Wireless Sensor Network, 2010, 2, 67-73
doi:10.4236/wsn.2010.21010 anuary 2010 (http://www.SciRP.org/journal/wsn/).
Published Online J
Comparison of Correlation Dimension and Fractal
Dimension in Estimating BIS index
Behzad AHMADI, Rassoul AMIRFATTAHI
Digital Signal Processing Research Laboratory, Department of Electrical and Computer Engineering,
Isfahan University of Technology, Isfahan, Iran
Received September 5, 2009; revised October 13, 2009; accepted October 20, 2009
Abstract
This paper compares the correlation dimension (D2) and Higuchi fractal dimension (HFD) approaches in
estimating BIS index based on of electroencephalogram (EEG). The single-channel EEG data was captured
in both ICU and operating room and different anesthetic drugs, including propofol and isoflurane were used.
For better analysis, application of adaptive segmentation on EEG signal for estimating BIS index is evalu-
ated and compared to fixed segmentation. Prediction probability (PK) is used as a measure of correlation
between the predictors and BIS index to evaluate the proposed methods. The results show the ability of these
algorithms (specifically HFD algorithm) in predicting BIS index. Also, evolving fixed and adaptive win-
dowing methods for segmentation of EEG reveals no meaningful difference in estimating BIS index.
Keywords: Adaptive Segmentation, Bispectral Index, Depth of Anesthesia, Correlation Dimension, Fractal
Dimension
1. Introduction
In the operating room, the anesthesiologist is essential in
providing optimal working conditions to surgeons, and in
ensuring patient safety and comfort. However, patient
awareness during surgery with the rate of 1:1000 [1] and
over dosing with anesthetic agents is of major clinical
concerns of anesthesia. Therefore, the necessity to assess
and monitor the depth of anesthesia (DOA) is obvious. In
conventional methods, DOA is measured based on the
monitoring of several physiological signals such as res-
piration pattern, blood pressure, body temperature, tear-
ing, sweating and heart rate [2], even though these sig-
nals are affected indirectly by anesthetic agents. On the
other hand, these agents have significant effects on the
electroencephalogram (EEG) waveform. Hence, it is
advocated that the EEG can provide a reliable basis for
deriving a surrogate measurement of anesthesia.
In recent years, the anesthesia community has wit-
nessed the development of a number of EEG-based algo-
rithms of consciousness. Several of the above methods
are available that have recently been reviewed by Freye
et al. [3] and Jameson et al. [4].
The earliest methods were based on the FFT analysis
of EEG signals. These approaches tend to find parame-
ters that describe spectrum characteristics. Peak power
frequency (PPF), median power frequency (MPF), and
spectral edge frequency (SEF) have been the first de-
scriptors in this field. Another parameter extracted from
spectrum was the ratio of power in two empirically de-
rived frequency bands [5]. In a work presented by Traast
et al. [6] the power of EEG in different frequency bands
was determined and the results indicate pronounced
changes in EEG during emergence from propo-
fol/sufentanil total intravenous anesthesia.
Ferenets et al [7] analyzed the performance of sev-
eral new measures based on the regularity and com-
plexity of the EEG signal. These measures consist of
spectral entropy (SpEn), approximate entropy (ApEn),
and Higuchi fractal dimension (HFD) and Lempel-Ziv
complexity (LZC). Their results show superior ability
of the mentioned measures to predict DOA. Due to the
arguments presented in their paper it is not feasible to
point out “the best” EEG measure for the assessment of
the depth of sedation Their results indicate that the
measures sensitive to both the power spectrum as well
as the amplitude distribution, i.e., the ApEn, LZC and
HFD, perform slightly better than the other two tested
measures. In the case of their tested measures, they
recommend window length of 20 s.

