Complexity of couplings in multivariate time series via ordinal persistent homology

We propose a new measure of the complexity of couplings in multivariate time series by combining the techniques of ordinal pattern analysis and topological data analysis. We construct an increasing sequence of simplicial complexes encoding the information about couplings among the components of a given multivariate time series through the intersection of ordinal patterns. The complexity measure is then defined by making use of the persistent homology groups. We validate the complexity measure both theoretically and numerically.