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

5.3 Testing the Modelalgorithm was done with a Viterbi path, generated by the following transition matrix:⎛T = ⎝0.98 0.01 0.010.01 0.98 0.010.01 0.01 0.98⎞⎠.First we did the estimation for the three states HMM parameter set⃗π gen =T gen =σ (1)gen =σ (2)gen =σ (3)gen =⎛ ⎞0.0⎝ 1.0 ⎠,0.0⎛⎞0.7 0.1 0.2⎝ 0.04 0.93 0.03 ⎠,0.77 0.23 1.3 · 10 −58( ) 1.0 0.0· 10 −35 ,0.0 1.0( ) 0.3 0.0· 10 −35 ,0.0 0.3( 163 0.00.0 163)· 10 −35 . (5.2)The re-estimated parameter set then is⃗π est =T est =σ (1)est =σ (2)est =σ (3)est =⎛ ⎞0.0⎝ 1.0 ⎠,0.0⎛⎞0.93 0.04 0.03⎝ 0.1 0.7 0.2 ⎠,0.23 0.77 9.1 · 10 −46( ) 0.3 0.0· 10 −35 ,0.0 0.3( ) 1.0 0.0· 10 −35 ,0.0 1.0( 162.8 0.00.0 163.4)· 10 −35 . (5.3)By comparison of 5.3 and 5.2, we observe the following results:• The transition matrix entries of the re-estimated parameter set for the states 1 and 3 areinterchanged.• The transition probabilities a 31 and a 32 in the third row of the re-estimated transition matrixare interchanged.• The noise intensities σ (q) with q ∈ {1,2,3} are estimated correctly.81