B. AHMADI ET AL. 68
Application of neural networks (NN) in estimating
DOA is reviewed by Robert et al [8]. They examined a
large number of EEG derived parameters as NN inputs
including spectral, entropy, complexity, bicoherence,
wavelet transformation derived, autoregressive modeling
and hemodynamic parameters as well as a great NN to-
pology such as MLP and Self-Organizing networks. Fi-
nally, they recommended a two hidden layers MLP
model or an ART model in which their weights are up-
dated after training phase continuously. Moreover the
use of qualitative parameters, besides quantitative ones,
as network inputs is recommended. In a recent work by
Lalitha et al [9] non-linear chaotic features and neural
network classifiers are used to detect anesthetic depth
levels. Chaotic features consist of correlation dimension,
Lyapunov exponent (LE) and Hurst exponent (HE) are
used as features and two neural network models, i.e.,
multi-layer perceptron network (feed forward model) and
Elman network (feedback model) are used for classifica-
tion. Their experimental results show that the Lyapunov
exponent feature with Elman network yields an overall
accuracy of 99% in detecting the anesthetic depth levels.
According to various mentioned methods, different
EEG monitors have been developed. The Narcotrend™
monitor (Monitor Technik, Bad Bramsted, Germany)
that is based on pattern recognition of the raw EEG and
classifies the EEG into different stages, introduces a di-
mensionless Narcotrend™ index from 100 (awake) to 0
(electrical silence). The algorithm uses parameters such
as amplitude measures, autoregressive modeling, fast
Fourier transform (FFT) and spectral parameters [10].
The SEDLine™ EEG monitor capable of calculating of
PSI™ index uses the shift in power between the frontal
and occipital areas. The mathematical analysis includes
EEG power, frequency and coherence between bilateral
brain regions [11]. Datex-Ohmeda™ s/5 entropy Module
uses entropy of EEG waves to predict DOA [4] and fi-
nally BIS™ (Aspect Medical Systems, Newton, MA),
that is the first monitor in the marketplace and has be-
come the benchmark comparator for all other monitors,
introduces the BIS™ index (that is a unit-less number
between 100 and 0) as a DOA indicator based on com-
bination of spectral, bispectral and temporal analysis [5].
Approximately 450 peer-reviewed publications between
1990 and 2006 have been examined the effectiveness,
accuracy and usefulness, both clinical and economical, of
the BIS™ monitor [4].
The interpretation of the EEG is complex since it is a
reflection of the brain state, the effects of the surgical
procedure and the influence of anesthetic drugs. Anes-
thetic agents have different effects on the EEG and do
not represent a straightforward solution concerning the
monitoring of depth of anesthesia. Thus, EEG processing
using linear methods such as power of frequency bands,
spectral edge frequency, etc. did not solve the problem.
Nevertheless, the ideas that the EEG informs the anes-
thesiologist directly about depth of anesthesia and the
inadequacy of linear measures to assess this depth under
different circumstances were the main motivation to in-
vestigate the application of non-linear mathematics to the
EEG. Consequently, nonlinear dynamical analysis has
emerged a novel method for the study of complex sys-
tems during the past few decades. Lyapunov exponent,
Hurst exponent, fractal dimension, information dimen-
sion, box dimension, and correlation dimension are some
of the methods by which the complexity of a system or a
signal could be quantified. Correlation dimension is by
the most popular due to its computational efficiency
compared to other fractal dimensions such as the infor-
mation dimension, capacity dimension, and pulse dimen-
sion [12].
A significant step forward was made by Grassberger
and Procaccia [13] who showed how the so called corre-
lation dimension could be used to estimate and bound the
fractal dimension of the strange attractor associated with
the nonlinear time signal at hand. The method was im-
mediately applied to many different types of time series
such as financial time series [14], physiological re-
cordings [12,15–17]. Specifically, in [18], the application
of correlation dimension to determine anesthetic drug
effect was described. Watt et al., demonstrated the
changes of phase space trajectories and dimensionality as
a result of changed depth of anesthesia [19]. Later,
Widman et al. investigated a modified version of D2, the
non-linear correlation index D*, as a measure of depth of
sevoflurane anesthesia [20]. Lee et al. found D2 to serve
as a better index for the depth of halothane anesthesia in
the rat compared to beta-power and median power fre-
quency [21]. All of the previous studies revealed the po-
tency of D2 or derived measures to measure the level of
consciousness and to support the presumption that shift-
ing from full consciousness to unconsciousness results in
more and more left out autonomic processes. Due to the
ability of correlation dimension in expressing this com-
plexity in a single number, it looks very attractive for the
purpose of an indicator of depth of anesthesia.
Fractal dimension, on the other hand, is a measure of
how ‘complicated’ a self-similar figure is [22]. In the
particular case of curves in a plane, while a topological
line is one-dimensional, a fractal curve has a fractal di-
mension D that is in the range 1 < D < 2. In general, the
compass fractal dimension represents a measure of the
degree of shape complexity or roughness of the curve.
Higuchi, Katz , Petrosian C, Petrocian D, Sevcik, Zero
set, Adapted box, compass, variogram are some methods
to calculate fractal dimension of signals.
The aim of this paper is to quantify the dynamic of the
system, which is affected by anesthetic drugs. Thus, cor-
relation dimension and fractal dimension of EEG signal
is calculated using optimized adaptive parameters. It is
further assumed that when DOA is changing gradually,
the EEG signal is composed of consecutive stationary

