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Multifractal analysis of heartbeat time series in human races
Appl. Comput. Harmon. Anal. 18 (2005) 329–335 www.elsevier.com/locate/acha
Letter to the Editor
Multifractal analysis of heartbeat time series in human races Eva Wesfreid a,c,∗ , Véronique L. Billat b , Yves Meyer a a CMLA, École Normale Supérieure de Cachan, 94235 Cachan, France b Département STAPS, Université d’Evry Val d’Essonne, France c LMPA, Université du Littoral, Côte d’Opale, 62228 Calais, France
Available online 19 February 2005 Communicated by Charles K. Chui on 18 March 2004
Abstract We present a multifractal analysis of heartbeat and heart rate time series in human races. In order to improve the training of athletes, we compare heart rate multifractal behavior in free and constant speed 10,000 m runnings. We analyze also marathon races, free pace 42.195 km running, we compare the first and the second half heartbeat signals to measure the effect of fatigue. We find that freedom for choosing the own pace variation could be the racing condition for keeping good health conditions in an exhausting exercise. 2005 Elsevier Inc. All rights reserved. Keywords: Multifractal analysis; Wavelets; Heartbeat signals
1. Introduction Best performances in middle-distance running (800–10,000 m) are actually characterized by speed variability. If one considers the last three world records for middle- and long-distance running, it can be observed that the velocity varies of 5% [4,5]. In races, the choice of speed and the variations of speed that will maximize an endurance athlete’s ability to succeed in winning a race involve a complex interplay of physiological and psychological factors [6]. Therefore, the exploration of physiological limits for exercise must also be approached in exercise model where the subject is free for regulating his pace [4,14].
* Corresponding author.
E-mail address: [email protected] (E. Wesfreid). 1063-5203/$ – see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.acha.2004.12.005
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E. Wesfreid et al. / Appl. Comput. Harmon. Anal. 18 (2005) 329–335
Fig. 1. ECG signal showing some R peaks.
Fig. 2. Athlete’s RR interval (ms) as function of time.
Most of the studies have described heart rate (HR) dynamics with classical linear models [15], while some have analyzed HR variability during exercise with Fourier transform [6]. Indeed, beyond the analysis of large trend, the main interest of such nonlinear dynamics analysis, in particular the multifractal method, is to appreciate the feature of healthy function to respond to unpredictable stimuli and stresses [16]. However, such analysis is complicated by the fact that biological signals are both irregular and nonstationary, that is, their statistical character changes slowly or intermittently as a result of variations in background influences [3]. Furthermore, very often there are ever smaller amplitude fluctuations at ever shorter time scales. Multifractal analysis is used for classifying signals which exhibit a rough behavior. This behavior is quantified by calculating pointwise Hölder exponents h(t) using multifractal analysis. Indeed, the human heart generates the heartbeat. A recording of the cardiac-induced skin potentials at the body’s surface, an electrocardiogram (ECG, Fig. 1), reveals information about atrial and ventricular electrical activity. Abnormalities in the temporal durations of the segments between deflections, or of the intervals betweens waves in the ECG, as well as their relative heights, serve to expose and distinguish cardiac dysfunction. Readily recognizable features of the ECG wave pattern are designated by the letters P-QRS-T; the wave itself is often referred to as the QRS complex. Aside from the significance of various features of the QRS complex, the timing of the sequence of the QRS complexes over thousands of heartbeats is significant. These intercomplex times are readily measured by recording the occurrence of the peaks of the large R waves which are the most distinctive feature of the normal ECG. Therefore two successive R peaks is the so-called “RR interval” (the heartbeat period in ms) while its inverse (the frequency) is the heart rate (in Hertz). The time interval between two R peaks are submitted to important fluctuations, even at rest. Figure 2 shows an athlete heartbeat in a marathon race.
