Fig. 2: System with mediator variable.
From: Decomposing causality into its synergistic, unique, and redundant components
(Left panel) Schematic of the functional dependence among variables and system equations, where Wi represents unobserved, stochastic forcing on the variable Qi. We use the notation \({Q}_{i}^{n}={Q}_{i}({t}_{n})\), where n indicates the time step. (Center and right panels) Results from SURD with redundant (R), unique (U) and synergistic (S) causalities in blue, red and yellow, respectively. The notation employed is such that R123 denotes \(\Delta {I}_{123\to j}^{R}\) and so on. The gray bar is the causality leak. The results from CGC, CTE, PCMCI, and CCM are depicted on the right. In all methods but CCM, the value of the bar represent the strength of the causal link. In CCM, a causal link is detected only when the value converges to 1 as the length of the time series increases, but not otherwise. CGC and CTE use the same normalization as SURD. The values for SURD, CTE and CCM are upper bounded by 1. The values for PCMCI represent conditional mutual information and are unbounded. The equations to quantify causality by each of the methods are in the Supplementary Materials.
