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State space reconstruction
. In order to reconstruct the state space using a
time series, it is resolved into coordinate values of a
d
-dimensional embedding
space by an embedding method. Let the state space be characterized by the
set of variables,
{
x
0
(
t
)
,x
1
(
t
)
, ..., x
d−
1
(
t
)
}
.Mostfrequentlyusedis
time delay
embedding
. Assume that only time series
x
0
(
t
) is available
2
. Then time-delayed
values of this series are used,
{
x
0
(
t
)
,x
0
(
t
+
τ
)
, ..., x
0
[
t
+(
d
−
1)
τ
]
}
.
This set is topologically equivalent to the original set of system variables, see
[18]. These variables are obtained by shifting the original time series by a fixed
time lag
τ
=
mΔt
where
m
is an integer and
Δt
is the interval between succes-
sive samplings. A most important problem of state space reconstruction is the
determination of the delay time
τ
and embedding dimension
d
for which several
methods are available. This result is known as Taken's famous 'Embedding The-
orem' which says: valuable information about the dynamics of the system can
be obtained, even if direct access to all the systems variables is impossible, as it
is common in cognitive neuroscience!
Characterization of the reconstructed attractor.
After reconstruction of
the attractor by embedding the next step is to characterize it. A common way
to do this is to visualize it with a phase portrait. A phase portrait is a two-
or three-dimensional plot of the reconstructed state space and the attractor.
The graph shown in Fig. 1 is an example of a two-dimensional phase portrait.
Other methods to display the reconstructed trajectories are Poincare sections
and recurrence plots [19].
Following embedding and visualization of the reconstructed attractor the next
step is to attempt to characterize it in a quantitative way. The classic measures
applied are correlation dimension, Lyapunov exponents and entropy mentioned
in Section 4, and new measures introduced frequently in the literature.
Checking the validity of the procedure.
The interpretation of nonlinear
measures is known to present problems sometimes since noisy time series can
give rise to the unwarranted impression of low-dimensional dynamics and chaos.
Therefore, the nonlinearity of the time series should be tested. It is customary
to do this by
surrogate data test
. The null hypothesis of the test is that the
original time series is generated from a linear stochastic process (possibly under-
going a nonlinear static transform). Demonstration of nonlinearity is important
since only nonlinear dynamical systems can have attractors other than a trivial
fixpoint attractor. Chaos in particular can only occur in nonlinear dynamical
systems.
5 Explanation by Nonlinear Dynamic Analysis
In the preceding sections the methodology of nonlinear dynamic analysis was
outlined roughly. We are now prepared to demonstrate how it is used in cognitive
2
x
0
may represent the electrical potential recorded by the EEG.
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