Causality in Time Series - ClopiNet

Robust Statistics for Causalityto fall off to zero with increasing lag magnitude, a novel coherency estimate is proposedbased on the cardinal sine and cosine functions, which also decay, as a compact basis:Ĉ i j (ω) = a i j,n∑︁C(Ω n ) + jb i j,n S(Ω n ) (34)∑︁p,q∈1..N{︁ai j ,b i j}︁= argmin(︁xi,p x j,q − a i j,n cosc(2πΩ k (t i,p − t j,q )) − b i j,n sinc(2πΩ n (t i,p − t j,q )) )︁ 2Where the sine cardinal is defined as sinc(x) = sin(πx)/x, and its Fourier transformis S( jω) = 1,| jω| < 1 and S( jω) = 0 otherwise. Also the Fourier transform of the cosinecardinal can be written as C( jω) = jω S ( jω). Although in principle we could chooseany complete basis as a means of Fourier transform estimation, the cardinal transformpreserves the odd-even function structure of the standard trigonometric pair. Computationallythis means that for autocorrelations, which are real valued and even, only sincneeds to be calculated and used, while for cross-correlation both functions are needed.As linear mixtures of independent signals only have symmetric cross-correlations, anynonzero values of the cosc coefficients would indicate the presence of dynamic interaction.Note that the Fast Fourier Transform earns its moniker thanks to the orthogonalityof sin and cos which allows us to avoid a matrix inversion. However their orthogonalityholds true only for infinite support, and slight correlations are found for finite windows -in practice this effect requires further computation (windowing) to counteract. The cardinalbasis is not orthogonal, requires full regression and may have demanding memoryrequirements. For moderate size data this not problematic and implementation detailswill be discussed elsewhere.8.2. Robustness evaluation based on the NOISE datasetA dataset named NOISE, intended as a benchmark for the bivariate case, has beenintroduced in the preceding NIPS workshop on causality Nolte et al. (2010) and canbe found online at www.causality.inf.ethz.ch/repository.php, alongwith the code that generated the data. It was awarded best dataset prize in the previousNIPS causality workshop and challenge Guyon et al. (2010). For further discussion ofCausality Workbench and current dataset usage see Guyon (2011). NOISE is createdby the summation of the output of a strictly causal VAR DGP and a non-causal SVARDGP which consists of mixed colored noise:y C,i =x N,i =K∑︁k=1K∑︁k=1[︃ ]︃a11 a 120 a 22[︃ ]︃a11 00 a 22C,kN,k(35)y C,i−k + w C,i (36)x N,i−k + w N,i55