Information theory for non-stationary processes with stationary increments

Carlos Granero-Belinchon (1, 2, 3) , Stéphane G. Roux (1) , Nicolas Garnier (1, 4)

1 Phys-ENS - Laboratoire de Physique de l'ENS Lyon
2 DOTA, ONERA, Université de Toulouse [Toulouse]
3 INRA - Institut National de la Recherche Agronomique
4 CNRS - Centre National de la Recherche Scientifique

We describe how to analyze the wide class of non stationary processes with stationary centered increments using Shannon information theory. To do so, we use a practical viewpoint and define ersatz quantities from time-averaged probability distributions. These ersatz versions of entropy, mutual information and entropy rate can be estimated when only a single realization of the process is available. We abundantly illustrate our approach by analyzing Gaussian and non-Gaussian self-similar signals, as well as multi-fractal signals. Using Gaussian signals allow us to check that our approach is robust in the sense that all quantities behave as expected from analytical derivations. Using the stationarity (independence on the integration time) of the ersatz entropy rate, we show that this quantity is not only able to fine probe the self-similarity of the process but also offers a new way to quantify the multi-fractality.

Domaines: Mécanique statistique [cond-mat.stat-mech]; Théorie de l'information et codage [math.IT]; Traitement du signal et de l'image [eess.SP]

Carlos Granero-Belinchon, Stéphane G. Roux, Nicolas Garnier. Information theory for non-stationary processes with stationary increments. Entropy, 2019, Information Theoretic Measures and Their Applications, 21 (12), pp.1223. ⟨10.3390/e21121223⟩. ⟨hal-02395022⟩