B. AHMADI ET AL. 69
N
segments which could be detected by adaptive segmenta-
tion methods. Thus, electroencephalograms were col-
lected from patients undergoing surgery and the corre-
sponding correlation dimension and fractal dimension
were extracted through different fixed and adaptive
windowing methods. Afterwards, correlation between
these parameters and BIS index was evaluated via proper
statistical analysis.
The organization of the rest of the paper is as follows:
In Section 2, methods and materials are described. The
results and conclusion are presented in Section 3, and
Section 4 includes the final discussion.
2. Methodology
In this section, the experiment, the data acquisition, and
the data analysis are described.
2.1. Patients
Following the approval of the ethical committee of the
medical school, eight coronary artery bypass graft sur-
gery candidates were selected (6 males, 2 females, of
average age 56.2 years and the average weight of 68.3kg)
and written informed consents were obtained from all
selected subjects. Inclusion criteria were absent of neu-
rological disorders such as cerebrovascular accidents and
convulsions. Preoperative neurological complications
(such as cerebral emboli and convulsion) caused exclu-
sion from the study. The anesthesiologist performed
Preoperative evaluation on the day before surgery. For
anxiolysis, the patients were premedicated by intramus-
cular morphine 0.1 mg/kg and promethazine 0.5 mg/kg,
30 minutes before transfer to operating room. After arri-
val in operating room, electrocardiogram, pulse oxy-
metry, depth of anesthesia, and invasive blood pressure
monitoring was established. The BIS-QUATTRO sen-
sorTM (Aspect Medical Systems, Newton, MA) applied
to the forehead of the patients before induction of anes-
thesia. Then 8 patients after preoxygenation with O2,
were anesthetized in the same manner by intravenous
thiopental sodium (5mg/kg), pancuronium bromide (0.1
mg/kg), fentanyl (5μg/kg), and lidocaine (1.5 mg/kg).
After the induction of anesthesia and until cardiopul-
monary bypass beginning, anesthesia continued by ad-
ministration of isoflurane (1 MAC), morphine (0.2
mg/kg) and O2 (100%). During coronary artery bypass
grafting under CPB, patients were anesthetized by pro-
pofol (50–150 μg/kg/min) under BIS control (40-60) and
O2 (80%). For organ protection during CPB, patients
were undergone mild hypothermia (31–33°C). After
coronary artery bypass grafting and patients rewarming
and obtaining standard CPB separation criteria, the pa-
tients gradually were weaned from CPB. After separation
from CPB, anesthesia was continued by isoflurane (1
MAC) and O2 (100%) administration to the end of sur-
gery. After surgery, patients were transported to ICU
under portable monitoring and manual ventilation. In the
ICU mechanical ventilation with 60% fractioned inspired
oxygen and standard homodynamic monitoring were
continued. In ICU and until complete recovery, the seda-
tive regimen was intravenous morphine (2mg) if needed.
In this study the raw EEG data and relative BIS index
were collected during whole period of operation from
operative room arrival to complete recovery in the inten-
sive care unit.
2.2. Data Acquisition
The EEG signal was collected by using a BIS-QUAT-
TRO Sensor™ that was composed of self-adhering flexi-
ble bands holding four electrodes, applied to the forehead
with a frontal-temporal montage.
The used EEG lead was Fpz-At1, and the reference
lead was placed at FP1. The sensor was connected to a
BIS-X-P Monitor and all binary data packets containing
raw EEG data wave signals and BIS index which is con-
verted to binary format using an A/D converter operating
with 128 Hz sampling frequency were recorded via an
RS232 interface on a laptop using a Bi-spectrum ana-
lyzer developed with C++ Builder by Satoshi Hagihira
[23]. The algorithms that are presented in this study were
tested on these raw EEG signals.
The sensor was attached to the patient’s forehead at
the beginning of anesthesia and the data were collected
continuously until he/she awoke at ICU. Therefore, in
this study a large amount of EEG data with their BIS
index was collected for each patient. Some other events
such as changes of anesthesia regimen, intubations and
applying CPB and transferring to ICU were recorded.
Because of short acting time of thiopental sodium (ap-
proximately 15–20 sec), this part of EEG data was not
analyzed.
2.3. Phase Space
Let for
)(ix 1, 2,3,...,i

