HEART RATE VARIABILITY (HRV) SIGNAL PROCESSING BY
USING WAVELET BASED MULTIFRACTAL ANALYSIS
SOUNDARARAJAN EZEKIEL
ROBERT A.HOVIS
Ohio Northern University
Ada, Ohio, USA
Abstract
Fractal geometry and wavelets are a new and
promising approach to analyze and characterize
non-stationary signals such as ECG, EEG and
stock price etc. Heart Rate Variability Signals,
derived from an ECG signal, are strongly related
to the activity of the autonomous nervous
system(ANS). HRV is usually investigated as
RR variability since the R wave is far easier to
detect due to its peaked shape. The classical
methods based on autocorrelation, thresholds or
derivatives, time domain methods and frequency
domain methods give a coarse quantification of
the variability, without distinguishing between
short-term and long-term fluctuations. In this
paper, we propose a new wavelet based method
to analyze Heart Rate Variability (HRV) signals.
The fractal dimension of the RR series can be
calculated by using wavelets, time being here
irrelevant. Another measure, Multifractal
Spectrum is computed with the help of a scaling
exponent. Using this strategy, we found that the
peak of the multifractal spectrum shifted to
higher dimensions and demonstrated increased
complexity and an
increasing amount of
“noise” for ANS regulations of HRV signals
during the tilt interval.
Keywords
Fractal dimension, Wavelet, Multifractal, Heart
Rate Variability, Scaling exponent
1. Introduction
Electrocardiogram (ECG) signals are curves
displaying heartbeat rhythms. ECG signals are
commonly used in medical care for monitoring,
diagnosis, and the treatment of patients suffering
from heart diseases. They are most usually
obtained by measuring the potential difference
between electodes which are placed on patients.
Electrical fields resulting from a patient's heart
beat are thus detected and the field variations
transferred to a voltage signal. This signal may
ABDULLAH A. ALSHEHRI
D.J.HEBERT
VLADIMIR SHUSTERMAN
University of Pittsburgh
Pittsburgh, PA, USA
be single-channel (if one electrode is used) or
multi-channel if several electodes are used.
Cardiologists can use minute features of these
signals to obtain important knowledge about the
function of their patient's heart. Heart Rate
Variability (HRV), derived from an ECG signal
is a measure of beat-to-beat(called RR interval)
changes in heart rate. The widely used method of
measuring HRV is Holter monitoring. In the
medical field, the following parameters are
generally taken into account: Heart rate, total
size of the signal, VLF (Very Low Frequency),
LF (Low Frequency representing slower changes
in heart rate), HF (High Frequency representing
quicker changes in heart rate), ApEn
(Approximate Entropy), Alphas (scaling
exponent) etc. For the last few decades, studies
have shown that a decrease in HRV is one of the
best predictors of arrhythmic events[10] or
sudden death after myocardial infarction[1].
Search for better methods of analysis has
therefore more than an academic interest. HRV
depends on various determinants including
baroreflex, cortical influences etc. We proposed
a
Multidimensional
Non-linear
Spectral
Estimation (MNSE) method to capture the
dynamics of the HRV signals. In section 2 we
recall some fundamental definitions of fractal
measure, multifractal analysis, and wavelets.
Section 3 introduces the MNSE method
Experimental results and discussions are given in
section 4. Section 5 concludes the paper.
2. Wavelet based Multifractal
Analysis
Multifractal theory was described for the first
time by B. B. Mandelbrot[2] in the context of
fully developed turbulence. Since then,
multifractals have increasingly been studied by
mathematicians. Multifractal analysis has
recently drawn much attention as a tool for
studying singular measures and functions in both
theory and in applications [3]. Multifractal
analysis has been extended as an application of
Choquet capacities in[4]. In the multifractal
scheme, the pointwise structure of a singular
measure is analyzed through the so called
''multifractal spectrum'', which gives either
geometrical or probabilistic information about
the distribution of points having the same degree
of singularity. Several definitions of a
multifractal spectrum exist. In this section, we
take an approach to multifractal analysis
pioneered by Levy Vehel et al[5][6] However,
instead of computing local singularity exponent
by using Choquet capacities [6], we develop an
efficient wavelet-based calculation of local
fractal dimension[10]. This method is explained
in Section 3. Using this method, we captured the
dynamics of HRV signals.
2.1 Wavelets
Wavelets are presently used in many
disciplines in science and engineering. In the
last few years, the wavelet transform has become
a cutting edge technology in the field of image
and signal processing. The concept of wavelets
was introduced by Jean Morlet and Alex
Grossmann. It was mainly developed by Y.
Meyer [7]. The first algorithm was developed by
Stephen Mallet in 1988[8]. After that, many
scientists like Ingrid Daubchies, Ronald Coifmen
and etc contributed to this field. A wavelet is a
waveform of effectively limited duration that has
an average value that is zero. So wavelet analysis
is done by breaking up of a signal into shifted
and scaled versions of the original (mother)
wavelet. From this, we can define a continuous
wavelet transform as the sum over all time of the
signal multiplied by a scaled and shifted version
of the wavelet function . i.e.
is the local dimension of  at x, thus E() is the
set of points at which  has local dimension .
