Spurious causalities with transfer entropy.

Transfer entropy (TE) seems currently to be the most widely used tool to characterize causal influences in ensembles of complex systems from observed time series. In particular, in an elemental case of two systems, nonzero TEs in both directions are usually interpreted as a sign of a bidirectional coupling. However, one often overlooks that both positive TEs may well be encountered for unidirectionally coupled systems so that a false detection of a causal influence on the basis of a nonzero TE is rather possible. This work highlights typical factors leading to such "spurious couplings": (i) unobserved state variables of the driving system, (ii) low temporal resolution, and (iii) observation errors. All are shown to be particular cases of a general problem: imperfect observations of states of the driving system. Importantly, exact values of TEs, rather than their statistical estimates, are computed here for selected benchmark systems. Conditions for a "spurious" TE to be large and even strongly exceed a "correct" TE are presented and discussed.