denote an observable proc-
ess generated by an unknown or unobservable system.
Following Brock (1986) [24], ()
x
iis said to have deter-
ministic explanation if:
:
(1) (())
F
MM
s
tFst

 (1)
where F is an unknown smoth map, and m is a d-di-
mensional manifold and:
:
()( ())
hM
x
thst

 (2)
where h is an unknown smoth map. Then, according to
Takens theorem [25], for an adequate choice of parame-
C opyright © 2010 SciRes. WSN

B. AHMADI ET AL. 70
ters and
m
E

, there exists a function such as: G
22
:
()(()
mm
EE
G
)
X
tGX 

 t
(3)
With
( )(( ),(),...,((1)))
m
Xixi xixiE

  (4)
And being the embedding dimension and
m
E

the
time delay which is usually fixed to one. The above vec-
tor is suggested by Taken for the extraction of informa-
tion in the dynamics of chaos by reconstructing the phase
space (state space) from time series of signal variable.
Thus, if ()
x
i
m
E
is a deterministic time series, then for
any pair of points ((and for an adequate
choice of , there exist arbitrary small and
),())yiyj
a0


so that:
() ()(())(())ifyiyjathenGyiGy j

 
(5)
where ab being the distances between the vector
and b. So, the images of close points are closed in
the phase space.
a
2.4. Correlation Dimension Estimation
The Grassberger and Proccacia algorithm using Theiler
method [26,27] considers spatial correlation between
pairs of points on a reconstructed attractor. For an
-dimensional phase space, the modified correlation
integral is defined as
m
E
()Cr
))()((
))(1(
2
)(
1







m
i
im
ij
jXiXr
mm
rC (6)
where (1)
m
mN E

 
m
E
is the number of embedded
points in an -dimensional space, is the length of
data series, and
N
r
is the radius of sphere centered on
or its box size, and
)(ix ()
x
 is the heaviside step
function as follows

1
() 0
x
xx  
0
0
(7)
If scales like
()Cr
() v Cr r (8)
Then, is called the correlation dimension of the
time series
v
x
or equivalently the slop of the log-log
plot of versus r. In this study correlation dimen-
sion is calculated while is fixed to 10.
(Cr)
r
The delay time

is commonly determined by using
the autocorrelation function (ACF) method (by finding
the place where ACF first attains zeros or below a small
value, e.g. 0.2 or 0.1), or mutual Information (MI)
method (Fraser& Swinney 1986) [28] (by finding the
place where MI first attains a minimum). In this study,

is changed manually from one to ten. The result
shows that smaller

s would yield in better correlation
between D2 and BIS index. So, it is fixed to one. Finally,
is fixed to 10 and correlation dimension is calcu-
lated.
m
E
Consequently, EEG is divided in to fixed (20 seconds)
segments and correlation dimension is calculated using
the above parameters. The above parameter is named
“fixed CD”.
2.5. Fractal Dimension
Nonlinear methods that assess signal complexity matter
if the signal itself is chaotic or deterministic.
Consider the time series (1), (2),...,()
x
xxN, where
is the total number of samples. The algorithm con-
structs new time series as:
Nk
() {(),(,(2),...,(()/.)})
m
X
kxmx xmkxmNmkk


mk
where km ,...,2,1

. The length,, is calculated as: )(kLm
1
1
()()(( 1)./
.
Nm
l
i
N
kxm kxm ikk
Nmk
k
 



i





 
m
L


















(9)
Higuchi algorithm calculates fractal dimension of a
time series directly in the time domain. It is based on a
measure of length, of the curve. An average
length is computed for all time series having the same
scale , as the mean of the length for
)(kL
k
1, 2,
k()
m
Lk
3,...,mk

