A new stochastic process to model Heart Rate series during exhaustive run and an estimator of its fractality parameter

In order to interpret and explain the physiological signal behaviors, it can be interesting to find some constants among the fluctuations of these data during all the effort or during different stages of the race (which can be detected using a change points detection method). Several recent papers have proposed the long-range dependence (Hurst) parameter as such a constant. However, their results induce two main problems. Firstly, DFA method is usually applied for estimating this parameter. Clearly, such a method does not provide the most efficient estimator and moreover it is not at all robust even in the case of smooth trends. Secondly, this method often gives estimated Hurst parameters larger than 1, which is the larger possible value for long memory stationary processes. In this article we propose solutions for both these problems and we define a new model allowing such estimated parameters. On the one hand, a wavelet-base estimator is applied to data. Such an estimator provides optimal convergence rates in a semiparametric context and can be used for smoothly trended processes. On the other hand, a new semiparametric model so-called locally fractional Gaussian noise is introduced and is characterized by a so-called parameter which can be larger than 1. Such semiparametric process is tested to be relevant for modeling HR data in the three characteristic phases of the race. It also shows an evolution of the local fractality parameter during the race confirming the results obtained by Peng et al. (1995) in their study regarding Hurst parameter of HR time series during the exercise for healthy adults (where the estimated parameter is close to that observed in the race beginning) and heart failure adults (where the estimated parameter is close to that observed in the end of race). So, this evolution, which can not be observed with DFA method, may be associated with fatigue appearing during the last phase of the marathon.