CHAPTER 10 PERMUTATION ENTROPY, MULTISCALE PERMUTATION ENTROPY AND WAVELET TRANSFORM ANALYSIS OF HEART RATE VARIABILITY

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1 157 CHAPTER 10 PERMUTATION ENTROPY, MULTISCALE PERMUTATION ENTROPY AND WAVELET TRANSFORM ANALYSIS OF HEART RATE VARIABILITY 10.1 INTRODUCTION Complexity parameters for time series are produced based on comparison of neighboring values. The definition applies to arbitrary real world data. For some well known chaotic dynamical systems, permutation entropy is similar as Lyapunov exponents, and is useful in the presence of dynamical or observational noise. Various measures of complexity were improved to compare time series and distinguish regular, (e.g. periodic), chaotic and random behavior (Steve Pincus and Burton H. Singer 1996) Among others, it has been indicated that complexity of heart (Supan Tungjitkusolmun 2004) can differentiate healthy and sick subjects and sometimes predict heart attack. The effects of noise are described. Nonlinear maps are presented. Lorenz attractor is given. Rossler attractor is presented. Runge-Kutta method is discussed. Permutation entropy is computed. Multiscale Permutation Entropy is computed. Denoising with noisy data is given.

2 158 defined. In introduction the permutation entropy of order n 2 can be The Lorenz attractor is a fractal structure corresponding to the longterm behaviour of the Lorenz oscillator. The Lorenz oscillator is a 3-dimensional dynamical system that exhibits chaotic flow. It is nonlinear and deterministic. The Rossler attractor is the attractor for the Rossler system, a system of three nonlinear ordinary differential equations. These differential equations define a continuous time dynamical system that exhibits chaotic dynamics associated with the fractal properties of the attractor. Multiscale permutation entropy is robust in presence of artifacts and divides pathological and healthy group. In conclusion Multiscale permutation entropy is implemented after artifact removed process which further increases the efficiency. Multiscale permutation entropy and the possibility of combining it with artifact removal increases its potentialities even more Wavelet Transform Analysis of HRV Signal The Discrete Wavelet Transform can be implemented with low-pass w(n) and high-pass v(n) filters and down-samplers to perform sub band coding. One utilizes db2 (Daubechies D4) as the mother wavelet. The a6 band is considered as the trend and, by set the parameters in a6 to zero and does the inverse DWT. One reconstructs the detrended HRV sequence.

3 METHODOLOGY Permutation Entropy of Heart Rate Variability Calculations Example: Let one take a series with seven values: x = (4; 7; 9; 10; 6; 11; 3) One organizes the six pairs of neighbors, according to their relative values, finding four pairs for which x t <x t+1 two pairs for which x t >x t+1. So four of six pairs of values are represented by the permutation 01 (x t <x t+1 ) and two of six are represented by 10. One defines the permutation entropy of order n = 2 as a measure of the probabilities of appearance of the permutations 01 and 10. So, (10.1) One goes another way, defining simple complexity measures which are easily calculated for any type of time series, be it regular, chaotic, noisy or from reality. A practical and a theoretical example were chosen to compare complexity with known concepts. Consider a time series One considers all n! permutations of order n which are considered as possible order types of n different numbers. For each ¼ in Sn one finds the relative frequency (10.2)

4 160 This estimates the frequency of ¼ as good as possible for a finite series of values. To determine p(¼) exactly, one has to assume an infinite time series {x1,x2,.}and take the limit for T> in the above formula. This limit exists with probability 1 when the underlying stochastic process fulfils a very weak stationarity condition: for n the probability for x(t) < x(t+k) should not depend on t. The permutation entropy of order n 2 is defined as (10.3) where the sum runs over all n! permutations ¼ of order n: (Christoph Bandt and Bernd Panpe 2002). This is the information contained in comparing n consecutive values of the time series. It is known that where the lower bound is obtained for an increasing or decreasing sequence of values, and the upper bound for a completely random system (i.i.d. sequence) where all n! possible permutations appear with the same probability. The time series presents some sort of dynamics when H (n) < log n!. Actually, in the experiments with chaotic and real world time series, H(n) did increase at most linearly with n. It seems useful to define the permutation entropy per symbol of order n; dividing by n since comparisons begin with the second value h n = H (n) / (n-1) (10.4) The effects of noise Permutation entropies have a practically important invariance property. If y(t) = f(x t ); where f is an arbitrary strictly increasing (or decreasing) real function, h n is the same for x(t) and y(t): Such non linear

