![](bg2.png)
different supplementary assumptions are made, and
validation is performed against large real datasets.
Rankine et al. separate two models having differ-
ent characteristics: seizure model and background
model, aiming to characterize different new-born real
EEGs (Figs. 1 and 2). We focus here on the (Rank-
50 100 150 200 250
Real EEG
Time (samples)
Figure 1: Background EEG signal.
50 100 150 200 250
Real EEG
Time (samples)
Figure 2: Seizure EEG signal.
ine et al., 2008) background EEG model, our aim be-
ing to find if it can be applied on adult surface and
depth data. We will asses its validity for both back-
ground and seizure signals. The different steps of the
cited model and employed methodological tools, are
described in more detail in the next section.
2.1 Background EEG Modelling
According to (Rankine et al., 2008) and the references
cited therein, the power spectrum of a background
surface EEG approximately follows a power law:
S( f) ≈
c
|f|
γ
(1)
where c is constant, f is frequency and γ is the power
law exponent
1
. If one wants to generate a simulated
EEG signal x(t), the first step is to express S( f) as
X( f)X
∗
( f), with X( f ) being the amplitude spectrum
of x(t), obtained by the Fourier transform:
X( f) =
√
c
|f|
γ
2
e
jθ( f)
, (2)
where θ( f) is the phase of the Fourier transform. In
order to obtain a more realistic signal, (Rankine et al.,
2008) proposes to generate several X
i
( f) using differ-
ent phase vectors θ
i
( f). Several x
i
(t) can be obtained
by inverse Fourier transform from X
i
( f), and the final
simulated background EEG signal is generated as
x(t) =
∑
i
F
−1
(X
i
( f)) (3)
1
Since real EEGs are non-stationary, γ is considered
constant for every epoch of 4 seconds (assuming a quasi-
stationary signal during one epoch).
As it can be seen, this model needs three parameters:
c, γ and θ(f). The amplitude c is of secondary impor-
tance, so we will focus only on the last two param-
eters. In order to use realistic values, they must be
extracted from real data.
2.1.1 Parameter Estimation
The method used in (Rankine et al., 2008) to estimate
the power law exponent γ exploits the linear relation-
ship between γ and the fractal dimension FD of a sig-
nal (Wornell and Oppenheim, 1992), expressed by:
FD =
5−γ
2
(4)
This step is useful because the FD can be estimated
from the real EEGs using one of the fractal dimension
estimation methods. Different fractal dimension esti-
mators such as Box-counting, Information and Corre-
lation dimensions (Ott, 2000) can be used, with quite
similar results on classical fractals. Higuchi’s FD es-
timation (Higuchi, 1988) is a particular example of
fractal dimension derived from box-counting. This al-
gorithm works directly in the time domain (analysing
the geometrical form of signal), so it can be used for
relatively short signal lengths (recall that EEG’s are
assumed stationary on short time intervals).
As said previously, in order to simulate realistic
signals, the needed parameters (FD and θ( f)) must
respect real signals characteristics. As in (Rankine
et al., 2008), we have estimated them using the fol-
lowing procedure, applied to a database of real adult
background EEG/SEEG signals:
• compute the FD and the phase for each signal
• assume that, over the database, FD follows a beta
distribution and estimate the distribution param-
eters (method of moments (NIST/SEMATECH,
2011)). Probability density function of a beta dis-
tribution with two parameters, α and β can be ex-
pressed as
f(x;α,β) =
Γ(α+ β)
Γ(α)Γ(β)
x
α−1
(1−x)
β−1
, (5)
where x ∈ [0,1] and Γ(z) =
Z
∞
0
t
z−1
e
−t
dt is the Γ
function.
• assume that the phase θ follows a uniform distri-
bution in [−π,π]
• test (Kolmogorov-Smirnov) the empirical distri-
butions against theoretical distributions generated
using the previously estimated parameters.
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