A fuzzy additive reasoning scheme for probabilistic Mamdani fuzzy systems

We introduce a type of probabilistic fuzzy system with a generalized Mamdani-type fuzzy rule base, and an additive reasoning scheme where conditional probabilities on fuzzy events are aggregated using an interpolation approach. In this way, probabilistic fuzzy outputs can be calculated for arbitrary crisp input vectors. If desired, the probabilistic fuzzy output can be made crisp using a defuzzification and averaging step. Besides introducing the architecture of the probabilistic fuzzy systems and the corresponding equations for calculating the input-output mapping, we summarize some key results from the probability theory and statistics on fuzzy sets. To show the working of the probabilistic fuzzy models introduced, we analyze a simulated GARCH time series using a data-driven approach. A probabilistic fuzzy rule-base is derived from the given data set containing rules that yield a rather good intuitive description of the underlying GARCH-process. Further, we show some additional results like the estimated regression plane and several (un)conditional probability distributions.