5 161 functions f occur, for example, when measuring physiological data with different equipment. However, the invariance of h n also implies its discontinuity near the constant time series x(t) = c where h(n) = 0: If this series is disturbed by an i.i.d. noise, no matter how small, then (10.5) the largest possible value Observational noise admits m times k! mi For time series of period m and n = km the disturbed time series permutations so that (10.6) as long as the noise preserves the order of the periodic orbit. The observational noise causes only a small increase of entropy. For the higher noise level, some peaks appear when r is below a band merging point, for example 3:5748; 3:5925; 3:6785. It is clear that the noise forecasts the band merging point but it seems surprising that the peak is greater than the entropy in the chaotic regime behind band combining Dynamical noise When this is added to x(t) during each step of the iteration, this produces better results. The entropy function hn(r) becomes smooth, approximating the entropy of the undisturbed time series for small noise level. There are examples of noise induced order where larger noise gives smaller hn but hn prolongs with noise level. The effects of small periods disappear

6 162 for larger noise level where the increase of entropy with growing r becomes clear Advantages Permutation entropies h n are appropriate complexity measures for chaotic time series, in the presence of dynamical and observational noise.the effect of amplitude of ECG data on the permutation entropy is very weak, and permutation entropy is less sensitive to noise. In contrast with all recognized complexity parameters, a small noise does not change the complexity of a chaotic signal. Permutation entropies can be calculated for arbitrary real world time series. Since the method is extremely fast and robust, it seems preferable when there are big data sets and no time for preprocessing and fine tuning of parameters Non Linear Maps Strange attractor An attractor is considered as strange if it has non-integer dimension or if the dynamics on it are chaotic. The term was floated by David Ruelle and Floris Takens to describe the attractor that resulted from a series of bifurcations of a system describing fluid flow. Strange attractors are differentiable in a few directions, but some are like a Cantor dust, and not differentiable Lorenz attractor The Lorenz attractor, named for Edward N. Lorenz, is a fractal structure corresponding to the long-term behavior of the Lorenz oscillator. The Lorenz oscillator is a 3-dimensional dynamical system that exhibits

7 163 chaotic flow, noted for its lemniscate shape. The map demonstrates how the state of a dynamical system (the three variables of a three-dimensional system) evolves over time in a complex, non-repeating pattern. The Lorenz oscillator is nonlinear and deterministic. In 2001 it was proven by Warwick Tucker that for a certain set of parameters the system exhibits chaotic behavior and displays what is today called a strange attractor. The strange attractor in this case is a fractal of Hausdorff dimension between 2 and 3. Grassberger P. (1983) has estimated the Hausdorff dimension to be 2.06 ± 0.01 and the correlation dimension to be 2.05 ± Differential equations (10.7) (10.8) (10.9) where is called the Prandtl number and is called the Rayleigh number. All,, > 0, but usually = 10, = 8/3 and is varied. The system exhibits chaotic behavior for = 28 but displays knotted periodic orbits for other values of. For example, with = it becomes a T(3,2) torus knot Rossler attractor The Rossler attractor is the attractor for the Rossler system, a system of three non-linear ordinary differential equations. These differential

$the fractal properties of the attractor.$ Some properties of the Rossler system can be attained via linear methods such as eigen vectors, but the

The defining equations are: (10.10) (10.11) (10.12) Rossler studied the chaotic attractor with a = 0.

