Chaos Analysis of Transcranial Doppler Signals for Feature
Extraction
Ali Ozturk
Computer Engineering Department, KTO Karatay University, Konya, Turkey
Havelsan Inc., Ankara, Turkey
Keywords: Chaos Analysis, TCD Signals, Correlation Dimension, Maximum Lyapunov Exponent, Recurrence Plots,
Chaotic Attractors, Space-Time Separation Plots.
Abstract: In this study, chaos theory tools were used for feature extraction from Transcranial Doppler (TCD) signals.
The surrogates data sets of the TCD signals which were used for the nonlinearity analysis were extracted as
the first feature set. The nonlinear cross prediction errors which were used for the stationary analysis were
also extracted for the TCD signals as another feature set. The chaotic invariant features like correlation
dimension, maximum Lyapunov exponent, recurrence quantification measures etc. give quantitative values
of complexity of the TCD signals. The correlation dimension and maximum Lyapunov exponent were
already used as features for classification of TCD signals in the literature. As another chaotic feature set, the
statistical quantitative values were extracted from the recurrence plots. The correct calculation of the time
delay and the minimum embedding dimension is crucial to correctly estimate all of the chaotic features.
These two data were calculated via mutual information and false nearest neighbours approaches,
respectively. The space-time separation plots were used in order to find the ideal dimension of Theiler
window w which is another important value for the correct estimate of chaotic measures. The reconstructed
chaotic attractors with 3-D embedding and 1-step time delay represent the visual phase space portrait of the
TCD signals. The attractors were also suggested as another candidate feature set.
1 INTRODUCTION
TCD study of the adult intracerebral circulation is
used to evaluate intracranial stenoses, cerebral
arteriovenous malformations, cerebral vasospasm
and cerebral hemodynamics in general (Evans et al..,
1989). The blood flow anomalies in the cerebral
vessels can be visually observed in the sonograms.
However, properly enabling the expert medical staff
to interpret TCD signals is difficult and this prevents
their wider and effective usage in the clinics.
In the literature, the linear features extracted
from Doppler signals via spectral analysis methods
were used for automatic medical diagnosis (Ubeyli
and Guler, 2005; Guler et al., 2002). The spectral
features of TCD signals were used for the
performance comparison of two different artificial
neural networks in (Serhatlioglu et al., 2003) for the
classification of the TCD signals.
There are various studies in which chaos theory
methods were used to analyse the Doppler signals
(Keunen et al., 1994; Vliegen et al., 1996). Keunen
et al., (1996) suggested that the TCD signals of
healthy subjects have an underlying nonlinear
dynamics. It was recognized by Visee et al., (1995)
that the nonlinear phenomena were lost in ischemic
cerebrovascular territory in patients with occlusive
cerebrovascular disease while there was nonlinearity
detected in noncompromised side.
Two chaotic invariant measures, i.e. the
correlation dimension and maximum Lyapunov
exponent, were used for the classification of TCD
signals in (Ozturk and Arslan, 2007; Ozturk et al.,
2008) to compare the performance of various
classifiers. The performance of the chaotic and
linear features were compared on a neuro-fuzzy
classifier in (Ozturk and Arslan, 2015).
If linear methods are used to analyse a time
series which is generated by a nonlinear process,
then some critical features of it can remain
undetected and most of it can be considered as noise.
The non-linear time series analysis (chaos theory)
provided some tools to quantitatively analyse a time
series which is generated by an underlying nonlinear
process. However, in order to apply nonlinear time
168
Ozturk, A.
Chaos Analysis of Transcranial Doppler Signals for Feature Extraction.
DOI: 10.5220/0005693701680174
In Proceedings of the 9th International Joint Conference on Biomedical Engineer ing Systems and Technologies (BIOSTEC 2016) - Volume 4: BIOSIGNALS, pages 168-174
ISBN: 978-989-758-170-0
Copyright
c
2016 by SCITEPRESS Science and Technology Publications, Lda. All rights reserved
series analysis (chaos theory) methods, it must be
proven that the signals are both nonlinear and
stationary. In (Ozturk and Arslan, 2007), the
surrogates data method was used to detect the
nonlinearity of TCD signals. In this study, The
surrogates data set of the TCD signals were
extracted as the first feature set. The stationarity of
the TCD signals is detected with nonlinear cross
prediction errors method in (Ozturk and Arslan,
2007). In this study, this method was used to extract
the non-linear cross prediction values as another
feature set.
The most famious chaotic features used in the
literature are the maximum Lyapunov exponent and
the correlation dimension. These chaotic measure
will eventually be different for a specific time series
generated by a specific natural phenomea, since
each chaotic attractor will have a different picture in
the embedded phase space. These two features were
also mentioned as a different feature set in this
study.
The recurrence plots are mainly used for
nonstationarity analysis and visualization of time
series. The visual data in the recurrence plots is hard
