Causality in Time Series - ClopiNet

Robust Statistics for Causalityi j* {i, j,i j}i* {i,i j}This means that we reorder the (identified) mixed-innovations system by placingthe target time series first and the driver series second, followed by all the rest. Thesame ordering, minus the driver is also useful. We define CSI asCS I( j → i|i j) log( ⊳ i j*D 11 ) − log( ⊳i* D 11 ) (25)CS I(i, j|i j) CS I( j → i|i j) −CS I(i → j|i j) (26)Where the D is upper-triangular form in each instance. This Granger Causality formulationrequires the identification of 3 different SVAR models, one for the entire timeseries vector, and one each for all elements except i and all elements except j. ViaCholesky decomposition, the logarithm of the top vertex of the triangle is proportionalto the entropy rate (conditional information) of the innovations process for the targetseries given all other (past and present) information including the innovations process.While this definition is clearly an interpretation of the core idea of Granger causality,it is, like DTF and PDC, not an independence statistic but a measure of (causal) informationflow among elements of a time-series vector. Note the anti-symmetry (bydefinition) of this information measure CS I(i, j|i j) = −CS I( j,i|i j) . Note also thatCS I( j → i|i j) and CS I(i → j|i j) may very conceivably have the same sign: the varioustriangular forms used to derive this measure are purely for calculation purposes,and do not carry intrinsic meaning. As a matter of fact other re-orderings and SVARforms may be employed for convenient calculation as well. In order to improve the explanatorypower of the CSI statistic the following normalization is proposed, mirroringthat defined in Equation (5) :F CS Ij→i|i j CS I(i, j|i j)log( ⊳i* D 11 ) + log( ⊳ j* D 11 ) + ζThis normalization effectively measures the ratio of causal to non-causal information,where ζ is a constant which depends on the number of dimensions and quantizationwidth and is necessary to transform continuous entropy to discrete entropy.8. Estimation of multivariate spectra and causality assignmentIn Section 6 and a series of causality assignment methods based on spectral decompositionof a multivariate signal were described. In this section spectral decompositionitself will be discussed, and a novel means of doing so for unevenly sampled data willbe introduced and evaluated along with the other methods for a bivariate benchmarkdata set.(27)53