Re: reply to Witlox Frank

Fred A Watkins (fwatkins@hyperlogic.com)
Wed, 8 May 1996 13:51:30 +0200


On 7 May 1996 16:01:56 GMT, you wrote:

>> From: "Witlox Frank" <FRW.WITLOX.F@ALPHA.UFSIA.AC.BE>
>> Date: Wed, 24 Apr 1996 10:48:46 +0200
>>
>> Does any one have references to the subject of MULTI-dimensional
>> membership function estimation?
>
>The feature of NN-driven Fuzzy Reasoning proposed in 1988 is to
>auto-design nonlinear multi-dimensional membership functions using
>a NN and to embed the NN into a fuzzy system as a membership
>value-generator. See:
> H. Takagi and I. Hayashi, ``NN-driven Fuzzy Reasoning,"
> Int'l Journal of Approximate Reasoning (Special Issue of
> IIZUKA'88), Vol.5, No.3, pp.191-213 (1991)
>
>The advantage of nonlinear multi-dimensional membership function is
>to be able to reduce the number of fuzzy rules. When we use the
>conventional one-dimensional membership function, the fuzzy
>partitioned rule area becomes hyper-cube. Suppose 2-D input space
>partitioned as:
> +----------------------------------+
> | |
> | * |
> | * * |
> | * * * |
> | ** * * |
> | * * * |
> | * |
> | * |
> | * |
> | *|
> | |
> | |
> +----------------------------------+
>These two area can be approximated used many hyper-cubes which are
>result of combination of two 1-D membership functions of x and y.
>The NN-driven Fuzzy Reasoning fuzzy-partitions the above input space
>into only two using a NN and involves the NN inside fuzzy systems.
>For example, the NN-driven Fuzzy Reasoning has only two fuzzy control
>rules for a car-pole pendulum which has three input variables: angle,
>angular velocity, and car position from center. See the detail in:
> I. Hayashi, H. Nomura, H. Yamasaki, and N. Wakami,
> ``Construction of fuzzy inference rules by NDF and NDFL,"
> Int'l Journal of Approximate Reasoning, Vol.6, No.4 (1992)
>
>The relation between general fuzzy inference and the above inference
>is described in the section 4 of:
> H. Takagi, N. Suzuki, T. Kouda, and Y. Kojima,
> ``Neural-networks designed on Approximate Reasoning
> Architecture and Its Applications," IEEE Trans. on
> Neural Networks, Vol.3, No.5, pp.752-760 (1992)
>that proposes the model whose direction is "fuzzy system for NN."
>
>FYI,
>
>///////////////////////////////--------------------------------
> Hideyuki TAKAGI,
> Associate Professor
> Kyushu Institute of Design
> Department of Acoustic Design
>4-9-1, Shiobaru, Minami-ku
>Fukuoka 815 Japan
>TEL & FAX <+81>92-553-4555
>FAX <+81>92-553-4569 (FAX in Dept. office for long pages)
>e-mail takagi@kyushu-id.ac.jp
>-------------------------------///////////////////////////////
>

Isn't it the case that you simply exchange complexity in the rulebase
for complexity in the membership functions? Isn't it true that the
data needed to operate this system in general grows geometrically with
the input dimension? If so, of what value is this? If not, where is
the proof? This is tantamount to the defeat of the problem of
combinatorial growth; is this what you claim?

Fred A Watkins, Ph.D.
HyperLogic Corporation
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