Analysis of Multifractal Behaviour of Electrocardiograms: DFA Method

Analysis of Multifractal Behaviour of Electrocardiograms: DFA Method

Sayanti Sankhari

Student Of Master Of Biomedical Engineering School Of Bio-science & Engineering, Jadavpur University, Kolkata-700032, India

Abstract

Attempts have been made in this paper to show the multifractal property of the electrocardiograms. With the aid of the detrended fluctuation analysis (DFA), the multifractal nature of the electrocardiograms(ECG) has been studied. Normal electrocardiograms are acquired through the polypara module from two subjects, whereas the diseased electrocardiogram of BIDMC congestive heart failure and atrial fibrillation are obtained from the physionet. Both the ECG signals have been analysed by the Hurst exponent, the partition function and the singular spectrum of multifractal detrended fluctuation analysis (MF-DFA). There is a little diversification between the spectrums of the normal ECGs and the diseased ECGs but the results show a strong degree of multifractality in the time series of the electrocardiograms. The results are all implemented upon the MATLAB platform.

1. Introduction

   whereas parasympathetic system decreases it [12]. For this type of intrinsically noisy system, the novel technique of detrended fluctuation analysis (DFA) has been developed to study the non-stationary behaviour of ECG signal [15]. Normal ECG signals have been acquired through the polypara module. The data of the diseased patient is collected from the patients suffering from BIDMC congestive heart failure and atrial fibrillation [16-17].

2. DFA Algorithm

   The DFA algorithm[2] quantifies fractal-like correlation properties by calculating the scaling property of the root-mean-square fluctuation of the integrated and detrended time series data. The steps of the DFA algorithm are as follows:

   Step 1: At first the ECG time series is taken which is denoted by {x(i)}. Then the profile {Y(i)} is determined.

   Life is one of the most complex non-linear systems and heart is the core of this life-cycle system. Fractals

   are fit for signal modelling in the real world, such as

   = < >

   =1

   Where <x> is the mean of the record

   (1)

   electroencephalograms (EEG), electrocardiograms (ECG), as well as turbulent flows, lightning strikes, DNA sequences, and geographical objects which represent some of many natural phenomena and are difficult to be characterized using traditional signal processing theory [1-4]. Electrocardiogram (ECG) is a graphical representation of cardiac activity. In general, ECG signals have unique morphological characteristics (P-QRS-T complex) and it is highly significant than other biological signals. Physiologic signals generate

   Step 2: The profile {Y(i)} is divided into []

   boxes of the same size S.

   Step 3: In each box, the integrated time series is fitted by using a polynomial function, (), which is the local trend.

   Step 4: The local trend is subtracted and the detrended fluctuation function is given by ,

   = (2)

   Step 5: In each box of size S, the variance is

   determined,

   complex fluctuations in their output signals that reflect

   the underlying dynamics [5-9]. The main features of this physiologic time series such as ECG are non-

   2 =

   1

   { 1 + }2 (3)

   stationarity, non-linearity and non-equilibrium phenomena. Human cardiac dynamics are driven by complex non-linear interactions of two competing forces: sympathetic stimulation increases the heart rate,

   =1

   Step 6: The qth order fluctuation function () is calculated,

   = 1 (2 ) 2

   (4)

   =1

   4. Results And Discussion

   In order to study the multifractal behaviour of the electrocardiograms, the singular spectrum, the Hurst

   Step 7: The procedure is repeated for different box sizes ( different scales).

   exponent and the partition function of both normal & diseased ECGs are studied. The diseased ECG is collected from the physionet.

   normal ecg

3. Parameters to study MF-DFA

   DFA [3] is a well-established method for determining data scaling behaviour in the presence of possible trends without knowing their origin & shape. Repeating the procedure for several scales, as discussed above, () will increase with increasing S. If the time series is of long range correlation, then

   () (5)

   The h(q) is called the genearalized Hurst exponent

   1. Generalised Hurst Exponent

      This parameter determines whether the time series is monofractal or multifractal. For monofractal time series

      h(q) is constant. On the other hand, for multifractal time series, h(q) depends on the value of q, the

      fluctuation function. Therefore, the exponent h(q) is

      0.4

      0.3

      0.2

      amplitude

      amplitude

      0.1

      0

      -0.1

      -0.2

      -0.3

      -0.4

      0 500 1000 1500

      sample index

      Figure 1: Normal ECG of subject1

      The generalized Hurst exponent h(q) of the ECG time series

      called the generalized Hurst exponent [5].

   2. Partition Function

      The partition function (q) is regarded as a characteristic function of the fractal behaviour. The partition function is given by,

      () = () 1 (6)

      If (q) versus q is linear, the time series is monofractal. If (q) versus q is convex, the time series has a multifractal property [5].