E. Wesfreid et al. / Appl. Comput. Harmon. Anal. 18 (2005) 329–335
331
It is now well documented that the fluctuations of the sequence of interbeat intervals can be used to assess the presence or likelihood of cardiovascular disease [11]. In healthy subjects, Ivanov et al. [8] demonstrated that at rest, when suitably rescaled using the wavelet-based time series analysis, the distributions of the variations in the beat-to-beat intervals for all healthy subjects were described by a single function stable over a wide range of time scales. This mean that there are ever smaller amplitude fluctuations at ever shorter time scales [3]. We hypothesized that during an exhaustive exercise, the modification of the heart rate variability could be a tool for detecting the presence of cardiac dysfunction during the marathon race. Sudden deaths due to cardiac pathologies are currently reported on marathon even if it has not been yet a matter of epidemiological serious study. We used the wavelet transform modulus maxima (WTMM) algorithm proposed by A. Arneodo, E. Bacry, and J.-F. Muzy [2], which estimates the spectrum of singularities of the Hölder exponents in a “multifractal formalism.” We show that the fatigue does not change significantly the scaling behavior readable in the estimated spectrum of singularities. This suggests that the fatigue appearing during the marathon races does not induce anomalies of the cardiac fluctuations.
2. WTMM algorithm The WTMM algorithm [2,13] estimates the spectrum of singularities of the Hölder exponents in a “multifractal formalism” [9,10]. The Hölder exponent [1,9,10,13] of a function f at a given value t0 of the time variable is (1) h(t0 ) = sup h: f (t) − Pm (t − t0 ) C|t − t0 |h , where Pm is a polynomial of degree not exceeding m. In other words h(t0 ) = lim inf t→t0
log |f (t) − Pm (t − t0 )| . log |t − t0 |
(2)
The spectrum of singularities of f is the Hausdorff dimension, D(α), of the sets Eα of points t0 where the Hölder exponent h(t0 ) takes a given value α. In a “multifractal formalism,” we suppose that the sets Eα of our physiological signal are fractals. The signal f is monofractal when the support of the singularity spectrum, D(α), is a point. It is well known that the fractional Brownian and the uniform Cantor measure are monofractal signals. The nonuniform Cantor measure is multifractal (see Fig. 3). The WTMM algorithm can be split into three steps: (a) Compute the partition function: Z(q, s) = p |Wf (up (s), s)|q , where the sum runs over the curves s → up (s). Here up (s) = arg max |Wf (u, s)| and these maxima are local maxima. (b) Estimate the slope of the line, log Z(q, s) vs log s. The scaling exponent, τ (q), measures the asymptotic decay of Z(q, s) at fine scales s: τ (q) = lim inf s→0
log Z(q, s) , log s
(3)
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E. Wesfreid et al. / Appl. Comput. Harmon. Anal. 18 (2005) 329–335
Fig. 3. Uniform and nonuniform Cantor measures.
which means that Z(q, s) Cε s τ (q)−ε for s → 0, ε > 0, s > 0 and that τ (q) is the largest exponent for which this holds. (c) Compute the singularity spectrum D(α) using the inverse of the Legendre transform 1 − τ (q) . (4) D(α) = min q α + q∈R 2 Figure 3 shows an approximation of the uniform and nonuniform Cantor measure (with 7 iterations and 8000 samples), at left, the scaling exponent τ (q) at middle and the singularity spectrum, D(α), at right.
3. Race data analysis We analyzed heart rate (HR) series, measured continuously beat-by-beat (by a cardio frequencemeter Polar 810S, Finland), during free and constant velocity, 10,000 m running and also in marathon RR interval signals, free pace 42.195 km races. In the 10,000 m runnings, each athlete performed two series, several times. In the first series, the subject was asked to run as fast as possible ( free speed). In the second series, the same distance was covered at a constant velocity, equal to the average velocity of the previous free-pace run (constant speed). In order to improve the training of athletes and to know whether it is preferable to ask the athletes to run at a constant or free pace, HR signals in free and constant speed runnings are compared. In the marathon races, we compare first and second half RR interval signals to measure the effect of fatigue on athlete’s performance.
