Probabilistic and Statistical Fuzzy Set Foundations of Competitive Exception Learning

Recently, a Competitive Exception Learning Algorithm (CELA) was introduced [1, 2]. This algorithm establishes an optimal mapping from a (continuous) M-dimensional input sample space to an N-dimensional (continuous) output sample space. CELA is aimed to discover regimes (i.e. local behavior in the input sample space) for which the conditional probability distribution in the output sample space systematically deviates from the average unconditional distribution. Previous papers on CELA dealt with the introduction of the algorithm by sketching its background and by describing the algorithmic sub-steps. The algorithm was tested successfully on both simulated and real world data, mainly in the field of financial markets. However, until now a precise and firm theoretical foundation of CELA is still lacking. The current paper resolves this imperfection. The contribution to be made here is twofold. First, we present, in section 2, a probability theory and statistics of fuzzy sets which in itself is interesting. Second, we re-formulate, in section 3, the CELA-algorithm within the probabilistic fuzzy framework introduced. We finalize with a discussion and outlook.