Decreasing Excess Fuzziness Included in Fuzzy Output Vectors from Neural Networks

Abstract

When a fuzzy vector is presented to a multi-layer feedforward neural network, the calculation of the corresponding fuzzy output vector is performed by fuzzy arithmetic. Since fuzzy arithmetic is independently executed at each unit of the neural network, the fuzziness of the fuzzy output from each unit is increased by the feedforward calculation in the neural network. In this paper, we first discuss such increase of the fuzziness from a viewpoint of the local application of the extension principle. In the local application, the extension principle is locally applied to the fuzzy input-output relation of each unit. On the other hand, theoretically the extension principle can be globally applied to the fuzzy input-output relation of the whole neural network. The global application is, however, very difficult in practice. Next we illustrate the increase of the fuzziness in the local application of the extension principle by computer simulations on numerical examples. Then we describe subdivision methods applied to the level sets of fuzzy input vectors for avoiding the increase of the fuzziness in fuzzy arithmetic. Finally we examine the effectiveness of the subdivision methods by applying them to rule extraction problems.