Innovation Approach
the Identification of Causal Models
Time Series Analysis

Innovation Approach

What causes the time-dependency
geophysical time series?

Three topics in time series

Dynamical System & Time Series Model

ExpAR Model

Natural frequency

Idea : Dynamic eigen-values

Make it non-explosive !

ExpAR Singular points


ExpAR Models
non-Gaussian distributions

Distribution of ExpAR process

Causal Models
discrete time and continuous time

Time discretizations

L.L. scheme

Examples of Exp(KtDt)

Innovation Approach

Three types of models



Non-Gaussian time series
nonlinear dynamics

Distribution of ExpAR process

Same Distribution
Different Dynamics


This implies
the validity of innovation approach

When residuals of your model
are non-Gaussian looking,
what would you do?


RBF-AR & RBF Neural Net

RBF-AR & RBF Neural Net

Application - (2)

How to identify ?

State Space Formulation

Frost & Kailath(1971)s theorem

Likelihood Calculation

Relations to Jazwinski(1970)s scheme

Two Choices for Approximation

Advantages of the L.L. Scheme

Numerical examples

Identification of the chaotic Rikitake model
(Ozaki et at. 2000)

Identification Results

Innovation of the estimated model

Three types of parameters

Initial values & Estimated States

Initial values & Innovations

Reality in Data Analysis

Obvious Choice

Application - (4)

Example : Data assimilation
in meteorology

Mutual understanding : on the way

Similar principles

Hidden approximations
perfect-model assumptions

Experience in Chaos

Non-penalized L.S. method(4D-VAR)
is even worse !

Prediction errors with assumptions

M.L.E. with L.L. Filtering

Innovation Approach to Spatial TimeSeries:

fMRI Machine

fMRI data

Innovations in spatial dynamics

Space-Temporal Model with stimulus

Estimated Model tells you something

Looking through

Example A

Slice 12

i=36 ~ 40

i=41 ~ 45

i=46 ~ 50

Innovation Maps

Spatial Impulse Response

Spatial ARX Simulation-1

Innovation Approach
could be useful in Space-Time

Thank you