Modélisation des fréquences cardiaques instantanées durant un marathon et estimation de leurs paramètres fractals

Bardet, Jean-Marc; Billat, Véronique; Kammoun, Imen

Journal de la société française de statistique, Tome 150 (2009) no. 1, pp. 101-126.

De nombreux papiers semblent indiquer que le paramètre de longue mémoire est une constante caractéristique de l’évolution des fréquences cardiaques instantanées. Des raffinements ont été également proposés en prenant en compte les comportements en basses et hautes fréquences et en associant à chacun de ces régimes des paramètres de type Hurst. Ces études ont cependant deux défauts : l’utilisation de la méthode DFA, non robuste, et le fait que les estimations des paramètres sont souvent supérieures à $1$. Pour résoudre ces deux problèmes nous proposons comme nouveau modèle pour ces données un processus formé par les accroissements d’un mouvement brownien fractionnaire multi-échelle avec deux régimes non nuls (un régime basse fréquence et un autre haute fréquence). Une analyse par ondelettes avec une ondelette mère vérifiant des propriétés de localisation en fréquence, permet d’estimer de manière convergente les deux paramètres fractals (à valeurs dans $ℝ$ et non simplement dans $(0,1)$) associés à ces deux régimes, et également de construire un test d’adéquation du modèle. Des simulations permettent de vérifier le bon comportement de l’estimateur par rapport à celui de la méthode DFA. Appliqué aux données de fréquences cardiaques instantanées relevées chez des athlètes courant le marathon, le modèle proposé est accepté lorsqu’on l’applique de façon différenciée aux trois périodes de course (début, milieu et fin de course). On montre ainsi une augmentation au cours de la course du paramètre basse fréquence, ce qui va dans le même sens que des résultats déjà obtenus : le coeur en début de course fonctionne comme celui de personnes en bonne santé, alors qu’en fin de course son comportement est proche de celui de malades cardiaques.

Numerous papers have shown that the Hurst parameter is a characteristic constant of heartbeat series. The estimation of two fractal type parameters was also proposed for differentiating low and high frequency behaviors. But those parameters are often estimated larger than $1$ with the DFA analysis which is a non robust estimation method. We propose a new model based on increments of a multiscale fractional Brownian motions with different low and high frequency behaviors for modeling heartbeat data, which are (for each runner) automatically stepped in several stages where the signal is nearly stationary with an adaptive change points detection method. Then, we consider a wavelet based estimator with a mother wavelet satisfying a frequency localization property. We prove that an adaptive version of this estimator is convergent for the proposed model and a chi-squared goodness-of-fit test is also built from this wavelet based estimator. Simulations show that this estimator is accurate while DFA estimator does not always converges. Moreover, the application of this method to Marathon heartbeat series indicates that the model fits well data in each stage of the race and that the low frequency fractal parameter increases during the race.

[1] Abry, P.; Flandrin, P.; Taqqu, M.; Veitch, D. Self-similarity and long-range dependence through the wavelets lens, Birkhäuser Boston, Boston, Massachusetts (2003), pp. 527-556 | MR | Zbl

[2] Absil, P.A.; Sepulchre, R.; Bilge, A.; Gérard, P. Nonlinear analysis of cardiac rhythm fluctuations using DFA method, Physica A (1999), pp. 235-244

[3] Bai, J. Least squares estimation of a shift in linear processes, J. Time Ser. Anal., Volume 5 (1998), pp. 453-472 | MR | Zbl

[4] Bardet, J.M. Statistical study of the wavelet analysis of fractional Brownian motion, IEEE Trans. Inform. Theory., Volume 48 (2002), pp. 991-999 | MR | Zbl

[5] Bardet, J.M.; Bertrand, P. Definition, properties and wavelet analysis of multiscale fractional Brownian motion, Fractals, Volume 15 (2007), pp. 73-87 | MR | Zbl

[6] Bardet, J.M.; Bertrand, P. Identification of the multiscale fractional Brownian motion with biomechanical applications, J. Time Ser. Anal., Volume 28 (2007), pp. 1-52 | MR | Zbl

