From a Gaussian mixture model to additive fuzzy systems

This work explores how a kind of probabilistic system, namely the Gaussian mixture model (GMM), can be translated to an additive fuzzy system. We will prove the mathematical equivalence between the conditional mean of a GMM, and the defuzzified output of a generalized fuzzy model (GFM). The relationship between a GMM and a GFM, and the conditions for GMM to GFM translation will be made explicit in the form of theorems. The work will then extend to special cases of the GFM, specifically the Mamdani-Larsen and Takagi-Sugeno fuzzy models. The possibility of reverse translation, that is, from a GFM to a GMM will also be discussed. Finally, we will consider the generality of a GMM, specifically how it can approximate other distribution functions.