A Stochastic Framework for Robust Fuzzy Filtering and Analysis of Signals—Part I

There are numerous applications across all the spectrum of scientific areas that demand the mathematical study of signals/data. The two typical study areas of theoretical research on signal/data processing are of modeling (i.e., understanding of signal's behavior) and of analysis (i.e., evaluation of given signal for finding its association to existing signal models). The objective of this paper is to provide a stochastic framework to design both fuzzy filtering and analysis algorithms in a unified manner. The signals are modeled via linear-in-parameters models (e.g., a type of Takagi-Sugeno fuzzy model) based on variational Bayes (VB) methodology. This gives rise to the “negative free energy maximizing” filtering algorithm. The issue of intractability was handled first by carefully choosing the priors as conjugate to the likelihood and then by using Stirling approximation for the Gamma function. This paper highlighted that it was analytically possible to maximize the information theoretic quantity, “mutual information,” exactly in the same manner as maximizing “negative free energy” in VB methodology. This gives rise to the “variational information maximizing” analysis algorithm. The robustness of the methodology against data outliers is achieved by modeling the noises with Student-t distributions. The framework takes into account the inputs noises as well apart from the usually considered output noise. The robustness of the adaptive filtering algorithm against noise is shown by a deterministic analysis where an upper bound on the magnitude of estimation errors is derived.