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PopescuFigure 1: SVAR causality and equivalence. Structural VAR equivalence and causality.A) direct structural Granger causality (both directions shown). z standsfor the delay operator. B) equivalent covariate innovations (left) and mixedoutput systems. Neither representation shows dynamic interaction C) sparse,one sided covariate innovations DAG is non sparse in the mixed output case(and vice-versa). D) upper triangular structure of the zero-lag matrix is notinformative in the 2 variable Gaussian case, and is equivalent to a full mixedoutput system.48
Robust Statistics for CausalityGranger, in his 1969 paper, suggests that ‘instantaneous’ (i.e. covariate) effects beignored and only the temporal structure be used. Whether or not we accept instantaneouscausality depends on prior knowledge: in the case of EEG, the mixing matrixcannot have any physical ‘causal’ explanation even if it is sparse. Without additionala priori assumptions, either we infer causality on unseen and presumably interactinghidden variables (mixed output form, the case of EEG/MEG) or we assume a a noncausalmixed innovations input. Note also that the zero-lag system appears to be causalbut can be written in a form which suggest the opposite difference causal influence(hence it is sometimes termed ‘spurious causality’). In short, since instantaneous interactionin the Gaussian case cannot be resolved causally purely in terms of predictionand conditional information (as intended by Wiener and Granger), it is proposed thatsuch interactions be accounted for but not given causal interpretation (as in ‘strong’Granger non-causality) .There are at least four distinct overall approaches to dealing with aliasing effectsin time series causality. 1) is to make prior assumptions about covariance matrices andlimit inference to domain relevant and interpretable posteriors, as in Bernanke et al.(2005) in economics and Valdes-Sosa et al. (2005) in neuroscience. 2) to allow forunconstrained graphical causal model type inference among covariate innovations, byeither assuming Gaussianity or non-Gaussianity, the latter allowing for stronger causalinferences (see Moneta et al. (2011) in this volume). One possible drawback of thisapproach is that DAG-type inference, at least in the Gaussian case in which there isso-called ’Markov equivalence’ among candidate graphs, is non-unique. 3) a physicallyinterpretable mixed output or co-variate innovations is assumed and the inferredsparsity structure (or the intersection thereof over the nonzero lag coefficient matrices)as the connection graph. Popescu (2008) implemented such an approach by using theminimum description length principle to provide a universal prior over rational-valuedcoefficients, and was able to recover structure in the majority of simulated co-variateinnovations processes of arbitrary sparsity. This approach is computationally laborious,as it is NP and non-convex, and moreover a system that is sparse in one form (covariateinnovations or mixed-ouput) is not necessarily sparse in another equivalent SVARform. Moreover completely dense SVAR systems may be non-causal (in the strongGC sense). 4) Causality is not interpreted as a binary value, but rather direction ofinteraction is determined as a continuous valued statistic, and one which is theoreticallyrobust to covariate innovations or mixtures. This is the principle of the recentlyintroduced phase slope index (PSI), which belongs to a class of methods based on spectraldecomposition and partition of coherency. Although auto-regressive, spectral andimpulse response convolution are theoretically equivalent representation of linear dynamics,they do differ numerically and spectral representations afford direct access tophase estimates which are crucial to the interpretation of lead and lag as it relates tocausal influence. These methods are reviewed in the next section.49
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Causality in Time SeriesVolume 5:Ca
Page 5: PrefaceThe following is a print ver
Page 9 and 10: JMLR: Workshop and Conference Proce
Page 11 and 12: Linking Granger Causality and the P
Page 39: Linking Granger Causality and the P
Page 42 and 43: Popescu1. IntroductionCausality is
Page 44 and 45: Popescugeneral terms, if we are pre
Page 46 and 47: PopescuType III error prob. γ = P
Page 48 and 49: Popescuwhich would correspond to a
Page 50 and 51: Popescuvalued vector. A Data Genera
Page 52 and 53: Popescuy i =K∑︁A k y i−k + Bu
Page 54 and 55: Popescuy i =K∑︁A k y i−k + Bu
Page 58 and 59: Popescu6. Spectral methods and phas
Page 60 and 61: PopescuH GCj→i|u = logD i − log
Page 62 and 63: Popescu8.1. The cardinal transform
Page 64 and 65: Popescuy N,i = Bx N,i (37)y = (1
Page 66 and 67: Popescu(a) Unmixed colored noise(b)
Page 68 and 69: PopescuAs we can see in both Figure
Page 70 and 71: Popescuβ > 0 y C,i =x N,i =⎡K∑
Page 72 and 73: Popescupretation of information flo
Page 74 and 75: Popescuit is suggested that a princ
Page 76 and 77: PopescuA. N. Kolmogorov and A. N. S
Page 78 and 79: PopescuH. White and X. Lu. Granger
Page 80 and 81: PopescuTable 6: α vs. ΨN * → 50
Page 82 and 83: Roebroeck Seth Valdes-Sosaincreasin
Page 84 and 85: Roebroeck Seth Valdes-SosaFigure 1:
Page 86 and 87: Roebroeck Seth Valdes-Sosasignal fr
Page 88 and 89: Roebroeck Seth Valdes-Sosablood flo
Page 90 and 91: Roebroeck Seth Valdes-Sosaat lower
Page 92 and 93: Roebroeck Seth Valdes-SosaTable 1:
Page 94 and 95: Roebroeck Seth Valdes-Sosa1984) and
Page 96 and 97: Roebroeck Seth Valdes-Sosaanalysis
Page 98 and 99: Roebroeck Seth Valdes-Sosamarginal
Page 100 and 101: Roebroeck Seth Valdes-SosaThe initi
Page 102 and 103: Roebroeck Seth Valdes-Sosaobserved
Page 104 and 105: Roebroeck Seth Valdes-Sosaand corti
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Roebroeck Seth Valdes-SosaDaniel Co
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Roebroeck Seth Valdes-SosaM. Havlic
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Roebroeck Seth Valdes-SosaA. Roebro
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Roebroeck Seth Valdes-SosaV. Walsh
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Moneta Chlass Entner HoyerA traditi
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Moneta Chlass Entner Hoyerto the
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Moneta Chlass Entner Hoyersecond st
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Moneta Chlass Entner HoyerThe Wald
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Moneta Chlass Entner HoyerStep 3 Ap
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Moneta Chlass Entner Hoyerthe struc
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Moneta Chlass Entner Hoyerstricted
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Moneta Chlass Entner Hoyerwhere h 1
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Moneta Chlass Entner Hoyerlatter re
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Moneta Chlass Entner Hoyercausal mo
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Moneta Chlass Entner HoyerL. Baring
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Moneta Chlass Entner HoyerS. Johans
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Moneta Chlass Entner HoyerA. Yatche
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Guyon Statnikov Aliferisis extremel
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Guyon Statnikov Aliferis4. A Data R
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Guyon Statnikov Aliferismethods on
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Guyon Statnikov Aliferisvolume 3, p
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