to interpret. Therefore, recurrence quantification
analysis is necessary to quantify the number and
duration of recurrences which is presented by the
state space trajectory of a dynamical system. The
other proposed feature set in this study is the
statistical quantitative values which are extracted
from the recurrence plots of TCD signals. The time
delay information and minimum embedding
dimension play important role for the correct
extraction of all chaotic features. This is also true for
recurrence quantification analysis. The other
important parameter which effects all of the results
is the Theiler window w and it is estimated from the
space-time separation plots.
The reconstructed 3-D chaotic attractors of the
TCD signals were also presented. These can be used
to extract a different set of features via image
processing methods.
2 MATERIALS AND METHOD
2.1 Hardware
The hardware of the system used for this study
involves a 2 Mhz ultrasound transducer, analog
Doppler unit (Multi Doppler Transducer XX, DWL
Gmb, Uberlingen, Germany), analog/digital
interface board (Sound Blaster Pro-16), and PIII
600 Mhz microprocessor PC with printer. The
Doppler unit is also equipped with imaging software
that makes it possible to focus the sample volume at
a desired location in the temporal region. The signal
obtained from the blood vessel is transferred to a PC
via a 16-bit sound card on an analog/digital interface
board (Ozturk et al., 2008). The signals were then
sampled to 0-255 interval as shown in Figure 1.
2.2 Surrogates Data Set
The method of Iterative Amplitude Adjusted Fourier
Transform (IAAFT) discussed in (Schreiber and
Schmitz, 1996) is used to generate surrogate data
sets. The Fourier-based surrogates depend on the
idea of creating constrained realizations. In this
approach, the measurable properties of the time
series are taken into account. The linear properties of
the time series are specified as in the following
2
1
0
/2
2
1
=
=
N
n
Nkni
nk
es
N
S
π
(1)
The surrogate time series are created by
multiplying the Fourier transform of the data by
random phases and then transforming back to the
time domain as:
Nkni
k
N
k
i
n
eSe
N
s
k
/2
1
0
1
π
α
=
=
(2)
Where 0≤α
k
<2π are independent uniform
random numbers.
2.3 Nonlinear Cross Prediction Errors
Data Set
The method used for the stationarity test of the TCD
signals in (Ozturk and Arslan, 2007) was utilized to
extract the non-linear cross prediction errors as
another feature set. This method divides the time
series into equal parts and the simple nonlinear
prediction algorithm (Hegger et al., 1999) is applied
to the segments to find the one-step ahead prediction
errors. The embedding vectors were obtained by
embedding the time series in 3-D phase space with a
time delay of 1. In the delay embedding space, all
neighbours of
are taken into account in order to
make a prediction at time +.
∆
=
|
(
)
|
∆
∈
(
)
(3)
Where
|
(
)
|
is the number of elements in the
neighbourhood
(
) of radius
3
є around the point
.
The time series was divided into segments S
i
,
i=1,..,N. For each two segments S
i
and S
j
, the root
Chaos Analysis of Transcranial Doppler Signals for Feature Extraction
169
mean squared error was computed using the
neighbours of S
i
to predict S
j
. For i=j, the cross
prediction errors will be smallest, since S
i
and S
j
are
identical.
For random time series such as white noise, the
nonlinear cross prediction errors are close to 1. On
the other hand, for a periodic signal which is
generated by sinus function, they are close to 0. For
the time series those are generated by natural
processes, they generally lie in between.
2.4 Recurrence Plots
The other two common methods in nonstationarity
analysis and visualization of time series are the
recurrence plots (Eckmann et al., 1987) (Casdagli,
1997) and the time-seperation plots (Provenzale et
al., 1997). It is difficult to interpret the recurrence
plots due to their complexity (Webber and Zbilut,
1994). The points obtained from a time series
belonging to a stationary process are spreaded over
the plot homogenously, while the points of a time
series belonging to non-stationary process are
grouped around the diagonal (Sprott, 2002). If there
are too many separated points in the plot, this
indicates randomness. This occurs when there is too
much noise in the time series or the embedding
dimension is insufficient. If the points over the
surface have no pattern then the process which
generates the time series has no or very poor
determinism (Kantz and Schreiber, 2005). Zbilut
et.al (1998), extracted some statistical quantitative
values from the recurrence plots. Some of these
values are the surface coverage rate of the points
(REC), the rate of the points parallel to the diagonal
(DET), the length distribution of the points which
form a straight line (ENT) indicating the rate of the
deterministic structures in the time series and the
regression coefficient (TREND) which represents
the relationship between the distance from the
diagonal and the recurrence number. The calculation
of these values takes too long for huge time series.
However, it was observed that for sub-sections of
TCD time series, these values do not differ
significantly and reflect the characteristics of the
original time series.
 =
,
,
(4)
Where N is the total number of points in a
recurrence plot and R
i,j
Є{0,1} depending of the
existence of a point on the i-j coordinates.
 =
()



