Causality in Time Series - ClopiNet

Popescu6. Spectral methods and phase estimationCross- and auto spectral densities of a time series, assuming zero-mean or de-trendedvalues, are defined as:ρ Li j (τ) = E (︁ y i (t)y j (t − τ) )︁S i j (ω) = F (ρ Li j (τ)) (12)Note that continuous, linear, raw correlation values are used in the above definitionas well as the continuous Fourier transform. Bivariate coherency is defined as:C i j (ω) =S i j (ω)√︀ S ii(ω)S j j (ω)(13)Which consists of a complex numerator and a real-valued denominator. The coherenceis the squared magnitude of the coherency:c i j (ω) = C i j (ω) * C i j (ω) (14)Besides various histogram and discrete (fast) Fourier transform methods availablefor the computation of coherence, AR methods may be also used, since they are alsolinear transforms, the Fourier transform of the delay operator being simply z k = e − j2πωτ Swhere τ S is the sampling time and k = ωτ S . Plugging this into Equation (9) we obtain:⎛⎞K∑︁X( jω) = ⎜⎝ A k e − j2πωτ S kk=1⎛⎞K∑︁Y( jω) = C ⎜⎝ I − A k e − j2πωτ S k⎟⎠k=1⎟⎠ X( jω) + BU( jω) + DY( jω) = CX( jω) (15)−1(BU( jω) + DW( jω)) (16)In terms of a SVAR therefore (as opposed to VAR) the mixing matrix C does notaffect stability, nor the dynamic response (i.e. the poles). The transfer functions fromith innovations to jth output are entries of the following matrix of functions:⎛⎞K∑︁H( jω) = C ⎜⎝ I − A k e − j2πωτ S k⎟⎠k=1−1D (17)The spectral matrix is simply (having already assumed independent unit Gaussiannoise):S ( jω) = H( jω) * H( jω) (18)The coherency as the coherence following definitions above. The partial coherenceconsiders the pair (i, j) of signals conditioned on all other signals, the (ordered) set ofwhich we denote (i, j):50