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

3 Theorydist(T ,T est ) = 0.002. (3.94)This is corresponding to a deviation of 0.2%, obviously an extraordinary good result.Taking a much larger sequence, namely this time a sequence of length 1000000 and thus alsoa trajectory of the same length led after 130 iteration steps, for the same transition matrix, to adistance between the true and the estimated transition matrices ofdist(T ,T est ) = 0.002, (3.95)which is in fact the same result as the one for a sequence with the length 100000. Obviously amuch longer sequence of hidden states does not change the result too much.On the other hand, the question, how the result changes, when taking a much shorter sequence,namely a sequence of 1000, we also wanted to investigate and got an estimated transition matrix⎛T est = ⎝0.99 4.5 · 10 −12 0.010.02 0.98 0.00.01 0.02 0.96⎞⎠. (3.96)One already is seeing by direct comparison of (3.90) with (3.96), that the estimation is not asgood, as it was for longer sequences. Figure (3.15) shows the convergence of the likelihood forthe estimation of the HMM-SDE data with a HMM sequence of length 1000.52