Distance preservation in state-space methods for detecting causal interactions in dynamical systems

TL;DR

This paper investigates how state-space methods for detecting causal interactions in dynamical systems rely on distance preservation, revealing limitations and conditions affecting their effectiveness.

Contribution

It provides theoretical guarantees and empirical analysis of distance preservation in state-space algorithms, highlighting their dependence on system properties.

Findings

Theoretical conditions under which distance preservation indicates causality.

Empirical evidence that many coupled systems do not satisfy distance preservation.

State-space methods' effectiveness depends on intrinsic system properties.

Abstract

We analyze the popular ``state-space'' class of algorithms for detecting casual interaction in coupled dynamical systems. These algorithms are often justified by Takens' embedding theorem, which provides conditions under which relationships involving attractors and their delay embeddings are continuous. In practice, however, state-space methods often do not directly test continuity, but rather the stronger property of how these relationships preserve inter-point distances. This paper theoretically and empirically explores state-space algorithms explicitly from the perspective of distance preservation. We first derive basic theoretical guarantees applicable to simple coupled systems, providing conditions under which the distance preservation of a certain map reveals underlying causal structure. Second, we demonstrate empirically that typical coupled systems do not satisfy distance preservation assumptions. Taken together, our results underline the dependence of state-space algorithms on intrinsic system properties and the relationship between the system and the function used to measure it -- properties that are not directly associated with causal interaction.