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

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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
3.12 Artificial Test Examples for Hidden Markov Model - Vector Auto Regression (HMM-VAR)Figure 3.15: Likelihood convergence 2. Convergence of the Likelihood for the estimation of atwo-dimensional trajectory with underlying HMM sequence of 1000 steps.Next we wanted to know, how estimation results are changing, when assigning the algorithm witha random sequence of the same length, but induced by a significantly different transition matrix,for example by a matrix like the following one:⎛T = ⎝0.5 0.25 0.250.25 0.5 0.250.25 0.25 0.5⎞⎠. (3.97)Obviously (3.97) creates a sequence in which the states are not holding as much time, as it is thecase for a sequence, induced by the matrix (3.90).Repeated evaluation of the estimation process delivered a very interesting result. Obviously estimationis as extraordinary good, as it was for the assignment with the correct sequence. Thedistance between the true transition matrix (3.90) and the estimated one is approximately 0.003,corresponding to a deviation of 0.3%, although the initialization of the algorithm was done bya sequence, generated by the transition matrix (3.97), significantly deviating from the true one,(3.90).In the following we will use this good result as a motivation, in order to go further and make someinvestigations on the estimation of the Diffusion constants. Since we have free Diffusion, thusthere is no potential, we just can concentrate on the covariance matrices and neglect the regression53
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Diploma ThesisDepartment for Theore
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AbstractAnomalous diffusion is a ub
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Zusammenfassung 1Anomale Diffusion
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Contents1 Motivation 12 Visual Tran
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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 and 60: 3.12 Artificial Test Examples for H
Page 63: 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
Page 111 and 112: A AppendixA.1 From Copernicus to Ne
Page 113 and 114: 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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