An Optimization Method for Training Generalized Hidden Markov Model based on Generalized Jensen Inequality

摘要

Recently a generalized hidden Markov model (GHMM) was proposed for solving the problems of aleatory uncertainty and epistemic uncertainty in engineering application. In GHMM, the aleraory uncertainty is derived by the probability measure while epistemic uncertainty is modelled by the generalized interval. Given any finite observation sequence as training data, the problem of training GHMM is often encountered. In this paper, an optimization method for training GHMM, as a generalization of Baum-Welch algorithm, is proposed. The generalized convex and concave functions based on the generalized interval are proposed for inferring the generalized Jensen inequality. With generalized Baum-Welch's auxiliary function and generalized Jensen inequality, similar to the multiple observations training, the GHMM parameters are estimated and updated by the lower and the bound observation sequences. A set of training equations and re-estimated formulas have been derived by optimizing the objective function. Similar to multiple observations (expectation maximization) EM algorithm, this method guarantees the local maximum of the lower and the upper bound and hence the convergence of the GHMM training process.