View of NONLINEAR FRACTAL ANALYSIS OF HEART RATE VARIABILITY FOR PROGNOSIS AND DETECTION OF CHF

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UGC APPROVED NO. 48767

Vol.03, Issue 04, April 2018, Available Online:www.ajeee.co.in/index.php/AJEEE

NONLINEAR FRACTAL ANALYSIS OF HEART RATE VARIABILITY FOR PROGNOSIS AND DETECTION OF CHF

BABITA SAXENA POOJA MISHRA

A.P. SIRT-S Bhopal Mtech Scholar

[email protected] [email protected] EC Dept.

Abstract: In this study we perform non linear fractal analysis of heart rate variability data of healthy and non healthy subjects this analysis is based on non linear methods recently developed such as Detrended Fluctuation Analysis (DFA), Rescaled Adjusted Range Statistics Plot (R/S), Wavelet Transform Modulus Maxima (WTMM), and Poincare Plot Aim of this work is extracting non linear features of h.r.v and comparing them, for prognosis & detection of congestive heart failure (CHF) the analysis is performed under MATLAB environment.

Keywords: Heart Rate Variability (HRV), Fractal Analysis, Nonlinear Dynamics.

1. INTRODUCTION

Conventional linear techniques using time and frequency domain features of heart rate (HR) variability fails in extracting important information coded into the ECG signals, as the originally the Heart Rate Dynamics follows a Non-linear model. Due to such reasons these techniques limits the prognosis and detection possibilityof Heart diseases. Recently a number of new techniques based on nonlinear system dynamics involving thechaotic behavior and fractal analysis have been used toanalyzethe HRV dynamics.Initial studies involving these techniques showing signs of their ability in providing information that can be used for prognosis and detection of heart disease and predicting the chances of vulnerability to life- threatening arrhythmias. This study analyzes the ability Of such methods to determine whether these new measurements of HRV behavior contains significant prognostic information in predicting and detecting the heart disease.The rest of the paper is arranges as the second section presents a brief literature review. The third section explains the non-linear analysis technique presents the fractal analysis techniques used to analyze the HRV. The fourth section provides the information about the classifier used. The fifth chapter proposed technique

2. LITERATURE REVIEW

A classification system to detect congestive heart failure (CHF) patients from normal (N) patients is presented by R.

A. Thuraisinghamet al [1], they used features from the second-order difference plot (SODP) obtained from Holter monitor cardiac RR intervals while the classification

procedure used was the k-nearest neighbor algorithm.

Evgeniya gospondinova et al [2] presented the wavelet- based multi-factral analysis of RR time series which can be used as non-invasive method is suitable for diagnostic, forecast and prevention of the pathological statuses.AbdelhaqOuelli et al [3] presented an automated detection method for cardiac arrhythmia. The feature values for each arrhythmia are extracted using autoregressive (AR) and multivariate autoregressive (MVAR) modeling of one-lead and two-lead electrocardiogram signals. The classification is performed using a quadratic discriminant function (QDF) and a multilayer perceptron (MLP). Detrended Fluctuation Analysis of Heart Rate Dynamics is presented by Jiun- Yang Chiang et al [4] they demonstrated that the Nonlinear HRV parameters such as detrended fluctuation analysis (DFA) can be used as an important outcome predictor in patients with cardiovascular diseases. Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series is presented by C.‐K.

Peng et al [5] they showed that, heart rate time series from patients with severe congestive heart failure show a breakdown of this long-range correlation behavior and describe a new method-detrended fluctuation analysis (DFA)-for quantifying this correlation property in nonstationary physiological time series. Application of this technique shows evidence for a crossover phenomenon associated with a change in short and long-range scaling exponents. Their method may be of use in distinguishing healthy from pathologic data sets based on differences in these scaling properties.