8 164 equations define a continuous-time dynamical system that exhibits chaotic dynamics associated with the fractal properties of the attractor. Some properties of the Rossler system can be attained via linear methods such as eigen vectors, but the main features of the system need non-linear methods such as Poincare maps and bifurcation diagrams. The defining equations are: (10.10) (10.11) (10.12) Rossler studied the chaotic attractor with a = 0.2, b = 0.2, and c = 5.7, though properties of a = 0.1, b = 0.1, and c = 14 have been more commonly used since. A Rossler attractor is shown in Figure Figure 10.1 Rossler attractor

9 Runge-kutta method In numerical analysis, the Runge Kutta methods are a significant family of implicit and explicit iterative methods for the approximation of solutions of ordinary differential equations. These techniques came into existence around 1900 by the German mathematicians Runge and Kutta The common fourth-order Runge Kutta method Let an initial value problem be specified as follows. (10.13) equations: Then, the RK4 method for this problem is given by the following (10.14) where y n + 1 is the RK4 approximation of y(t n + 1 ), and (10.15) Thus, the next value (y n + 1 ) is determined by the present value (y n ) plus the product of the size of the interval (h) and an estimated slope. The slope is a weighted average of slopes:

10 166 k 1 is the slope at the beginning of the interval; k 2 is the slope at the midpoint of the interval, using slope k 1 to determine the value of y at the point t n + h / 2 using Euler's method; k 3 is again the slope at the midpoint, but now using the slope k 2 to determine the y-value; k 4 is the slope at the end of the interval, with its y-value determined using k 3. the midpoint: In averaging the four slopes, greater weight is given to the slopes at The RK4 method is a fourth-order method, meaning that the error per step is on the order of h 5, while the total accumulated error has order h 4. Note that the above formulae are valid for both scalar- and vectorvalued functions (i.e., y can be a vector and f an operator). For example one can integrate Schrödinger's equation using the Hamiltonian operator as function f.

11 Implementation Flow-chart for the Runge-Kutta method Start Develop slopes k1s for all variables at initial point Make prediction of dependent variable at mid points Mid point values are used to compute set of slopes at mid point(k2) With these new slopes another set of mid point prediction is done which leads to new slope prediction(k3) These are then employed to make prediction at the end of the interval that are used to develop slopes at the end of the interval (k4) K s are combined into a set of increment functions Brought back to the beginning to make final prediction Stop Figure 10.2 Flow chart for Runge Kutta method A flow chart for Runge-Kutta method is shown in Figure 10.2.

168 Figure 10.3 Flow chart for permutation entropy A flow chart for Permutation Entropy is shown in Figure 10.3. 10.3 MULTISCALE PERMUTATION ENTROPY One has developed a novel method to calculate Multiscale Permutation Entropy from complex signals.

The MPE method demonstrates that correlated random signals colored noised are more complex than uncorrelated random signals white noised.

12 168 Figure 10.3 Flow chart for permutation entropy A flow chart for Permutation Entropy is shown in Figure MULTISCALE PERMUTATION ENTROPY One has developed a novel method to calculate Multiscale Permutation Entropy from complex signals. Multiscale permutation entropy (MPE) is robust in presence of artifacts and divides pathological and healthy groups. The MPE method demonstrates that correlated random signals colored noised are more complex than uncorrelated random signals white noised. Compared to traditional complexity measures, MPE has the advantage of being applicable to both physiologic and physical signals of finite length Calculation of MPE The Multiscale Permutation Entropy (Wajid Aziz and Muhammad Arif 2005) projects to segregate short-range from long-range correlations in the input signal. Entropy values are computed for different scales of the signal. The rationale is that pathological states do not alter the variability of the heart rate for all ranges of correlations.

13 RESULTS in Table Permutation entropy for Congestive Heart Failure patients is shown Table 10.1 Congestive heart failure Age Sex Monitori- ng duration Permutation entropy 55 M 12 hours F 12 hours F 12 hours M 12 hours F 12 hours Ave. : SD. : Permutation entropy for Normal Sinus Rhythm subjects is indicated in Table Table 10.2 Normal sinus rhythm Age Sex Monitori-ng duration Permutation entropy 64 F 12 hours M 12 hours F 12 hours M 12 hours Ave. : SD :

