Novel Fuzzy Inference System (FIS) Analysis and Design Based on Lattice Theory

We introduce novel (set- and lattice-theoretic) perspectives and tools for the analysis and design of fuzzy inference systems (FISs). We present an FIS, including both fuzzification and defuzzification, as a device for implementing a function f: R^Nrarr R^M. The family of FIS functions has cardinality aleph₂=2^aleph1, where aleph₁ is the cardinality of the set R of real numbers. Hence the FIS family is much larger than polynomials, neural networks, etc.; furthermore a FIS has a capacity for local generalization. A formulation in the context of lattice theory allows us to define the set F^* of fuzzy interval numbers (FINs), which includes both (fuzzy) numbers and intervals. We present a metric d_K on F^*, which can introduce tunable nonlinearities. FIS design based on d_K has advantages such as: an alleviation of the curse of dimensionality problem and a potential for improved computer memory utilization. We present a new FIS classifier, namely granular self-organizing map (grSOM), which we apply to an industrial fertilizer modeling application