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.

Classification : 62P10, 60G22, 62M07, 62M09
Mot clés : Série de fréquences cardiaques instantanées, Analyse par ondelettes, Detrended Fluctuation Analysis, Paramètre de Hurst, Processus longue mémoire, Bruit gaussien fractionnaire.
Keywords: Heartbeat data, Wavelet analysis, Detrended Fluctuation Analysis, Hurst parameter, Fractional processes

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