Maximum likelihood estimation of aggregated Markov processes

We present a maximum likelihood method for the modelling of aggregated Markov processes. The method utilizes the joint probability density of the observed dwell time sequence as likelihood. A forward-backward recursive procedure is developed for efficient computation of the likelihood function and its derivatives with respect to the model parameters. Based on the calculated forward and backward vectors, analytical formulae for the derivatives of the likelihood function are derived. The method exploits the variable metric optimizer for search of the likelihood space. It converges rapidly and is numerically stable. Numerical examples are given to show the effectiveness of the method.