E. Wesfreid et al. / Appl. Comput. Harmon. Anal. 18 (2005) 329–335
333
Fig. 4. Free and constant heart rate scaling law behavior.
Fig. 5. Heart rate (left: singularity spectrum, right: scaling exponent).
3.1. Free vs constant 10,000 m races Figure 4 shows log2 Z(q, s) vs log2 s, where Z(q, s) is the partition function defined previously. This representation is less satisfactory for negatives values of q, such behavior can be a consequence of signal length (too small). We observed that the free pace race: • induced a lower average heart rate than the constant one (165 ± 14 vs 169 ± 13 bpm), bpm meaning beats per minute; • consumed less running energy than the constant one, running energy measures the ratio of oxygen uptake over, speed times athletes weight (4.263 ± 0.168 vs 4.83 ± 0.237 kJ kg−1 km−1 ); • in most cases, has smaller heartbeat D(α) support and less convex scaling exponent function q → τ (q). Heart rate in free speed seems to be less multifractal. Figure 5 shows this behavior for one race.
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E. Wesfreid et al. / Appl. Comput. Harmon. Anal. 18 (2005) 329–335
Fig. 6. Heart rate (left: singularity spectrum, right: scaling exponent).
3.2. First and second half parts of marathon races We observed that: • the average heart rates on the first and second half marathon were not significantly different (144±22 vs 144 ± 14 bpm); • in most races the singularity spectra D(α) and the scaling exponents τ (q) of the first and second half RR interval are almost the same. Figure 6 shows the singularity spectrum, D(h), at left and the scaling exponent τ (h) at right. ◦ is for the first half and + is for the second half. The Hölder exponent hmax with maximal Hausdorff dimension (D(h)) for the two halves are not significantly different. This behavior was observed for most of the marathon races. RR interval multifractal characteristics were not significantly different between the first and second part of the race. This suggests that the speed variation allows a good adaptation to the fatigue. Indeed, the RR interval kept a multifractal behavior, which is believed to be an indicator of good health [7]. Figures 3–6 were obtained with the Last Wave software [12].
4. Conclusion We proposed here a multifractal analysis of heart rate time series in free and constant speed 10,000 m runnings and in marathon races, free pace. The fatigue appearing during the marathon does not induce anomalies of the cardiac fluctuation readable on the multifractal behavior.
E. Wesfreid et al. / Appl. Comput. Harmon. Anal. 18 (2005) 329–335
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This suggests that freedom for choosing the own pace variation could be the racing condition for keeping good health conditions in an exhausting exercise. However, the results we obtained can be questioned. Indeed • there are other tools for processing this kind of signals and these could be compared with multifractal analysis; • the shape of multifractal spectrum does not provide a precise information; and • the relation between variability and fatigue needs to be further explored.
