Commutativity as Prior Knowledge in Fuzzy Modeling

In fuzzy modeling (FM), the quantity and quality of the training set is crucial to properly grasp the behavior of the system being modeled. However, the available data is often not large enough or is deficiently distributed along the input space, not revealing the system behavior completely. In such cases, the consideration of any prior knowledge about the system can be decisive for the accuracy achieved by the fuzzy modeling. This paper faces with the integration of mathematical properties satisfied by a system as prior knowledge in FM, focusing on the commutativity property as a starting point. With this aim, several measures are developed to evaluate the commutativity in a fuzzy environment dealing with different elements involved in FM. Then, several approaches are proposed to measure the commutativity degrees of a fuzzy rule with respect to the training set and a simple method is presented to integrate these degrees into the FM task. The experimental results shows the accuracy improvement gained by the proposed method.