Jensen-distance rate for stationary time series based on cross-spectral methods

Entropy and mutual information rates are key concepts in information theory that measure the average uncertainty and statistical dependence growth between two stochastic processes, respectively. This paper introduces a distance rate measure for discrepancy growth between two stationary processes, termed the Jensen-distance rate (JDR), which is based on spectral and cross-spectral densities. I examine fractional noise as a specific case of a weakly stationary process, where the asymptotic JDR is computed, and numerical results demonstrate the method's performance. Additionally, I propose a JDR estimator based on the Blackman–Tukey spectral estimator for samples. Finally, an application to an ozone monitoring network showcases the estimated JDR for time series data, highlighting the practical utility of the proposed distance rate in time series analysis, including maximum/minimum concentrations and intra-daily seasonality.