Conditional Density Estimation Using Probabilistic Fuzzy Systems

We consider conditional density approximation by fuzzy systems. Fuzzy systems are typically used to approximate deterministic functions in which the stochastic uncertainty is ignored. We propose probabilistic fuzzy systems (PFSs), in which the probabilistic nature of uncertainty is taken into account. These systems take also fuzzy uncertainty into account by their fuzzy partitioning of input and output spaces. We discuss an additive reasoning scheme for PFSs that leads to the estimation of conditional probability densities and prove how such fuzzy systems compute the expected value of this conditional density function. We show that some of the most commonly used fuzzy systems can compute the same expected output value, and we derive how their parameters should be selected in order to achieve this goal. The additional information and process understanding provided by the different interpretations of the PFS models are illustrated using a real-world example.