1. SYNOPSIS
ON
CHAOS ANALYSIS OF HEART RATE VARIABILITY
Submitted by
Name of Project Members Roll No
MONODIP SINGHA ROY 11900310004
SHAIBAL CHAKRABORTY 11900310005
SUKRIT GHORUI 11900310023
BIKASH RASAILY 11900310047
Under the guidance of
MR. SUBHOJIT SARKER
ASSISTANT PROFESSOR
ECE DEPARTMENT
DEPARTMENT OF ELECTRONICS & COMMUNICATION ENGINEERING
SILIGURI INSTITUTE OF TECHNOLOGY
PO: SUKNA, SILIGURI, PIN: 734 009, WEST BENGAL
2013-2014

2. Chaos Analysis of Heart Rate Variability
Abstract:
This paper presents a classification method based on chaotic analysis of heart rate
variability. Heart rate variability (HRV) is a reliable reflection of the many physiological
factors modulating the normal rhythm of the heart. Heart rate (HR) is a non stationary
signal, has been conventionally analyzed with time & frequency domain methods, which
measure the overall magnitude of R-R interval. It has been proposed that the normal
heartbeat is associated with complex nonlinear dynamics and chaos This paper uses
non-linear parameters for analyzing of chaos in heart rate variability, the parameters
used are: Lyapunov exponent, Fractal dimension(FD), Poincarẻ plot, Entropy
(Approximate entropy(ApEn), Sample entropy(SampEn), Multiscale entropy(MSEn)),
Detrended fluctuation analysis(DFA). Lyapunov exponent is a chaotic analysis which
measures the rate at which the trajectories separate from one another. Fractal
dimension (FD) that refers to a non-integer or fractional dimension originates from
fractal geometry. Poincaré plot is a visual tool in which each RR interval is plotted as a
function of previous RR
Interval. ApEn, a non-linear complexity index, to quantify the randomness of
physiological time-series. SampEn, measuring complexity and regularity of clinical and
experimental time-series data and compared it with ApEn. The DFA method, used to
quantify the scaling behavior of a time series was previously described and used to
quantify the fractal-like scaling properties of R-R interval data.
INTRODUCTION
eart Rate Variability (HRV) refers to changes in the interval or distance between one
beat of the heart and the next. These variations are produced by the influence of
the sympathetic and Para-sympathetic branches of the Autonomic Nervous System
(ANS) on the sino-atrial (SA) node.
Fig. : R-R Interval Variation
Analysis of HRV by non-linear dynamics can significantly improve the identification
Cardiac disease. The rationale in the emergence of non-linear measures of HRV is that
the heart is not a periodic oscillator under normal physiological conditions and
complicated feedback control of the heart may give rise to non-linear dynamics that are
not well reflected by conventional measures of HRV. The software Kubios is used
H

3. for analyzing Heart Rate Variability. Kubios HRV is an advanced tool for
studying the variability of heart rate intervals, due to its wide variety of
different analysis options and the easy to use interface. The software is
mainly designed for the analysis of normal human HRV, but it is also
usable for animal researches.
CHAOS:
What exactly is chaos?
The name "chaos theory" comes from the fact that the systems that the
theory describes are apparently disordered, but chaos theory is really about finding the
underlying order in apparently random data.
In general, a chaotic system is a deterministic system with
random fluctuations. It has a bounded steady-state behavior that is quasi-periodic and
not at equilibrium. Its spectrum exhibits the character of a continuous and broadband
signal. These attributes are characteristic of electrocardiographic signals and suggest the
use of chaotic theory for analysis of their variation in time.
Chaos theory studies the behavior of dynamical systems that are highly sensitive to
initial condition, an effect which is popularly referred to as “BUTTERFLY EFFECT”,
because the flapping of the wings of butterfly represents a small change in initial
condition of the system, which causes a chain of events leading to large scale
phenomenon. But if the butterfly not flapped the wings, the trajectories of the system
have been vastly different.
Chaos In Heart Rate Variability:
Many people confuse Heart Rate with Heart Rate Variability. The human heart is a bio-
electrical pump beating at an ever changing rate: it is not like a clock that beats at a
Steady, unchanging rate. This variability in heart rate is an adaptive quality in a healthy
body.
Heart Rate Variability (HRV):
The human heart is a bio-electrical pump beating an ever changing rate. By variability
we mean changes in the interval or distance between one heart and the next. The inter
beat interval (IBI) is the time between the R- wave and the next in millisecond. The IBI is
highly variable within any given time period. IBI or HRV have relevance for physical,
emotional and mental function. HRV has been analyzed with time & frequency domain
methods, which measure the overall magnitude of R-R interval fluctuation around its
mean value or the magnitude of fluctuation in some predetermined frequencies.
There are two primary fluctuations for HRV:
1. Respiratory Arrhythmia- This HRV is associated with respiratory & faithfully
tracks the respiratory rate across range of frequencies.
2. Low Frequency oscillation- This HRV is associated with Mayer waves (Traube-
Hering-Mayer waves) of blood pressure & is usually at a frequency of 0.1Hz or a
10 sec period.

