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Nonlinear time series analysis of human alpha rhythm
G. Nolte1, T. Sander2, A. Lueschow3, B. A. Pearlmutter1
1Dept. of Computer Science, University of New Mexico, Albuquerque, USA; 2 Physikalisch-Technische Bundesanstalt,
Berlin, Germany; 3Neurophysics Group, Universitätsklinikum Benjamin Franklin, Berlin, Germany
Abstract
Nonlinearity is often deduced by showing that a dataset signi£cantly deviates from its phase randomized versions, i.e.
surrogate data. For real data, however, non-stationarities like artifacts and onsets and offsets of rhythmic activity will
cause false positives. We propose a new test which detects dynamical nonlinearity by measuring time-asymmetry, using
surrogate data merely to estimate the standard deviation of the process.
The method is applied to multi-channel MEG measurements of ongoing alpha-band activity modulated by a simple visual
memory task involving motor activity. The signal to noise ratio was enhanced using ICA, and the analysis was performed
on a single separated source. We found that, if the peak at 10 Hz is accompanied by a substantial higher harmonic, time
asymmetry can be detected signi£cantly in virtually any epoch of 3 second duration. Finally, we applied our recently pro-
posed method to estimate correlation dimension for noisy data. We found very satisfactory scaling plots with dimension
around 1.5. As a byproduct, we showed that the nondeterministic fraction can be explained almost completely by external
noise.
1 Introduction
We £rst propose and apply a new method that detects non-
linear dynamics in a time series. The main goal of the con-
struction is to avoid false positives from non-stationarities.
In contrast to the more common tests for inconsistency
with colored noise [1], which are based on comparison
of the data with phase randomized versions (’surrogate
data’) and are sensitive e.g. to spiky artifacts or activity on-
set/offset, our test is based on the detection of time asym-
metry, as proposed by [2] and others. A severe problem
is that without reference to surrogate data, the statistical
analysis implicitly assumes mutual independence of all an-
alyzed data segments. Enforcing this [2] leads (for sys-
tems with long autocorrelation, as in the alpha rhythm) to
a disastrous reduction of usable data. Apart from the ac-
tual de£nition of the discriminating statistic, which de£nes
’difference’, our new approach is to refer to surrogate data
only to estimate its standard deviation, while still measur-
ing its absolute difference (to zero) and not its relative dif-
ference (to the corresponding value for the surrogate data).
This method is applied to human alpha activity present in
two MEG datasets.
The second goal is estimation of the correlation dimension
(D2) of the alpha rhythm, which in turn is an estimate of
the number of degrees of freedom of the underlying dy-
namical system. In practice, the noisy signals obtained
from real-world systems make a meaningful estimate of
the correlation dimension extremely dif£cult to achieve.
Here, we apply a recently proposed method [3] to remove
noise bias from the scaling plots.
2 Measurements and pre-processing
In 8 MEG measurements (from a visual stimulation
paradigm designed for other purposes, sampled at 500 Hz)
of neural responses to visual stimulation, the underlying
generators were separated using Independent Component
Analysis [4]. Separated sources with alpha activity (de-
£ned by a 10 Hz spectral peak) could be clearly identi£ed.
We found two subjects with a substantial energy in the
20 Hz harmonic. For the others, the energy at 20 Hz was
detectable but too low for further analysis. Since a single
frequency is always consistent with linear dynamics, only
these two respective time series were further analyzed, af-
ter being passed through a 0.5 Hz–40 Hz bandpass £lter.
Dataset A was measured using a 49-channel planar MEG
system, where the sensor array was located tangentially
over position T6 of the 10–20 electrode position standard.
The alpha generator was localized to the occipital lobe,
and the rhythm is hence identi£ed as ’occipital alpha.’ We
found the rhythm to be very stable, and active about 70%
of the time. For the dimension analysis below we picked a
single time window of 16 seconds of stable alpha activity.
Dataset B was measured using an 80-channel whole head
MEG system. The alpha generator localized to the central
region, and was suppressed by £nger movement. It was
hence associated with rolandic mu rhythm [5]. Due to the
motor-related suppression we found only a few relatively
short, but still strong, bursts of alpha activity. Since a di-
mension analysis requires a relatively long time series, we
concatenated the four bursts into a single 11.3 second time
series.
