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I am a bit busy and I cannot get too much into details. However,
I can contribute a bit of references to the Fuzzy Approximation
Theorem. You may already know some or all. They are just background
material, but they contain many references.
You can look at: http://sipi.usc.edu/~kosko/
There you will be able to download:
- B. Kosko, S. Mitaim, "Adaptive Joint Fuzzy Sets for Function
Proceedings of the 1997 International Conference on Neural Networks
(ICNN-97), pp. 537-542, June 1997
- B. Kosko, S. Mitaim, "What is the Best Shape for a Fuzzy Set in
Approximation?", Proceedings of the 5th IEEE International Conference
on Fuzzy Systems (FUZZ-96), pp. 1237-1243, September 1996
Both of them contain information related to FAT.
Other papers listed in http://sipi.usc.edu/~kosko/ should
be of interest to you.
I also suggest that you look at:
- B. Kosko, "Neural Networks and Fuzzy Systems: A Dynamical Systems
Approach to Machine Intelligence", Englewood Cliffs, NJ: Prentice
Hall, 1992, ISBN 0 13 611 435 0
- Bart Kosko, "Fuzzy Engineering, 1/e", 1996 Prentice Hall,
- Bart Kosko, "Fuzzy Systems as Universal Approximators",
Proceedings of the First IEEE Conference on Fuzzy Systems
ZZ-92), pages 1153-62, San Diego, March 1992
- C.H. Chen, "Fuzzy Logic and Neural Network Handbook"
McGraw-Hill Series on Computer Engineering. New York, NY:
McGraw-Hill, 1996. ISBN 0 07 011189 8
(Chapter 9 of this book is authored by B. Kosko and
has as title "Additive fuzzy systems: from function
approximation to learning").
As a last resource, my (growing) web pages of links:
(A frame of: http://www3.sympatico.ca/nuptek/)
(A frame of: http://www.ecf.utoronto.ca/~bruno/)
I remember just now that "Fuzzy Logic Toolbox
For Use with MATLAB","User's Guide", Version 2 or more recent,
describes how to create a fuzzy inference system from data,
using ANFIS (adaptive neuro-fuzzy inference
system). The system is a Sugeno system. The method is
described from page 2-86 to page 2-119. While this will
not help you with the proof, if you have MATLAB, it will
help you to "play" with the concept and to come up with
some idea for the proof. There is also a list of
references at the end of the chapter.
If you tell us (all people in this newsgroup) something more
in detail about what you are doing, somebody may come up with
some better idea.
Bruno Di Stefano
Vitaliy Kolodyazhniy wrote:
> Dear all,
> I am a 2-nd year PhD student. Currently, I am summarizing all my research
> and working on my thesis. The title of the thesis is "Control of stochastic
> systems using fuzzy-neural models". I have developed a number of adaptive
> control schemes based on the 1st-order Sugeno models, and now I need
> mathematical proofs of approximation power of such models. As it seems to
> me, most existing proofs deal with the 0-th order Sugeno-type or alike fuzzy
> systems, among them are the results obtained by L.-X. Wang, B. Kosko, V.
> Kreinovich, and many others. Intuitively, it is quite clear that the
> 1st-order Sugeno systems possess much better approximation capability than
> the 0th-order ones, but so far I haven't found any mathematical work on this
> Another important thing (in close connection with the first one) is
> estimation of the approximation accuracy with a given number of clusters
> (rules) with given parameters.
> Could you give me any links to downloadable papers? Though I would
> appreciate any help.
> Thanks in advance.
> Vitaliy Kolodyazhniy
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