Analysis of non-stationary Data. Illia Horenko
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1 Analysis of non-stationary Data Illia Horenko Scientific Computing Institute of Mathematics Freie Universität Berlin (FU) DFG Research Center MATHEON Mathematics in key technologies
2 Motivation global stationarity, local stationarity, examples Non-Stationarity for continuous state space processes: Fuzzy Clustering with Regression Models (FCRM) Finite Element Clustering for continuos state space processes Non-Stationarity for discrete state space processes: Kernel Filtering Methods Single Trend Model FEM-Clustering of Markov-chain output Example I: analysis of historical weather patterns Example II: analysis of the historical temperatures ( )
3 Non-Stationarity Market Phase 2 Fluid Mechanics Market Phase 1 Meteorology/Climate Computational Finance Biophysics/Drug Design (Local) weak stationarity: (local) time independence of mean values and covariances Not fulfilled in many cases
4 Non-Stationarity in Continuous State Space Time Series
5 Example: weather data analysis Global Historical Data Local Historical Data Non-stationarity because of the climate change => standard data-analysis methods are not applicable
6 Memo I: K-Means clustering Assumption: time-independence of cluster centers local stationarity
7 Toy Example: Non-stationary Extension of K-Means: FCRM
8 Non-stationary Extension of K-Means: FCRM Exercise 1: Derive an expression for the optimal estimator of the regression parameters θ (x and γ are fixed). What happens to this estimate if R is growing? How to define the optimal R? Suggest and discuss the possible numerical solutions.
9 Non-stationary Extension of K-Means: FCRM Toy Example: Proper identification of persistent clusters impossible (try the FEM-Clustering?)
10 FEM: Regularized Clustering Functional Regularized clustering functional:
11 FEM: Regularized Clustering Functional Iterative Subspace Minimization: sparse QP canbeused
12 Toy Example: FEM: Regularized Clustering Functional
13 FEM: Regularized Clustering Functional
14 FEM: Regularized Clustering Functional
15 Non-Stationarity in Discrete State Space Time Series
16 Example:Weather in UK( ) Historical Circulation Data: weather regimes (Data from the Univ. of East Anglia) 3 atmospherical states considered
17 Memo II: Markov Chain Log-Likelihood Observed Time Series:, Markov-Property: Log-Likelihood: Maximization problem is ill-posed => regularization necessary
18 Circulation Patterns for UK( ) Historical Circulation Data: 27 Lamb regimes (Data from the Univ. of East Anglia) 3 atmospherical states considered Regularization: global stationarity assumption anticyclonic anticyclonic other cyclonic other cyclonic
19 Locally Stationary Markov Chain Regularization: local stationarity assumption inside of the window Gaussian Window: Approximate Log-Likelihood: Gaussian Kernel Filtering: Standard statistical methods applicable to calculate the conf. intervals
20 Locally Stationary Markov Chain Regularization: local stationarity assumption inside of the window Gaussian Window: Approximate Log-Likelihood: Gaussian Kernel Filtering: Exercise 2: proof that this formula is true
21 Circulation Patterns for UK( ) Historical Circulation Data: 27 Lamb regimes (Data from the Univ. of East Anglia) 3 atmospherical states considered Gaussian Window vs. Homogenous Estimator anticyclonic anticyclonic other cyclonic cyclonic other
22 Circulation Patterns for UK( ) Historical Circulation Data: 27 Lamb regimes (Data from the Univ. of East Anglia) 3 atmospherical states considered
23 Parametric Estimation (H. 08, to appear in J. of Atmos. Sci.) Markovian Trend Model: Log-Likelihood: Numerics: Nelder-Mead Optimization Algorithm
24 Circulation Patterns for UK( ) Historical Circulation Data: 27 Lamb regimes (Data from the Univ. of East Anglia) 3 atmospherical states considered Polynomial Trend Model anticyclonic anticyclonic other cyclonic cyclonic other
25 Circulation Patterns for UK( ) Historical Circulation Data: 27 Lamb regimes (Data from the Univ. of East Anglia) 3 atmospherical states considered Polynomial Trend Model
26 FEM: Regularized Clustering Functional Regularized clustering functional:
27 FEM: Regularized Clustering Functional Regularized clustering functional: (H. 08, to appear in J. of Atmos. Sci.)
28 FEM: Regularized Clustering Functional Regularized clustering functional: (H. 08, to appear in J. of Atmos. Sci.) Iterative Subspace Minimization: sparse QP canbeused
29 Circulation Patterns for UK( ) Historical Circulation Data: 28 Lamb regimes (Data from the Univ. of East Anglia) 3 atmospherical states considered anticyclonic anticyclonic other cyclonic cyclonic other
30 Circulation Patterns for UK( ) Historical Circulation Data: 28 Lamb regimes (Data from the Univ. of East Anglia) 3 atmospherical states considered Gaussian Kernel Estimator FEM-Clustering vs. Single Trend
31 Example: analysis of hystorical temperatures ( )
32 Temperature Data Analysis Global Historical Temperatures Historical Temperatures in Europe Data from Boulder Center, USA Data from Ch. Franzke, BAS Cambridge Temperature Data on the 2D-Grid ( )
33 European Data Analysis
34 European Data Analysis
35 European Data Analysis Cluster 1 Cluster 2 Cluster 3
36 European Data Analysis Cluster 1 Cluster 2 Cluster 3
37 European Data Analysis Cluster 1 Cluster 2 Cluster 3
38 European Data Analysis Cluster 1 Cluster 2 Cluster 3
39 European Data Analysis
40 Global Data Analysis Global Historical Temperature Data (80x120 grid, daily values ) (Data from the NCAR, Boulder, US )
41 Global Data Analysis Global Historical Temperature Data (80x120 grid, daily values ) (Data from the NCAR, Boulder, US )
42 Take-Home-Messages 1. Different methods for analysis of non-stationary data were presented 2. One can use similar approaches in both continuous and discrete data analysis 3. Issue of non-stationarity is important in analysis of historical time series
43 Thank you for attention!
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