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CHAPTER 9

WAVELET ENTROPY ANALYSIS OF

HEART RATE VARIABILITY

9.1 INTRODUCTION

The transform of a signal is another form of indicating the signal. It

does not change the information content existing in the signal. The Wavelet

Transform gives a time-frequency representation of the signal. It was

developed to comeacross the short coming of the Short Time Fourier

Transform (STFT), which can be used to analyze non-stationary signals.

While STFT indicates a constant resolution at all frequencies, the Wavelet

Transform utilizes multi-resolution technique by which different frequencies

are analyzed with different resolutions.

A wave is an oscillating function of time or space and is periodic.

In contrast, wavelets are localized waves. They have energy pointed in time or

space and are fitted with analysis of transient signals. While Fourier

Transform and STFT utilize waves to analyze signals, the Wavelet Transform

utilizes wavelets of finite energy. Demonstration of a wave and a wavelet is

shown in Figure 9.1.

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(a) (b)

Figure 9.1 Demonstration of (a) a Wave and (b) a Wavelet

The wavelet analysis (Rajendra Acharya et al 2002) is done like the

STFT analysis. The signal to be analyzed is multiplied with a wavelet

function as it is multiplied with a window function in STFT, and the

transform is computed for each segment produced. Unlike STFT, in Wavelet

Transform (Provaznik et al 2000; Bunluechokchai and English 2003) the

width of the wavelet function changes with every spectral component. The

Wavelet Transform, at high frequencies, indicates good time resolution and

poor frequency resolution, while at low frequencies, the Wavelet Transform

indicates good frequency resolution and poor time resolution.

Definition of a wavelet is given. The Disctrete Wavelet Transform

(DWT) is introduced. Filter banks are explained. Wavelet families are

indicated. Daubechies Wavelet Transform is given. Wavelet Entropy is

described.

Given a stochastic process s(t) its jointed signal is assumed to be

given by the sampled values S = [ s(n), n = 1, ---, M]. Its wavelet expansion

has associated wavelet coefficients. The set of wavelet coefficient at level j,

{Cj (k)}k is also a stochastic process where k represents the discrete time

variable. Wavelet coefficient entropy is given.

In this chapter, wavelet entropy analysis of Heart rate variability is

done.

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9.1.1 Definition of a Wavelet

Regardless of scale and magnitude, a function is admissible as a

wavelet if and only if

(9.1)

for which it is sufficient that its mean disappears :

(9.2)

This condition is not very forcive, though it differentiates wavelets

(band-pass filters) from low- or high-pass filters.

9.1.2 The Discrete Wavelet Transform

The Wavelet Series is a sampled replica of Continuous Wavelet

Transform (CWT) and computation may consume significant amount of time

and resources, depending on the resolution needed. The Discrete Wavelet

Transform (DWT), which is dependent on sub-band coding is enquired to

yield a fast computation of Wavelet Transform. It is not difficult to implement

and decreases the computation time and resources needed.

The foundations of DWT return to 1976 when techniques to

decompose discrete time signals were implemented. Similar work was done in

speech signal coding which was called sub-band coding. In 1983, a technique

similar to sub-band coding was developed which was called pyramidal

coding. Later many improvements were made to these coding schemes which

produced efficient multi-resolution analysis schemes.

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In the case of DWT, a time-scale representation of the digital signal

is achieved using digital filtering techniques. The signal to be analyzed has

gone through filters with different cutoff frequencies at different scales.

9.1.3 Filter Banks

9.1.3.1 Multi-resolution analysis using filter banks

Filters are one of the most universally used signal processing

functions. Wavelets can be implemented by iteration of filters with rescaling

(Martin Vetterli 1992). The resolution of the signal, which is a measure of the

amount of detail information in the signal, is obtained by the filtering

operations, and the scale is determined by upsampling and downsampling

(subsampling) operations.

