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

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3 Theory• β i (t) = ∑ N j=1 a i, jb j (o t+1 )β j (t + 1)• P(O|λ) = ∑ N i=1 β i(1)π i b i (o 1 )We proceed and compute the probability γ i (t) of being in state i at time t for the state sequence Obyγ i (t) = P(Q t = i|O,λ)and express it with respect to the forward- and backward-variables. First:P(Q t = i|O,λ) = P(O,Q t = i|λ)P(O|λ)Furthermore because of the conditional independence it is:= P(O,Q t = i|λ)∑ N j=1 P(Q,Q t = j|λ) .α i (t)β i (t) = P(o 1 ,...,o t ,Q t = i|λ)P(o t+1 ,...,o T |Q t = i,λ) = P(O,Q t = i|λ)and thus the definition of γ i (t) in terms of α i (t) and β i (t) is:γ i (t) =α i(t)β i (t)∑ N j=1 α j(t)β j (t)The probability of being in state i at time t and being in state j at time t + 1 is given asξ i, j (t) = P(Q t = i,Q t+1 = j,O|λ)P(O|λ)=α i (t) · a i, j b j (o t+1 )β(t + 1)∑ N i=1 ∑ N j=1 α i(t) · a i, j b j (o t+1 )β(t + 1) .Through the summation over all time steps t one obtains the expected total number of transitionsaway from state i for O :Tγ i (t).∑t=1With the same assumptions one obtains for the expected number of transitions from state i to statej for O :T −1ξ i, j (t).For the estimation of HMM we get for the relative frequency spent in state i at time 1:∑t=1π i = γ i (1).The quantity a i j which is an entry of the transition matrix, is the expected number of transitionfrom state i to state j, relative to the expected total number of transition away from state i.a i, j =And for discrete distributions, the quantityb i (k) =∑T−1t=1 ξ i, j(t)∑ T −1t=1 γ i(t)∑T−1t=1 δ o t, ,v kγ i (t)∑ T −1t=1 γ i(t)is the expected number of times that the output observations have been equal to v k while being instate i, relative to the expected total number of times in state i.26
3.8 Two Approaches on Stochastic Systems with Equivalent Results3.8 Two Approaches on Stochastic Systems with EquivalentResultsBefore we will go deeper into considerations of HMMs and the estimation processes of its parameters,the purpose of this thesis needs considerations on the physical phenomenon of diffusion, tobe examined in the next section. This, as we will see can be considered by two different ways,grounding on two distinct philosophical and physical preconditions, but leading to the same results.These two preconditions and the conditions of their equivalence are presented in and justified bythe following axiom:Axiom 3.1 (Theoretical aproaches based on dynamics of one single system versus theoreticalaproaches based on an ensemble of similar systems - Equivalence of the two assumptionsjusitfied by the Quasi Ergodic Hypothesis).Consider a stochastic system, that we can examine and for which we can set up a theoreticalmodel. Then two models can be formulated, everyone of which based on one pre-condition:1. Model 1 based on the following assumption:The dynamics of a single system in time will be described by means of a single trajectory.2. Model 2 based on the following assumption:The dynamics of a single system is no more a matter of interest, moreover the dynamics ofan ensemble, consisting of similar systems (trajectories), will be described.Then the two models are equivalent in delivering the same results, denoted byModel 1 ⇐⇒ Model 2, (3.32)provided that the Quasi Ergodic Hypothesis is fulfilled. The Quasi Ergodic Hypothesis is testifyingthat a trajectory within the phase space of the system reaches every point in the phase spacearbitrary close. The consequence is that the time average is equal to the ensemble average.This basic separation is a central property of the so called thermodynamical systems. This meanslarge systems, consisting of N ≥ 10 23 particles. Here the question for the dynamical state (velocities˙q k and momenta p k for every single particle k) faces the question for the thermodynamicalstate (temperature T , volume V and pressure P).”The knowledge of the thermodynamical state is by no means sufficient for the determinationof the dynamical state. Studying the thermodynamical state of a homogeneousfluid of given temperature, we observe that there is an infinite number of statesof molecular motion that correspond to it. With increasing time, the system existssuccessively in all these states that correspond to the given thermodynamical state.From this point of view we may say that a thermodynamical state is the ensemble of27
Page 1: Diploma ThesisDepartment for Theore
Page 5 and 6: AbstractAnomalous diffusion is a ub
Page 7 and 8: 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: 3.7 Baum-Welch-AlgorithmAlgorithm 3
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 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
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5.5 Estimation of the Noise Intensi
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5.6 Estimation on the Basis of Diff
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Page 101 and 102:
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6 Conclusion and OutlookThe main as
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7 Bibliography[1] R. C. Aster, B. B
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[36] C. U. M. Smith: Elements of Mo
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Index11-cis retinal isomer, 87TM se
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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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