Fuzzy approximation via grid point sampling and singular value decomposition

This paper introduces a new approach for fuzzy approximation of continuous function on a compact domain. The approach calls for sampling the function over a set of rectangular grid points and applying singular value decomposition to the sample matrix. The resulting quantities are then tailored to become rule consequents and membership functions via the conditions of sum normalization (SN) and non-negativeness (NN), The inference paradigm of product-sum-gravity (PSG) is apparent from the structure of the decomposition equation, The present approach differs from previous works as it imposes no presupposition on the shapes of the membership functions, the rule consequents, and the particular inference paradigm of the fuzzy approximator, All information are extracted directly from the function samples. The present approach yields a class of equivalent fuzzy approximator to a given function, A tight bounding technique to facilitate normal or close-to-normal membership functions is also formulated, The fuzzy output approximates the given function to within an error which is dependent on the sampling intervals and the singular values discarded from the approximation process. Tradeoff between the number of membership functions and the desired approximation accuracy is also discussed, The present approach is applicable to functions of a general number of input variables. Several numerical examples are included to illustrate its effectiveness.