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Peer Reviewed and Refereed Journal IMPACT FACTOR: 2.104(INTERNATIONAL JOURNAL) UGC APPROVED NO. 48767

3. THE NON-LINEAR ANALYSIS TECHNIQUES In recent years, the nonlinear analysis is constantly gaining the attention, because the approximating and analyzing the non-linear measurements in linear way misses the considerable part of information. The same is applicable for HRVmeasurements which are non-linear and nonstationary and aconsiderable amount of information is coded in the dynamics oftheir fluctuation in different time periods. These important characteristics of signal dynamics aremissed when analyze through conventional (linear) mathematical methods.Therefore, development and implementation of newnon-linear mathematical methods and knowledge, based on thefractal, multifractal and wavelet theory will allow scientists toidentify new reasons for HRV fluctuations. This research isconducted on selected control groups of patients with cardiovascular diseases, forming part of an information database of 300 patients of a cardiological clinic; each with 24-hour Holter recordings. Each recordcontains information ofaround 100,000 heartbeats. The first problem is based on thefact that fluctuation of the physiological signals possesseshidden information in the form of self-similarity, scalestructure, monofractality and multifractality, through theapplication of these methods [8-13]. The fractal, multifractaland wavelet-based multifractal analysis of the fluctuations isuseful not only for getting the comprehensive information forphysiological signals of patients, but also provide a possibilityfor foresight, prognosis and prevention of the pathologicalstatuses. The second problem is due to the large volume ofresearch information is extremely important the correctdetermination the trend of the disease of the patients forrelatively large period of time - several years, for instance.Such tests are performed periodically to compare the graphiccharacteristics of images from clinical studies undertaken as aresult of treatment and give an idea of the patient's conditionand treatment quality.

There a large number of these methods have been developed however in this section only the techniques listed below are discussed

 Rescaled adjusted the range statistics plot (R/S),

 Detrended Fluctuation Analysis (DFA)

 Wavelet Transform Modulus Maxima (WTMM)

 Poincaré plot

Rescaled adjusted the range statistics plot (R/S): Rescaled range analysis (R/S) was developed by Harold E. Hurst while working asa water engineer in Egypt (Hurst 1951) and was later applied to financial time seriesby Mandelbrot

and van Ness (1968), Mandelbrot (1970). The basic idea behind R/Sanalysis is that a range, which is taken as a measure of dispersion of the series, followsa scaling law. If a process is random, the measure of dispersion scales according tothe squareroot law so that a power in the scaling law is equal to 0.5. Such value isconnected to Hurst exponent of 0.5.In the procedure, one takes returns of the time series of length 𝑇 and divides theminto 𝑁 adjacent sub-periods of length 𝜈 while𝑁𝜈 = 𝑇. Each sub-period is labeled as𝐼_𝑛with 𝑛 = 1,2 … . . , 𝑁. Moreover, each element in 𝐼_𝑛 is labeled 𝑟_𝑘,𝑛with 𝑘 = 1,2, … , 𝜈. For each sub-period, one calculates an average value and constructs new series of accumulated deviations from the arithmetic mean values (a profile).The procedure follows in calculation of the range, which is defined as a differencebetween a maximum and a minimum value of the profile 𝑋_𝑘,𝑛, and a standard deviationof the original returns series for each sub-period In. Each range 𝑅𝐼_𝑛 is standardized bythe corresponding standard deviation 𝑆𝐼_𝑛 and forms a rescaled range as

𝑅/𝑆 _𝐼_𝑛 =𝑅_𝐼_𝑛

𝑆_𝐼_𝑛, (1)

The process is repeated for each sub-period of length 𝜈.

We get average rescaled ranges ^𝑅

𝑆 𝜈 for each sub-interval of length 𝜈.

