Measuring Connectivity in Linear Multivariate Processes: Definitions, Interpretation, and Practical Analysis

Table 1

Connectivity definitions and conditions for their existence. (See text for details).

Definition		MV closed-loop process	MVAR process, time domain	MVAR process, Frequency domain
Direct causality	𝑦𝑗→𝑦𝑖	Knowledge of 𝑌𝑗 improves prediction of 𝑦𝑖 (n)	𝑎𝑖𝑗(𝑘)≠0	π ij(𝑓)≠0
Causality	𝑦𝑗⇒𝑦𝑖	𝑦𝑗→𝑦𝑚⋯→𝑦𝑖	𝑎𝑚𝑗(𝑘)≠0,…,𝑎𝑖𝑚(𝑘)≠0	γ ij(𝑓)≠0
Direct coupling	𝑦𝑖↔𝑦𝑗	𝑦𝑖→𝑦𝑗 or 𝑦𝑗→𝑦𝑖	𝑎𝑗𝑖(𝑘)≠0 or 𝑎𝑖𝑗(𝑘)≠0	Πij (𝑓)≠0
Spurious direct coupling		𝑦𝑖→𝑦𝑚 and 𝑦𝑗→𝑦𝑚	𝑎𝑚𝑖(𝑘)≠0 and 𝑎𝑚𝑗(𝑘)≠0
Coupling	𝑦𝑖⇔𝑦𝑗	𝑦𝑖⇒𝑦𝑗 or 𝑦𝑗⇒𝑦𝑖	𝑎𝑚𝑖(𝑘)≠0,…,𝑎𝑗𝑚(𝑘)≠0; or 𝑎𝑚𝑗(k)≠0,…,𝑎𝑖𝑚(𝑘)≠0	Γ𝑖𝑗(𝑓)≠0
Spurious coupling		𝑦𝑚⇒𝑦𝑖 and 𝑦𝑚⇒𝑦𝑗	𝑎𝑠𝑚(𝑘)≠0,…,𝑎𝑖𝑠(𝑘)≠0; and 𝑎𝑠𝑚(𝑘)≠0,…,𝑎𝑗𝑠(𝑘)≠0