Paper:

# Multi-Level Interpolation for Inference with Sparse Fuzzy Rules: An Extended Way of Generating Multi-Level Points

## Kiyohiko Uehara^{*} and Kaoru Hirota^{**}

^{*}Ibaraki University, 4-12-1 Nakanarusawa-cho, Hitachi 316-8511, Japan

^{**}Tokyo Institute of Technology, G3-49, 4259 Nagatsuta, Midori-ku, Yokohama 226-8502, Japan

As an extended way for inference based on multi-level interpolation, the number of multi-level points, generated with the α-cuts of given facts, is increased by making larger the number of levels of α. The conventional way uses the number of the levels adopted to define each given fact by α-cuts. A basic study is performed with triangular membership functions, where it is examined how the accuracy of mapping with fuzzy rules changes in increasing the number of the levels. It is also examined how deduced consequences behave when the number is increased. Moreover, convergent core sets of consequences are theoretically derived in the increase by the effective use of non-adaptive inference operations for core sets. They are used as references in simulation studies. Increasing the number of the levels provides nonlinear mapping with more precise reflection of distribution forms of sparse fuzzy rules to consequences. The basic study here contributes to improving the reflection accuracy. In simulations for the basic study, it is found that the mapping accuracy improves when the number of levels of α is increased. It is also confirmed that deduced core sets converge to those theoretically derived. Support sets are also found to converge in increasing the number of the levels. The core sets and support sets of deduced consequences do not, however, monotonically converge. This property causes difficulty in determining the optimized number of the levels so as to satisfy required mapping accuracy. In order to solve this problem, further discussions may be possible to theoretically derive convergent consequences and to use them in practical fields, in accordance with the theoretically derived core sets mentioned above.

*J. Adv. Comput. Intell. Intell. Inform.*, Vol.17, No.2, pp. 127-148, 2013.

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