Statistics for differential topological properties between datasets with an application to reservoir computers

Recording (multidimensional) time series from dynamical systems in experiments or numerical studies on coupled systems is common. In many cases, we want to find relationships between systems or subsystems, but multidimensional data prevent simple graphical tests. We here develop statistics that test the data for basic, fundamental relationships, like the existence of continuous functions between the subsystems. These are common fundamental questions that should be answered before diving into function fitting and other data handling schemes since lacking continuity and, possibly, smoothness in data relations guarantees that other more specific data fitting attempts will fail or be meaningless. We develop useful testing statistics by creating data-driven versions of basic topological concepts including continuity, differentiability, point set distance comparisons, diffeomorphisms, and embeddings. As an application we show how such statistics can aid in the analysis of the dynamics of reservoir computer systems, many of which rely on there being specific relations (like embeddings) between the drive and the reservoir computer.