Distinguishing Anticipation from Causality: Anticipatory Bias in the Estimation of Information Flow

We report that transfer entropy estimates obtained from low-resolution and/or small data sets show net information flow away from a purely anticipatory element whereas transfer entropy calculated using exact distributions show the flow towards it. This means that for real-world data sets anticipatory elements can appear to be strongly driving the network dynamics even when there is no possibility of such an influence. Furthermore, we show that in the low-resolution limit there is no statistic that can distinguish anticipatory elements from causal ones.

Figure 1

Transfer entropy ($m=2$) versus anticipation $\alpha$ in unidirectionally coupled ($\xi =0.3$) logistic maps: (a) 1-bit measurement resolution, (b) 2-bit resolution, and (c) 3-bit resolution. For coupling strength $\xi =0.3$, reliably determining the correct direction of information flow using transfer entropy requires at least 3 bits of measurement resolution.

Figure 2

Transfer entropies ($m=2$) of the unidirectionally coupled ($\xi =0.4$) maps without anticipation ($\alpha =0$, dashed lines) and with full anticipation ($\alpha =1$, solid lines) plotted versus kernel diameter $d$. For a nonanticipatory response system $T_{x\to y}>T_{y\to x}$ for all kernel diameters. For the anticipatory response system there is a critical kernel diameter $d^{*}\approx 0.34$ above which the inferred information flow is in the wrong direction ($T_{x\to y}<T_{y\to x}$).

Figure 3

Crossover diameter $d^{*}$ versus coupling strength computed with $m$ past values. Taking into account more past values lowers the crossover diameter and therefore increases the resolution needed to detect anticipatory dynamics.