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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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