The DWT is computed by successive lowpass and highpass

filtering of the discrete time-domain signal. This is known as the Mallat

algorithm or Mallat-tree decomposition. Its importance is in the manner it

connects the continuous-time mutiresolution to discrete-time filters. In the

Figure 9.2, the signal is shown by the sequence x[n], where n is an integer.

The low pass filter is shown by G0

while the high pass filter is indicated by

H0. At each level, the high pass filter results in detail information, d[n], while

the low pass filter associated with scaling function results in coarse

approximations, a[n].

Figure 9.2 Three-level wavelet decomposition tree

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At every decomposition level, the half band filters result in signals

spreading only half the frequency band. This doubles the frequency resolution

as the uncertainity in frequency is reduced by half. In accordance with

Nyquist’s rule if the original signal has a highest frequency of , which needs

a sampling frequency of 2 radians, it has a highest frequency of /2 radians.

It can be sampled at a frequency of radians thus removing half the samples

with no loss of information. This decimation by 2 halves the time resolution

as the complete signal is represented by only half the number of samples.

Thus, while the half band low pass filtering discards half of the frequencies

and thus halves the resolution, the decimation by 2 doubles the scale.

With this approach, the time resolution becomes arbitrarily good at

high frequencies, while the frequency resolution appears arbitrarily good at

low frequencies. The filtering and decimation process is continued until the

desired level is obtained. The maximum number of levels bends on on the

length of the signal. The DWT of the original signal is then obtained by

concatenating all the coefficients, a[n] and d[n], starting from the last level of

decomposition.

Figure 9.3 Three-level wavelet reconstruction tree

Figure 9.3 shows the reconstruction of the original signal from the

wavelet coefficients. Basically, the reconstruction is the reverse process of

decomposition. The approximation and detail coefficients at each level are

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upsampled by two, passed through the low pass and high pass synthesis filters

and then added. This process is passed through the same number of levels as

in the decomposition process to achieve the original signal. The Mallat

algorithm works well if the analysis filters G0

and H0 are interchanged with

the synthesis filters G1

and H1.

9.1.4 Wavelet Families

There are a number of basis functions that can be utilized as the

mother wavelet (Ahuja et al 2005) for Wavelet Transformation. Since the

mother wavelet results in all wavelet functions utilized in the transformation

through translation and scaling, it produces the characteristics of the resulting

Wavelet Transform. The details of the particular application should be taken

into account and the appropriate mother wavelet should be chosen in order to

utilize the Wavelet Transform efficiently.

Figure 9.4 Wavelet families (a) Haar (b) Daubechies4 (c) Coiflet1

(d) Symlet2 (e) Meyer (f) Morlet (g) Mexican hat

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Figure 9.4. shows some of the commonly used wavelet functions.

Haar wavelet is one of the oldest and simplest wavelet. Daubechies wavelets

are the most popular wavelets. They indicate the foundations of wavelet

signal processing and are used in numerous applications. These are also called

Maxflat wavelets as their frequency responses have maximum flatness at

frequencies 0 and . This is a very acceptable property in some applications.

The Haar, Daubechies, Symlets and Coiflets are compactly sided by

orthogonal wavelets. These wavelets along with Meyer wavelets produces

perfect reconstruction. The Meyer, Morlet and Mexican Hat wavelets are

symmetric in shape. The wavelets are selected depending on their shape and

their ability to analyze the signal in a particular application.

It is well known that the choice of mother wavelet and scaling

function is application-dependent. The proper choice of mother wavelet and

father scaling function is important in diverse applications. The method of

choosing an appropriate basis for choice of mother wavelet has been that of

trial and error. The choice of wavelet can affect the results of the analysis, but

that so far there is no known means of selecting a suitable basis, other than

experience. Wavelets of finite and infinite support find elaborative use in

diverse applications. Any signal that is non-stationary is a candidate for

wavelet analysis. For producing wavelet algorithms, common desirable

properties of the scaling function and wavelet are: finite small support

(to reduce computational chore); explicit and simple expression;

symmetry/antisymmetry (or linear phase); orthogonality or biorthogonality;

high regularity (smoothness); low approximation error, and good time-

frequency localisation.

