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JACIII Vol.15 No.3 pp. 264-287
doi: 10.20965/jaciii.2011.p0264
(2011)

Paper:

# Inference for Nonlinear Mapping with Sparse Fuzzy Rules Based on Multi-Level Interpolation

## Kiyohiko Uehara*, Shun Sato*, and Kaoru Hirota**

*Ibaraki University, Hitachi 316-8511, Japan

**Tokyo Institute of Technology, Yokohama 226-8502, Japan

June 21, 2010
Accepted:
December 27, 2010
Published:
May 20, 2011
Keywords:
fuzzy inference, sparse rule base, nonlinear mapping, convex fuzzy set, α-cut
Abstract
An inference method is proposed for sparse fuzzy rules on the basis of interpolations at a number of points determined by α-cuts of given facts. The proposed method can perform nonlinear mapping even with sparse rule bases when each given fact activates a number of fuzzy rules which represent nonlinear relations. The operations for the nonlinear mapping are exactly the same as for the case when given facts activate no fuzzy rules due to the sparseness of rule bases. Such nonlinear mapping cannot be provided by conventional methods for sparse fuzzy rules. In evaluating the proposed method, mean square errors are adopted to indicate difference between deduced consequences and fuzzy sets transformed by nonlinear fuzzy-valued functions to be represented with sparse fuzzy rules. Simulation results show that the proposed method can follow the nonlinear fuzzy-valued functions. The proposed method contributes to both reducing the number of fuzzy rules and providing nonlinear mapping with sparse rule bases.
K. Uehara, S. Sato, and K. Hirota, “Inference for Nonlinear Mapping with Sparse Fuzzy Rules Based on Multi-Level Interpolation,” J. Adv. Comput. Intell. Intell. Inform., Vol.15 No.3, pp. 264-287, 2011.
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