The length 𝜈 is increased and the whole process is repeated. We use the procedureused in recent papers so that we use the length u equal to the power of a set integervalue. Thus, we set a basis b, a minimum power 𝑝𝑚𝑖𝑛 and a maximum power 𝑝𝑚𝑎𝑥 sothat we get 𝜈 = 𝑏^{𝑝𝑚𝑖𝑛}, 𝑏^{𝑝𝑚𝑖𝑛 +1}, … , 𝑏^{𝑝𝑚𝑎𝑥}, where 𝑏^{𝑝𝑚𝑎𝑥} ≤ 𝑇.Rescaled range then scales as

𝑅/𝑆 _𝐼_𝑛 = 𝑐𝜈^𝐻, (2)

Where,𝑐is a finite constant independent of 𝜈.A linear relationship in doublelogarithmic scale indicates a power scaling.To uncover the scaling law, we use an ordinary least squares regression onlogarithms of each side of (2).

We suggest using logarithm with basis equal to b. Thus, we get

log_𝑏 𝑅

𝑆 𝜈~ log_𝑏𝑐 + 𝐻 ∙ 𝑙𝑜𝑔_𝑏𝜈 (3) Where,𝐻is Hurst exponent.

Detrended Fluctuation Analysis (DFA):

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Detrended fluctuation analysis (DFA) was firstly proposed by Peng et al. (1994) whileexamining series of DNA nucleotides. Compared to the R/S analysis examined above,DFA uses different measure of dispersion—squared fluctuations around trend of thesignal. As DFA is based on detrending of the sub-periods, it can be used for no stationary time series contrary to R/S.Starting steps of the procedure are the same as the ones of R/S analysis as thewhole series is divided into non-overlapping periods of length u which is again seton the same basis as in the mentioned procedure and the series profile is constructed.The following steps are based on Grech and Mazur (2005). Polynomial fit 𝑋_𝜈,𝑙 of theprofile is estimated for each sub-period In. The choice of order 𝑙 of the polynomialis rather a rule of thumb but is mostly set as the first or the second order polynomialtrend as higher orders do not add any significant information (Vandewalle et al.

1997).

The procedure is then labeled as DFA-0, DFA-1 and DFA- 2 according to an order ofthe filtering trend (Hu et al.

2001). We stick to the linear trend filtering and thus useDFA-1 in the paper. A detrended signal 𝑌𝜈,𝑙 is then constructed as

𝑌𝜈,𝑙 𝜈, 𝑙 = 𝑋 𝑡 − 𝑋_𝜈,𝑙, (4) Fluctuation𝐹_𝐷𝐹𝐴² (𝜈, 𝑙), which is defined as

𝐹_𝐷𝐹𝐴² 𝜈, 𝑙 =1 𝜈 𝑌_𝜈,𝑙²

𝜈

𝑡=1

𝑡 , (5)

Scales as

𝐹_𝐷𝐹𝐴² 𝜈, 𝑙 ~𝑐𝜈^{2𝐻 𝑙}, (6) where again 𝑐 is a constant independent of 𝜈.We again run an ordinary least squares regression on logarithms of (6) and estimateHurst exponent 𝐻(𝑙) for set 𝑙-degree of polynomial trend in same way as for 𝑅/𝑆 as

log_𝑏𝐹_𝐷𝐹𝐴 𝜈, 𝑙 ~ log_𝑏𝑐 + 𝐻(𝑙) ∙ 𝑙𝑜𝑔_𝑏𝜈 (7) DFA can be adjusted and various filtering functions 𝑋_𝜈,_𝑙can be used.

Wavelet Transform Modulus Maxima (WTMM):

The wavelet transform of a continuous time signal, 𝑥(𝑡), is defined as:

𝑇 𝑎, 𝑏 = 1

𝑎 𝑥 𝑡 𝜓^∗ 𝑡 − 𝑏 𝑎 𝑑𝑡,

∞

−∞

(8) where𝜓^∗(𝑡)is the complex conjugate of the wavelet function 𝜓^∗(𝑡), 𝑎is the dilation parameter of the wavelet and 𝑏is the location parameter of the wavelet. In order to be classified as a wavelet, a function it must satisfy certain mathematical criteria. These are:

 A wavelet must have finite energy: i.e.:

𝐸 = 𝜓 𝑡 ²𝑑𝑡 < ∞

∞

−∞

, (9)

 If 𝜓 (𝑓)is the Fourier transform of𝜓(𝑡), i.e.:

𝜓 𝜔 = 𝜓 𝑡 𝑒^{𝑖 𝜔 𝑡}𝑑𝑡, (10)

∞

−∞

Then the following condition must hold:

𝐶_𝑔 = 𝜓 𝜔 ²

𝜔 𝑑𝜔 < ∞, (11)

∞

0

This implies that the wavelet has no zero frequency component, i.e.𝜓 0 = 0, or, to put it another way, it must have a zero mean. Equation 11 is known as the admissibility condition and 𝐶_𝑔is called the admissibility constant. The value of 𝐶_𝑔depends on the chosen wavelet.

 For complex (or analytic) wavelets, the Fourier transform must both be real and vanish for negative

frequencies.

The contribution to the signal energy at the specific 𝑎 scale and 𝑏location is given by the two-dimensional wavelet energy density function known as the scalogram:

𝐸 𝑎, 𝑏 = 𝑇 𝑎, 𝑏 ², (12) The total energy in the signal may be found from its wavelet transform as follows:

𝐸 = 1 𝐶𝑔

1

𝑎² 𝑇 𝑎, 𝑏 ²𝑑𝑎𝑑𝑏

∞

−∞

= 𝑥 𝑡 ²𝑑𝑡,

∞

−∞ (13) The contribution to the total energy distributioncontained within the signal at a specific a scale is given by

𝐸 𝑎 = 𝑇 𝑎, 𝑏 ^∞ ²𝑑𝑏

−∞

, (14) Modulus Maxima:

Wavelet modulus maxima are defined as

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𝑑 𝑇 𝑎, 𝑏 ²

𝑑𝑏 = 0, (15) and are used for locating and characterizing singularities in the signal. (Note that equation 15 also includes inflection points with zero gradient. These can be easily removed when implementing the modulus maxima method in practice.). Although relatively new, continuous modulus maxima-based methods in have recently been used by our group to analyses some engineering and medical signals [11], [12].

Poincaré plot:

The Poincaré plot analysis is a geometrical and non-linear method to assess the dynamics of heart rate variability (HRV).

The Poincaré plot is a representation of a time series into a phase space, where the values of each pair of successive elements of the time series define a point in the plot. The theoretical background that supports the use of a phase space

is the Takens theorem [8]. According to Takens, it is possible

to reconstruct the attractor of a dynamical system by mapping

a scalar measurement into a phase space using a given time delay and embedding dimension [9]. The Poincaré plot in HRV is a scatter plot of the current R-R interval plotted against the preceding R-R interval.However, the assessment and standardizationof these qualitative classifications are difficult because theyare highly subjective. A quantitative analysis of the HRVattractor displayed by the Poincaré plot can be made byadjusting it to an ellipse. For the performance analysis, theSD1 (Standard Deviation1), SD2 (Standard Deviation 2) andarea of ellipse are used as evaluation parameters [9].

Thedefinitions are given in the next:

Fig. 1 Poincaré plot of a healthy patient (left) and unhealthy patient (right)

Standard Deviation 1(SD1):

Is the standard deviation (SD) of the instantaneous (short term) beat-to-beat R-R interval variability (minor axis of the

ellipse or SD1). SD1 can be calculated as

𝑆𝐷1 = 𝑣𝑎𝑟(𝑥1), (16) Standard Deviation 2(SD2):

Is the standard deviation (SD) of the long term R-R interval

variability (major axis of the ellipse or SD2). SD2 can be calculated as:

𝑆𝐷2 = 𝑣𝑎𝑟(𝑥₂), (17) where𝑣𝑎𝑟(𝑥) is the variance of variable 𝑥, and

𝑥₁=𝑅𝑅_𝑖− 𝑅𝑅_𝑖+1

2 , 18

𝑥₁=𝑅𝑅_𝑖+ 𝑅𝑅_𝑖+1

2 , 19 𝑅𝑅_𝑖and𝑅𝑅_(𝑖+1)are vectors defined as:

𝑅𝑅_𝑖= 𝑅𝑅₁, 𝑅𝑅₂, … , 𝑅𝑅_𝑁−1 , (20) 𝑅𝑅_𝑖+1= 𝑅𝑅₂, 𝑅𝑅₃, … , 𝑅𝑅_𝑁 , (21)

Thus, it means, the 𝑥1 and𝑥2correspond to the rotation of𝑅𝑅𝑖 and 𝑅𝑅𝑖+1 by angle^𝜋₄.

𝑥₁

𝑥₂ = cos 𝜋

4 −sin 𝜋 4 sin 𝜋

4 cos 𝜋 4

𝑅𝑅_𝑖

𝑅𝑅_𝑖+1 , (22)

4. SVM classification

In machine learning, support vector machines (SVMs) are supervised learning models used for classification and regression analysis. Given a set of training examples, each marked for belonging to one of two categories, an SVM training algorithm builds a model that assigns new examples into one category or the other, making it a non- probabilistic binary linear classifier.

Let 𝑥_𝑖 ∈ 𝑅^𝑚be a feature vector or a set of input variables and let 𝑦_𝑖 ∈ {+1, −1}be a corresponding class label, where 𝑚is the dimension of the feature vector. In linearly separable cases a separating hyper-plane satisfies [8].

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Figure 2: Block Diagram of the Proposed System.

Figure 3: Maximum-margin hyper-plane and margins for an SVM trained with samples from two classes. Samples on the margin are called the support vectors.

𝑦𝑖 𝑤. 𝑥_𝑖 + 𝑏 ≥ 1, 𝑖 = 1 … 𝑛, … . 4

Where the hyper-plane is denoted by a vector of weights 𝑤and a bias term 𝑏. The optimal separating hyper-plane, when classes have equal loss-functions, maximizes the margin between the hyper-plane and the closest samples of classes. The margin is given

SVM training always finds a global minimum, and their simple geometric interpretation provides fertile ground for further investigation.

Most often Gaussian kernels are used, when the resulted SVM corresponds to an RBF network with Gaussian radial basis functions. As the SVM approach ―automatically‖

solves the network complexity problem, the size of the hidden layer is obtained as the result of the QP procedure.

Hidden neurons and support vectors correspond to each other, so the center problems of the RBF network is also solved, as the support vectors serve as the basis function centers.Classical learning systems like neural networks suffer from their theoretical weakness, e.g. back- propagation usually converges only to locally optimal solutions. Here SVMs can provide a significant improvement. The absence of local minima from the above

algorithms marks a major departure from traditional systems such as neural networks.SVMs have been developed in the reverse order to the development of neural networks (SVMs). SVMs evolved from the sound theory to the implementation and experiments, while the SVMs followed more heuristic path, from applications and extensive experimentation to the theory.

.4 Proposed Work

In proposed work we will perform filtering to reduce unwanted noise which may affect the RR interval measurement we will also calculate features vectors of all ecg signals (we get from BIDMS CHF databse available at physiconet.org) using non linear fractal analysis techniques such as Detrended fluctuation analysis (DFA) ,Rescaled Adjusted Range statistics plot (R/S) , Wavelet Transform modulus maxima (WTMM) , & Poincare Plot. After that we will separate these features vectors in two groups healthy & unhealthy than we classify these vectors using classification techniques like clustering and support vector machine. By this analysis we compare the capabilty &

efficiency of these techniques for obtaining results.

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Trained Classifier SVM Clustering

Train Classifier Feature Extractor

Testing Data Training Data

Evaluate Results Feature Extractor

Group & Label

Collect Decision

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