14 170 Table Permutation entropy for Arrhythmia patients is shown in Table 10.3 Arrhythmia Age Sex Monitoring duration Permutation entropy 64 F 12 hours M 12 hours F 12 hours M 12 hours M 12 hours Ave. : SD : in Table Permutation entropy for Sudden Cardiac Death patients is indicated Table 10.4 Sudden cardiac death Age Sex Monitoring duration Permutation entropy 43 M 12 hours F 12 hours M 12 hours F 12 hours Ave. : SD :

15 171 Permutation entropy (PE) for NSR subjects is higher compared with CHF patients (P= ) PE for CHF patients is lower compared with NSR subjects (P=0.946). PE for Arrhythmia patients is lower compared with NSR subjects (P=0.991). PE for SCD patients is lowercomparedwith NSR subjects (P=0.997). Permutation entropies for nonlinear maps i.e., Lorenz map, Kaplan-Yorke map, Rossler attractor and Henon map are indicated in Table Table 10.5 Nonlinear maps Non linear maps Lorenz Permutation Entropies X Y Z No of Values Kaplan-Yorke Rossler X Y Z Henon

16 Attractors of Nonlinear Maps Figure 10.4 Lorenz attractor X Vs T For Lorenz Attractor X Vs T, time is shown in Figure Points in space with time is shown in Figure It is like spiral. Figure 10.5 Points in space with time

17 173 Kaplan - Yorke Attractor Y Vs X for Kaplan-Yorke Attractor is indicated in Figure It is like inverted parabola. Figure 10.6 Kaplan -york Attractor Y Vs X Rossler attractor T in seconds Figure 10.7 Rossler attractor X Vs T

18 174 distorted sine wave. X Vs T, time is shown in Figure 10.7 for Rossler Attractor. It is like Points in space with time are shown in Figure It is a 3- dimensional (XYZ) curve. Figure 10.8 Points in space with time The main advantage of Sample Entropy SpEn and Pemuatation Entropy (PE) is that they are independent of datalength over certain scales.the analysis is done for two subjects in healthy condition (Normal Sinus Ryhthm) using PE.This is shown in the Figure 10.9.

19 175 E n t r o p y Scale Figure 10.9 Entropy vs Scale for subjects 2238 (Nsr 002) and 2298 (Nsr 050) Figure Entropy vs scale for series 1 and series 2 Figure Entropy Vs Scale for series 1 and series 2 is plotted in

20 176 Outliers and artifacts affect the entropy values at lower scales because they change time series standard deviation and therefore, the value of similarity criterion. The MPE analysis of CHF condition of two different patients of different ages using HRV denoised using wavelet analysis is shownin Figure Scale Figure Entropy vs scale for 3234 (chf 201) denoise and 3255 (chf 221) denoise Entropy Vs Scale for 3234 (chf 201) denoise and 3255 (chf 221) denoise is plotted in Figure Normal Sinus Rythm The MPE analysis of four different healthy subjects of different ages using HRV contaminated with certain trends(artifacts) and denoised is shown in Figure

21 177 Scale Figure Entropy vs scale for Normal sinus rhythm subjects becomes constant. One can observe that MPE initially increases with scale and later Arrhythmia The MPE analysis of health condition of two different patients of different ages suffering with arrythmia is shown in Figure Scale Figure Entropy vs scale for Arrhythmia patients

22 178 Above figure shows HRV analysis using both noisy and denoisy data. At lower scales, the entropy was very high for noisy data which would lead to a misconception of health condition. MPE analysis of all types of heart conditions The MSE analysis of Different conditions (Congestive Heart Failure, Arrhythmia, Healthy) is shown in the Figure This result shows that finally that complexity of signal for the healthy one is higher at higher scales but lower at smaller scales. The complexity of diseased patients heart signal decreases gradually with scale with a local shift at scale 7 due to respiratory modulation of heart rate (Respiratory sinus arrhythmia) which has higher amplitude in healthy subjects than in CHF subjects. E n t o p y Figure Entropy for Arrhythmia patients, NSR subjects and CHF patients Table Multiscale Permutation Entropy is shown for CHF patients in

