🤖 AI Summary
This study addresses the challenge of accurately identifying directional lead-lag relationships among variables in multivariate time series. It proposes a novel framework grounded in temporal irreversibility, rigorously defining directionality as behavioral asymmetry under time reversal. Central to this approach is the introduction of a circulation matrix derived from lagged covariances as the core measure of directionality. The methodology incorporates an unbiased estimator, cross-fitting debiasing, delete-block jackknife standard errors, and an exact randomization test under a block-wise null hypothesis, with extensions to nonlinear settings. Evaluated on four simulated systems with known directional structures, the proposed method substantially outperforms conventional approaches—including correlation networks, Granger causality, and transfer entropy—demonstrating markedly reduced misidentification of directional relationships.
📝 Abstract
Many questions across the sciences take the same form: several coupled series are observed together, and the analyst wants to know not merely that they move together but which one moves first, and how strongly. This paper sets out a complete method built on one organising idea: the direction of a coupled system is exactly the part of its behaviour that changes when the record is played backwards. Tools built on contemporaneous covariance alone (correlation matrices, distance measures, spanning trees, undirected centralities, principal components) carry no information about direction: a reversible system and a circulating one can share identical covariance at every sampling of the same point-in-time record. Formally, direction is a circulation matrix carried by the lagged covariance. Its vanishing is exactly statistical time reversibility for linear systems, feature maps carry the characterisation to nonlinear ones, and under the Gaussian benchmark its magnitude is an entropy-production functional of the identified circulation, the quadratic component of the divergence per unit time between the forward and reversed records. Around this estimand we build a cross-fitted estimator removing first-order bias, delete-block jackknife standard errors, and a randomisation test exact under its stated block null, with a familywise correction and a nonlinear extension. A sampling theory says when the arrow is measurable at all, and a design layer separates transmission from the ordering of clocks. A laboratory of four systems with known answers compares the method with correlation networks, Granger causality, transfer entropy, and connectedness indices, reporting the failures of each when read as a measure of direction, including our own. Complete algorithms and worked examples in two languages make the paper the base reference for a series of applications.