A basic condition for the mother wavelet and scaling function in

any superresolution algorithm is that they should be finitely encouraged.

Every desirable property of the father scaling function and the mother wavelet

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in both implementation and approximation aspects is considered in the

subsections below.

The orthogonality of wavelets to the scaling functions

(9.3)

gives the equation

(9.4)

having a solution of the form

(9.5)

Another condition of the orthogonality of wavelets to all

polynomials up to the power (M 1), defining its regularity and oscillatory

behavior

(9.6)

provides the relation

(9.7)

giving rise to

(9.8)

when the formula (9.5) is taken into account.

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The normalization condition

(9.9)

can be rewritten as another equation for hk:

(9.10)

Let us write down the Equations (9.4),(9.8), (9.10) for M = 2

explicitly:

(9.11)

The solution of this system is

(9.12)

that, in the case of the minus sign for h3, corresponds to the well known filter

(9.13)

These coefficients define the simplest D4 (or 2 ) wavelet from the

famous family of orthonormal Daubechies wavelets (Mahmoodabadi et al

2005) with finite support. It is shown in the below Figure 9.5 by the dotted

line with the corresponding scaling function shown by the solid line. Some

other higher rank wavelets are shown there. It is evident from Figure 9.5

(especially, for D4) that wavelets are smoother in some points than in others.

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M=2

Figure 9.5 Mother wavelet and scaling function of Daubechies wavelet,

for M=2 & M=4

9.1.5 Wavelet Transform

Figure 9.6 How an infinite set of wavelets is replaced by one scaling

function

In Figure 9.6, how an infinite set of wavelets is replaced by one

scaling function is shown.

9.1.5.1 Sub band coding

A wavelet transform can be considered as a filter bank, and

wavelet transforming a signal as passing the signal through this filter bank.

The outputs of the different filter stages are the wavelet- and scaling function

transform coefficients. This idea is called subband coding. Splitting the signal

spectrum with an iterated filter bank is shown in Figure 9.7.

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Figure 9.7 Splitting the signal spectrum with an iterated filter bank

By subband coding , every time iteration is done in the filter bank,

the number of samples for the next stage is halved so that in the end just one

sample is deleted.

9.1.5.2 Daubechies Wavelet Transform

Figure 9.8 The Daubchies D4 function

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Every step of the wavelet transform provides the scaling function to the data input. If the original data set has N values, the scaling function will be applied in the wavelet transform step to calculate N/2 smoothed values. In the ordered wavelet transform the smoothed values are stored in the lower half of the N element input vector. A Daubchies D4 function is shown in Figure 9.8.

The wavelet function coefficient values are:

g0 = h3 ;g1 = -h2 ;g2 = h1 ;g3 = -h0 (9.14)

The scaling and wavelet functions are calculated by taking the inner product of the coefficients and four data values. The equations are shown below:

Daubechies D4 scaling function:

(9.15)

(9.16)

Daubechies D4 scaling function:

(9.17)

(9.18)

Figure 9.9 Daubechies D4 forward transform matrix for an 8 element

signal

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Figure 9.10 Daubechies D4 inverse transform matrix for an 8 element

transform result

Daubechies D4 forward transform matrix for an 8 element signal is

shown in Figure 9.9. Daubechies D4 inverse transform matrix for an 8

element transform result is shown in Figure 9.10.

9.1.5.3 Wavelet Applications

Some areas of application

1. Image processing : Increasing of quality image, image

compression (wavelets are base of MPEG4), FBI fingerprint

compression

2. Signal processing : Noise reduction, Data compression.

Other examples of wavelet applications are in astronomy, stock

market, medicine, nuclear engineering, neurophysiology, music, optics etc.

9.1.5.4 Denoising with Noisy Data

transformed the signal to the wavelet domain using Daubechies

D4 wavelet.

