Quantitative analysis of EEG signals: Time-frequency methods and ...

unknown topology it is necessary to calculate C(R) for several embedding dimensions 6 .
Then, for deterministic signals, the correlation dimension D 2 should converge towards
a saturation value, for some n>n o .
Singular Value Decomposition (SVD)
In many occasions the amount ofnoisein a signal makes impossible the calculation
of D 2 . One way of reducing the noise from the signal is by using the SVD method.
The main idea is to form a matrix from the embedding vectors (eq. 43) and then, after
calculating the eigenvectors and eigenvalues, to make a reconstruction of the signal
by rotating the matrix to a reduced base composed by the eigenvectors considered
signicant. Since the eigenvectors with relative small eigenvalues are supposed to be
dominated by noise, this phase space reduction is a way of eliminating noise from the
signal.
5.2.4 Problems arising when calculating the Correlation Dimension
In order to make the phase space reconstruction described in eq. 43 an appropriate time
lag and an appropriate embedding dimension n should be chosen. Since an unfortunate
election could lead to wrong results, several criteria were proposed (Blanco et al., 1996a
Abarbanel et al., 1993).
Concerning the time lag, if it is to small, the components in eq. 43 will have nearly the
same value and they will not properly span the phase space (redundancy). In the extreme
case, the attractor will be constricted to a diagonal line and its topology will be lost. On
the other hand, if the time lag is very high, the components of each embedding vector
will become totally unrelated to each other, thus becoming meaningless (irrelevance).
Several methods were developed for estimating the optimal time lag, among them, the
rst zero of the autocorrelation function has been the most used. This criterion assures
linear independence of the components. Some variations were also proposed as for
example the value of a certain decay of the autocorrelation function. Furthermore, for
taking into account non linear eects, criteria based on non-linear correlations were
dened (average of mutual information) (Abarbanel et al., 1993 Pjin, 1990). Other
approach was given by Rosenstein et al. (1994), who proposed a geometrical-based
method. The main idea is to look for a time lag that gives an optimal expansion of the
embedding vectors with respect to the diagonal in the phase space.
The election of a proper n was also subtle to discussion. If n is too small, the attractor
will be not completely unfolded and on the other hand, if n is too high, calculations
6 In fact, following Takens's proposal the dimension n of the embedding phase-space should be chosen
at least twice the dimension of the attractor.
71