POINTER ADAPTATION AND PRUNING OF MIN-MAX FUZZY INFERENCE AND ESTIMATION

A new technique for adaptation of fuzzy membership functions in a fuzz y inference system is proposed, The painter technique relies upon the isolation of the specific membership functions that contributed to the final decision, followed by the updating of these functions' paramete rs using steepest descent, The error measure used is thus backpropagat ed from output to input, through the min and max operators used during the inference stage, This occurs because the operations of min and ma x are continuous differentiable functions and, therefore, can be place d in a chain of partial derivatives for steepest descent backpropagati on adaptation, Interestingly, the partials of min and max act as ''poi nters'' with the result that only the function that gave rise to the m in or max is adapted; the others are not, To illustrate, let alpha = m ax [beta(1), beta(2), ..., beta(N)]. Then partial derivative alpha/par tial derivative beta(n) = 1 when beta(n) is the maximum and is otherwi se zero, We apply this property to the fine tuning of membership funct ions of fuzzy min-max decision processes and illustrate with an estima tion example, The adaptation process can reveal the need for reducing the number of membership functions, Under the assumption that the infe rence surface is in some sense smooth, the process of adaptation can r eveal overdetermination of the fuzzy system in two ways, First, if two membership functions come sufficiently close to each other, they can be fused into a single membership function, Second, if a membership fu nction becomes too narrow, it can be deleted, In both cases, the numbe r of fuzzy IF-THEN rules is reduced, In certain cases, the overall per formance of the fuzzy system ran be improved by this adaptive pruning.