Multiscale entropy rates: A study on different stochastic processes

In this paper, we propose to analyze the behavior of the entropy rate (ER) when applied to a signal and its coarse-grained versions. The “multiscale entropy rate” (MSER) is deduced by storing in a vector the resulting ERs. Our contribution consists in studying the MSER calculated on different stochastic processes. When dealing with Gaussian complex or real moving average (MA) processes or autoregressive (AR) processes, which can be seen as the filtering of a white Gaussian driving process, the MSER depends on the variances of the driving processes of the corresponding minimum-phase ARMA process at each scale. More particularly, we derive the analytical expression of the MSER for $1^{st}$-order MA or AR processes using different approaches. This study allows us to better understand what each scale brings in and to describe the behavior of the MSER for these types of processes. We also show that there is a mapping between the stochastic-process parameters and the ER computed at different scales. Finally, we show that the multiscale procedure is not relevant for a sum of complex exponentials disturbed by an additive white Gaussian noise.