Combining fuzzy systems

The paper shows how to combine any number of fuzzy systems and compares this new method with a weighted arithmetic mean. The new method combines the throughputs of the fuzzy systems and thus combines before it defuzzifies. The mean combines just the defuzzified outputs. The new method adds the output fuzzy set of each fuzzy system and then defuzzifies this sum in a higher-level additive fuzzy system. The combined fuzzy systems need not be additive fuzzy systems. The paper derives the exact form of the additive combiner. It reduces to an unweighted mean of centroids when all fuzzy sets have the same volume and all systems either have the same credibility weights or have binary credibility weights. The weighted mean ignores the inherent weighting information in the relative volumes. Its constant normalizer leads to scale defects when it combines weighted fuzzy systems even when all the weights are the same. The paper derives the first-order and second-order statistics of the general additive system and shows how fuzzy systems act as adaptive model-free statistical estimators