Mutual information matrix based on asymmetric Shannon entropy for nonlinear interactions of time series

Zhao et al. (Nonlin. Dyn. 88, 477-487, 2017) presented the mutual information matrix (MIM) analysis for the study of nonlinear interactions in multivariate time series as an extension of Random Matrix Theory analysis. They considered the histogram estimation of mutual information based on Shannon entropy for discrete distributions. This paper is motivated by the latter, extending MIM analysis from a nonparametric and probabilistic discrete approach to a parametric and probabilistic continuous approach. Specifically, this paper presents the MIM based on Maximum Likelihood Estimators (MLEs) for flexible and tractable families of continuous multivariate distributions, called multivariate skew-elliptical families of distributions. This method focus on multivariate skew-Gaussian and skew-t distributions that allow modeling skewness and heavy-tails, respectively. Performance of the proposed methodology is illustrated by numerical results given by sinusoidal and vector autoregressive fractionally integrated moving-average models, and applied to a meteorological monitoring network data set. Results show that the consideration of skewness and heavy-tails in the transformed ozone time series produced some differences in the MIM estimations compared with those obtained by applying histogram estimations to transformed data. Given that mutual information index (MII) increases in line with the number of bins for the histogram estimator, the proposed methodology based on MLEs considered more robust estimators with respect to the histogram ones to determine the MII of multivariate time series.