A Bayesian approach to consequent parameter estimation in probabilistic fuzzy systems and its application to bearing fault classification

A bearing is an essential component in rotating machinery, one of its principal cause of failure, and its health condition is directly related to the safety and effective operation of such machinery. To the best of our knowledge, it is the first time that a probabilistic fuzzy system is applied to bearing fault classification. The type of probabilistic fuzzy classifier considered is a parsimonious fuzzy rule based model where each rule can diagnose a set of faults each one with its probability. For this kind of real world application, it is desirable to develop interpretable and accurate MIMO fuzzy systems, able to deal with the dimensionality and uncertainty present in data (vibration signals). For parameter estimation we adopt a two steps sequential state-of-the-art data-driven method. First, the antecedents of the rules are estimated using an iterative supervised clustering algorithm. Based on the antecedents the consequent parameters are then estimated. For this, a new method for consequent estimation is proposed. This is based on the observation that for defining a rule consequent not all training data within the region of applicability of that rule are equality relevant. Two criteria for selecting a relevant region in the feature space for consequent parameter estimation are proposed. Results show a statistically significant improvement on the performance of probabilistic fuzzy diagnosers trained with the proposed method, independently of the criterion used for defining the relevant region, when compared with the above mentioned state-of-the-art method. Moreover, the proposed consequent parameter estimation method practically has no overhead on the overall training of the diagnoser. Results show that an equilibrium can be found between the model level of detail and its accuracy. However, when accuracy is the sole comparison criterion, the proposed probabilistic fuzzy systems systematically matches the performance of other data-driven models like distance based methods (K-nearest neighbors), connectionists (probabilistic neural networks), or maximum margin classifiers (support vector machines).