,

,
(5)
Where
(
)
is the histogram of the length of
the diagonal lines.
 =
(
)
ln()


(6)
Where () corresponds to the diagonal line length.
 =
∑(

/
)(



〉)


∑(

/
)


(7)
Where
is the maximal number of diagonals
parallel to the main diagonal line.

indicates
the average of the recurrence points 
.
The values of the time delay, the minimum
embedding dimension and the Theiler window are
important for the calculation of the statistical
quantitative measures mentioned above.
2.5 Space-time Separation Plots
In recurrence plots, the graph of the points which are
closer than a specific ε distance value are obtained
with absolute time. The space-time separation plots
are obtained with relative time. In this kind of plots,
if a vector on the reconstructed attractor has at least
one neighbour in a specific δt interval and Δd
distance, then it is marked as δt – Δd point. By
means of these plots, it is possible to identify the
temporary correlations and to find the dimension of
the Theiler window w which is used in correlation
dimension algorithm of Theiler (1990) and in
Lyapunov exponent estimation of Kantz algorithm
(Kantz, 1994). The first peak point which is close to
the general height in the space-time separation point
generally gives the ideal dimension of Theiler
window w.
2.6 Correlation Dimension
The correlation dimension is computed most
efficiently by the correlation sum (Grassberger and
Procaccia, 1983):
)(
1
),(
<=
Θ=
N
wjk
kj
N
mj
pairs
ss
N
mC
εε
(8)
Where s
i
are m-dimensional delay vectors,
N
pairs
=(N-m+1)(N-m-w+1)/2 is the number of pairs of
points covered by the sums, Θ is the Heaviside step
function and w is the Theiler window (Theiler,
1990). The following power law exists between
embedding dimension m, ball radius ε and
correlation dimension D
2
2
),(
D
mC
εε
(9)
The correlation dimension D
2
can be defined as
D=lim
→
lim
→
(, ) (10)
BIOSIGNALS 2016 - 9th International Conference on Bio-inspired Systems and Signal Processing
170
Where
(
,
)
=
(,)

(11)
The minimum embedding dimension and the time
delay between embedding vectors are crucial in
order to estimate the correlation dimension correctly.
For the calculation the time delay, the mutual
information method suggested by Fraser and
Swinney (1986) was used. In this method, the
mutual information S is computed for different τ
delay values,
=
ij
ji
ij
ij
pp
p
pS
)(
ln)(
τ
τ
(12)
Where p
i
is the probability to find a time series
value in the i-th interval, and p
ij
(τ) is the joint
probability that an observation falls into the i-th
interval and the observation time τ later falls into the
j-th interval. The time delay τ where the mutual
information S takes the first minimum value is the
optimum delay and is used for embedding.
We used the false nearest neighbors method
proposed in (Hegger et al., 1999) to find the
minimum embedding dimension. This method
includes some small changes to the original
algorithm proposed by (Kennel et al., 1992) to avoid
the wrong results due to the noise in the time series.
Assuming that the standard deviation of the time
series is σ, the threshold of false nearest neighbors is
r and the distance between the vectors of the phase
space is found according to maximum difference,
the false nearest neighbors statistics is calculated as
in the following:
(13)
Where, is the nearest neighbour of the
vector S
n
and k(n) is the index of the time series
which is different than n and supplying the condition
of
|
−
|
being minimum. The second Heaviside
function in the nominator is used to eliminate the
vectors of which initial distances are higher than σ/r.
The same function also exists in the denominator for
the same reason.
2.7 The Maximal Lyapunov Exponent
In the phase space, the distances between embedding
vectors on attractor do not grow everywhere with the
same rate. They may shrink locally. Therefore, the
maximal Lyapunov exponent calculation will be the
average of the local divergence rates over the whole
embedding vectors.
The algorithm developed by Rosenstein et. al.
(1993) was used to find the maximal Lyapunov
exponent. This algorithm computes the local
divergence rates of the state space distances over the
whole time series data. The stretching factor S is
found for different N values as in following

