The goal of this computation is to use both persistent homology to recognize stable patterns in trajectories and integrated geometrical information (a branch of the integrated information theory) to track inherent information processes leading to a cause-effect repertoire, which cannot be easily accessed by an external observer and which cannot be trivially reduced to mechanistic causal effects.
For the evaluation of the persistent homology we used the function gridDiag from the R-package TDA (https://cran.r-project.org/web/packages/TDA/vignettes/article.pdf), which evaluates a given real valued function over a triangulated grid, constructs a filtration of simplices using the values of the function, and computes the persistent homology of the filtration.
A similar approach based on persistent homology has been used for the extraction of system inter-variability in populations with complex-multiscale systems (https://loop-impact.frontiersin.org/impact/article/465982#totalviews/views). In such works the Geometrical Integrated Information Theory and the analysis of topological persistence for each individual in different time events are quite similar. This could imply that the implemented methodology can be used to analyze causality chains in different time scales and assess in this way how autonomous a system is.