One problem that arises is how big the sets E()
are for various ’s. There are two approaches
for this: a) we may consider (E()) as  varies
and b) find the Hausdorff dimension of E().
The function
is termed the multifractal spectrum of .
3. Multidimensional Non-Linear
Spectral Estimation (MNSE)
The two signals shown in Figures 1 and 2
look quite similar. They have approximately the
same statistical properties, such as mean,
standard deviation, and variance. But they are
quite different.
Figure 1
where scaling means stretching(or compressing)
and position means shifting the wavelet.
2.2. Fractal Measure
The basic approach of Vehel et al. is as
follows. A measure  on Rn(with 0<( Rn)<)
can give rise to a hierarchy of fractal sets. For
0 we define sets
where
Figure 2
The one on the top is random and the one on the
bottom is derived from Xn+1=CXn(1-Xn), where
C is any constant number (say 3.95). Many
signals look random. Examples are heart rate,
blood pressure in the arteries, and stock prices. It
has always been assumed that these fluctuations
can be described by random processes. However,
if these fluctuations are not random, we might
then be able to understand these mechanisms and
to control them. This will lead to an increase in
our understanding of physiological systems. To
analyze such kinds of signals we develop a new
wavelet based approach to estimate a multifractal
spectrum and its important parameters. We call
this method Multidimensional Non-linear
Spectral Estimation (MNSE).
last parts describe the normal function. The other
three features each demonstrate abnormal wave
types, namely, tilted, recovery, and tilted with
drug administered.
3.1 Methodology
Our method is a wavelet based method,
because wavelets are useful in many frameworks
for approximation and they are also a good tool
to analyze physiological signals. First, we start
with the original HRV signal which has some
abnormal values (spikes) and then truncate the
abnormal values or remove those values. Let n
be the length of the truncated signal. Apply a
continuous wavelet transformation for multilevel
say l ( 32 levels ) to get wavelet coefficients Cji
where i varies from 1 to n and j varies from 1 to
l. Secondly, we construct the slope signal S as
follows: For each i, set:
Fit Linear regression Y=a+bX and set Si=b.
Next, cluster the slope signal S into N segments,
where each segment consists of about 600
elements. For each segment, compute E() for
different  (say 15 bins), and the following
measures: fractal dimension(FD)[9], average
fractal dimension, mean alpha, second moment
about zero for alpha, third moment about zero
for alpha, standard deviation for alpha,
maximum alpha, and maximum fractal
dimension etc,. Finally, draw multifractal
spectrum by plotting fractal dimension of each
set E().
In this method, the ECG is registered for 24
hours. Then the sequence of RR-intervals are
registered. Figure 3 a-f shows an original signal
with five different classes of HRV Tachograms
from an experiment in which the subject is lying
on a platform which can be tilted. The first and
Figure 3 (a)
Figure 3 (b)
Figure 3 (c)
spectral exponent into 15 bins for which a fractal
dimension of each signal fragment was
computed. For each pattern, we computed
AFD(Average Fractal Dimension), Mean of
alpha, SD of Alpha, Second and Third moment
of alpha, Maximum of alpha, and Maximum FD.
Figure 4 (a)-(f) shows sample plots of one
pattern for one patient.
Figure 3 (d)
Figure 4 (a)
Figure 3 (e)
Figure 4 (b)
Figure 3 (f)
4. Results and Discussions
Twelve patients (age:48 19,36%male, 8%
heart disease) were included in the study (n=12).
We used the ECG data collected over a 24-hour
period for the patterns. The time-domain and
frequency-domain parameters of HRV were
obtained and then the Multifractal spectrum was
computed by using multilevel (32 levels) wavelet
transformed coefficients of HRV by MNSE
methodology. Here we clustered the slope of the
wavelet coefficients according to their local
Figure 4 (c)
5. Conclusion
Figure 4 (d)
A framework using wavelets and multifractal
spectrum was presented. This framework is
suitable to analyze non-stationary signals
especially physiological signals. Although in
principle the techniques of non-linear dynamics
have been shown to be powerful tools for
characterization
of
various
autonomic
regulations of the cardiovascular system, no
major breakthrough has yet been achieved by
their application to biomedical data, including
HRV analysis. However, further experimental
analysis need to be carried on to fine the tune
various parameters in this method.
6. References
Figure 4 (e)
Figure 4 (f)
We found the peak of the multifractal
spectrum. It shifted to higher dimensions and
demonstrated an increased complexity with an
increasing amount of noise for ANS regulation
of HRV signals during tilt. This method can also
provide a significant amount of additional
information which is not covered by the standard
measures.
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