. If scale like, (Lk)()
f
D
Lk 
f
D
k
, then
the curve is said to show fractal dimension [29].
The fractal dimension of EEG signal is calculated via
above method while fixed windowing method (with the
window length of 20 seconds) which is named “fixed
HFD”.
2.6. Adaptive Segmentation of EEG Data
It would be desirable to adapt the analysis window to
changes in the given signal, allowing the window to be
as long as possible while the signal remains stationary,
and to start a new window at the exact instant when the
signal changes its characteristics [30]. In order to per-
form the described approach, two adaptive windowing

B. AHMADI ET AL. 71
methods have been used, adaptive variance detection and
ACF (Auto-Correlation Function) distance methods. In
both approaches, a reference window is extracted at the
beginning of each scan, and the given EEG signal is ob-
served through a moving window. In each of the above
algorithms, the performance of the reference window
length L= 1 second has been evaluated.
In adaptive variance detection method, a segment
boundary is drawn when the variance of the moving
window becomes m times greater or larger than the vari-
ance of the reference window. By considering the
changes of EEG variance, 7k

and are tested
and finally was considered better for applying
the above procedure (The CD and HFD values obtained
using this method of windowing are named “CD_var”
and “HFD_var”, respectively).
10k
10k
On the other hand, for implementing ACF distance
method, let ()
Rk

be the ACF of the reference window
at the beginning of a new segmentation step, where k is
the lag or delay. Let (, )
Tnk

be the ACF of the test and
sliding window positioned at time instant n. A normal-
ized power distance dp (n) between ACFs is computed
as:
(,0) (0)
() min {(,0),(0)}
TR
p
TR
n
dn n