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3 Detecting nonlinearity
We will detect nonlinearity within the framework of cor-
relation sums and correlation dimensions. Our £rst step
is to embed a single valued time series x = (x1, · · · , xn)
into an m-dimensional space by constructing vectors ~xi =
(xi, xi+τ , · · · , xi+(m−1)τ )T The time delay, set here to
τ = 6, should be chosen suf£ciently large to avoid es-
sential redundancies but otherwise as small as possible.
From these vectors a correlation sum C(²,m) at scale ² is
de£ned by
C(²,m) = 1N
∑
i<j−W
f(||~xi − ~xj ||/²) (1)
where we take f to be a gaussian kernel f(x) =
exp(−x2/4). N is the total number of terms to be summed
over, || · || is the euclidian norm, and W is a parameter set
to exclude pairs of vectors which are too close in time to
be regarded as independent [1]. The correlation dimension
can be estimated from
d(²,m) = d lnC(²,m)d ln ² (2)
If d(²,m) is constant as a function of both ² (scaling) and
m (saturation) in suf£ciently large regions, this constant
is called the correlation dimension D2 of the underlying
dynamical system.
To detect dynamical nonlinearities, we test whether the
correlation dimension is invariant under local reversal of
time as de£ned by replacing ~xj , the second vector in
f(||~xi − ~xj ||/²), by its ¤ipped version. As a measure of
time-asymmetry we choose a weighted average of the dif-
ference of d1(²,m) and d2(²,m), calculated for the origi-
nal and the locally time reverted time series, respectively:
∆(x) =
M
∑
m=1
∫
d(ln ²)d1(²,m)− d2(²,m)S(²,m)
²2
²2 + σ20
(3)
where S(²,m) = (C−11 (²,m) + C−12 (²,m))1/2 is a
(rough) estimate of the standard deviation of d1(²,m) −
d2(²,m), the maximal embedding dimension is M = 10,
and ²2/(²2 + σ20) is an additional heuristic weighting fac-
tor with a free parameter σ0. We found in simulations that
after normalizing all time series to unit standard deviation
the setting σ0 = 1 generally improved the results. The
standard deviation of ∆(x) is then estimated as
σ(x) =
(
1
K
K
∑
k=1
(
∆(xPk )
)2
)1/2
(4)
where xPk is the k-th version of the phase randomized data,
and we set K = 20. The number of standard deviations
by which x deviates from the null hypothesis (being time
symmetric) is estimated as the ratio of ∆(x)/s(x). Note
that in the de£nition of σ(x) the mean was not subtracted.
0 5 10 15
−150
−100
−50
0
50
100
150
Time [s]
A
m
pl
itu
de
[a
.u.
]
0 5 10 15
−5
0
5
10
Time [s]
Si
gn
ifi
ca
nc
e
Figure 1: Top: selected time window with a short and a
long alpha burst. Bottom: signi£cance of time asymme-
try (solid) and deviation from colored noise (dotted) as a
function of the center of a 3 second window.
This accounts for the possibility that remnants of autocor-
relation can bias∆(x), which is compensated for by a cor-
responding increase of the estimated standard deviation.
For comparison we also calculate the signi£cance value
of the ’pure phase randomization test’ similar to Theiler’s
[1]. Let ∆P (x, xPk ) be the dimension difference (3) of
x and the k-th realization of the phase randomized data.
Signi£cance is then given by the ratio of the mean and the
standard deviation of ∆P (x, xPk ).
We tested this method on simulated data and found that,
in contrast to the pure phase randomization test, a strong
deviation from stationarity (here: a sudden increase of the
amplitude of gaussian white noise by a factor of 3 in the
middle of the epoch) did not lead to false positives, while
our test was more sensitive to true nonlinearities (here: the
Henon-map).