23 179 Table 10.6 Multiscale permutation entropy of CHF patients Patient Multiscale Permutation Entropy Chf Chf Chf Chf Chf Chf Chf Chf Chf Chf Chf Chf Chf Chf Ave. : Table Multiscale Permutation Entropy is indicated for Arrhythmia patients in Table 10.7 Multiscale permutation entropy of Arrhythmia patients Patient Multiscale Permutation Entropy Ave. :

24 180 Multiscale Permutation entropy is indicated in Table 10.8 for Normal Sinus Rhythm subjects. Table 10.8 Multiscale permutation entropy of Normal Sinus Rhythm subjects Subject Multiscale Permutation Entropy Nsr Nsr Nsr Nsr Nsr Nsr Nsr Nsr Nsr Nsr Nsr Nsr Nsr Nsr Nsr Ave. : The values near to average value are correct. The values away from average value are correct with error.

25 181 Wavelet function coefficients of HRV using Daubechies wavelet(d4) Figure Wavelet function coefficients of HRV using Daubechies wavelet (D4) Wavelet function coefficients of HRV using Daubechies wavelet (D4) are shown in Figure The figures are plotted for CHF.

26 182 De noised signal(chf) De noised Black Original Red Figure De noised signal (CHF) Denoised and original signals are shown in Figure For congestive heart failure patients, one male aged 55 has permutation entropy of It is valid. One male aged 62 has permutation entropy of It is valid. Table 10.1 permutation entropy of CHF patients is shown. For NSR subjects, are female aged 64 has permutation entropy of It is valid. A male aged 59 has permutation entropy of It is valid. In Table permutation entropy of NSR subjects is indicated. For Arrhythmia patients, a male aged 67 has permutation entropy of It is incorrect. A male aged 59 has permutation entropy of It is correct. In Table 10.3 permutation entropy of arrhythmia patients is shown. For sudden cardiac death, a male aged 43 has permutation entropy of It is correct. A female aged 72 has permutation entropy of

27 which is not correct. In Table 10.4 permutation entropy of SCD patients is indicated. For CHF patients, CHF 219 has MSPE of which is correct. In Table 10.6, MSPE of CHF patients is shown. For Arrhythmic data base, 115 has MSPE of which is correct. In Table 10.7, MSPE of arrhythmia patients is indicated. For NSR, NSR 019 has MSPE 0f which is correct. In Table 10.8, MSPE of NSR subjects is shown. Multiscale permutation entropy is implemented after artifact removed process which further increases the efficiency of multiscale permutation entropy and of combining it with Artifact Removal increases its possibilities even more CONCLUSIONS It is taken care of the objective of examining the usefulness of entropy as a HRV parameter. Entropy of HRV has been computed with data of different patients. The Permutation Entropy has been computed employing various probabilities taken from the MIT-BIH Arrhythmia Database. One of the key observations is that a low value of entropy shows the risk of sudden death. The permutation entropy values of the standard nonlinear series can be directed to compare the real time data of a patient and predict the condition of the patient. Lower values in the case of Congestive heart failure and sudden cardiac death report that the patient is not in a normal health condition and provides medical supervision.

28 184 It is aimed at examining Multiscale Permutation Entropy as method to evaluate HRV. Performance analysis of multiscale permutation entropy was done over a large set of data and its efficiency in evaluating HRV over traditional schemes are observed. Multiscale Permutation Entropy is found to be suitable when compared to Sample Entropy and other techniques with which it is compared with. How the HRV varied for patients of different age groups, and different conditions of the heart are studied. The results are highly appreciated when artifact removal process is included. Multiscale permutation entropy is implemented after artifact removed process which further increases the efficiency. Multiscale permutation entropy and the possibility of combining it with artifact removal increases its potentialities even more. Permutation entropies are computed for model inputs like Lorenz attractor, Kaplan-Yorke attractor, Rossler attractor and Henon map. They resemble those of CHF patients, sudden cardiac death patients and NSR subjects.

Permutation entropy a natural complexity measure for time series

Permutation entropy a natural complexity measure for time series Christoph Bandt and Bernd Pompe Institute of Mathematics and Institute of Physics, University of Greifswald, Germany (February 18, 2002)