Identification of Possibilistic Linear Systems by Quadratic Membership Functions of Fuzzy Parameters

We have already formalized several models of the possibilistic linear regression analysis, where it is assumed that possibilistic parameters are non-interactive, i.e., the joint possibilistic distribution of parameters is defined by minimum operators. In this paper, we will deal with the interactive case in which quadratic membership functions defined by A. Celmins are considered. With the same view as described in our former studies, the possibilistic linear system with quadratic membership functions can be obtained by solving linear programming problems. The joint possibilistic distribution of parameters defined by a positive definite matrix is interactive. This point is different from our former works, where fuzzy parameters are non-interactive. The fact that some parameters become crisp in our former works is criticized by A. Celmins. In response to his concern, we will propose the new method for data analysis with quadratic membership functions. Our problem under consideration is similar to one considered by A. Celmins, but the proposed approach is simpler and more understandable than his approach based on the least-squares fitting with a complex algorism. Furthermore, similar problems are discussed by H. Bandemer, where fuzzy observations are transferred into fuzzy parameters. Our main concern is on a method of obtaining interactive fuzzy parameters into which fuzziness of data can be converted. Since our approach resorts to the linear programming, it is easy to apply our approach to real data analysis. This is very important for users who intend to analyse fuzzy data. Numerical examples are shown to explain our approach.