Causality in Time Series - ClopiNet

PopescuH GCj→i|u = logD i − logD ( j)i(21)The Shannon entropy of a Gaussian random variable is the logarithm of its standarddeviation plus a constant. Notice than in this paper the definition of Granger Causalityis slightly different than the literature in that it relates to the innovations process of amixed output SVAR system of closest rotation and not a regular MVAR. The secondterm D ( j)iis formed by computing a reduced SVAR system which omits the jth variable.Recently Barrett et al. have proposed an extension of GC, based on prior work byGeweke (1982) from interaction among pairs of variables to groups of variables, termedmultivariate Granger Causality (MVGC) Barrett et al. (2010). The above definition isstraightforwardly extensible to the group case, where I ad J are subsets of 1..D, sincetotal entropy of independent variables is the sum of individual entropies.H GCJ→I|u = ∑︁i∈I(︂logD i − logD (J)iThe Granger entropy can be calculated directly from the transfer function, using theShannon-Hartley theorem:H GCHj→i)︂(22)⎛ ⃒∑︁ ⃒⃒Hi= − ∆ω ln⎜⎝ 1 − j (ω) ⃒ 2 ⎞⃒S ii (ω)⎟⎠ (23)ωFinally Nolte (Nolte et al., 2008) introduced a method called Phase Slope Indexwhich evaluates bilateral causal interaction and is robust to mixing effects (i.e. zerolag, observation or innovations covariance matrices that depart from MVAR):⎛⎞∑︁PS I i j→i = Im⎜⎝C i * j (ω) C i j(ω + dω) ⎟⎠ (24)ωPSI, as a method is based on the observation that pure mixing (that is to say, alleffects stochastically equivalent to output mixing as outlined above) does not affectthe imaginary part of the coherency C i j just as (equivalently) it does not affect theantisymmetric part of the auto-correlation of a signal. It does not place a measure thephase relationship per se, but rather the slope of the coherency phase weighted by themagnitude of the coherency.7. Causal Structural InformationCurrently, Granger causality estimation based on linear VAR modeling has been shownto be susceptible to mixed noise, in the presence of which it may produce false causalityassignment Nolte et al. (2010). In order to allow for accurate causality assignment inthe presence of instantaneous interaction and aliasing the Causal Structural Information(CSI) method and statistic for causality assignment is introduced below.Consider the SVAR lower triangular form in (10) for a set of observations y. Theinformation transfer from i to j may be obtained by first defining the index re-orderings:52