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applied a threshold at wavelet coefficients and

inverse-transform to the signal domain.

9.1.6 Detrending With Wavelet Filter

As stated earlier, trend (noise in HRV) can be removed using

denoising technique. The trend in an HRV sequence produces slow variation

of the sequence with frequency much lower than the LF band of HRV.

The DWT can be implemented with low-pass w(n) and high-pass

v(n) filters and down-samplers 2, as depicted in Figure 9.11. One utilizes

db3 as the mother wavelet. After applying six levels of DWT, the sub band

component with the lower frequency range is a6 which corresponds to

0~0.0313 Hz for the HRV interpolated and re-sampled to 4 Hz. The a6 band

is treated as the trend and, by set the parameters in a6 to zero and perform the

inverse DWT, one reconstructs the detrended HRV sequence.

Figure 9.11 Detrending with wavelet filter

Detrending with wavelet filter is shown in Figure 9.11.

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9.2 METHODOLOGY

9.2.1 Wavelet Entropy

The advantages of projecting an arbitrary continuous stochastic

process in a discrete wavelet space are known. The wavelet time-frequency

representation does not have any assumptions about signal stationarity and is

capable of detecting dynamic changes due to its localization properties.

Unlike the harmonic base functions of the Fourier analysis, which are

localized in frequency but infinitely elongate in time, wavelets are well

localized in both time and frequency. Moreover, the computational time is

significantly shorter since the algorithm has the utility of fast wavelet

transform in a multiresolution framework. Wavelet entropy (Zunino et al

2008 ; Alcaraz and Rieta 2007 and Putrock et al 2004) is introduced.

9.2.2 Wavelet Coefficient Entropy

In the following, given a stochastic process s(t) its jointed signal is

assumed to be given by the sampled values S = {s(n), n = 1, · · · ,M}. Its

wavelet expansion has associated wavelet coefficients given by

(9.19)

with j = N, · · · , 1, and N = log2M. The number of coefficients at each

resolution level is Nj = 2jM. Note that this correlation gives information on

the signal at scale 2 j and time 2 jk. The set of wavelet coefficients at level j,

{Cj(k)}k, is also a stochastic process where k represents the discrete time

variable. Wavelet coefficient entropy is given by

Wavelet coefficient entropy (9.20)

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9.2.3 Wavelet Sample Entropy

The Wavelet sample entropy analysis (Alcaraz and Rieta 2008) of

the HRV from the R-R intervals available in the PHYSIONET is as follows.

Firstly the R-R intervals available are converted into heart rates (beats per

minute) and from heart rate, HRV is calculated. Next, four levels of wavelet

decomposition are applied to the HRV signals. Regarding the wavelet family

selection, there are no established rules for the choice of wavelet functions.

Thereby, Daubechies 4 orthogonal wavelet families are tested, because only

in an orthogonal basis any signal can be uniquely decomposed and the

decomposition can be inverted without loosing information. The wavelet

coefficients vector belonging to the scale containing the dominant HRV, those

with the largest R-R interval difference, is linearly interpolated by the factor

2m 1, being m the discrete wavelet scale. A vector of wavelet coefficients

with a number of samples equal to the original signal is achieved for the

chosen scale. Considering that different scales indicate wavelet coefficients

vectors with diverse number of samples, this interpolation is necessary.

Moreover, unsuccessful results are achieved when noninterpolated wavelet

coefficients vectors are analyzed. This combination of WT and SpEN has

been named Wavelet Sample Entropy (WSE) and allows one to detect

regularity variations in the HRV signal that would be left shadowed in other

cases.

Given a signal x(n)=x(1), x(2),…, x(N), where N is the total

number of data points, SpEn algorithm can be summarized as given in 7.2.2.