4. Typesof Heart Rate are:
1. Neurogenic Heart Rate.
2. Mynogenic Heart Rate.
Autonomic influences of Heart Rate:
 Cardiac automatically intrinsic to various pacemaker tissues.
 Heart rate & rhythm are largely under control of the autonomic nervous
system.
Measurement of Heart Rate Variability:
Heart rate variability is the difference between the highest heart rate and the lowest
heart rate within each cardiac cycle measured in beats per minute. The index is called
“HR max – HR min”. A second index of HRV is widely used in medical research is the
Standard Deviation (SD) of “N-N interval”. The N-N interval is the normalized beat to
beat interval.
Heart Rate Variability Artifact:
Errors in the location of the instantaneous heart beat will result in errors in the
calculation of the HRV. HRV is highly sensitive to artifact & errors are as low as even 2%
of the data will result in unwanted biases in HRV calculation.
Chaos In HRV:
The nature of Heart Rate Variability (HRV) is not fully understood yet.
Recently many studies have indicated that HRV is chaotic. However most of these
studies used return maps or correlation dimension estimates to reach this conclusion. But
none of these methods discriminate chaos from fractal signal.
By variability we mean changes in the interval or distance between one beat of the
heart and the next. The interbeat interval (IBI) is the time between one R-wave (or heart
beat) and the next, in milliseconds. The IBI is highly variable within any given time
period. So, we can say that there is chaos in Herat Rate Variability (HRV).
Chaotic Analysis Of Heart Rate Variability:
The beat-to-beat variability of heart rate (HRV) arises in both
healthy persons and patients with organic heart diseases. Analysis of HRV from recorded
Ambulatory electrocardiograms (ECG) can help identify patients who are at high risk of
sudden cardiac death. Although a great deal of work has been devoted to the analysis of
HRV, a reliable method of evaluation has not emerged. These methods can be classified
into four major categories: statistical analysis, spectral analysis, and model analysis and
chaotic analysis. In general, a chaotic system is a deterministic system with random
fluctuations. It has a bounded steady-state behavior that is quasi-periodic and not at
equilibrium. Its spectrum exhibits the character of a continuous and broadband signal.
These attributes are characteristic of electrocardiographic signals and suggest the use of
chaotic theory for analysis of their variation in time.

5. There are six types of non-linear parameters that are used for the chaos analysis of
heart rate variability, they are:
1. Lyapunov Exponent.
2. Entropy (Approximate Entropy, Sample Entropy, Multiscale Entropy).
3. Poincare Plot.
4. Fractal Dimension.
5. Correlation Dimension.
6. Detrended Fluctuation Analysis.
Lyapunov Exponent:
The method of Lyapunov characteristic exponents serves as a useful tool to
quantify chaos. Specifically, Lyapunov exponents measure the rates of convergence or
divergence of nearby trajectories .Negative Lyapunov exponents indicates convergence,
while positive Lyapunov exponents demonstrate divergence and chaos. The magnitude
of the Lyapunov exponent is an indicator of the time scale on which chaotic behavior
respectively.
Physically, the Lyapunov exponent is a measure of how rapidly nearby trajectories
converges or diverges.
To determine Lyapunov exponent, two nearby points x0 and x0+ Δ x0 will generate
orbits of its own and separation Δx0 will be function of time Δ x0(x0, t) . For chaotic data
the mean exponential rate of divergence of two initially close orbits is characterized by:
𝜆 = lim
𝑡→∝
1
𝑡
𝑙𝑛
│∆𝑥( 𝑋0, 𝑡)│
│𝛥𝑋0│
Maximum positive λ is chosen as it quantifies sensitivity of the system to initial
conditions and gives a measure of predictability.
Approximate Entropy (ApEn):
The ApEn is a measure of system complexity, which quantifies the
unpredictability of fluctuations in a time series such as heart rate time series. The
method examines the time series for similar epochs (period marked by distinctive
Character). The more frequent and more similar epoch lead to lower value of ApEn.
Informally, given N points, the family of statistics ApEn (m, r, N) is approximately equal
to the negative average natural logarithm of the conditional probability that two
sequences that are similar for m points remain similar within a tolerance r, at the next
point. If ‘N’ is length of time series and ‘m’ is length of sequence to be compared and ‘r’
is tolerance for accepting matches. It is convenient to set tolerance = r*SD. The standard
deviation (SD) of data has been set at SD =1.
Define vector u (j) for time series of N data points:
u ( j) :1 ≤ j ≤ N (9)
This series forms N − m +1 number of xm (i) vectors where, xm (i):1 ≤ i ≤ N − m +1 and
xm (i) has data length of m points defined as [u (i + k): 0 ≤ k ≤ m −1 ] from
u (i) tou(i + m −1) . If Bi is number of vectors xm ( j) within ‘r’ of xm (i) , ( xm (i) –
template and xm ( j) -template match) and Ai is number of vectors xm+1 ( j) within ‘r’ of
xm+1 (i) .

