Double Fuzzy Point Extension of the Two-step Fuzzy Rule Interpolation Methods

The “Double Fuzzy Point” rule representation opens a new dimension for expressing changes of fuzziness in fuzzy rule-based systems. In the case of standard “Fuzzy Point” rule representations, it is difficult to describe fuzzy functions in which crisp observations are required to have fuzzy conclusions, or in which an increase in the fuzziness of observations leads to reduced fuzziness in conclusions. These problems are mainly due to a lack of information. A fuzzy point rule determines the connection between a pair of fuzzy sets taken from the domain and the range of the rule. Expressing the fuzzy function through a set of fuzzy points and fuzzy interpolation between pairs of those points, each fuzzy point can be considered as a node point with given location and fuzziness. In common, sparse rule-base definitions, these node points are usually disjunctive on the domain, defining only single antecedent-consequent fuzziness connections at the location of the fuzzy points. However, this kind of information is insufficient when the goal is to express changes in the fuzziness of a given location in the domain. One solution to this problem is the double fuzzy point rule representation concept. Double fuzzy points are pairs of fuzzy points which share the same reference locations, but have different fuzziness properties. The existence of two different fuzziness values in a single location within the domain creates new possibilities for introducing fuzzy interpolation methods capable of interpolating not only between locations, but between changes in local fuzziness values as well. The main goal of this paper is to discuss how two-step Fuzzy Rule Interpolation methods can be adapted to be able to handle the double fuzzy point concept. To this end, an approach referred to as the Generalized Double Fuzzy Point Methodology (GDFPM) is proposed.