References [1] P. Abry, P. Gonçalvès, J. Lévy Véhel, Lois d’échelle, fractales et ondelettes, Hermès Science, ISBN 2-7462-0410-X, 2002. [2] A. Arneodo, F. Argoul, E. Bacry, J. Elezgaray, J.F. Muzzy, Ondelettes multifractales et turbulence, Diderot Éditeur, Paris, 1995. [3] J.B. Bassingthwaighte, L.S. Liebovitch, B.J. West, Fractal Physiology, American Physiology Society, Oxford Univ. Press, 1994, p. 364. [4] V.L. Billat, J. Slawinski, M. Danel, J.P. Koralsztein, Effect of free versus constant pace on performance and oxygen kinetics in running, Med. Sci. Sports Exerc. 33 (2001) 2082–2088. [5] F. Cottin, Y. Papelier, F. Durbin, J.P. Koralsztein, V.L. Billat, Effect of fatigue on spontaneous velocity variations in human middle-distance running: use of short-term Fourier transformation, Eur. J. Appl. Physiol. 87 (2002) 17–27. [6] Y. Fukuba, B.J. Whipp, A metabolic limit on the ability to make up for lost time in endurance events, J. Appl. Physiol. 87 (1999) 853–861. [7] S. Havlin, L.A.N. Amaral, Y. Ashkenazy, A.L. Goldberger, P.Ch. Ivanov, C.K. Peng, H.E. Stanley, Application of statistical physics to heartbeat diagnosis, Physica A 274 (1999) 99–110. [8] P.Ch. Ivanov, M.G. Rosenblum, C.-K. Reng, J. Mietus, S. Havlin, H.E. Stanley, A.L. Goldberger, Scaling behavior of heartbeat intervals obtained by wavelet-based time-series analysis, Nature 383 (1996) 323–327. [9] S. Jaffard, Y. Meyer, R.D. Ryan, Wavelets Tools for Science and Technology, SIAM, ISBN 0-89871-448-6, 2001. [10] S. Jaffard, Multifractal formalism for functions Part I: results valid for all functions, SIAM J. Math. Anal. 28 (4) (1997) 944–970; S. Jaffard, Multifractal formalism for functions Part II: self-similar functions, SIAM J. Math. Anal. 28 (4) (1997) 971–998. [11] R. Kitney, D. Linkens, A. Selman, A. McDonald, The interaction between heart rate and respiration: Part II—nonlinear analysis based on computer modeling, Automedica 4 (1982) 141–153. [12] http://wave.cmap.polytechnique.fr/soft/LastWave.html. [13] S. Mallat, A Wavelet Tour of Signal Processing, Academic Press, ISBN 0-12-466606-X, 1999. [14] M.A. Townsend, Road-racing strategies, Med. Sci. Sports Exerc. 14 (1982) 235–243. [15] F. Xu, E.C. Rhodes, Oxygen uptake kinetics during exercises, Sports Med. 27 (1999) 313–327. [16] Y. Yamamoto, J.O. Fortrat, R.L. Hughson, On the fractal nature of heart rate variability in humans: effect of respiratory sinus arythmia, Am. J. Physiol. 269 (1995) H480–H486.
Letter to the Editor
Multifractal analysis of heartbeat time series in human races Eva Wesfreid a,c,∗ , Véronique L. Billat b , Yves Meyer a a CMLA, École Normale Supérieure de Cachan, 94235 Cachan, France b Département STAPS, Université d’Evry Val d’Essonne, France c LMPA, Université du Littoral, Côte d’Opale, 62228 Calais, France
Available online 19 February 2005 Communicated by Charles K. Chui on 18 March 2004
Abstract We present a multifractal analysis of heartbeat and heart rate time series in human races. In order to improve the training of athletes, we compare heart rate multifractal behavior in free and constant speed 10,000 m runnings. We analyze also marathon races, free pace 42.195 km running, we compare the first and the second half heartbeat signals to measure the effect of fatigue. We find that freedom for choosing the own pace variation could be the racing condition for keeping good health conditions in an exhausting exercise. 2005 Elsevier Inc. All rights reserved. Keywords: Multifractal analysis; Wavelets; Heartbeat signals
1. Introduction Best performances in middle-distance running (800–10,000 m) are actually characterized by speed variability. If one considers the last three world records for middle- and long-distance running, it can be observed that the velocity varies of 5% [4,5]. In races, the choice of speed and the variations of speed that will maximize an endurance athlete’s ability to succeed in winning a race involve a complex interplay of physiological and psychological factors [6]. Therefore, the exploration of physiological limits for exercise must also be approached in exercise model where the subject is free for regulating his pace [4,14].
* Corresponding author.
E-mail address: [email protected] (E. Wesfreid). 1063-5203/$ – see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.acha.2004.12.005
330
E. Wesfreid et al. / Appl. Comput. Harmon. Anal. 18 (2005) 329–335
Fig. 1. ECG signal showing some R peaks.
Fig. 2. Athlete’s RR interval (ms) as function of time.