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9.3 RESULTS

Table 9.1 Wavelet coefficient entropy of CHF patients

S.No. Patient Wavelet Coefficient Entropy

1 Chf201 0.1533712 Chf202 0.5276503 Chf203 0.4338824 Chf204 0.5106875 Chf205 0.0609526 Chf206 0.1533467 Chf207 0.0325958 Chf208 0.0343749 Chf209 0.13948510 Chf210 0.15967811 Chf211 0.15361712 Chf212 0.01281913 Chf213 0.08655114 Chf214 0.10707415 Chf215 0.04070016 Chf216 0.15755317 Chf217 0.02581718 Chf218 0.11985819 Chf219 0.02879720 Chf220 0.02879721 Chf221 0.04058322 Chf222 0.12875623 Chf223 0.15808224 Chf224 0.17169225 Chf225 0.06609026 Chf226 0.10694927 Chf228 0.12381928 Chf229 0.073392

Avg. : 0.137035 SD : 0.132924

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Table 9.2 Wavelet coefficient entropy of Arrhythmia patients

S.No. Patient Wavelet Coefficient Entropy 1 100 0.1387032 101 0.1096303 102 0.1096304 103 0.0323175 104 0.1597506 105 0.1051787 106 0.1592908 107 0.1456969 108 0.566690

10 110 0.09427911 111 0.33480412 112 0.12978913 113 0.13286114 114 0.118779

Avg. : 0.166957

SD : 0.127611

Table 9.3 Wavelet coefficient entropy of NSR subjects

S.No. Subject Wavelet Coefficient Entropy 1 Nsr01 0.1458752 Nsr02 0.0894373 Nsr04 0.2307884 Nsr05 0.0971185 Nsr06 0.0428676 Nsr07 0.1357237 Nsr08 0.0784768 Nsr09 0.0064189 Nsr10 0.08386210 Nsr11 0.08525111 Nsr12 0.14935612 Nsr13 0.15082413 Nsr14 0.32449914 Nsr15 0.10152915 Nsr16 0.091908

Avg. : 0.120929

SD : 0.074156

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Table 9.4 Wavelet sample entropy CHF patients

S.No. Patient Wavelet Sample Entropy

1 Chf201 1.8227532 Chf202 1.8431713 Chf203 1.7297034 Chf204 1.9101885 Chf205 0.0436926 Chf206 0.4870787 Chf207 1.8558558 Chf208 1.9216799 Chf209 1.729703

10 Chf210 1.77863111 Chf211 1.88767212 Chf212 1.93423013 Chf213 1.86400914 Chf214 1.91690415 Chf215 1.885848

Ave : 1.640740

SD : 0.549032

Table 9.5 Wavelet sample entropy of Arrhythmia patients

S.No. Patient Wavelet Sample Entropy

1 100 1.840182 2 101 1.143007 3 102 1.820431 4 103 1.660509 5 104 1.822823 6 105 1.822807 7 106 1.832837 8 107 1.820757 9 108 1.817141 10 110 1.840198 11 111 1.822132 12 112 1.665962 13 113 1.760799 14 114 1.796088

Avg : 1.747548

SD : 0.177096

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Table 9.6 Wavelet sample entropy of NSR subjects

S.No. Subject Wavelet Sample Entropy 1 Nsr01 1.905585

2 Nsr02 1.842789

3 Nsr04 1.904265

4 Nsr05 1.936459

5 Nsr06 1.885816

6 Nsr07 1.867522

7 Nsr08 1.880762

8 Nsr09 1.892008

9 Nsr10 1.905420

10 Nsr11 1.911907

11 Nsr12 1.820959

12 Nsr13 1.916766

13 Nsr14 1.783927

14 Nsr15 1.846876

15 Nsr16 1.912966 Avg. : 1.880935

SD : 0.040044

For CHF patients, CHF 202 has Wavelet Coefficient Entropy of

0.527650. It is valid. CHF 209 has wavelet Coefficient Entropy of 0.139485.

It is valid. CHF 212 has Wavelet Coefficient Entropy is 0.012819. It is valid.

In Table 9.1 Wavelet coefficient entropy of CHF patients is indicated.

