IR attempts to determine inductively behavioral relations between input and output variables of an unknown system, i.e., IR tries to model an unknown system.
The task of modeling an unknown system can be reduced to solving an optimization problem. IR discretizes the search space in order to keep the amount of computation in the solution of this optimization problem limited.
Unfortunately, valuable information about the system to be modeled is lost in the process of discretization. To keep the loss of information small, the search space must be discretized using a fine granularity.
The aim of this project was the design of a fuzzy extension of the IR methodology. The fuzzy membership functions make it possible to keep the loss of information associated with the discretization much smaller. For this reason, the number of discrete classes can be kept much smaller in the discretization, i.e., the search space can be discretized using a much coarser granularity. The discretized search space is thereby kept much smaller.
The fuzzy discretization exhibits yet another desired side effect. Unfortunately, not only the search space, but also the solution space is discrete after the discretization of the problem. Consequently, the solution can only be found with the resolution accuracy of the discretization. However, when a fuzzy discretization is employed, also the solution space is fuzzy. Therefore, the fuzzy membership functions of the solution space can be used to interpolate between neighboring discrete solutions. In this way, a real-valued solution can be found, i.e., the fuzzy solution space is dense, in spite of the fact that it has been discretized.
The fuzzy extension of the IR methodology that has been realized in FIR was developed and implemented by Donghui Li of the University of Arizona. Although George Klir and his students at the State University of New York at Binghamton developed another fuzzy extension of the IR methodology simultaneously with us, the two extensions differ in a few important details.
Klir in his fuzzy extension mapped real-valued variables onto the commonly used fuzzy doubles that consist of a discrete class value and a real-valued fuzzy membership function. In contrast, Li decided to make use of fuzzy triples. Li's fuzzy triples contain, beside from the usual class and membership values, a third ternary piece of information: the side value. The side value determines, whether the original real-valued data point lies to the left, at the center, or to the right of the maximum of the fuzzy membership function. The side value makes the mapping unique, which proves beneficial during the process of interpolation.