6. Then Cm (r) i - probability that xm ( j) is within ‘r’ of xm (i) Cm (r), i = ( Bi )/( N − m +1)
The average of the natural logarithm of Cm (r), the average of the natural logarithm of
Ci
m (r) is,
∅ 𝑚 ( 𝑟) =
1
(𝑁−𝑚+1)
∑ ln[𝑐𝑖
𝑚( 𝑟)]𝑁−𝑚+1
𝑖=1
The ApEn for fixed parameters of n, m & r is:-
ApEn (m, r, N) = Φm(r-Φm+1(r))
After algebraic manipulation:
𝐴𝑝𝐸𝑛( 𝑚, 𝑟, 𝑁) = (
1
𝑁−𝑚+1
)∑ ln[ 𝑐𝑖
𝑚( 𝑟)]−
1
(𝑁−𝑚)
𝑁−𝑚+1
𝑖=1 ∑ ln[𝑐𝑖
𝑚+1
(𝑟)𝑁−𝑚
𝑖=1 ]
[When N is large]
𝐴𝑝𝐸𝑛( 𝑚, 𝑟, 𝑁) ≅ (
1
𝑁−𝑚
)∑ [− ln(
𝐴𝑖
𝐵𝑖
)]𝑁−𝑚
𝑖=1
Approximate entropy is used for chaos analysis for low dimensional signals of heart; the
analysis is performed by transforming data to a phase space in which there are ‘N’
dimensional space, where ‘N’ is the number of dynamical variable.
ApEn represents the chaos as the complex degree of a data sequence and
its value is proportional to the complex degree. Usually, the value of approximate
entropy is a non negative number. More complex the sequence is, larger the value of
the approximate entropy is. Approximate entropy has a strong ability to characterize the
complexity of the signals, such as the chaotic signals and the random signals. The
amount of the data required to calculate the approximate entropy is small, so it is
suitable for analyzing short time data. Besides, approximate entropy has a strong
anti-noise capability.
Sample Entropy (SampEn):
Sample Entropy is the negative logarithm of the conditional
probability that a point which repeats itself with a tolerance of ε in an m dimensional
phase space will repeat itself in an m+1 dimensional phase space. Self-matches are not
included in calculating the probability, described a reduction in SampEn of neonatal HR
prior to the clinical diagnosis of sepsis and sepsis-like illness. The SampEn was found to
be significantly reduced before the onset of atrial fibrillation.
Mathematical Representation :
𝑆𝑎𝑚𝑝𝐸𝑛( 𝑚, ∈) = −log(
𝐶( 𝑚+1,∈)
𝐶( 𝑚,∈)
)
Limitations of SampEn:
Stationary is required; higher pattern length requires an increased number of
data points; evaluates regularity on one scale only; outliers (missed beats, artifacts) may
affect the entropy values.