[7] Bardet, J.M.; Bertrand, P.R. Estimation of the spectral density of a process observed at random times (2009) (Preprint)

[8] Bardet, H. J.M.and Bibi; Jouini, A. Adaptive wavelet based estimator of the memory parameter for stationary Gaussian processes, Bernoulli, Volume 14 (2008), pp. 691-724 | Zbl

[9] Bardet, J.M.; Kammoun, I. Asymptotic Properties of the Detrended Fluctuation Analysis of Long Range Dependence Processes, IEEE Trans. Inform. Theory., Volume 54 (2008), pp. 2041-2052 | Zbl

[10] Bardet, J.M.; Kammoun, I. Detecting changes in the fluctuations of a Gaussian process and an application to heartbeat time series (2008) (Preprint)

[11] Bardet, J.M.; Lang, G.; Moulines, E.; Soulier, P. Wavelet estimator of long-range dependent processes, Stat. Inference Stoch. Process., Volume 3 (2000), pp. 85-99 | Zbl

[12] Birgé, L.; Massart, P. Minimal penalties for Gaussian model selection, Probab. Theory Related Fields, Volume 138 (2007), pp. 33-73 | Zbl

[13] Bai, J.; Perron, P. Estimating and testing linear models with multiple structural changes, Econometrica, Volume 66 (1998), pp. 47-78 | Zbl

[14] Bertrand, P.R.; Teyssière, G.; Boudet; Chamoux, A. Detection of Change–Points in the Spectral Density. With Applications to ECG Data. (2009) (Preprint)

[15] Theory and applications of long-range dependence (Doukhan, P.; Oppenheim, G.; Taqqu, M.S., eds.), Birkhäuser, Boston, 2003 | Zbl

[16] Flandrin, P. Wavelet analysis and synthesis of fractional Brownian motion, IEEE Trans. on Inform. Theory, Volume 38 (1992), pp. 910-917 | Zbl

[17] Fox, R.; Taqqu, M.S. Large-sample properties of parameter estimates for strongly dependent Gaussian time series, Ann. Statist., Volume 14 (1986), pp. 517-532 | Zbl

[18] Goldberger, A.L.; Amaral, L.A.N.; Hausdorff, J.M.; Ivanov, P.C.; Peng, C.K.; Stanley, H.E. Fractal Dynamics in Physiology : Alterations with Disease and Aging, PNAS, Volume 99 (2002), pp. 2466-2472

[19] Gough, N.A.J. Fractals, chaos and fetal heart rate, Lancet, Volume 339 (1992), pp. 182-183

[20] Gough, N.A.J. Fractal analysis of fetal heart rate variability, Physiol. Meas., Volume 14 (1993), pp. 309-315

[21] Havlin, S.; Amaral, L.A.N.; Ashkenazy, Y.; Goldberger, A.L.; Ivanov, P.Ch.; Peng, C.K.; Stanley, H.E. Application of statistical physics to heartbeat diagnosis, Physica A, Volume 274 (1999), pp. 99-110

[22] Ivanov, P.C.; Amaral, L.A.N.; Goldberger, A.L.; Havlin, S.; Rosenblum, M.G.; Stanley, H.E.; Struzik, Z. From 1/f Noise to Multifractal Cascades in Heartbeat Dyamics, Chaos, Volume 11 (2001), pp. 641-652 | Zbl

[23] Ivanov, P.C.; Rosenblum, M.G.; Nunes Amaral, L.A.; Struzik, Z.S.; Havlin, S.; Goldberger, A.L.; Stanley, H.E. Multifractality in human heartbeat dynamics, Nature, Volume 399 (1999), pp. 461-465

[24] Ivanov, P.C.; Rosenblum, M.G.; Peng, C.K.; Mietus, J.; Havlin, S.; Stanley, H.E.; Goldberger, A.L. Scaling Behaviour of Heartbeat Intervals obtained by Wavelet-Based Time-Series Analysis, Nature, Volume 383 (1996), pp. 323-327

