Generalized hidden Markov models. I. Theoretical frameworks

In this paper, we present the theoretical framework for the generalization of classical hidden Markov models using fuzzy measures and fuzzy integrals. The main characteristic of the generalization is the relaxation of the usual additivity constraint of probability measures. Fuzzy integrals are defined with respect to fuzzy measures, whose key property is monotonicity with respect to set inclusion. This property is far weaker than the usual additivity property of probability measures. As a result of the new formulation, the statistical independence assumption of the classical hidden Markov models is relaxed. Two attractive properties of this generalization are: the generalized hidden Markov model reduces to the classical hidden Markov model if we used the Choquet fuzzy integral and probability measures; and the establishment of a relation between the generalized hidden Markov model and the classical nonstationary hidden Markov model in which the transitional parameters vary with time.