Detecting Causal Relations between Time Series with the Cross Markov Matrix Technique

A new measure for the characterization of interconnected dynamical systems coupling is proposed. The proposed measure, called Cross Markov Matrix, starts from the idea introduced by Campanhao et al (PLoS ONE 6, e23378 (2011)) of transforming time series into complex networks based on the transition probability in a Markov model. Our approach extends this idea to coupled time series by mean of an interleaving technique which allows the construction of a complex network from a pair of time series. The topological structure of the complex network changes with the strength of the coupling between the time series, the coupling strength. The topological changes may be assessed by monitoring the entropy of the adjacency matrix, represented as an image or by mean of the other indicators as, for example, the medium articulation. The properties of the Cross Markov Matrix have been investigated with the help of a systematic series of numerical tests using synthetic data. The potential of the approach is then substantiated by the analysis of various real-life examples, ranging from environmental and global climate problems to the mutual influence between media coverage of Brexit and the pound-euro exchange.