   3. Singular Spectrum

The singular spectrum is also an important tool for fractal investigation in time series. It is denoted by

(). In fact, the curve of the spectrum is single- humped for a multifractal, while it reduces to a point for monofractal time series [6]. For a multifractal, the maximum of the spectrum denotes the dominant fractal exponent, and the width of the spectrum provides the range of the fractal exponents. The singular spectrum is calculated by Legendre Transform:

0.22

0.2

0.18

h(q)

h(q)

0.16

0.14

0.12

0.1

0 10 20 30 40 50 60 70 80 90 100

q

Figure 2: The generalized Hurst exponent of normal electrocardiograms of subject1

= ( q )

(7)

= q (8)

where () = () 1 is the partition function

-0.2

-0.4

-0.6

-0.8

tau(q)

tau(q)

-1

-1.2

-1.4

-1.6

-1.8

-2

The partition function of the ECG time series

3000

2800

2600

amplitude

amplitude

2400

2200

2000

1800

1600

1400

normal ecg

0 500 1000 1500

sample index

0 20 40 60 80 100 120

q

Figure 3: The partition function of the normal electrocardiograms of subject1

Figure 5: Normal electrocardiogram of subject 2

1

0.98

0.96

0.94

f(alpha)

f(alpha)

0.92

0.9

0.88

0.86

The singular spectrum of the ECG time series

0.6

0.55

0.5

0.45

0.4

h(q)

h(q)

0.35

0.3

0.25

The generalized Hurst exponent h(q) of the ECG time series

0.84

0.82

0.8

0 10 20 30 40 50 60 70 80 90 100

alpha

0.2

0.15

0.1

0 10 20 30 40 50 60 70 80 90 100

q

Figure 4: The singular spectrum of normal electrocardiogramof subject1

Figure 6: The generalized Hurst exponent of normal electrocardiograms of subject2

0

-0.5

-1

tau(q)

tau(q)

-1.5

-2

-2.5

-3

The partition function of the ECG time series

300

250

amplitude

amplitude

200

150

100

50

Diseased ecg

-3.5

0 20 40 60 80 100 120

q

0

0 500 1000 1500

sample index

Figure 7: The partition function of the normal electrocardiograms of subject2

Figure 9: Diseased electrocardiogram of BIDMC congestive failure (chf01)

1

0.9

0.8

f(alpha)

f(alpha)

0.7

0.6

The singular spectrum of the ECG time series

1.8

1.6

1.4

1.2

1

h(q)

h(q)

0.8

0.6

0.4

The generalized Hurst exponent h(q) of the diseased ECG time series

0.5

0.4

0 10 20 30 40 50 60 70 80 90 100

alpha

0.2

0

-0.2

0 10 20 30 40 50 60 70 80 90 100

q

Figure 8: The singular spectrum of normal electrocardiogram of subject2

Figure 10 : The generalized Hurst exponent of diseased electrocardiograms (chf01)

The partition function of the diseased ECG time series

5

4

3

2

tau(q)

tau(q)

1

0

-1

-2

-3

0 20 40 60 80 100 120

q

Figure 11: The partition function of diseased electrocardiograms (chf01)

The singular spectrum of the diseased ECG time series

2.5

Figure 14: The generalized Hurst exponent of diseased electrocardiograms (n01)

2

1.5

f(alpha)

f(alpha)

1

0.5

0

-0.5

The partition function of the diseased ECG time series

4

3

2

tau(q)

tau(q)

1

0

-1

0 10 20 30 40 50 60 70 80 90 100 -1

alpha

Figure 12: The singular spectrum of diseased electrocardiograms (chf01)

Diseased ecg

300

-2

0 20 40 60 80 100 120

q

Figure 15: The partition function of diseased electrocardiograms (n01)

250

amplitude

amplitude

200

150

100

50

2

1.5

1

f(alpha)

f(alpha)

0.5

The singular spectrum of the diseased ECG time series

0

0 500 1000 1500

sample index

Figure 13: Diseased electrocardiogram of atrial fibrillation (n01)

0

-0.5

The generalized Hurst exponent h(q) of the diseased ECG time series

1.2

-1

0 10 20 30 40 50 60 70 80 90 100

alpha

1

0.8

h(q)

h(q)

0.6

0.4

0.2

0

-0.2

0 10 20 30 40 50 60 70 80 90 100

q

Figure 16: The singular spectrum of diseased electrocardiograms (n01)

Figure 1 shows the normal ECG signal of subject1, Figure 2 shows the generalized Hurst exponent of the normal electrocardiograms of subject1, Figure 3 shows the partition function of the normal electrocardiograms of subject1 and Figure 4 shows the singular spectrum of the normal electrocardiograms of subject1. Similarly, Figure 5, Figure 6, Figure 7 and Figure 8 shows the respective results of normal electrocardiograms of subject2. The diseased electrocardiogram is taken from the physionet. It contains the data of BIDMC congestive heart failure

and atrial fibrillation. Figure 9 shows the diseased electrocardiogram of BIDMC congestive heart failure, Figure 10 shows the generalised Hurst exponent of the diseased electrocardiogram of atrial fibrillation, Figure

11 shows the partition function of the diseased electrocardiogram of atrial fibrillation and Figure 12 shows the singular spectrum of the diseased electrocardiogram of atrial fibrillation. Similarly, Figure 13 shows the diseased electrocardiograms of atrial fibrillation, Figure 14, Figure 15 and Figure 16 shows the respective results of the diseased electrocardiograms of atrial fibrillation.

