SHAPES RECOGNITION WITH NEURAL NETWORKS

Abstract

In real-word applications, training data are not always complete. Some of these data may be incomplete or missing and others may be uncertain or ambiguous. First of all, one of the simplest and most realistic manners for -representing incomplete or missing data is the interval representation, this means that the inputs are given by intervals instead of real numbers. Two approaches have been proposed before in [3], [13] to solve such problems. In the first approach [3], an architecture of interval neural networks and a learning algorithm as an extension of standard BP(Back-Propagation) algorithm have been proposed. However, in [3], a very complicated cost function was used in the learning algorithm, and the algorithm was applied only to the disjoint classes, and it had not been tested using overlapping classes. In this work, we have implemented the same algorithm of [3] for interval inputs, but using a simple cost function, very similar to that used in the standard BP algorithm [1, 2]. Using such simple cost function, we have obtained the same results as in [3]. Moreover, we have tested the developed algorithm using overlapping classes, and the testing results indicated that the algorithm is useful only for disjoint classes and it is not good for overlapping classes. In the second approach f13], a neural networl{s, that can utilize, the expert knowledge represented by fuzzy if-then rules as well as the numerical data, in the learning have been proposed. Although the proposed algorithm in [13] was dealing mainly with fuzzy numbers as inputs, the most important parts of fuzzy set .theorem, resolution principle using a-cuts and the extension principle •were not mentioned or studied in [13]. Moreover, the a-cuts were not introduced in the developed algorithm of [13]. Thus, in the present work, in the second approach, we have developed using the same algorithm in [13], we have studied in details, the concepts of a-cuts, resolution principle and the extension principle. On the other hand, we introduce a-cuts explicitly in our implementation. The testing results indicated that the a-cuts have a great influence on the output separation regions. Testing results indicated also that the developed algorithm has a very good classification capability for separating the overlapping fuzzy input data.