# Fuzzy Classification

Rafa/l (pop@gumbeers.elka.pg.gda.pl)
Sun, 14 Jan 1996 07:25:28 +0100

Hello,

I am a student of Technical University of Gdansk in Poland and I am
interested in applying Fuzzy Logic in "Pattern and Signal Recognition".
I've read some articles partially concerning that problem, among them:

"FUNNY (FuzzY or Neural Net) METHODS FOR ADAPTIVE PROCESS CONTROL"
by H.Bersini and V.Gorrini
"Self-Structuring Fuzzy Systems for Function Approximation"
by H.Bersini, V.Gorrini, T.Salome
"MLP, RBF, FLC: What's the difference"
by H.Bersini and V.Gorrini

And I found in one of them the following:
" .. It is necessary to transform the input patterns somehow from their
initial physically derived format into another representation form in
witch patterns requiring similar (output) responses are indeed similar to
one another... This transformation can create a new vector space in witch
the relative distances among the input vectors are different from those
in original vector space, essentially rearranging the points. ...".
Text above and the rest of article suggest that the problem of
classification is equal to:

(a) problem of finding hyper-space (non-linear) witch separate sets of
patterns requiring similar (output) responses <- (1) in appropriate
n-dimensional vector space <- (2).
(b) problem of finding non-linear functional witch separate sets (1).
(c) problem of finding transformation of space (2) into another vector space
where images of sets (1) are linearly separable.
..
In all of above non-linear mappings like neural nets and fuzzy systems do
the job. But there is a known fact from Functional
Analysis that: any two disjoint convex sets can be separated by a linear
functional (witch can be determined, for instance in R^n), so any finite
number of convex sets can be separated by a set of linear
functionals. In case when
convex-ions of sets (simplexes) (1) aren't disjoint, one can imagine a
devision of the input patterns classes in such a way that:
(i) "new" output-classes are created and any two of them are either
into the same "old" output-class or in different "old" output-classes
(ii) there is an relation of equivalence among those "new"
output-classes witch belongs to the same "old" output-class.

For instance to solve XOR - problem with this method we have to separate the
following points:
*(I) *(II)

*(II) *(I)

The simplex containing class (I) is not disjoint with that containing
class (II), so we have to define a new devision. One of the possible is:

*(I) *(II)

*(III) *(I)

And equivalence relation : (III) :=: (II). The simplexes corresponding to
classes (I),(II),(III) are disjoint, so that we can find a set of linear
functionals witch separate those simplexes what gives the final solution.

Thus in order to solve any classification problem with given set of instance
patterns first check if simplexes (1) are disjoint if are not then
construct subdivision f:X->R what (X-set of input patterns) gives
disjointness of "new" simplexes (1). The next step is to find set of
linear functionals witch separate those simplexes.

I really like the concept of automatic tuning parameters and a
self-structuring in Surgeno's fuzzy systems. Does there exist any
different from gradient method for tuning the parameters in such systems?

Where can I find in cyber-space articles concerning "Pattern and Signal
Recognition" for instance in Post-Script format?