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We analyze the popular ``state-space'' class of algorithms for detectingcasual interaction in coupled dynamical systems. These algorithms are oftenjustified by Takens' embedding theorem, which provides conditions under whichrelationships involving attractors and their delay embeddings are continuous.In practice, however, state-space methods often do not directly testcontinuity, but rather the stronger property of how these relationshipspreserve inter-point distances. This paper theoretically and empiricallyexplores state-space algorithms explicitly from the perspective of distancepreservation. We first derive basic theoretical guarantees applicable to simplecoupled systems, providing conditions under which the distance preservation ofa certain map reveals underlying causal structure. Second, we demonstrateempirically that typical coupled systems do not satisfy distance preservationassumptions. Taken together, our results underline the dependence ofstate-space algorithms on intrinsic system properties and the relationshipbetween the system and the function used to measure it -- properties that arenot directly associated with causal interaction.

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