Diffusion Processes with Hidden States from ... - FU Berlin, FB MI

3.12 Artificial Test Examples for Hidden Markov Model - Vector Auto Regression (HMM-VAR)algorithm, meaning that it stops when the change of the new likelihood in comparison with the lastone is not bigger than this number. The algorithm has to be initialized with an initial Viterbi path,in order to get started. We first decided to initialize it with a random sequence of three distinctstates, generated by the same transition matrix, as the sequence which has to be estimated. Thequestion, how the estimation result will be when initializing with a random sequence, created byanother transition matrix, we will investigate later.The procedure of estimation ended after approximately 100 iteration steps. Figure 3.14 shows theconvergence of the likelihood. The estimated transition matrix that we obtained is⎛T est = ⎝0.98 0.01 0.010.01 0.98 0.010.01 0.01 0.98⎞⎠, (3.93)obviously not to distinguish from the true one, given by (3.90).Figure 3.14: Likelihood convergence 1. Convergence of the Likelihood for the estimation of atwo-dimensional trajectory with underlying HMM sequence of 100000 steps.We computed the distance between the two matrices, the true transition matrix (3.90) and theestimated one (3.93) which is equal to the square root of the summed squared distances betweencorresponding entries:51