For arrhythmia patients, 108 has Wavelet coefficient Entropy of

0.566690. It is correct. 104 has Wavelet Coefficient Entropy of 0.159750. It is

correct. In Table 9.2, Wavelet coefficient entropy of arrhythmia patients is

shown.

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In NSR subjects, NSR -6 has Wavelet Coefficient Entropy of

0.042867. It is valid. NSR 07 has Wavelet Coefficient Entropy of 0.135723. It

is valid. In Table 9.3 Wavelet coefficient entropy of NSR subjects is

indicated.

For CHF patients, CHF 205 has Wavelet Sample Entropy of

0.043492. It is valid. CHF 209 has Wavelet Sample Entropy of 1.729703. It is

valid. In Table 9.4, Wavelet sample entropy of CHF patients is shown.

For Arrhythmia data base, 101 has Wavelet Sample Entropy of

1-143007. It is valid. 110 has Wavelet Sample Entropy of 1.840198. It is

valid. 113 has Wavelet Sample Entropy of 1.760799. It is valid. In Table 9.5,

Wavelet sample entropy of arrhythmia patients is shown.

For NSR subjects, NSR 05 has Wavelet Sample Entropy of

1.936459. It is correct. NSR 08 has Wavelet Sample Entropy of 1.880762. It

is correct. NSR 14 has Wavelet Sample Entropy of 1.783927. It is correct. In

Table 9.6 Wavelet sample entropy of NSR subjects is indicated.

The values near to average value are correct. The values far away

from average value are correct with error.

Wavelet coefficient entropy (WCE) for NSR subjects is lower compared with

CHF patients (P=0.87).

WCE for CHF patients is higher compared with NSR subjects (P=0.74).

WCE for Arrhythmia patients is higher compared with NSR subjects

(P=0.093).

Wavelet sample entropy (WSE) for NSR subjects is higher compared with

CHF patients (P=0.0).

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WSE for CHF patients is lower compared with NSR subjects (P=0.94).

WSE for Arrhythmia patients is lower compared with NSR subjects

(P=0.993).

New analysis methods of HR behaviour motivated by nonlinear

dynamics have been developed to quantify the dynamics of HR

9.4 DISCUSSION

Wavelet Coefficient Entropy is less for Normal Sinus Rhythm

subjects when compared with that for Congested Heart Failure and Arrythmia

patients. Higher value for Wavelet Coefficient Entropy indicates Arrythmia

after Myocardial Infarction. Wavelet Sample Entropy is more for Normal

Sinus Rhythm subjects when compared with that for Congested Heart Failure

and Arrythmia patients. Lesser value for Wavelet Sample Entropy indicates

Arrythmia after Myocardial Infarction.

9.5 CONCLUSIONS

The usefulness of discrete wavelet transform for the analysis of

heart rate variability studied. The analysis is done by using one mother

wavelet, Daubechies and using two differentiate methods, wavelet coefficient

entropy and wavelet sample entropy. The analysis gives a better idea about

mortality rate of healthy subject and diseased conditions i.e. Arrhythmia and

CHF. The wavelet sample entropy gives better results compared with wavelet

coefficient entropy.

Non linear method is more atachable to the complex HRV analysis.

Fourier entropy is another method of non linear HRV analysis. This study

takes care two diseased cases, arrhythmia and congestive heart failure. A

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future study comparing these diverse techniques can give the most suitable

diagnostic method to stratify the risk associated with heart failure.

Higher value for wavelet coefficient entropy indicates arrhythmia

after Myocardial infarction or diseased states. Lesser value for wavelet sample

entropy indicates arrhythmia after Myocardial infarction or diseased states.

WCE and WSE are not calculated for SCD patients.

In the next chapter, next nonlinear parameters, Permutation entropy

and Multiscale Permutation entropy are computed.

The WSE has low values for diseased patients compared with NSR

subjects. WCE has high values for diseased patients compared with NSR

subjects. So WSE gives better results compared to WCE.