Adaptive filter theory (2nd ed.)
Adaptive filter theory (2nd ed.)
Time Series Analysis, Forecasting and Control
Time Series Analysis, Forecasting and Control
Application for deinterlacing method using edge direction classification and fuzzy inference system
Multimedia Tools and Applications
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Fuzzy systems are unique in that they possess both linguistic input/output descriptions (the fuzzy if then rules) and numerical input/output descriptions (the input/output functions they implement). Adaptive fuzzy systems, also known as neural fuzzy systems, learn the functions they implement through training with numerical data. Precision--evaluated by the mean square error--has been the most important performance index. The linguistic meanings of the fuzzy if-then rules appear to be insignificant for adaptive fuzzy systems. We show that the linguistic meanings of system parameters can be used to set better initial values, thus reduce the mean square error. On the other hand, for adaptive fuzzy systems with Gaussian-type membership functions, there exist a trade-off between how meaningful the fuzzy if-then rules are and how precise the implemented function is. The fuzzy if-then rules may fail to provide a coarse description of the input/output function after training even when the mean square error is small. We propose a "mean square distance" that measures the meaning of the fuzzy system in terms of the discrepancy between the fuzzy if-then rules and the implemented numerical function. Performance of adaptive fuzzy systems are evaluated using both the mean square error (precision) and the mean square distance (meaning) in time series prediction experiments with Mackey Glass and Box Jenkins data. The effects of weight adaptation methods and varying Gaussian widths are studied. Adaptive fuzzy systems with Gaussian-type membership functions that perform well in precision (have small mean square error) may not have meaningful fuzzy if-then rules anymore.