Re: Q: Fuzzy singleton?

Scott Dick (dick@morden.csee.usf.edu)
Sat, 24 Jul 1999 02:17:19 +0200 (MET DST)

"jg.campbell" wrote:
>
> 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.

One thing missing here. The relation Rij is not merely the intersection
of Ai and Bj, but must be a fuzzy implication. There are a number of
possibilities, one being the Kleene-Dienes implication

max((1-mAi(x)), mBj(u))

These implications have different properties; one useful reference
describing these differences is

D. Park, Z. Cao, A. Kandel, "Investigations on the Applicability of
Fuzzy Inference," Fuzzy Sets and Systems, Vol. 49, pp. 151-169, 1992.

>
> 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/
>
> Sent via Deja.com http://www.deja.com/
> Share what you know. Learn what you don't.

-- 
***********************************************************************
*                               *                                     *
*  Scott Dick                   *                                     *
*  Research Assistant           *       Cool & brilliant thought      *		
*  USF Computer Science         *       still under contruction       *
*  dick@morden.csee.usf.edu     *                                     *
*                               *                                     *
***********************************************************************

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