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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