Cartesian hidden Markov models with applications

The author introduces the concept of a Cartesian hidden Markov model (CHMM), which consists of a Markov chain assuming values in the Cartesian product of a finite number of elementary state sets. The states are observed via a multivariable probabilistic mapping, again assuming values in a Cartesian product of finite sets of observables. The CHMM can be reduced to an ordinary (i.e., scalar) HMM by conventional nonlinear techniques. The forms of the forward-backward algorithm which gives the fixed-interval smoothed maximum a posteriori (MAP) estimates of the states and the Viterbi algorithm which gives the MAP fixed-interval sequence are straightforward generalizations of the scalar case. Two applications of CHMMs in the area of frequency tracking are briefly indicated