Most of the studies have described heart rate (HR) dynamics with classical linear models [15], while some have analyzed HR variability during exercise with Fourier transform [6]. Indeed, beyond the analysis of large trend, the main interest of such nonlinear dynamics analysis, in particular the multifractal method, is to appreciate the feature of healthy function to respond to unpredictable stimuli and stresses [16]. However, such analysis is complicated by the fact that biological signals are both irregular and nonstationary, that is, their statistical character changes slowly or intermittently as a result of variations in background influences [3]. Furthermore, very often there are ever smaller amplitude fluctuations at ever shorter time scales. Multifractal analysis is used for classifying signals which exhibit a rough behavior. This behavior is quantified by calculating pointwise Hölder exponents h(t) using multifractal analysis. Indeed, the human heart generates the heartbeat. A recording of the cardiac-induced skin potentials at the body’s surface, an electrocardiogram (ECG, Fig. 1), reveals information about atrial and ventricular electrical activity. Abnormalities in the temporal durations of the segments between deflections, or of the intervals betweens waves in the ECG, as well as their relative heights, serve to expose and distinguish cardiac dysfunction. Readily recognizable features of the ECG wave pattern are designated by the letters P-QRS-T; the wave itself is often referred to as the QRS complex. Aside from the significance of various features of the QRS complex, the timing of the sequence of the QRS complexes over thousands of heartbeats is significant. These intercomplex times are readily measured by recording the occurrence of the peaks of the large R waves which are the most distinctive feature of the normal ECG. Therefore two successive R peaks is the so-called “RR interval” (the heartbeat period in ms) while its inverse (the frequency) is the heart rate (in Hertz). The time interval between two R peaks are submitted to important fluctuations, even at rest. Figure 2 shows an athlete heartbeat in a marathon race.
E. Wesfreid et al. / Appl. Comput. Harmon. Anal. 18 (2005) 329–335
331
It is now well documented that the fluctuations of the sequence of interbeat intervals can be used to assess the presence or likelihood of cardiovascular disease [11]. In healthy subjects, Ivanov et al. [8] demonstrated that at rest, when suitably rescaled using the wavelet-based time series analysis, the distributions of the variations in the beat-to-beat intervals for all healthy subjects were described by a single function stable over a wide range of time scales. This mean that there are ever smaller amplitude fluctuations at ever shorter time scales [3]. We hypothesized that during an exhaustive exercise, the modification of the heart rate variability could be a tool for detecting the presence of cardiac dysfunction during the marathon race. Sudden deaths due to cardiac pathologies are currently reported on marathon even if it has not been yet a matter of epidemiological serious study. We used the wavelet transform modulus maxima (WTMM) algorithm proposed by A. Arneodo, E. Bacry, and J.-F. Muzy [2], which estimates the spectrum of singularities of the Hölder exponents in a “multifractal formalism.” We show that the fatigue does not change significantly the scaling behavior readable in the estimated spectrum of singularities. This suggests that the fatigue appearing during the marathon races does not induce anomalies of the cardiac fluctuations.
2. WTMM algorithm The WTMM algorithm [2,13] estimates the spectrum of singularities of the Hölder exponents in a “multifractal formalism” [9,10]. The Hölder exponent [1,9,10,13] of a function f at a given value t0 of the time variable is (1) h(t0 ) = sup h: f (t) − Pm (t − t0 ) C|t − t0 |h , where Pm is a polynomial of degree not exceeding m. In other words h(t0 ) = lim inf t→t0
log |f (t) − Pm (t − t0 )| . log |t − t0 |
(2)
The spectrum of singularities of f is the Hausdorff dimension, D(α), of the sets Eα of points t0 where the Hölder exponent h(t0 ) takes a given value α. In a “multifractal formalism,” we suppose that the sets Eα of our physiological signal are fractals. The signal f is monofractal when the support of the singularity spectrum, D(α), is a point. It is well known that the fractional Brownian and the uniform Cantor measure are monofractal signals. The nonuniform Cantor measure is multifractal (see Fig. 3). The WTMM algorithm can be split into three steps: (a) Compute the partition function: Z(q, s) = p |Wf (up (s), s)|q , where the sum runs over the curves s → up (s). Here up (s) = arg max |Wf (u, s)| and these maxima are local maxima. (b) Estimate the slope of the line, log Z(q, s) vs log s. The scaling exponent, τ (q), measures the asymptotic decay of Z(q, s) at fine scales s: τ (q) = lim inf s→0
log Z(q, s) , log s
(3)
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E. Wesfreid et al. / Appl. Comput. Harmon. Anal. 18 (2005) 329–335
Fig. 3. Uniform and nonuniform Cantor measures.