For the real data, we measured nonlinearity in the two time
series (A and B) over a sliding 3 second window and set
W = 500. For dataset A we found signi£cant time asym-
metry in 71% of the epochs, which approximately coin-
cide with those epochs containing alpha rhythm. Data
set B consists mainly of noise, with a few strong alpha
bursts. Fig. 1 shows the signi£cance, as a function of the
center of the moving window, for both this new test and
the conventional one. The new test gives clear detection
of time-asymmetry over the long alpha burst, while the
conventional test is almost uninterpretable. Note that for
the short burst any detection of signi£cant time asymmetry
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100
0
1
2
3
4
5
ε
D
im
en
si
on
A
m=8..12
100
0
1
2
3
4
5
ε
D
im
en
si
on
B
m=7..11
Figure 2: Estimated dimension as a function of scale ² for
various embedding dimensions m for datasets A and B. In
contrast to the uncorrected estimates (dotted) we observe
very satisfactory scaling and saturation for the corrected
estimates (solid) calculated with (5).
could not be interpreted as a property of the alpha dynam-
ics, since the burst is shorter than the minimum delay W .
4 Dimension and noise level
In [3] we showed that
d′(²,m) = d(²,m)−m(d(²,m+ 1)− d(²,m))1− (d(²,m+ 1)− d(²,m)) (5)
is a noise-robust estimator of the correlation dimension
when the correlation sums are calculated using gaussian
kernels.
Fig. 2 shows scaling plots within a selected range of em-
bedding dimensions for data sets A and B, for both the
uncorrected and corrected dimension estimates. For both
data sets the uncorrected dimensions neither scale nor sat-
urate to any non-trivial value. For dataset A we see very
clear scaling for the corrected estimate over about a fac-
tor of 4 in scale, and we can identify a correlation dimen-
sion of D2 ≈ 1.6. The scaling plot for data set B is less
convincing, but after noise correction a scaling region can
again be clearly identi£ed, giving an estimated correlation
dimension of D2 ≈ 1.4.
In [3] we also presented a similar method to estimate the
noise level. Normalizing the standard deviations of the
time series to one we found a ‘total noise level’ of 0.34
and 0.38 for A and B, respectively. An independent esti-
mate of the additive noise is available by comparing time
windows with and without alpha, leading to noise level es-
timates of 0.30 and 0.34 for A nd B, respectively. Hence,
the ’total noise level’ can be almost completely explained
by additive noise independent of alpha, implying that the
alpha generator appears to be very deterministic.
5 Conclusion
We propose a new method to detect dynamical nonlinear-
ity in a time series. In contrast to the commonly used com-
parison of the data to phase randomized versions, which
tests whether the data are inconsistent with colored but
stationary noise, this test was based on detecting asym-
metry with respect to local time reversal. The important
new idea is that, in order to estimate statistical proper-
ties, we still refer to phase randomized data, but use them
only to estimate the standard deviation of the measure. As
a consequence, the test is extremely robust against non-
stationarities and, surprisingly, was found to be even more
sensitive to true nonlinear dynamics. Application to MEG
data of human alpha rhythm with a substantial 20 Hz har-
monic resulted in signi£cant time-asymmetry in virtually
all epochs which contain alpha.
We also applied a new method to remove the noise bias
from scaling plots in the estimation of the correlation
dimension of a dynamical system. After correction we
found excellent scaling properties at a correlation dimen-
sion D2 ≈ 1.5. Comparison of ’total’ and additive noise
showed that the alpha generator is essentially determinis-
tic.
Acknowledgements
Stimulating discussions with L. Trahms, M. Burghoff and
G. Curio are gratefully acknowledged. Supported by the
MIND Institute/USA and DFG grant/Germany number
Ma 1782/1-4.
References
[1] J. Theiler and P.E. Rapp; Electroencephalography
and clinical Neurophysiology 98; 213–222; 1996.
[2] M.J. van der Heyden, C. Diks, J.P.M. Pijn and D.N.
Velis; Physics Letters A 216; 283–288; 1996.
[3] Nolte G, Ziehe A, Müller K-R; Phys. Rev. E 64;
016112 (1–10); 2001.
[4] T.H. Sander, G. Wübbeler, A. Lueschow, G. Curio,
and L. Trahms; IEEE Trans. Biomed. Eng. 49, 345–
54; 2002.
[5] R. Salmelin and R. Hari; Elec. clin. Neurophys. 91,
237–248, 1994.
