Synergy, redundancy and unnormalized Granger causality

Abstract

Abstract-We analyze by means of Granger causality the effect of synergy and redundancy in the inference (from time series data) of the information flow between subsystems of a complex network. Whilst fully conditioned Granger causality is not affected by synergy, the pairwise analysis fails to put in evidence synergetic effects. We show that maximization of the total Granger causality to a given target, over all the possible partitions of the set of driving variables, puts in evidence redundant multiplets of variables influencing the target, provided that an unnormalized definition of Granger causality is adopted. Along the same lines we also introduce a pairwise index of synergy (w.r.t. to information flow to a third variable) which is zero when two independent sources additively influence a common target; thus, this definion differs from previous definitions of synergy.

Enhanced prefrontal functional–structural networks to support postural control deficits after traumatic brain injury in a pediatric population

“…One step that severely affects connectivity matrices is regressing (or not regressing) out the global signal. Here, in agreement with previous work ( Alonso-Montes et al, 2015 ; Amor et al, 2015 ; Diez, Bonifazi, et al, 2015 ; Diez, Erramuzpe, et al, 2015 ; Mäki-Marttunen et al, 2013 ; Marinazzo et al, 2014 ; Stramaglia, Angelini, Cortes, Marinazzo, 2015 ; Stramaglia et al, 2016 ), we regressed from each individual time series the global signal, which is well-known to produce more negative correlations in functional connectivity matrices ( Murphy, Birn, Handwerker, Jones, Bandettini, 2009 ; Saad et al, 2012 ). After we repeated the entire analysis without regressing out the global signal, Figure 3 did not change, but the results for the task-positive network differed from those shown in Figure 4 .…”

Functional connectivity change as shared signal dynamics

“…It will be important to determine which of the existing directional/effective connectivity methods (Friston et al 2003; Roebroeck et al 2005; Nolte et al 2008; Ramsey et al 2011; Smith et al 2011b) involve variance normalization, and if all of them do, then it will be important to develop new approaches that do not involve this analysis step for studies examining effective connectivity change. Importantly, there is already evidence of advantages when using an unnormalized version of a popular form of effective connectivity, Granger causality (Angelini et al 2010; Stramaglia et al 2015). …”

“…In the future it may be useful to identify partial correlation-like approaches that are adapted to not include variance-based normalization. One promising possibility is to use multivariate Granger causality without variance-based normalization (Angelini et al 2010; Stramaglia et al 2015), which estimates all time series simultaneously to achieve the main benefits of partial correlation in the context of directional connectivity. Note, however, that potential issues with using Granger causality with fMRI have been identified (Smith et al 2011a; 2011b), such that this approach may be best applied to other modalities such as EEG.…”

Effect of variations of the complexity of the target variable on the assessment of Wiener–Granger causality in cardiovascular control studies

“…This hypothesis deserves to be tested because recently the appeal of un-normalized WGC indexes, such as the UPD, has been amplified by the proof that they are more suitable than normalized WGC markers for the assessment of the redundant/synergistic behaviors of source signals in transferring the information to the destination one (Barrett 2015). In addition, the UPD has two extra advantages: (i) it is directly linked to the decomposition of the total predictability of the target signal in the complete universe of knowledge into the portion assessed in the restricted universe of knowledge and the one genuinely associated to the introduction of the presumed cause (Porta et al 2015b); (ii) the UPD from a group of independent sources to a given destination can be decomposed into partial UPDs due to the single sources (Angelini et al 2010, Stramaglia et al 2015.…”