Fuzzy Relations Extraction from Experimental Data

In this chapter, a problem of fuzzy genetic object identification expressed mathematically in terms of fuzzy relational equations is considered.

Fuzzy relational calculus [1, 2] provides a powerful theoretical background for knowledge extraction from data. Some fuzzy rule base is modelled by a fuzzy relational matrix, discovering the structure of the data set [3 - 5]. Fuzzy relational equations, which connect membership functions of input and output variables, are built on the basis of a fuzzy relational matrix and Zadeh’s compositional rule of inference [6, 7]. The identification problem consists of extraction of an unknown relational matrix which can be translated as a set of fuzzy IF-THEN rules. In fuzzy relational calculus this type of problem relates to inverse problem resolution for the composite fuzzy relational equations [2]. Solvability and approximate solvability conditions of the composite fuzzy relational equations are considered in [2, 8, 9]. While the theoretical foundations of fuzzy relational equations are well developed, they call for more efficient use of their potential in system modeling. The non-optimizing approach [10] is widely used for fuzzy relational identification. Such adaptive recursive techniques are of interest for the most of on-line applications [11 - 13]. Under general conditions, an optimization environment is the convenient tool for fuzzy relational identification [14]. An approach for identification of fuzzy relational models by fuzzy neural networks is proposed in [15 - 17].