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

will be dominated by noise. One rst criteria is to take n > 2D 2 (Takens, 1981). In
this case, the attractor will be completely unfolded but this criterion assumes a previous
estimation of D 2 . One solution is then to repeat the calculations for dierent sets of
embedding dimensions and in case of nding convergence, then to cheek if this criterion
is fullled. However, this is sometimes dicult to implement because the calculations
do not depend solely on or on n, on the contrary, they depend on the combination
of both (Broomhead and King, 1986 Mees et al., 1987 Albano et al., 1988 Palus and
Dvorak, 1992). Then, an estimation of a minimum n would be helpful. In this respect,
Kennel et al. (1992) proposed the false nearest neighbors method. The main idea is to
calculate if for a certain n nearest neighbors in the phase space still remain close for a
dimension n +1. If this is not the case, then the attractor was not completely unfolded
and the embedding dimension must be higher. The procedure is repeated for increasing
embedding dimensions until neighbors remain close.
Another important problem arises when choosing the linear region in the plots of
log C(R) vs. log R and also in how to calculate the slope. Up to the moment there is
no unique solution to this question and dierent groups have dierent approaches for
obtaining the values of D 2 from the log C(R) vs. log R plots. Some researchers prefer to
show directly the plots (Pjin et al., 1997) or to limit the calculation to the correlation
integral (eq. 44).
5.2.5 Lyapunov Exponents and Kolmogorov Entropy
Another useful tool for characterizing the attractor are the Lyapunov exponents. Lyapunov
exponents provide a quantitative indication of the level of chaos of a system.
They measure the mean exponential divergence of initially close phase space trajectories
with time. As more rapidly two trajectories diverges for a certain period of time,
more chaotic is the system and more sensitive to initial conditions.
Let us consider a small spherical hypervolume in the phase space. After a short
time, as trajectories evolve, the sphere will have an ellipsoid shape with its axes deformed
according the Lyapunov exponents. If the system is known to be dissipative, the volume
in the phase space will tend to contract and the sum of the Lyapunov exponents will be
negative. The longest axis of the ellipsoid will correspond to the most unstable direction,
determined by the largest Lyapunov exponent. Usually only this exponent is computed.
If it is positive, trajectories will diverge otherwise, they will get closer reaching a non
chaotic attractor. Following this argument, a necessary condition for a system to be
chaotic is that at least one of the exponents (the largest one) is positive. Lyapunov
exponents also give an indication of the period of time in which predictions are possible
and this is strongly related with the concept of information theory and entropy. In fact,
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