Generalized Hidden Markov Models — Part I : Theoretical Frameworks

This is the first paper in a series of two papers describing a novel generalization of classical hidden Markov models using fuzzy measures and fuzzy integrals. In this paper, we present the theoretical framework for the generalization and, in the second paper, we describe an application of the generalized hidden Markov models to handwritten word recognition. 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. An attractive property of this generalization is that the generalized hidden Markov model reduces to the classical hidden Markov model if we used the Choquet fuzzy integral and probability measures. Another interesting property of the generalization is 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.