Interpolation in homogenous fuzzy signature rule bases

Abstract

Fuzzy signature sets (FSigSets) are extensions of the original fuzzy set concept, and also of the Vector Valued Fuzzy Set notion. In a FSigSet rule base the (input) universe of discourse X is mapped into a set of hierarchically grouped fuzzy sets, and each element of X has a “membership degree” consisting of a rooted tree with membership degrees at each leaf and aggregations at the intermediate vertices. The structure of the tree is identical for each element in the case of homogenous FSigSets, and so are the aggregations, depending only on the position of the vertex. Interpolation in fuzzy rule bases allows the calculation of a conclusion in the output universe Y belonging to an observation even if there are gaps in the rule base and the observation does not intersect with any of the antecedent sets. The key question here is how to determine the degree of similarity, or inversely, the distance, of any observation from the surrounding antecedents of the rules in the base, so that the distance incorporates the information involved with the close connection of the features in the sub-groups, and the aggregations expressing the form of this connection. A solution is proposed, and a pair of numerical examples is presented.