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

**What causes the
time-dependency
in
geophysical time series?**

**Dynamical System & Time
Series Model**

**ExpAR Models
&
non-Gaussian distributions**

**Causal Models
in
discrete time and continuous time**

**Time
discretizations
of
dx=f(x)dt+dw(t)**

**Application-(1)
Non-Gaussian time series
and
nonlinear dynamics**

**Same
Distribution
Different Dynamics**

**This implies
the
validity of innovation approach**

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

**Application-(3)
RBF-AR & RBF Neural Net**

**Frost & Kailath(1971)’s
theorem**

**Relations to
Jazwinski(1970)’s scheme**

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

**Innovation of the estimated
model**

**Initial values &
Estimated States**

**Example : Data
assimilation
in
meteorology**

**Mutual understanding : on
the way**

**Hidden approximations
behind
perfect-model assumptions**

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

**Prediction errors with
assumptions**

**Innovation Ａｐｐｒｏａｃｈ
to Ｓｐａｔｉａｌ ＴｉｍｅSeries:**

**Innovations in spatial
dynamics**

**Space-Temporal Model with
stimulus**

**Estimated Model tells you
something**