Abstract
The inference of causal relations between observable phenomena is paramount across scientific disciplines; however, the means for such enterprise without experimental manipulation are limited. A commonly applied principle is that of the cause preceding and predicting the effect, taking into account other circumstances. Intuitively, when the temporal order of events is reverted, one would expect the cause and effect to apparently switch roles. This was previously demonstrated in bivariate linear systems and used in design of improved causal inference scores, while such behaviour in linear systems has been put in contrast with nonlinear chaotic systems where the inferred causal direction appears unchanged under time reversal. The presented work explores the conditions under which the causal reversal happens—either perfectly, approximately, or not at all—using theoretical analysis, low-dimensional examples, and network simulations, focusing on the simplified yet illustrative linear vector autoregressive process of order one. We start with a theoretical analysis that demonstrates that a perfect coupling reversal under time reversal occurs only under very specific conditions, followed up by constructing low-dimensional examples where indeed the dominant causal direction is even conserved rather than reversed. Finally, simulations of random as well as realistically motivated network coupling patterns from brain and climate show that level of coupling reversal and conservation can be well predicted by asymmetry and anormality indices introduced based on the theoretical analysis of the problem. The consequences for causal inference are discussed.
Highlights
The temporal symmetry of physical processes, as well as the common fundamental lack of it, is among the most fascinating natural phenomena with deep theoretical consequences
The results show that already the linear approximations of both brain and climate systems show imperfect causal structure reversal under time reversal, while the extent to which this property is broken is closely predicted by deviation of the coupling matrix from normality in both realizations of randomly connected networks as well as these more realistic connectivity structure scenarios
We showed that somewhat counter-intuitively, even in linear processes the time reversal has nontrivial effect on the coupling structure—in a general case it neither conserves nor reverses it, while these two special phenomena appear under quite specific conditions of normality (or more generally under fulfillment of the conditions given by Equation (10)) and symmetry (14), respectively
Summary
The temporal symmetry of physical processes, as well as the common fundamental lack of it, is among the most fascinating natural phenomena with deep theoretical consequences. In the following we limit the analysis to the simplest but most commonly treated case of a system with delayed interactions with lag of one time step (p = 1) and with white noise, i.e., A0−1 = I ( Xt = −A1Xt−1 + Yt) In this case, the causal structure is given by a single matrix A := −A0−1A1 = −A1, which we shall further denote as the coupling matrix. We found normality and symmetry of the coupling matrix as sufficient conditions for reversal and conservation of the coupling under time-reversal, albeit in principle these may not be necessary depending on the existence of other eligible solutions for Equations (10) and (14)—we leave the characterization of the full set of solutions for these matrix equations under the additional requirements of Definition 1 and Theorem 1 as an open problem for further work, based on preliminary analysis we conjecture the solutions might be equivalent with the normality and symmetry condition, respectively
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