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with
x
as a transition variable that can be described by an AR(1) model with the
parameters
D
and
D
and with the nonlinear component
fx
(
)(
M O
x
).
td
t
1
There is also the
ESTAR model
fx
(
)
1 [
kx
(
k
) ]
2
td
td
with
k
> 0, and the
LSTAR model
1
fx
(
)
.
td
2
1exp[ (
kx
k
)]
td
State-space modelling
of nonlinear time series relies on the theory of
first-
order Markov chains
in the
n
-dimensional state space, where the observation
vector is represented by
and the nonlinear time series model
is represented by the
stochastic difference equation
T
xxx
(,
,
x
),
t
t
t
1
t
n
1
xSx
H
(,
.
.
t
t
t
1
Alternatively, the nonlinear state space can be used for modelling the nonlinear
time series, relying on transition probability
P x
{
A
x
x
x
,
x
d
x
,
j
i
}
t
1,
i
1
i
1
t
t
1,
j
t
1
where
x
denotes the (
i
+1)th component of
x
1
.
t
1,
i
1
2.5.9 Chaotic Time Series Models
In the last two decades or so, research in the field of
chaotic time series analysis
has steadily grown and it is today an interesting field of work for mathematicians
and engineers. Initially, the research interest was in estimating the dimension of the
underlying
attractor
and the
Liapunov exponents
of the chaotic systems that
characterize the space-filling properties and the stability of dynamic systems. The
attention was later focused on the techniques of chaotic time series modelling and
on prediction of future time series values using most frequently the
nonlinear
autoregressive model
for the state vector
x
(
t
).
xt
(
d
)
F xt xt d
[ ( ),
(
),
xt d
(
),...,
xt d
(
)]
1
2
n
1
where
d
is the
delay factor
between the individual observations and
n
is the
number of observations considered. Here, the nonlinear time series model is
required when the model should hold globally. Otherwise, for local considerations,
a
local linear model
is preferred.
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