# Re: number of membership functions?

Rainer Holve (holve@forwiss.uni-erlangen.de)
Sun, 7 Jun 1998 02:08:22 +0200 (MET DST)

EarlCox wrote:
>
> Whoa!
>
> The problem with using many, many membership functions is simply this: a
> membership function by itself simply defines one semantic partitioning of the
> variables Universe of Discourse. A MF onlyparticipates in a model when we have
> a rule that descibes the mdoel behavior through the interaction of these MF.
> So..if we define 10,000 MF (to pick some really absurd number) then we would
> need at least 10,000 rules to describe the state of the model for each MF. So
> if I have three variables with fifteen MF's each, then the rules are 5**15 or
> approx 3.05x10**10. Which is alot. ;-)
>
> Of course Bill Comb's method of rule Union side steps the exponential growth of
> rules with variables, but the basic problem still exists.

Does it really?
I just posted some remarks on Comb's and Andrew's paper in the Fuzzy
Pattern Recognition
thread, but since it appears inthis thread as well, I just repost it
here
..
IEEE Trans. Fuzzy Sys 6:1, pp 1-11, 1998
Combinatorial Rule Explosion Eliminated by a Fuzzy Rule Configuration
W.E. Combs and J.E. Andrews

I have a problem with this paper, because it somehow suggests, that
everything that can be
done with the traditional rulebase style can also be done with the
proposed URM though
with far fewer rules. Well, the traditional fuzzy systems can be viewed
as universal
function approximators and I doubt that this holds for the URM as well.

Maybe I just didn't get the point of that approach but could someone
point out a
URM for the 2-d XOR-problem i.e. (in traditional rules)

if X is LOW and Y is LOW then Z is LOW
if X is LOW and Y is HIGH then Z is HIGH
if X is HIGH and Y is LOW then Z is HIGH
if X is HIGH and Y is HIGH then Z is LOW

I really don't see how this could be modeled with a disjunction of
any number of one-dimensional rules.

THere is this strange constraint 2 in the paper which says

"Everey input and output universal set relation must be monotonic."
It's explained in the Appendix as follows:
If Input P has the fuzzy sets P1<= P2<=P3<= ... <=Pn (Constraint 1) and
the
output R has the fuzzy sets R1<=R2<=R3<= ... <= Rn (also constraint 1)
then if
"If R2 is greater than R1 and P1 relates to to R1 and P2 relates to R2,
then P3 must relate to an element of R >= R2, etc."

This sounds to me as if the method is only able to model globally
monotonic functions,
i.e. wherever you are in the problem space, if you look parallel to of
one of the
input axes, the function your URM models is always increasing
(or decreasing if you turn around :)

If this is the case, I will happily agree that the URM is the best
suited way to
model such functions (but only those). One might ask, how many real
world
problems there are that fit into this form of monotony but I think that
the proposed approach
is clearly unsuited for e.g. pattern recognition where you want to learm
some fuzzy rules from data
to capture an unknown function you know nothing about.

Rainer

```--
Rainer Holve.......................................FORWISS
Am Weichselgarten 7..............D-91058 Erlangen, Germany
Email: ...................................holve@forwiss.de
URL:........................http://www.forwiss.de/~rrholve
Telefon: +49-9131-691-257............Fax: +49-9131-691-185

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