International Journal of Signal Processing Systems Vol. 4, No. 2, April 2016 The Effect of Chi Meditation on the Multifractal Nature of Heart Rate Variability Ram Gopal Reddy Lekkala National Institute of Technology, Warangal, India Email: [email protected] Srinivas Kuntamalla Kakatiya Institute of Technology and Science, Warangal, India Email: [email protected] Abstract—Heart rate variability (HRV) analysis is fast scale invariant variability is also known as fractal and the emerging as a noninvasive research and clinical tool for methodology employed to evaluate it is often called assessing cardiac and autonomic nervous system function. fractal analysis. In geometrical terms a fractal object is a The variations in the heart rate are the consequences of self-similar structure, which means that looking closely at multiple and complex mechanisms, making the heart rate smaller regions reveals a scaled version of the whole dynamics nonlinear in nature. So, nonlinear analysis of HRV provides more appropriate information for object. The word ‘fractal’ was first used by Mandelbrot understanding and interpretation of the physiological [4]. In case of a time series, the similarity is not structural problems associated with cardiovascular system. In this but statistical. The fractal time series look same when paper, multifractal detrended fluctuation analysis of HRV viewed in different time scales (scale invariant). signals pertaining to pre-meditation and during meditation Analysis of fractal scaling exponents by detrended conditions is worked out. We observed a right shift in the fluctuation analysis (DFA) is one such method which peaks of multifractal spectra for the subjects during describes the fractal correlation properties of biological meditation, which represents an increase in multifractal time series. Breakdown of short-term fractal organization nature of HRV. This result clearly shows that there will be an improvement of health in the cardiovascular system in human Heart rate (HR) dynamics, has been observed in functioning during meditation. various disease states, such as in heart failure [5], [6] and during atrial fibrillation [7]. Fractal analysis methods Index Terms—heart rate variability, multifractal detrended differ from the traditional measures of HRV because they fluctuation, meditation, nonlinear analysis, multifractal measure the qualitative characteristics and correlation spectra features of HR behavior instead of the magnitude of variability. Briefly, a scaling exponent obtained by the DFA method quantifies the relations of HR fluctuation at I. INTRODUCTION different scales. Low-exponent values correspond to Biological signals are both nonlinear and non- dynamics where the magnitude of beat-to-beat HR stationary, i.e., their statistical character changes slowly variability is close to the magnitude of long-term or intermittently as a result of variations in background variability. Conversely, high-exponent values correspond influences [1]. Furthermore, very often there exist smaller to dynamics where the magnitude of long-term variability amplitude fluctuations at shorter time scales. The idea of is substantially higher than the beat-to-beat variability [8]. having hidden information in physiological time series Fractals can be classified into two categories: created a growing interest in applying the concepts and monofractals and multifractals. Monofractals are those, techniques from statistical physics, for a wide range of whose scaling properties are the same in different regions biomedical problems [2], [3]. of the systems and multifractals are complicated self- Heart rate variability (HRV) analysis is fast emerging similar objects consisting of differently weighted fractals as a noninvasive research and clinical tool for assessing with different non-integer dimensions. As a result, cardiac and autonomic nervous system function, and it multifractal system is a generalization of a fractal system refers to the beat-to-beat alterations in the heart rate. in which a single scaling exponent is not enough to Various investigations confirmed that long-term heart describe its dynamics; instead a continuous spectrum of rate variability (HRV) fluctuations are not random, but exponents (the so called singularity spectrum) is needed exhibit long-term correlations that do not exhibit any [2], [9], [10]. characteristic scale, but are rather “scale invariant”. Scale The width and shape of the multifractal spectrum can invariance of a time series means that there is no specific also differentiate between the heart rate variability from scale of time can be identified in the data. This type of patients with heart diseases like ventricular tachycardia, ventricular fibrillation and congestive heart failure [11], Manuscript received December 15, 2014; revised March 20, 2015. [12]. The multifractality has also been reported in heart ©2016 Int. J. Sig. Process. Syst. 128 doi: 10.12720/ijsps.4.2.128-132 International Journal of Signal Processing Systems Vol. 4, No. 2, April 2016 rate fluctuations of healthy individuals [13]. The for each segment v, v = 1, …, Ns and multifractal analysis has become a useful tool to detect 푠 1 long-range correlation in heartbeat fluctuations for the 퐹2(푣, 푠) = ∑{푌[(푁 − (푣 − 푁 )푠 + 푖] − 푦 (푖)}2 (4) 푠 푠 푣 diagnosis of heart failure [14]. The multifractal structure =1 of heart rate variability is therefore suggested to reflect for v = N + 1, …, 2N . important properties of the autonomic regulation of the s s where yv(i) is the least square fitted polynomial in heart rate. segment v. 