[25] Lavielle, M. Detection of multiple changes in a sequence of random variables, Stoch. Process Appl. (1999), pp. 79-102 | Zbl

[26] Lavielle, M.; Ludeña, C. The multiple change-points problem for the spectral distribution, Bernoulli, Volume 6 (2000), pp. 845-869 | Zbl

[27] Lavielle, M.; Moulines, E. Least-squares estimation of an unknown number of shifts in a time series, J. Time Ser. Anal., Volume 21 (2000), pp. 33-59 | Zbl

[28] Martinis, M.; Knezevic, H.; Krstacic, G.; Vargovic, E. Changes in the Hurst exponent of heartbeat intervals during physical activities, Phys. Rev. E, Volume 70 (2004) | DOI

[29] Moulines, E.; Roueff, F.; Taqqu, M.S. On the spectral density of the wavelet coefficients of long memory time series with application to the log-regression estimation of the memory parameter, J. Time Ser. Anal., Volume 28 (2007), pp. 155-187 | Zbl

[30] Peng, C.K.; Buldyrev, S.V.; Havlin, S.; Simons, M.; Stanley, H.E.; Goldberger, A.L. Long-Range Anticorrelations and Non-Gaussian Behavior of the Heartbeat, Physical Rev. E, Volume 49 (1994), pp. 1685-1689

[31] Peng, C.K.; Havlin, S.; Stanley, H.E.; Goldberger, A.L. Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series, Chaos, Volume 5 (1995) no. 1, pp. 82-87 | DOI

[32] Pikkujämsä, S.M.; Mäkikallio, T.H.; Airaksinen, K.E.; Huikuri, H.V. Determinants and interindividual variation of R-R interval dynamics in healthy middle-aged subjects, Am. J. Physiol. Heart Circ. Physiol., Volume 280 (2001), pp. 1400-1406

[33] Peng, C.K; Mietus, J.; Hausdorff, J.; Havlin, S.; Stanley, H.E.; Goldberger, A.L. Long-Range Anticorrelations and Non-Gaussian Behavior of the Heartbeat, Phys. Rev. Lett., Volume 70 (1993), pp. 1343-1346

[34] Robinson, P.M. Gaussian semiparametric estimation of long range dependence, Ann. Statist., Volume 23 (1995), pp. 1630-1661 | Zbl

[35] Roueff, F.; Taqqu, M.S. Asymptotic normality of wavelet estimators of the memory parameter for linear processes, J. Time Ser. Anal., Volume 30 (2009), pp. 534-558 | Zbl

[36] Staudacher, M.; Telser, S.; Amann, A.; Hinterhuber, H.; Ritsch-Marte, M. A new method for change-point detection developed for on-line analysis of the heartbeat variability during sleep, Physica A, Volume 349 (2006), pp. 582-596

[37] Heart rate variability. Standards of measurement, physiological interpretation, and clinical use (1996), pp. 1043-1065 (Technical report)

[38] Taqqu, M.S.; Teverovsky, V.; Willinger, W. Estimators for long-range dependence : an empirical study, Fractals, Volume 3 (1995), pp. 785-798 | Zbl

[39] Veitch, D.; Abry, P.; Taqqu, M.S. On the Automatic Selection of the Onset of Scaling, Fractals, Volume 11 (2003), pp. 377-390 | Zbl

[40] Vedel, B.; Wendt, H.; Abry, P.; Jaffard, S. On the impact of the number of vanishing moments on the dependence structures of compound Poisson motion and fractional Brownian motion in multifractal time (2008) (Preprint)

[41] Wesfreid, E.; Billat, V.; Meyer, Y. Multifractal analysis of heartbeat time series in human races, Appl. Comput. Harmon. Anal., Volume 18 (2005), pp. 329-335 | Zbl

[42] Yeh, R.G.; Shieh, J.; Han, Y.; Wang, Y.; Tseng, S. Detrended fluctuation analyses of short-term heart rate variability in surgical intensive care units, Biomed Eng.- Appl, Volume 18 (2006), pp. 67-72