Novel fuzzy inference system (FIS) analysis and design based on lattice theory. Part I: Working principles

This work substantiates novel perspectives and tools for analysis and design of Fuzzy Inference Systems (FIS). It is shown rigorously that the cardinality of the set F of fuzzy numbers equals א1, hence a FIS can implement “in principle” א2 functions, where א2 = 2א1>א1 and א1 is the cardinality of the set R of real numbers; furthermore a FIS is endowed with a capacity for local generalization. A formulation in the context of lattice theory introduces a tunable metric distance dK between fuzzy numbers. Implied advantages include: (1) an alleviation of the curse-of-dimensionality problem, regarding the number of rules, (2) a capacity to cope rigorously with heterogeneous data including (fuzzy) numbers and intervals, and (3) a capacity to introduce systematically useful nonlinearities. Extensive evidence from the literature appears to corroborate the proposed novel perspectives. Computational experiments demonstrate the utility of the proposed tools. A real-world industrial application is also described.