which means that Z(q, s) Cε s τ (q)−ε for s → 0, ε > 0, s > 0 and that τ (q) is the largest exponent for which this holds. (c) Compute the singularity spectrum D(α) using the inverse of the Legendre transform 1 − τ (q) . (4) D(α) = min q α + q∈R 2 Figure 3 shows an approximation of the uniform and nonuniform Cantor measure (with 7 iterations and 8000 samples), at left, the scaling exponent τ (q) at middle and the singularity spectrum, D(α), at right.
3. Race data analysis We analyzed heart rate (HR) series, measured continuously beat-by-beat (by a cardio frequencemeter Polar 810S, Finland), during free and constant velocity, 10,000 m running and also in marathon RR interval signals, free pace 42.195 km races. In the 10,000 m runnings, each athlete performed two series, several times. In the first series, the subject was asked to run as fast as possible ( free speed). In the second series, the same distance was covered at a constant velocity, equal to the average velocity of the previous free-pace run (constant speed). In order to improve the training of athletes and to know whether it is preferable to ask the athletes to run at a constant or free pace, HR signals in free and constant speed runnings are compared. In the marathon races, we compare first and second half RR interval signals to measure the effect of fatigue on athlete’s performance.
E. Wesfreid et al. / Appl. Comput. Harmon. Anal. 18 (2005) 329–335
333
Fig. 4. Free and constant heart rate scaling law behavior.
Fig. 5. Heart rate (left: singularity spectrum, right: scaling exponent).
3.1. Free vs constant 10,000 m races Figure 4 shows log2 Z(q, s) vs log2 s, where Z(q, s) is the partition function defined previously. This representation is less satisfactory for negatives values of q, such behavior can be a consequence of signal length (too small). We observed that the free pace race: • induced a lower average heart rate than the constant one (165 ± 14 vs 169 ± 13 bpm), bpm meaning beats per minute; • consumed less running energy than the constant one, running energy measures the ratio of oxygen uptake over, speed times athletes weight (4.263 ± 0.168 vs 4.83 ± 0.237 kJ kg−1 km−1 ); • in most cases, has smaller heartbeat D(α) support and less convex scaling exponent function q → τ (q). Heart rate in free speed seems to be less multifractal. Figure 5 shows this behavior for one race.
334
E. Wesfreid et al. / Appl. Comput. Harmon. Anal. 18 (2005) 329–335
Fig. 6. Heart rate (left: singularity spectrum, right: scaling exponent).
3.2. First and second half parts of marathon races We observed that: • the average heart rates on the first and second half marathon were not significantly different (144±22 vs 144 ± 14 bpm); • in most races the singularity spectra D(α) and the scaling exponents τ (q) of the first and second half RR interval are almost the same. Figure 6 shows the singularity spectrum, D(h), at left and the scaling exponent τ (h) at right. ◦ is for the first half and + is for the second half. The Hölder exponent hmax with maximal Hausdorff dimension (D(h)) for the two halves are not significantly different. This behavior was observed for most of the marathon races. RR interval multifractal characteristics were not significantly different between the first and second part of the race. This suggests that the speed variation allows a good adaptation to the fatigue. Indeed, the RR interval kept a multifractal behavior, which is believed to be an indicator of good health [7]. Figures 3–6 were obtained with the Last Wave software [12].
