Consensus and selectivity in fuzzy rule based interpolation

Many fuzzy logic systems, in effect, perform interpolation on fuzzy X-Y graphs. Each If-then rule represents a fuzzy point on the graph; the job of the system is to balance two criteria that sometimes are in conflict. One criterion is selectivity, which asks that the behavior of the function at an intermediate point should resemble its behavior at the nearest given point. The other criterion is consensus, which asks that behavior at an intermediate point should reflect information from all nearby rules. The present paper examines the relationship between the parameter value and the behavior of the family of S-implication operators derived from the Schweizer-Sklar family of T-norms. The context is the interaction between rules, not just the behavior of an isolated rule. Subsequent research will examine other families of operators such as R-implications, Q-implications, and Mamdani pseudo-implications. The key consideration will be the effects of the parameter on the balance between consensus and selectivity. A fuller understanding of this relationship will help system designers to produce a system with desired characteristics by taking advantage of an important dimension of the flexibility that is the raison d´etre of fuzzy systems