=
=
N
n
nn
Xn
XX
N
S
o
1
0
0
1
ln
1
ϑ
(14)
Where,
is an embedding vector on the
attractor,
are the neighboring vectors within
diameter ϵ and
0
Xn
ϑ
is the number of these
neighbors.
The first slope of the curve obtained by plotting
S values for various N values on x-y coordinate
system gives the maximal Lyapunov exponent.
3 EXPERIMENTAL RESULTS
AND DISCUSSION
The profile of the subjects from which the TCD
signals were obtained is given in the following table.
Table 1. The profile of the subjects used in this study.
Males Females
Age
Range
Avg.
Age
Cerebral
aneurysm
12 8 55-65 59.5±0.5
Brain
Hemorrhage
4 6 21-36 27.0±0.5
Cerebral
Oedema
11 11 3-40 25.0±0.5
Brain Tumor 12 18 12-41 29.5±0.5
Healthy 15 8 23-65 31.5±0.5
Summary 54 51 3-65 35.0±0.5
The Iterative Amplitude Adjusted Fourier
Transform (IAAFT) method is used to generate the
surrogate data sets for each TCD signal. A sample
surrogate data set for the TCD signal of a patient
with cerebral aneurysm (Figure 1) is given in Figure
2. It is not easy to visually distinguish the surrogate
data sets from the original TCD signal, but the
surrogates are created with the linear properties via
Fourier transform.
=
=
++
Θ
Θ
Θ
=
1
1
)(
)(
)(
)(
)(
)(
1
1
)(
)(
)(
)1(
)(
)1(
)(
mN
n
m
nk
m
n
m
nk
m
n
mN
n
m
nk
m
n
m
nk
m
n
SYK
SS
r
SS
r
r
SS
SS
rX
σ
σ
)(
)(
m
nk
S
Chaos Analysis of Transcranial Doppler Signals for Feature Extraction
171
Figure 1: The TCD signal of a patient with cerebral
aneurysm.
Figure 2: The surrogate time series generated for the TCD
signal given in Figure 1.
In the following figure, it can be seen that the
nonlinear cross-prediction error values for the TCD
signals lie between pure deterministic (sinus signal)
and random (Gaussian) time series. These values can
be used as another feature set for the classification of
the TCD signals.
Figure 3: The non-linear prediction errors for sinus,
Gaussian random and one for each TCD signals time
series.
In the following figure, the recurrence-plots
of the TCD signals belonging to different patient
groups are given. As can be seen from the figures,
each patient has a different structure within its
recurrence-plot. The statistical quantitative values
like REC, DET, ENT and TREND were calculated
using the recurrence-plots of the TCD signals. These
values constitute the other chaotic feature set
In the following figure, the space-time
separation plots of the TCD signals belonging to
different patient groups are given. In these plots, the
first peak point which is close to the
a)
b)
Figure 4: Recurrence plots of the TCD signals belonging
to a patient of (a) Hemorrhage and (b) Healthy subject.
general height in the space-time separation
point was used to identify the ideal dimension of
Theiler window w.
a)
b)
Figure 5: The space-time separation plot of the TCD
signal with patient groups a) Oedema (w=50) b) Tumor
(w=40).
BIOSIGNALS 2016 - 9th International Conference on Bio-inspired Systems and Signal Processing
172
The chaotic attractors embedded in 3-
dimensions for each patient group are given in the
following figure. The first 300 points of TCD signals
were used to draw the attractors in order to enhance
the visuality. The time delay used to draw the
attractors was 1. These chaotic attractor pictures can
be used to extract another feature set for the TCD
signals.
a)
b)
Figure 6: The chaotic attractors of the TCD signals
embedded in 3-dimensions with 1 time delay a) Healthy b)
Oedema.
In the following figure, the correlation
dimension estimation for the TCD time series of a
patient with brain hemorrhage is given as an
example.
Figure 7: Correlation dimension estimation of a patient
with brain hemorrhage.
For all of the chaotic features, we need time
delay information which is obtained by mutual
information method. According to this method, the
first delay value τ at which time delayed mutual
information takes the minimum is a good candidate
for a reasonable time delay. In the following figure,
the time delay estimation for the TCD signals with
brain oedema is given.
Figure 8. Time delay value estimation for the patients with
brain oedema.
In the following figure, it is shown how the
maximal Lyapunov is calculated for a patient with
cerebral aneurism.
Figure 9: Maximal Lyapunov exponent estimation for the
TCD signal of a patient with cerebral aneurism.
In the following figure, the minimum
embedding dimension estimation for the TCD
signals with brain tumor is given.
Figure 10: Minimum embedding dimension estimation for
some of the patients with brain tumor.
Chaos Analysis of Transcranial Doppler Signals for Feature Extraction
173
4 CONCLUSIONS
Besides the chaotic invariant measures such as
correlation dimension and maximal Lyapunov
exponent, some other feature sets which can be
extracted with non-linear time series analysis may be
used to further evaluate some of the brain vessel
diseases. The chaotic invariant measures may be
supported with these feature sets. These features
alone or with chaotic measures together may be used
to train a classifier. After the generalization, the
classifier may be used to make automatic diagnosis
of the brain diseases.
The non-linear cross prediction errors of the
TCD signals and the statistical quantitative values
extracted from the recurrence plots can also be used
to train various classifiers in order to make
automated diagnosis of the brain vessel diseases.
The reconstructed 3-D chaotic attractor
pictures can be used to extract another feature set for
the TCD signals.
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