Re: Fuzziness: what am I missing?

Dimitri Lisin (dima@wpi.edu)
Mon, 6 Apr 1998 00:50:27 +0200 (MET DST)

On Mon, 30 Mar 1998, Anthony Cowden wrote:

> Dimitri Lisin wrote:
> > What I want to know, though, is if you have a normal mathematical function
> > given to you, how do you build a fuzzy inferencing system that best
> > approximates it (for some given sampling)? I am trying to figure this out
> > mathematically, but the math gets rather messy. Can anyone help?
>
> Why would you want to do this in practice? Mathematical functions
> (including fuzzy systems) are a symbolic way of representing something
> else. Why would you want to represent a representation with another
> representation? This only seems to remove you farther from the
> problem. One of the nice features of fuzzy logic is that it tends to
> move the function closer to the semantics underlying the descritpion of
> the problem than most mathematical functions allow.

There several reasons to do this in practice. The first one, is
theoretical. If fuzzy systems are another way to represent a function
then there should be a way to build a fuzzy system that would be
equivalent to a function defined in a conventional way. This is similar
to establishing an equivalence between deterministic and non-deterministic
Turing machines.

A more practical reason, is that approximating a regular function is a
good test case for a fuzzy system. Wouldn't it be nice to know just how
well a fuzzy system can approximate, say y = x^2, given some set of
data points (x,y pairs)? Wouldn't it also be nice to know the shapes of
the membership functions of the fuzzy sets that give you the best
approximation for a given set of data points?

And finally the most practical reason, the reason I asked the question in
the first place. For the project that I am working on
(http://www.wpi.edu/~dima/ummed/proposal/) I take an image and treat each
pixel in it as a point object having a mass, which may be proportional to
its brightness, or may be specified in a different way (using fuzzy rules,
for example). Thus each pixel exerts a gravitational force on all the
other pixels in the image. For my purposes, sometimes it is usefull for
the force to fall off as 1/r^2, and sometimes it is usefull for it to fall
off as 1/r. What I am hoping to be able to do, is to build a fuzzy system
that approximates 1/r^2 and a fuzzy system that approximates 1/r, and
somehow come up with a way to gradually adjust a fuzzy system to go from
one function to the other. Is this at all possible?

Going along the same lines, here is another question. Let us say you have
a quantity f, which depends on x and y. Suppose you have an intuitive
understandig of the dependency of f on y, and you have an exact
mathematical formulation of how f depends on x. How can this be
represented by a fuzzy system? Is it possible to represent a fuzzy
dependency on one variable, and a crisp dependency on the other? In this
case approximating a "normal" function with a fuzzy system may come in
handy.

Dimitri Lisin
dima@wpi.edu
http://www.wpi.edu/~dima/

P.S. Anthony, I noticed your company has some openings in Middletown, RI.
How far is that from Worcester, MA? In any case, I am finishing up my
Master's, and I'll be looking for a job soon. If you, or anyone in you
company is interested, my resume is at
http://www.wpi.edu/~dima/resume.html
Thank you.