7. Multiscale Entropy:
Multiscale entropy is a new method for measuring the complexity of a
finite length time series. This computation tool can be applied both to the physical and
physiological data sets, and can be used with a variety of measures of entropy, the heart
period changes on a beat to beat basis & its variations occurs over a large set pf
temporal scale. Due to such multiscale behavior HRV cannot be completely
characterized on a single time scale & cardiac interbeat time series have a complex
temporal structure with multiscale correlation under healthy conditions. Although ApEn
& SampEn for measuring the complexity of physiological time series have been widely
used & proved to be useful in discriminating b/w health & disease states, but due to
multiple scale mechanism of HR some result may lead to misleading conclusion.
Multiscale Entropy is used for the chaos analysis with the help of
white noise, fractal Brownian noise & the 1/f noise. It is seen that the entropy measure
for the deterministic chaotic time series increase on small scale & the gradually
decrease indicating the reduction of complexity on the large scales. This trend of the
variation of the SampEn with scale is entirely different from the white-fractal Brownian-
1/f noises. Moreover the variations of SampEn for all chaotic data sets showed a similar
behavior. This established the fact that the MSE analysis of chaos can be used to detect
the determinism in a time series.
Poincarẻ Plot:
A Poincare plot named after Henri Poincare, is used to quantify self-similarity in
process, usually periodic function.Poincaré plot is a visual tool in which each RR interval
is plotted as a function of previous RR interval.Poincaré plot provides summary
information as well as detailed beat-to-beat information on the behavior of heart. The
problem regarding Poincaré plot use has been lack of obvious quantitative measures
that characterize the salient features of Poincaré plots. To quantitatively characterize
the plot, a number of techniques like converting the two-dimensional plot into various
one-dimensional views; the fitting of an ellipse to the plot shape; and measuring the
correlation coefficient of the plot have been suggested The width of the plot (sd2)
corresponds to the level of short term HRV while the length of the plot(sd1)
corresponds to level of long term HRV.The advantage of Poincare plot is their ability to
identify beat-to-beat cycle and patterns in data that are difficult to identify with spectral
analysis.
The Poincare Plot is a graphical representation of the correlation between
successive RR intervals, i.e. plot of RRj+1 as a function of RRj as described in Fig. 4.
The shape of the plot is the essential feature. A common approach to parameterize the
shape is to fit an ellipse to the plot as shown in Fig . The ellipse is oriented according to
the line-of-identity (RRj = RRj+1). The standard deviation of the point’s perpendicular to
the line-of identity denoted by SD1 describes short-term variability which is mainly
caused by RSA. The standard deviation along the line-of-identity denoted by SD2, on the
other hand, describes long-term variabilitySD1 related to the fast beat-to-beat variability
in the data, while SD2describes the longer-term variability of R–R. The ratio SD1/SD2
may also be computed to describe the relationship between these components.

8. Fig: Poincare Plot with its descriptors SD1 & SD2
Fractal dimension:
A fractal dimension is a ratio providing a statical index of complexity
comparing how detail in a pattern changes with the scale at which it is measured. It is
an index for characterizing fractal patterns or sets by quantifying their complexity as a
ratio of change in the detail tocharatrize a broad spectrum objects ranging from the
abstract to practical phenomena, including turbulence, human physiology.
Chaos theory has evolved into the study of the fractal behavior of physical
systems that at first seem entirely random but in fact are not entirely so. A fractal is a
set of points that when looked at smaller scales, resembles the whole set. The concept
of fractal dimension (FD) that refers to a non-integer or fractional dimension originates
from fractal geometry. The FD emerges to provide a measure of how much space an
object occupies between Euclidean dimensions. The FD of a waveform represents a
powerful tool for transient detection. The fractal dimension computes the number of
degrees of freedom of the data in local areas of attractors using q neighbours of
representative point (hence its name local intrinsic dimension) then performs an
average of all these LID on the whole attractor to obtain the estimate. It is relatively
immune to noise, by opposition to the correlation dimension estimation , and since it
requires a large embedding dimension, it frees the user from its parameters are very
difficult to tune ,which redirects its application to chaos detection only.
Mathematical representation
𝐷𝑜 = lim
𝑒→0
logN(𝑒)
𝑙𝑜𝑔1/𝑒
Correlation dimension:-
The correlation dimension is a measure of dimensionality of the space occupied
by a set of random point. It describes the dimensionality of the process in relation to its
geometrical reconstruction in phase space. It is one of the most widely used measures
of fractal dimension and is usefull indicator for various chaotic analysis of HRV.
For an R-R interval series having N data points, the phase space plot is
constructed with heart rate [n] on the x-axis and heart rate. The idea of correlation
dimension is to construct a probability function c(n) such that two arbitrary points on
the orbit . The CD can be calculated using distance between each pair of points .
We could then define the correlation dimension as
CD = lim
𝑟→0
log(c( 𝑟))
log(𝑟)

