Causality in Time Series - ClopiNet

Roebroeck Seth Valdes-SosaTable 1: Types of Influence defined by absence of the corresponding independence relations.See text for acronym definitions.Global( All horizons)Local(Immediate future)ContemporaneousStrong(Probability Distribution)By absence ofstrong, conditional, global independence:X 2 (t)SCGi X 1 (t)||X 3 (t)By absence ofstrong, conditional, local independence:X 2 (t)SCLi X 1 (t)||X 3 (t)By absence ofstrong, conditional, contemporaneousindependence:X 2 (t)SCCi X 1 (t)||X 3 (t)Weak(Expectation)By absence ofweak, conditional, global independence:X 2 (t)WCGi X 1 (t)||X 3 (t)By absence ofweak, conditional, local independence:X 2 (t)WCLi X 1 (t)||X 3 (t)By absence ofweak, conditional, contemporaneousindependence:X 2 (t)WCCi X 1 (t)||X 3 (t)this definition is appropriate for point processes, discrete and continuous time series,even for categorical (qualitative valued) time series. The only problem with this formulationis that it calls on the whole probability distribution and therefore its practicalassessment requires the use of measures such as mutual information that estimate theprobability densities nonparametrically.As an alternative, weak concepts of influence can be defined based on expectations.Consider weak conditional local independence in discrete time, which is defined:E [ X 1 [t + ∆t]| X 1 [t,−∞], X 2 [t,−∞], X 3 [t,−∞]] = E [ X 1 [t + ∆t]| X 1 [t,−∞], X 3 [t,−∞]](7)When this condition does not hold we say X 2 weakly, conditionally and locally influences(WCLi) X 1 given X 3 . To make the implementation this definition insightful,consider a discrete first-order vector auto-regressive (VAR) model for X = [X 1 X 2 X 3 ]:X [t + ∆t] = AX [t] + e[t + ∆t] (8)For this case E [ X[t + ∆t]| X[t,−∞]] = AX [t], and analyzing influence reduces to findingwhich of the autoregressive coefficients are zero. Thus, many proposed operationaltests of WAGS influence, particularly in fMRI analysis, have been formulated as testsof discrete autoregressive coefficients, although not always of order 1. Within the samemodel one can operationalize weak conditional instantaneous independence in dis-84