reply to Witlox Frank
Tue, 7 May 1996 17:56:13 +0200

> 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."


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)