comment, the
suspicions always arise: (a) is the question so dumb or irrelavant that
everyone just being too nice to say that it's dumb; or (b) is it really
difficult. In this case, I think I may not be unique in the confusion
that existed in my mind.]

I have solved the problem. My model of a fuzzy rule-base system was at
variance with that in the literature; specifically my idea of what
constituted 'fuzzification'.

Let us have fuzzy sets Ai on the input universe X; this easily
generalises to n-dimensional vector x, but I don't want to get carried
away with subscripts.

Likewise fuzzy sets Bj on output universe U.

The rule:

if Ai then Bj

may be expressed as a relation Rij on X x U

Assume that we use AND (or T-norm -- min, product etc.) for implication;
therefore

Rij(x,u) = mAi(x).mBj(u) (product)
where m... signifies the membership of the fuzzy set.

Now (in the correct interpretation), we have an input x'; this is
fuzzified -- in general to a fuzzy set A' on X (mA'(x)).

Now to generate a fuzzy set on U, we compose:

mA'(x)oRij(x,u) -> B'(u)

This is done 'sup-star'.

However, usually, mA'(.) is created as a 'singleton' set, i.e. mA'(x) =
1, x=x'; mA'(x)=0, otherwise. This allows much simplification to (2), in
particular, we simply pick out one row of Rij(.,.).

My confusion was that my _incorrect_ model (based on my software
implementation -- which works okay) was as follows:

- fuzzification = determination of membership of x' in the various
Ai(.)s.

- rule-base = mappings between Ais and
Bjs.

Actually, this model is adequate in many cases -- but it's different
from the commonly agreed one.

Apologies -- especially to those to whom I've spread this confusion! In
due course, I'll be correcting parts of reports on my website. [If
anyone wants further repentance, or wants to comment, I'd be happy to
discuss further, either here or by private email].

Best regards,

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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