9. It is another tool for detecting chaotic motion is the fractal (Hussdroff) dimension. The
correlation dimension measures spatial correlation between the points in a
reconstructed multi dimensional space, depending on the value of ED. According to the
theory chaos and the characteristic behavior of systems the relationship between the
steady behavior of system and its dimension can be summarized as
Steady state systems Dimensions
Equilibrium point 0
1- periodic 1
K-periodic K
Chaotic Non-integer
If the dimension is a non- integer then system is chaotic. The Correlation dimension
value will be high for the chaotic data and it decreases as the variation of R-R signal
becomes less or rhythmic.
DeterndedFluctuation Analysis:
In the DFA method, the variation of each point in a time series from its mean
value is treated as a step in a random walk. These variations are partially integrated to
obtain the random walk time series. It permits detection of intrinsic self-similarity in a
seemingly non stationary time series and avoids spurious detection of apparent self-
similarity which may be due to artifact extrinsic trends. DFA provides quantitative
method for determining the degree to which a time series is random at the one extreme
and correlated at the other. DFA ranges in values from 0.5(random) to 1.5(correlated),
which normal values of just over 1.
The detrended fluctuation analysis (DFA) is a chaotic measure that can be used to
quantify the fractal scaling properties of HR signals of short interval. Therefore, we apply
DFA to the analysis of RR interval time-series data, which calculates the root mean
square fluctuation of integrated and detrended time series, permits the detection of
intrinsic self similarity, embedded in a time series, and also avoids the spurious
detection of apparent self-similarity. DFA is scaling analysis technique proposed, to
detect long-range correlations in the time-series having non-stationeries. DFA was
developed specifically to distinguish between intrinsic fluctuations generated by the
complex systems and those caused by the external or environmental stimuli. Another
principal advantage of DFA is that it is able to detect long-range correlation in time-
series having non-stationarities. DFA was found to carry additional information about
smoothness of time series that was not provided by the traditional time and frequency
HRV measures.
We can take it as
𝐹( 𝑙) = [
1
𝐿
∑ (𝑦𝑖 − 𝑖𝑎 − 𝑏)2
]1/2𝐿
𝑗=1
So that, xt is a bounded time series.
Where Xt is called cumulative sum, where Xt is divided into time windows .Yi of length L
sample and a,b are slope and intercept parameter respectively. And L is window size.

10. Work Done:
 Detail study about Chaos Analysis, Heart Rate variability and
Non-linear Parameters.
 Analyzed application of chaos analysis in HRV.
 Having the detailed concept of non-linear parameters of
Heart Rate Variability Studied application of non- linear
parameter in chaos analysis.
Work To Be Done:
 Analysis of non-linear parameters using Kubios software.
 Using software we have to calculate the non-linear
parameters of hrv from database.
 Compare the theoretical result from the obtained result from
analysis of database.
Conclusion:
Heart rate variability has considerable potential to assess the role
of autonomic nervous system fluctuations in normal healthy individuals
and in patients with various cardiovascular and non-cardiovascular
disorders.
Till now we have concluded that it is possible to perform chaos
analysis of heart rate variability using non-linear parameters.
References:
1. http://en.wikipedia.org/w/index.php?title=Heart_rate_variability&oldid=556325
566
2. Multiscale Analysis of Complex Time Series: Integration oj.
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Wenwen Tung and Jing Hu.
3. Www.IEEExplore.com
4. 2nd International Conference on Biomedical Engineering & Assistive
Technologies, by Dr Butta Singh and Dr Dilbag Singh.
5. On Establishment of Power Law Slope (PLS) as an Effective Tool in Non-Linear
Analysis of Heart Rate Variability , by Subhojit Sarker, Dr. Ankur Ganguly and D.N.
Tibarewala.

11. 6. Chaos, Complexity, and Entropy, by Michel Beranger.
7. Complexity quantification of dense array EEG using Sample Entropy, by Pravitha
8. Recent Trends in Nonlinear Methods of HRV Analysis: A Review by Ramesh K.
Sunkaria
9. On the Presence of Deterministic Chaos in HRV Signals by I RadojiCiC, D Mandie,
D Vulie
10. Linear and non-linear analyses of heart rate variability: a mini review by Pascale
Mansier ,Jean Clairambault ,Nathalie Charlotte, Claire Medigue ,Christophe
Vermeiren ,Gilles LePape , Fraqois Car& , Athanassia Gounaropoulou,
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