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

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3 TheoryFigure 3.11: Three-dimensional diffusion with potential and 1000 time steps. X, Y, Z denote thethree spatial coordinates.As we can see in the Figures 3.9 and 3.10, the generated trajectory is representing a standardBrownian motion, while Figure 3.11 is showing a diffusional movement, which is not free, butseemingly influenced by a potential force, that is causing a drift of the process in direction of thebottom with subsequent diffusion on the vicinity of the bottom.Just in the framework, in which these three-dimensional trajectories are generated, we will proceedand generate two-dimensional trajectories, with underlying Hidden Markov Sequences, in orderto estimate afterwards these observation trajectory data. This will be done in the next subsection.48
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
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Diploma ThesisDepartment for Theore
Page 5 and 6:
AbstractAnomalous diffusion is a ub
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Zusammenfassung 1Anomale Diffusion
Page 9 and 10: Contents1 Motivation 12 Visual Tran
Page 11 and 12: List of Figures2.1 Anatomy of the e
Page 13 and 14: 1 Motivation„NEC FASCES, NEC OPES
Page 15 and 16: Theoretically we profit from enormo
Page 17 and 18: 2 Visual Transduction”The eye owe
Page 19 and 20: (a)(b)(c)Figure 2.3: (a) The anatom
Page 21 and 22: effect of generating the photorecep
Page 23 and 24: 3 Theory”The theory is the net, t
Page 25 and 26: 3.2 Markov Chains, Markov Processes
Page 31 and 32: 3.3 Hidden Markov ModelsP[X(t + τ)
Page 33 and 34: 3.4 Maximum Likelihood Principlewit
Page 35 and 36: 3.5 Optimization3.5 OptimizationAcc
Page 37 and 38: 3.7 Baum-Welch-AlgorithmAlgorithm 3
Page 39 and 40: 3.8 Two Approaches on Stochastic Sy
Page 41 and 42: 3.9 DiffusionConsider a basin fille
Page 43 and 44: 3.9 Diffusion169].Rearranging (3.34
Page 45 and 46: 3.9 Diffusionwith τ = t −t ′ .
Page 47 and 48: 3.9 DiffusionD = 2k2 B T 2σ . (3.4
Page 49 and 50: 3.10 Hidden Markov Models with Stoc
Page 55 and 56: 3.11 Hidden Markov Model - Vector A
Page 57 and 58: 3.11 Hidden Markov Model - Vector A
Page 59: 3.12 Artificial Test Examples for H
Page 63 and 64: 3.12 Artificial Test Examples for H
Page 69 and 70: 3.13 Global Optimization Methods3.1
Page 71 and 72: 3.13 Global Optimization MethodsEve
Page 73 and 74: 4 Fluorescence Tracking Experiments
Page 75 and 76: 4.2 Fluorescence SpectroscopyAfter
Page 77 and 78: 4.3 Single Molecule Tracking via Wi
Page 79 and 80: 4.4 Total Internal Reflection Fluor
Page 81 and 82: 4.4 Total Internal Reflection Fluor
Page 83 and 84: 4.6 The expected range for the Tran
Page 85 and 86: 5 Modeling of the Experiment”The
Page 87 and 88: 5.1 Experimental Data(a)(b)(c)(d)Fi
Page 89 and 90: 5.2 Model Ansatz and Estimation of
Page 91 and 92: 5.2 Model Ansatz and Estimation of
Page 93 and 94: 5.3 Testing the Modelalgorithm was
Page 95 and 96: 5.5 Estimation of the Noise Intensi
Page 97 and 98: 5.6 Estimation on the Basis of Diff
Page 103 and 104: 6 Conclusion and OutlookThe main as
Page 105 and 106: 7 Bibliography[1] R. C. Aster, B. B
Page 107 and 108: [36] C. U. M. Smith: Elements of Mo
Page 109 and 110: Index11-cis retinal isomer, 87TM se
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A AppendixA.1 From Copernicus to Ne
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A.2 Important Papers on the Rhodops
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A.3 Probability TheoryDefinition A.
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A.4 Definitions for OptimizationDef
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A.4 Definitions for OptimizationDef
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A.7 The Fluctuation-Dissipation-The
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A.7 The Fluctuation-Dissipation-The
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A.8 Kramers-Moyal Forward Expansion
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A.9 Deriving the Fokker-Planck Equa
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A.9 Deriving the Fokker-Planck Equa
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DanksagungenIch möchte folgenden M
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AffirmationHereby I, Arash Azhand,
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Diffusion Processes with Hidden States from ... - FU Berlin, FB MI

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