 (16)
The condition where dp (n) becomes larger than a spe-
cific threshold, ThP, is considered to represent a signifi-
cant change in ACF, and used to mark a segment bound-
ary [30]. Again, due to the variation of dp (n) values,
three different ThPs (25000, 30000, and 35000), are
tested and is used to evaluate the de-
scribed approach (The CD and HFD values obtained
using this method of windowing are named “CD_ACF”
and “HFD_ACF”, respectively).
30000
P
Th 
2.7. Statistical Analysis
The correlation between BIS index and the extracted
sub-parameters was investigated with the model-inde-
pendent Prediction Probability (Pk) [31]. As a nonpara-
metric measure, the Pk is independent of scale units and
does not require knowledge of underlying distributions
or efforts to linearize or otherwise transform scales. A Pk
value of 1 means that the predicting variables (“fixed
HFD”, “HFD_var”, “HFD_ACF”, “fixed CD”, “CD_
var”, and “CD_ACF”) always predict the value of the
predicted variable (e.g., BIS index) correctly. Pk value of
0.5 means that predictors predict no better than only by
chance. The Pk values were calculated on a spreadsheet
using the Excel 2003 software program and the PKMA
CRO written by Warren Smith [31]. In the case of in-
verse proportionality between indicator and indicated
parameters, the actual measured Pk value is 1-Pk. An-
other statistical analysis used in this study was ordinal
logistic regression. This regression examines the rela-
tionship between one or more predictors and an ordinal
response. The index that determines the efficiency of this
regression model is called “Concordant”, which shows
the percentage of values predicted successfully with the
model.
Our results are presented in three different groups;
during propofol and isoflurane anesthesia in operating
room and in ICU under sedative regimens. Higuchi frac-
tal dimension and correlation dimesion were calculated
based on the algorithms described in the previous sec-
tions and different parameters were obtained. Correlation
between our extracted parameters (“fixed HFD”, “HFD_
var”, “HFD_ACF”, “fixed CD”, “CD_var”, and “CD_
ACF”) and BIS index were calculated via proper statis-
tical analysis. In order to apply logistic regression analy-
sis the total BIS index range was divided into six
non-overlapping groups: 0-25, 25-40, 40-50, 50-60,
60-80, and 80-100.Then, the best model was determined
to identify how good a predictor could predict the BIS
level.
3. Results and Conclusions
According to Figure 1, HFD reveals higher Pk values than
CD in the ICU group. Thus, HFD could predict BIS index
much more acceptable in comparison to CD. Also, it could
be concluded from Figure 1 that application of adaptive
segmentation could neither improve the estimation of
BIS index based on HFD nor based on CD methods.
According to Figure 2, HFD parameters (“fixed HFD”,
“HFD_var”, and “HFD_ACF”) are more capable of es-
timating BIS index than CD parameters (“fixed CD”,
“CD_var”, and “CD_ACF”) in propofol group, due to
their higher Pk values. Adaptive segmentation methods
could improve the performance of CD in predicting BIS
Prediction Probability (Pk)
CD_ACFCD_varfixed CDHFD _ AC FHFD_varfix ed HF D
0.85
0.80
0.75
0.70
0.65
0.60
0.55
0.50
Figure 1. Comparison of different parameters in ICU group
using Pk values.
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B. AHMADI ET AL. 72
Prediction Probability (Pk)
CD_ACFCD_varfixed CDHFD _ACFHFD_v arfixed HFD
0.80
0.75
0.70
0.65
0.60
0.55
0.50
Figure 2. Comparison of different parameters in isoflurane
group using Pk values .
Pre dici
t
ion Probabili
t
y (P
k
)
CD_ACFCD_varfixed CDHFD_AC FHF D_varfixed HFD
0.72
0.69
0.66
0.63
0.60
Figure 3. Comparison of different parameters in propofol
group using Pk values.
index, especially using variance method. Of course this
is not true for none of the remaining parameters.
Based on Figure 3, HFD parameters (“fixed HFD”,
“HFD_var”, and “HFD_ACF”) are approximately as
capable as CD parameters (“fixed CD”, “CD_var”, and
“CD_ACF”) in estimating BIS index in isoflurane group.
Among different extracted parameters in this group,
“CD_var” could predict BIS index better than other
parameters. Although adaptive segmentation methods
couldn’t improve the performance of CD in predicting
BIS index, they are able to improve estimating of BIS
index based on correlation dimension calculation.
4. Discussions
In this study, raw EEG signals with relative BIS index of
8 patients undergoing coronary artery bypass graft sur-
gery have been acquired. Besides, a method based on
Higuchi fractal dimension, correlation dimension and
adaptive segmentation is proposed for estimating BIS
index. In order to investigate the capability of the ex-
tracted parameters in predicting BIS index, appropriate
statistical analyses were used in three different groups.
Due to the fact that HFD and CD are capable of quan-
tifying signals and systems complexity, high Pk values
were obtained while using these parameters as measures
of DOA. Of course, adaptive segmentation methods were
not able to improve this process, except when CD was
computed using variance method in isoflurane and pro-
pofol groups, which seems a little bit unusual due to the
ability of adaotive segmentation methos in tracking sig-
nal changes. The main result was the superiority of vari-
ous HFD parameters to those of CDs in predicting the
BIS index in most of the groups and methods which is
evidenced by the high correlation of those parameters
with BIS index. In other words, Higuchi fractal dimen-
sion is more capable than correlation dimension in esti-
mating BIS index.
Furthermore, in order to improve the accuracy of the
depth of anesthesia estimation, comparison against seda-
tion scales (such as OAA/S) and drug levels is needed.
The reason is that BIS is not equal to depth of anesthesia
but needs to be validated for DOA assessment itself.
The work reported is preliminary. Although the results
are significant, wide patient population is necessary for
better evaluation. Therefore, obtained claims and out-
comes should be confirmed by working on a larger group
of patients and drugs and through various surgeries. To
sum up, the approaches proposed in this study based on
the application of adaptive algorithms, including ACF
and variance methods for segmentation, fractal dimen-
sion and correlation dimension (and mainly fractal di-
mension) were sensible and meaningful in estimating
BIS index.
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