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

6.2 Theoretical Background
Although the denition of this time evolving entropy can be done from any timefrequency
representation, as Gabor Transform or dierent type of wavelets, I will describe
the method from the wavelet coecients C ij (where i denotes time and j are the
dierent scales) obtained after applying the multiresolution decomposition method (see
sec. x4.2.3).
Once the coecients C ij are known, the energy for each time i and level j can be
calculated as
E ij = C 2 ij (51)
Since the number of components for each resolution level is dierent, I will redene
the energy by calculating, for each level, its mean value in successive time windows
(t = 128ms) denoted by the index k which will now give the time evolution. Then,
the energy will be:
E kj = 1 N
i 0
X+t
i=i 0
E ij (52)
where i 0 is the starting value of the time window (i 0 =1 1+t 1+2t : : :) and N is
the number of components in the time window for each resolution level. For every time
window k, the total energy can be calculated as:
E k = X j
E kj (53)
and we can dene the quantity
p kj = E kj
E k
(54)
as a probability distribution associated with the scale level j. Clearly, for each time
window k, P j p kj = 1 and then, following the denition of entropy given by Shannon
(1948), the time varying Wavelet-entropy can be dened as (for further details see Blanco
et al., 1998a):
WS k = ; X j
p kj log 2
p kj (55)
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