Multiscale Entropy Rate: A Variant of the Multiscale Entropy

The multiscale entropy (MSE), which consists in estimating the sample entropies of the signal and its coarsegrained (CG) versions, is used in a wide range of applications to characterise and classify signals. Several variants were proposed and different studies were carried out on their properties when they are applied to a white noise, a 1/f process and a discrete fractional Gaussian noise (dfGn). In this paper, we propose a variant based on the concept of entropy rate. To this end, two vectors respectively storing $m+1$ and a consecutive samples of the process under study are considered. Under Gaussian assumptions, the expressions of the corresponding entropies are derived. Their difference corresponds to the entropy rate when $m$ increases and tends to infinity. The latter is finally computed with the signal and its CG versions. Shannon and Rényi entropies are considered. After motivating the introduction of the CG-based multiscale approach, applications with moving average, autoregressive and dfGn processes are presented.