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Innovation Approach to the Identification of Causal Models in Time Series Analysis

Innovation Approach to the Identification of Causal Models in Time Series Analysis. T. Ozaki Institute of Statistical Mathematics. ( Wold, Kolmogorov, Wiener, Kalman, Kailath). Innovation Approach. ( Box-Jenkins , Akaike etc.). Causal Model. Ozaki(1985, 1992, 1995, 2000).

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Innovation Approach to the Identification of Causal Models in Time Series Analysis

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  1. Innovation Approach to the Identification of Causal ModelsinTime Series Analysis T. Ozaki Institute of Statistical Mathematics

  2. (Wold, Kolmogorov, Wiener, Kalman, Kailath) Innovation Approach (Box-Jenkins , Akaike etc.) Causal Model Ozaki(1985, 1992, 1995, 2000)

  3. What causes the time-dependencyingeophysical time series? Geophysical Dynamical System

  4. Three topics in time series 1. Nonlinear time series Dynamical Systems 2. Non-Gaussian time series Dynamical Systems & Shot Noise 3. Spatial time series Spatial Dynamics Innovation Approach

  5. Ozaki&Oda(1976) Dynamical System & Time Series Model Restoring force ∝ W・GM sin x W・GM sin x  x : for small x W・GM sin x  x - 1/6 x3 : for large x xt D.S. nt

  6. ExpAR Model • Smaller prediction errors ｔｈａｎ AR models. 2. Ｍｅｃｈａｎｉｃａｌ interpretation Ozaki&Oda(1976)

  7. Natural frequency n(t) x(t) nt xt f0

  8. Idea : Dynamic eigen-values 1. Duffing equation 2. van der Pol equation

  9. Make it non-explosive ! Explosive ! Non-explosive ! Make it stay inside for large xt-1 !  

  10. Ozaki(1985) ExpAR Singular points

  11. Ozaki(1985) ExpAR-Chaos

  12. ExpAR Models & non-Gaussian distributions Ozaki(1985) : Stable singular points : Unstable singular points

  13. Distribution of ExpAR process nt+1 ExpAR xt+1 non-Gaussian Gaussian white noise

  14. Causal Models indiscrete time and continuous time ? ?

  15. Time discretizationsofdx=f(x)dt+dw(t) Euler scheme Heun scheme Runge-Kutta scheme Explosive nonlinear AR model ! Any non-explosive scheme ?

  16. i). Ozaki(1984) L.L. scheme ii). Ozaki(1985) iii). Biscay et al.(1996) iv). Shoji & Ozaki(1997) Characteristics 1. Simple 2. A-stable

  17. Examples of Exp(KtDt)

  18. (Wold, Kolmogorov, Wiener, Kalman, Kailath) Innovation Approach (Box-Jenkins , Akaike etc.) Causal Model Ozaki(1985, 1992, 1995, 2000)

  19. Data Three types of models Time Series Models (ExpAR, neural net etc.) Dynamic Phenomena Stochastic Dynamical Systems Dynamical Systems

  20. Applications 1. Non-Gaussian time series and nonlinear dynamics 2. Estimation of dx=f(x)dt+dw(t) dx/dt=f(x) 3. RBF-Neural Net vs ExpAR, RBF-AR modeling 4. Spatial time series modeling

  21. Application-(1)Non-Gaussian time seriesandnonlinear dynamics Does non-Gaussian-distributed time series mean non-Gaussian prediction errors? Not Necessarily !