4. Conclusion We proposed here a multifractal analysis of heart rate time series in free and constant speed 10,000 m runnings and in marathon races, free pace. The fatigue appearing during the marathon does not induce anomalies of the cardiac fluctuation readable on the multifractal behavior.
E. Wesfreid et al. / Appl. Comput. Harmon. Anal. 18 (2005) 329–335
335
This suggests that freedom for choosing the own pace variation could be the racing condition for keeping good health conditions in an exhausting exercise. However, the results we obtained can be questioned. Indeed • there are other tools for processing this kind of signals and these could be compared with multifractal analysis; • the shape of multifractal spectrum does not provide a precise information; and • the relation between variability and fatigue needs to be further explored.
References [1] P. Abry, P. Gonçalvès, J. Lévy Véhel, Lois d’échelle, fractales et ondelettes, Hermès Science, ISBN 2-7462-0410-X, 2002. [2] A. Arneodo, F. Argoul, E. Bacry, J. Elezgaray, J.F. Muzzy, Ondelettes multifractales et turbulence, Diderot Éditeur, Paris, 1995. [3] J.B. Bassingthwaighte, L.S. Liebovitch, B.J. West, Fractal Physiology, American Physiology Society, Oxford Univ. Press, 1994, p. 364. [4] V.L. Billat, J. Slawinski, M. Danel, J.P. Koralsztein, Effect of free versus constant pace on performance and oxygen kinetics in running, Med. Sci. Sports Exerc. 33 (2001) 2082–2088. [5] F. Cottin, Y. Papelier, F. Durbin, J.P. Koralsztein, V.L. Billat, Effect of fatigue on spontaneous velocity variations in human middle-distance running: use of short-term Fourier transformation, Eur. J. Appl. Physiol. 87 (2002) 17–27. [6] Y. Fukuba, B.J. Whipp, A metabolic limit on the ability to make up for lost time in endurance events, J. Appl. Physiol. 87 (1999) 853–861. [7] S. Havlin, L.A.N. Amaral, Y. Ashkenazy, A.L. Goldberger, P.Ch. Ivanov, C.K. Peng, H.E. Stanley, Application of statistical physics to heartbeat diagnosis, Physica A 274 (1999) 99–110. [8] P.Ch. Ivanov, M.G. Rosenblum, C.-K. Reng, J. Mietus, S. Havlin, H.E. Stanley, A.L. Goldberger, Scaling behavior of heartbeat intervals obtained by wavelet-based time-series analysis, Nature 383 (1996) 323–327. [9] S. Jaffard, Y. Meyer, R.D. Ryan, Wavelets Tools for Science and Technology, SIAM, ISBN 0-89871-448-6, 2001. [10] S. Jaffard, Multifractal formalism for functions Part I: results valid for all functions, SIAM J. Math. Anal. 28 (4) (1997) 944–970; S. Jaffard, Multifractal formalism for functions Part II: self-similar functions, SIAM J. Math. Anal. 28 (4) (1997) 971–998. [11] R. Kitney, D. Linkens, A. Selman, A. McDonald, The interaction between heart rate and respiration: Part II—nonlinear analysis based on computer modeling, Automedica 4 (1982) 141–153. [12] http://wave.cmap.polytechnique.fr/soft/LastWave.html. [13] S. Mallat, A Wavelet Tour of Signal Processing, Academic Press, ISBN 0-12-466606-X, 1999. [14] M.A. Townsend, Road-racing strategies, Med. Sci. Sports Exerc. 14 (1982) 235–243. [15] F. Xu, E.C. Rhodes, Oxygen uptake kinetics during exercises, Sports Med. 27 (1999) 313–327. [16] Y. Yamamoto, J.O. Fortrat, R.L. Hughson, On the fractal nature of heart rate variability in humans: effect of respiratory sinus arythmia, Am. J. Physiol. 269 (1995) H480–H486.