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

ln d(t) ln C + t (50)
This gives a set of parallel lines for the dierent embedding dimensions, and the
largest Lyapunov Exponent can be calculated as the mean slope, averaging all the
embedding vectors.
Lyapunov exponents are very sensible to the election of the time lag, the embedding
dimension and especially to the election of the evolution time. If the evolution time is too
short, neighbor vectors will not evolve enough in order to obtain relevant information.
If the evolution time is too large, vectors will \jump" to other trajectories thus giving
unreliable results.
5.3 Stationarity
Due to the fact that Chaos methods require stationary signals and EEGs are known to
be highly non stationary, I will briey discuss this topic and I will propose a criterion of
stationarity. In principle, non-stationarity means that characteristics of the time series,
such as the mean, variance or power spectra, change with time. More technically, if we
have a time series of discrete observed values fx 1 x 2 :::x N g, stationarity means that
the joint probability distribution function f 12 (x 1 x 2 ) depends only on the time dierences
jt 1 ; t 2 j and not on the absolute values t 1 and t 2 (Jenkins and Watts, 1968). Statistical
tests of stationarity have revealed a variety of results in EEGs, and estimates of
stationary epochs range from some seconds to several minutes (Lopes da Silva, 1993a).
However, whether or not the same data segment is considered stationary, depends on
the problem being studied and the type of analysis to be performed.
In the case of EEGs, due to the large amount of data needed for the application of
the non linear dynamic methods, strict stationarity is almost impossible to achieve. This
problem has brought agreatvariety of results exposed by dierent authors (Basar, 1990
Basar and Bullock, 1989). A less restrictive requirement, called \weak stationarity" of
order n, is that the moments up to some order n are fairly stable with time. If the
probability distribution of a signal is Gaussian, it can be completely characterized by its
mean ( m ), its variance ( 2 ) and its autocorrelation function. In this case, second order
stationarity ( n = 2 ), plus an assumption of normality, is enough to assure complete
stationarity (Jenkins and Watts, 1968). Consequently, in order to check for stationarity
of the EEG segments to be used for analysis, the following procedure was used (Blanco
et al., 1995a).
1. The total EEG time series was divided in bins with a xed number of data. The
election of the bin length depends on the type of data and on the analysis to be
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