  22. Distribution of ExpAR process nt+1 ExpAR xt+1 non-Gaussian Gaussian white noise

  23. Ozaki(1985,1990) Same DistributionDifferent Dynamics a=3,b=1 i) a=3,b=1 ii) a=3,b=1 iii)

  24. (Ozaki, 1985,1992) Gamma-distributed Process(Type II) Mechanism • x(t) is generated from i) Gaussian white noise ii) Nonlinear Dynamics iii)Variable Transformation

  25. This implies the validity of innovation approach Non-Gaussian time series Gaussian white noise

  26. When residuals of your model arenon-Gaussianlooking,what would you do? 1.Introducenon-Gaussian noisemodel Bayesian Approach 2.Improve theCausal Model so that it produces Gaussian residuals Innovation Approach

  27. Application-(3)RBF-AR & RBF Neural Net ExpAR model is not suffucient More complicated dynamics, i.e. fk(x(t-1)) RBF-AR y-dependent system characteristics, i.e. fk(x(t-1), y(t-1)) RBF-ARX

  28. RBF-AR & RBF Neural Net More complicated fk(x(t-1)) f(xt ) RBF-AR(p,d,m) 1 xt Z1 0 Z2 RBF- Neural Net (p,d,m)

  29. Application - (2) Estimation of Numerical examples 1. Rikitake chaos, ( Geophysics) 2. Zetterberg Model (Brain Science) 3. Dynamic Market model ( Finance)

  30. How to identify ? from

  31. State Space Formulation

  32. Frost & Kailath(1971)’s theorem : Gaussian white noise Non-Gaussian time series Nonlinear Filter Gaussian white noise

  33. Likelihood Calculation Innovation Approach : Gaussian white noise Frost & Kailath(1971) How to obtain nt and snt2 ? Nonlinear Filter

  34. Relations to Jazwinski(1970)’s scheme Innovation Approach Calculate p(xk|zk) & p(zk|zk-1) etc. by Local Gauss model. L.L.

  35. Two Choices for Approximation 1) Local Gauss 2) Local non-Gauss Use of Fokker-Planck equation. Computationally 1) is super more efficient than 2)

  36. Advantages of the L.L. Scheme See B.L.S.Prakasa Rao(1999) Statistical Inference for Diffusion Type Processes H.Schurz(1999) A Brief Introduction To Numerical Analysis of (Ordinary) Stochastic Differential Equations Without Tears Essence is Stability & Efficiency

  37. Numerical examples 1. Rikitake chaos ( Geophysics) 2. Zetterberg Model (Brain Science) 3. Dynamic Market model ( Finance)

  38. Identification of the chaotic Rikitake model(Ozaki et at. 2000) Simulation obtained by the LL scheme Rikitake(1957), Ito(1982)

  39. Identification Results Simulated and estimated states

  40. Innovation of the estimated model P>0.95

  41. Three types of parameters Structured-parameter optimization for M.L.E. method 1. Peng & Ozaki(2001) 2. 3.

  42. Initial values & Estimated States : optimized : optimized : not optimized : optimized

  43. Initial values & Innovations : optimized : optimized : not optimized : optimized

  44. Reality in Data Analysis 1.Zero prediction errors are not possible to attain even with nonlinear causal (chaos) models. • Time - inhomogeneous residuals • Gaussian white residuals + a few outliers 2.Residuals of ARIMA models are usually almost Gaussian, but not always. • Generally distributed residuals are rare ! (We don’t see bi-modally distributed residuals)

  45. Whitening filter-1 Whitening filter-2 Obvious Choice A question is whether it is possible to find a perfect deterministic model for the data x1,x2,…,xN Causal Model-1 Causal Model-2 (Stochastic Model) (Deterministic Model)

  46. Application - (4) Spatial time series modeling

  47. Example : Data assimilationin meteorology States on the lattice points (i,j) Estimation Observations

  48. Rabier et al(1993) Mutual understanding : on the way Variationalmethod (4D-Var ) Ide & Ghil(1997)  Closed system  Perfect model with Observation errors Penalized Least Squares method •  Open system • Model errors as well as Observation errors

  49. Similar principles Variational method (4D-Var )  Closed system  Perfect model with observation errors Penalized Least Squares method  Open system  Model errors as well as observation errors Sequential method (Extended Kalman Filter  MLE )  Open system  Model errors as well as observation errors (similar to P.L.S. but not the same!)

  50. Hidden approximations behind perfect-model assumptions Infinite dimensional state  Finite dimension ( Model error) Open universe  Closed system ( Model error) Nonlinear dynamics  Numerical Approximation ( Model error)

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