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

5 Modeling of the Experimentvalues that are lying in the magnitude of 10 −35 . Also there we pointed out, that rescaling can be away to overcome this problem, provided that we know in which magnitude the according motionis taking place. In the case of the transducin motion we know that the range of motion is lying inthe magnitude of micrometers, thus we can rescale the observation and the according estimatednoise intensity parameters with 10 −35 since diffusion coefficients in the magnitude of micrometersare leading to noise intensities in the magnitude of 10 −35 (see section 4.6 on page 71).The estimation delivers the following results for a HMM-SDE model with three hidden states:⃗ π est =T est =σ (1)est =σ (2)est =σ (3)est =⎛ ⎞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 0.35( 162.5 0.00.0 163.4)· 10 −35 .Here we have a probability of one that the hidden state sequence is starting with state two. Furthermorethe estimated transition matrix is assigning a holding probability of 0.93 to state two.State one has a holding probability of 0.77, while the holding probability of state three is nearlyzero.The noise intensities are leading to the following three diffusion coefficientsD (1) =D (2) =D (3) =( ) 1.6 0.0 µm2,0.0 1.6 s( ) 5.4 0.0 µm2,0.0 5.4 s( ) 0.01 0.0 µm2.0.0 0.01 sIn section 4.6 on page 71 we stated that our diffusion coefficients for transducin should lie in arange, justified by recent works on the rhodopsin-transducin system, given by ( 4.3 on page 72)D T = 0.8 µm2s,...,7.0 µm2 .sThus our estimation results for the first and the second state diffusion coefficient is lying directwithin this range, while the third one is too low.78