Source
Lizier, Prokopenko & Zomaya (2008), “Local information transfer as a spatiotemporal filter for complex systems”, arXiv 0809.3275v1 (Phys. Rev. E 77, 026110), LIT-1175, read in full in NOTE-943.
What was actually shown
An identity. T_{Y→X} = ⟨t_{Y→X}(n + 1, k, l)⟩, with t = log p(x_{n+1} | x⁽ᵏ⁾_n, y⁽ˡ⁾n)/p(x{n+1} | x⁽ᵏ⁾_n) (Eq. 9). The local value is unbounded above and can be negative.
History length. With k = 1, local TE does not separate rule 110’s gliders from the background (Fig. 2). With k ≥ 6 it does, and with k = 16 it concentrates on the gliders’ leading edges (Figs. 3, 5). For a periodic domain of period p, k ≥ p − 1 gives zero transfer there; with gliders no finite k is enough.
Apparent against complete. Complete TE conditions on the destination’s other neighbours (Eq. 14) and is nonnegative in deterministic systems. In rule 18’s domain it is about 1 bit at alternate sites from each channel, where apparent TE is 0. Checked: there the rule is L ⊕ R.
Particles. In rules 110, 54, 18 and 146, gliders and domain walls are where local TE is high, in their direction of motion.
What could have come out otherwise: long-history local TE could have highlighted the background domain as much as the gliders, as k = 1 does.
What this does not say
- That transfer entropy measures a flow from one source. Rule 18’s domain has 1 bit determined jointly by two sources; apparent TE gives it to neither and complete TE to each (CLAIM-335).
- That complete TE is available outside models. It needs the causal parents.
- That a finite k is correct. k is bounded by sample size, and the averages have not converged at k = 16.
- Anything about continuous or stochastic systems, which are asserted to be within reach and not tested.
- That high local TE is causal influence. The paper’s own perspective is “predictive or computational”, and it defers comparison with Ay and Polani’s interventional flow.