푚 II. METHODOLOGY AND DATA 푚−푘 푦푣(푖) = ∑ 퐶(푘)푖 (5) A. Multifractal Detrended Fluctuation Analysis 푘=0 The simplest type of multifractal analysis is based Here C(k) are the set of coefficients. Linear, quadratic, upon the standard partition function multifractal cubic, or higher order polynomials can be used in the formalism, which has been developed for the multifractal fitting procedure (conventionally called DFA1, DFA2, characterization of normalized, stationary measures [15]– DFA3, …). [17]. Unfortunately, this standard formalism does not Step 4: Averaging all the segments to obtain the qth give correct results for nonstationary time series that are order fluctuation function affected by trends or that cannot be normalized. Thus, in 1/푞 2푁푠 the early 1990s an improved multifractal formalism has 1 2 푞/2 been developed, the wavelet transform modulus maxima 퐹푞(푠) = { ∑[퐹 (푣, 푠)] } (6) 2푁푠 (WTMM) method [18], which is based on wavelet 푣=1 analysis and involves tracing the maxima lines in the where, the index variable q can take any real value except continuous wavelet transform over all scales. An zero (for q = 0, see step 5). For q = 2, the standard DFA alternative approach based on a generalization of the procedure is retrieved. As we are interested in how Fq(s) DFA method is proposed by Kantelhardt et al [19]. This (generalized q dependent fluctuation functions) depend multifractal DFA does not require the modulus maxima on the time scale s for different values of q. Hence, we procedure, and hence does not involve more effort in must repeat steps 2 to 4 for several time scales s. It is programming than the conventional DFA. apparent that Fq(s) will increase with increasing s. Of The generalized multifractal DFA (MFDFA) procedure course, Fq(s) depends on the DFA order m. By consists of five steps. The first three steps are essentially construction, Fq(s) is only defined for s ≥ m + 2. identical to the conventional DFA procedure [20], [21]. Step 5: The scaling behavior of the fluctuation The procedure of MFDFA as described by Kanthelhardt, functions by analyzing log–log plots Fq(s) versus s for et al. [19] is as follows. each value of q is determined as shown in Figure 1. If the Let us suppose that x(i) is a non-stationary time series series x(i) are long-range power-law correlated, Fq(s) of length N. The mean of the series x(i) is given by increases, for large values of s, as a power-law, 푁 ℎ(푞) 1 퐹푞(푠) = 푠 (7) 〈푥〉 = ∑ 푥(푖) (1) 푁 If such a scaling exists ln Fq(s) will depend linearly on =1 ln s, with h(q) as the slope. For very large scales, Step 1: Compute the integrated time series (profile) 푠 > 푁/4, Fq(s) becomes statistically unreliable because the number of segments Ns for the averaging procedure in 푌(푖) = ∑[푥(푘) − 〈푥〉] 푖 = 1, … , 푁. (2) step 4 becomes very small. Thus, scales 푠 > 푁/4 are 푘=1 excluded from the fitting procedure to determine h(q), usually. In general, the exponent h(q) in Eq. (7) depends Subtraction of the mean 〈푥〉 is not compulsory, since it would be eliminated by the later detrending in the third on q. For stationary time series, h(2) is identical to the step well-known Hurst exponent H [15]. Thus, the function h(q) can be called as generalized Hurst exponent. The Step 2: The profile Y(i) is divided into Ns = int(N/s) non-overlapping segments of equal length s. plot of q versus h(q) is shown in Fig. 1(D). Since the length N of the series is often not a multiple The value of h(0), which corresponds to the limit h(q) of the considered time scale s, a short part at the end of for q→0, cannot be determined directly using the the profile may remain. In order not to disregard this part averaging procedure in Eq. (6) because of the diverging of the series, the same procedure is repeated starting from exponent. Instead, a logarithmic averaging procedure has to be employed, the opposite end. Thereby, 2Ns segments are obtained altogether. 2푁푠 1 2 ℎ(0) Step 3: The local trend for each of the 2Ns segments is 퐹0(푠) = exp { ∑ ln [퐹 (푣, 푠)]} ~ 푠 (8) 4푁푠 calculated by a least –square fit of the series. Then 푣=1 determine the variance Note that h(0) cannot be defined for time series with 푠 1 fractal support, where h(q) diverges for q→0.