Continuous Time HMM
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In the previous chapters, we considered HMMs which are probabilistic functions of discrete Markov chains. In the continuous time HMM, observations depend on states of a continuous time Markov process. The continuous time models assume that during a small time interval the probability of a state change is also small. Therefore, if the time interval is small, the state transition probability matrixPis close to a unit matrixI.When we estimated the model parameters on the basis of experimental data in Sec. 1.5, we discovered that the diagonal elements of the transition probability matrices are close to one. For that reason, it was convenient to represent the models with the matricesR = P — I.The other consideration which must be taken into account is the accuracy of all the computations: we would rather express all the equations containingPin terms ofRto avoid computations with numbers having different precision. These and other considerations suggest that we might make another step and consider a limiting case of an HMM when the time unit becomes infinitesimally small. In the limit we obtain a continuous time HMM.
KeywordsTransition Probability Matrix Interarrival Time Continuous Time Model Discrete Time Markov Chain Single Server Queue
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