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

3.12 Artificial Test Examples for Hidden Markov Model - Vector Auto Regression (HMM-VAR)3.12.2 Two-Dimensional Free Diffusion with a Hidden Markov ModelSequence of three hidden States(a)(b)Figure 3.12: Random sequence of three states. Graphical representation of a random Hidden Statesequence with 3 states and a sequence length of 100.000. (a) The first 2000 and (b)the first 10.000 steps.We later will investigate experimentally observed trajectories of single molecules, in this specialcase single molecule tracking data of transducin proteins (see section 4.5 on page 70 and chapter5 on page 73). These trajectory data will be two-dimensional projections of originally threedimensionaldiffusional molecular movements. This fact is grounding on limitations of modernWide-Field Fluorescence Spectroscopy in general (see section 4.3 on page 65). Thus we willapply in this section our investigations, considered the estimation of model parameter sets, toartificial examples of two dimensions. First we developed a Java based program, which generateda 100.000 steps long random sequence of three hidden states (see figure 3.12), based on thefollowing transition matrix:⎛T = ⎝0.98 0.01 0.010.01 0.98 0.010.01 0.01 0.98⎞⎠, (3.90)and afterwards for every step of the hidden state path one diffusion-based trajectory point on thex-y-plane, based on the following parameter set:λ q = {F q , µ q , ,σ q }, (3.91)with49