Numerics of Stochastic Processes II ( ) Illia Horenko
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1 Numerics of Stochastic Processes II ( ) Illia Horenko Research Group Computational Time Series Analysis Institute of Mathematics Freie Universität Berlin (FU) DFG Research Center MATHEON Mathematics in key technologies
2 Short Reminder Discrete State Space, Discrete Time: Markov Chain Discrete State Space, Continuous Time: Markov Process Last Time Stochastic Processes Continuous State Space, Discrete Time: Markov Process Continuous State Space, Continuous Time: Stochastic Differential Equation
3 Short Reminder: Markov Processes Infenitisimal Generator: Markov Process Dynamics: This Equation is Deterministic: ODE! 1) Numerical Methods from ODEs like Runge-Kutta-Method become available for stochatic processes 2) Monte-Carlo-Sampling just at the end! Concepts from the Theory of Dynamical Systems are Applicable
4 Short Reminder Discrete State Space, Discrete Time: Markov Chain Discrete State Space, Continuous Time: Markov Process Stochastic Processes Continuous State Space, Discrete Time: Markov Process Continuous State Space, Continuous Time: Stochastic Differential Equation Today
5 Markov Process in R Realizations of the process: Process: Next-Step Probability: Next-Step Distribution: This Equation is Deterministic!
6 Markov Process in R Realizations of the process: Process: Next-Step Probability: Next-Step Distribution: This Equation is Deterministic!
7 Markov Process in R Infenitisimal Generator: Markov Process Dynamics: This Equation is Deterministic: PDE! 1) Numerical Methods from PDEs like Finite-Element-Methods (FEM) become available for stochatic processes 2) Monte-Carlo-Sampling just at the end! Concepts from the Theory of Dynamical Systems are Applicable
8 Short Trip in the World of Dynamical Systems
9 Dynamical systems viewpoint Essential Dynamics of many dynamical systems is defined by attractors. Attractor stays invariant under the flow operator Figures from Separation of informative (attractor) and non-informative (rest) parts of the space leads to dimension reduction
10 How to localize the attractor? Attractors can have very complex, even fractal geometry Problem: Euclidean distance cannot be used as a measure for the relative neighbourghood of attractor elements. Strategy: the data have to be embedded into Eucllidean space Whitney embedding theorem (Whitney, 1936) : sufficiently smooth connected -dimensional manifolds can be smoothly embedded in -dimensional Euclidean space.
11 How to construct the embedding? Takens Embedding (Takens, 1981) and is a smooth map. Assume that has an attractor with dimension. Let, be a proper measurement process. Then is a -dimensional embedding in Euclidean space
12 Illustration of Takens Embedding The image of is unfolded in Problems: - how to filter out the attractive subspace? - how to connect the changes in attractive subspace with hidden phase? Strategy: Let. Define a new variable. Let the attractor for each of the hidden phases be contained in a distinct linear manifold defined via
13 Topological dimension reduction (H. 07): for a given time series we look for a minimum of the reconstruction error μ T
14 Topological dimension reduction (H. 07): for a given time series we look for a minimum of the reconstruction error μ T TT (X-μ) t T (X-μ) t x t
15 Topological dimension reduction (H. 07): for a given time series we look for a minimum of the reconstruction error μ T TT (X-μ) t T (X-μ) t x t
16 Analysis of embedded data PCA + Takens Embedding (Broomhead&King, 1986) Let. Define a new variable. Let the attractor from Takens Theorem be in a linear manifold defined via. Then Reconstruction from m essential coordinates:
17 PCA+Takens: data compression/reconstruction
18 Example: Lorenz Lorenz-Oszillator with measurement noise
19 Example: Lorenz Reconstructed trajectory (red, for m=2)
20 Themen 1) Markov-Prozesse (I): Dynamische Systeme und Identifikation der metastabilen Mengen (Maren) 2) Markov-Prozesse (II): Generatorschätzung (??) 3) Large Deviations Theory (Olga) 4) Dimensionsreduktion in Datenanalyse: graphentheoretische Zugänge I (Paul) 5) Dimensionsreduktion in Datenanalyse: graphentheoretische Zugänge II(Ilja) 6) Prozesse mit Errinerung (I): opt. Steuerung, Kalman-Filter und ARMAX-Model (??) 7) Prozesse mit Errinerung (II): Change-Point analysis (Peter) 8) Varianz von Parameterschätzungen (Bootstrap-Algorithmus) (Beatrice) 9) Modelvergleich in Finanzsektor (Lars)
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