Causality in Time Series - ClopiNet

Popescu8.1. The cardinal transform of the autocorrelationCurrently there are few commonly used methods for cross- power spectrum estimation(i.e. multivariate spectral power estimation) as opposed to univariate power spectrumestimation, and these methods average over repeated, or shifting, time windows andtherefore require a lot of data points. Furthermore all commonly used multivariate spectralpower estimation methods rely on synchronous, evenly spaced sampling, despitethe fact that much of available data is unevenly sampled, has missing values, and canbe composed of relatively short sequences. Therefore a novel method is presented belowfor multivariate spectral power estimation which can be estimated on asynchronousdata.Returning to the definition of coherency as the Fourier transform of the auto-correlation,which are both continuous transforms, we may extend the conceptual means of its estimationin the discrete sense as a regression problem (as a discrete Fourier transform,DFT) in the evenly sampled case as:Ω n n, n = −⌊N/2⌋...⌊N/2⌋ (28)2τ 0 (N − 1)Ĉ i j (ω)| ω=Ω = a i j,n + jb i j,n (29)ρ ji (−kτ) = ρ i j (kτ) = E(x i (t)x j (t + kτ)) (30)ρ i j (kτ 0 ) 1N − k∑︁q=1:N−kx i (q)x j (q + k) (31){︁ }︁ ∑︁N/2ai j ,b i j = argmink=−N/2(︁ρi j (kτ 0 ) − a i j,n cos(2πΩ n τ 0 k) − b i j,n sin(2πΩ n τ 0 k) )︁ 2(32)where τ 0 is the sampling interval. Note that for an odd number of points the regressionabove is actually a well determined set of equations, corresponding to the 2-sidedDFT. Note also that by replacing the expectation with the geometric mean, the aboveequation can also be written (with a slight change in weighting at individual lags) as:{︁ai j ,b i j}︁= argmin∑︁(︁xi,p x j,q − a i j,n cos(2πΩ k (t i,p − t j,q )) − b i j,n sin(2πΩ n (t i,p − t j,q )) )︁ 2p,q∈1..N(33)The above equation holds even for time series sampled at unequal (but overlapping)times (x i ,t i ) and (x j ,t j ) as long as the frequency basis definition is adjusted (for exampleτ 0 = 1). It represents a discrete, finite approximation of the continuous, infinite autoregressionfunction of an infinitely long random process. It is a regression on the outerproduct of the vectors x i and x j . Since autocorrelations for finite memory systems tend54