It is obvious from the results that the relation between (q) and q of the normal and diseased both electrocardiograms is non-linear which shows that the ECG time series is multifractal in nature. Also, the singular spectrum of the normal and diseased electrocardiograms are more-or-less single-humped, which shows the multifractality of the ECG time series.

5. Conclusion

In this paper, a detailed analysis of the fractal properties of the ECG time series is studied, which is analysed by the Hurst exponent, partition function and the singular spectrum. The results from all these methods show that the ECG time series shows multifractal properties both in the case of normal and diseased electrocardiograms.

References

1. Richard F. Voss, Characterization and Measurement of Random Fractals, Physica Scripta, Vol. T13, 1986, pp. 27-32

2. Hong Zhang, Wenguo Li, Qiang Yu, Multifractality Of Financial Time Series, 2009 International Conference on Future BioMedical Information Engineering, IEEE.

3. Ping Zhou, Miao Yang, Ning Li, The Multifractal Characterization of R-R Intervals In Atrial Fibrillation, ICSP 2006 Proceedings, IEEE.

4. Ganng Xiong, Shuning Zhang, Qiang Liu, The Time- Singularity Multifractal Spectrum Distribution, Physica A 391, 2012, pp.4727-4739.

5. Rong-Guan Yeh, Jiann-Shing Shieh, Yin-Yi Han, Yu- Jungwang, Shih-Chun Tseng, Detrended Fluctuation Analyses Of Shortterm Heart Rate Variablity In Surgical Intensive Care Units, Vol. 18 ,No. 2, April 2006.

6. P. Abry, H. Helgason, P. Goncalves, E. Pereira, P. Gaucherand, M. Doret, Multifractal Analysis Of Ecg For Intrapartum Diagnosis Of Fetal Asphyxia, 2010,IEEE.

7. Jun Wang, Xinbao Ning, Ying Chen, Multifractal Analysis Of Electronic Cardiogram Taken From Healthy And Unhealthy Adult Subjects, Physica A 323, 2003, pp. 561- 568.

8. Xiaodong Yang, Xinbao Ning, Jun Wang, Multifractal Analysis Of Human Synchronous 12-Lead ECG Signals Using Multiple Scale Factors, Physica A 384, 2007, pp. 413- 422.

9. H.E. Stanley, L.A.N. Amaral, A.L. Goldberger, S. Havlin, P.Ch. Ivanov, C.-K. Peng, Statistical physics and physiology: Monofractal and multifractal approaches, Physica A 270, 1990, pp. 309-324.

10. M. Meyer, O. Stiedl, Self-Affine Fractal Variability Of Human Heartbeat Interval Dynamics In Health And Disease Eur J Appl Physiol 90, 2003, pp. 305-316.

11. Paolo Castiglioni, Luc Quintin, Andrei Civijian, Gianfranco Parati, and Marco Di Rienzo, Local-Scale Analysis of Cardiovascular Signals by Detrended Fluctuations Analysis: Effects of Posture and Exercise Proceedings of the 29th Annual International Conference of the IEEE EMBS, City Internationale, Lyon, France, August 23-26, 2007.

12. Danuta Makowiec, Rafal Galaska, Andrzej Rynkiewicz, Joanna Wdowczyk-Szulc, Multifractal estimators of short- time autonomic control of the heart rate, Proceedings of the International Multiconference on Computer Science and Information Technology, 2009, IEEE, pp. 405-411.

13. R Krishnam, H Nazeran, S Chatlapalli, E Haltiwanger, Y Pamula, Detrended Fluctuation Analysis: A Suitable Long- term Measure of HRV Signals in Children with Sleep Disordered Breathing, Proceedings of the 27th Annual IEEE Conference Shanghai, China, September 1-4, 2005.

14. R Magrans, P Gomis, P Caminal, G Wagner, Multifractal Properties of the Heart Rate Dynamics during Acute Myocardial Ischemia, 2009, pp. 417-420.

15. Kantelhardt JW, et al., Multifractal detrended fluctuation analysis of nonstationary time series, Physica A 316 , 2002, pp. 87-114.

16. Xiaodong Yang, Tongfeng Sun, Shanhan Ma, Yong Zhou, Multiscale Detection in ECG Multifractal Structure, 2010 IEEE.

17. Xiaodong Yang, Aijun He, Xinbao Ning, Multiscale Multifractality Analysis of Human Healthy and Unhealthy Heartbeat Time Series, International Conference on Complex Medical Engineering, 2007 IEEE