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

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A AppendixDefinition A.6 (Independent Random Variables).The countably valued random variables X 1 ,...,X n are said to be independent if for any sequenceof real numbers x 1 ,...,x n we haveP(X 1 = x 1 ,...,X n = x n ) = P(X 1 = x 1 )P(X 2 = x 2 )···P(X n = x n ).A.4 Definitions for OptimizationThe following definitions are all taken from [46].Definition A.7 (Objective Function).An objective function f : X ↦→ Y with Y ⊆ R is a mathematical function, which is subjectto optimization.Definition A.8 (Problem Space).The problem space X (phenome) of an optimization problem is the set containing all elementsx which could be the solution of the optimization problem.Definition A.9 (Local Maximum).A (local) maximum ˆx l ∈ X of one (objective) function f : X ↦→ R is an input element withf ( ˆx l ) ≥ f (x) for all x neighboring ˆx l .Definition A.10 (Local Minimum).A (local) minimum ˇx l ∈ X of one (objective) function f : X ↦→ R is an input element withf ( ˇx l ) ≤ f (x) for all x neighboring ˇx l .104
A.4 Definitions for OptimizationDefinition A.11 (Local Optimum).A (local) optimum x ∗ l∈ X of one (objective) function f : X ↦→ R is either a local maximumor a local minimum.Definition A.12 (Global Maximum).A global maximum ˆx ∈ X of one (objective) function f : X ↦→ R is an input element withf ( ˆx) ≥ f (x) for all x ∈ X.Definition A.13 (Global Minimum).A global minimum ˇx ∈ X of one (objective) function f : X ↦→ R is an input element withf ( ˇx) ≤ f (x) for all x ∈ X.Definition A.14 (Global Optimum).A global optimum x ∗ ∈ X of one (objective) function f : X ↦→ R is either a global maximum or aglobal minimum.Definition A.15 (Optimal Set).The optimal set X ∗ is the set that contains all optimal elements.Definition A.16 (Solution Candidate).A solution candidate x is an element of the problem space X of a certain optimizationproblem.105
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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
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1 Motivation„NEC FASCES, NEC OPES
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Theoretically we profit from enormo
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2 Visual Transduction”The eye owe
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(a)(b)(c)Figure 2.3: (a) The anatom
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effect of generating the photorecep
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3 Theory”The theory is the net, t
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3.2 Markov Chains, Markov Processes
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3.3 Hidden Markov ModelsP[X(t + τ)
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3.4 Maximum Likelihood Principlewit
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3.5 Optimization3.5 OptimizationAcc
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3.7 Baum-Welch-AlgorithmAlgorithm 3
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3.8 Two Approaches on Stochastic Sy
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3.9 DiffusionConsider a basin fille
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3.9 Diffusion169].Rearranging (3.34
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3.9 Diffusionwith τ = t −t ′ .
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3.9 DiffusionD = 2k2 B T 2σ . (3.4
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3.10 Hidden Markov Models with Stoc
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3.11 Hidden Markov Model - Vector A
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3.11 Hidden Markov Model - Vector A
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3.12 Artificial Test Examples for H
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Page 65 and 66: 3.12 Artificial Test Examples for H
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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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Page 119 and 120: A.4 Definitions for OptimizationDef
Page 121 and 122: A.7 The Fluctuation-Dissipation-The
Page 123 and 124: A.7 The Fluctuation-Dissipation-The
Page 125 and 126: A.8 Kramers-Moyal Forward Expansion
Page 127 and 128: A.9 Deriving the Fokker-Planck Equa
Page 129 and 130: A.9 Deriving the Fokker-Planck Equa
Page 131 and 132: DanksagungenIch möchte folgenden M
Page 133: AffirmationHereby I, Arash Azhand,
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