3.1 Large-scale Granger causality analysis

We use large-scale Granger causality (lsGC) [3] to obtain a measure of directed multivariate information flow between every pair of regional time-series in the resting-state human brain. The principle of lsGC is derived from that of Granger causality [5, 41], which states that if the prediction quality of time-series x_s improves in the presence of time-series x_r, then x_r Granger causes x_s. Large-scale Granger causality comprises of estimating a time point in the future, using vector auto-regressive (VAR) modelling (of order d), in a dimension-reduced space defined by retaining the first c Principal Components (PC) of the time-series ensemble. These predictions are transformed back to the original time-series ensemble space using the inverse of the linear transformation function in the initial rotation of the axes to the PC space. The prediction errors, i.e. residuals, are used to obtain the lsGC coefficients. We calculate the variance of the residuals, when the full ensemble was used for obtaining the PCs, and compare these with those residuals obtained when information of one time-series is removed. We thus obtain a matrix of lsGC coefficients, containing measures of directed multivariate information flow scores for every pair of time series.

Consider an ensemble of time-series X  ∈ ℝ^N×T, where N and T are the number of time-series and the number of temporal samples respectively. Let X = (x₁, x₂, x₃, …, x_N)^T be the whole multidimensional system, x_n  ∈ ℝ^T a single time-series with n ∈ {1,2, …, N}, where x_n = (x_n(1), x_n(2), …, x_n(T)). Large-scale Granger causality differs from cGC by performing Granger causality analysis in an embedded lower dimension space. This is carried out by obtaining a multivariate GC score by first decomposing X into its first c high-variance principal components Z  ∈ ℝ^c×Tusing Principal Component Analysis (PCA), i.e.,

Subsequently, Z is modelled using Vector Auto-Regressive (VAR) modelling of order d and the estimate Z^ is computed by using the d auto-regression parameter matrices, each of size c × c. To obtain the influence of time-series x_r on all other time-series, we remove the information of x_r from the transformation matrix W, obtain Z_X\xr and model Z^X\xr as its VAR estimate.

The two estimates Z^ and Z^X\xr are projected into the original high-dimensional space using the respective inverse PCA transformation. Following this, the errors, in the original high dimensional space, E and E_X\xr, are computed after which we quantify the prediction quality by comparing the variance of the prediction errors obtained with and without consideration of x_r. If the variance of the prediction error of a given time series, say x_s, decreases with using x_r, then we say that x_r Granger-causes x_s [4].

F_xr→xsis the GC index for the influence of x_r on x_s, which is stored in the affinity matrix A at position A_{(s, r).} e_xs,X\xris the error in predicting x_s, when x_r was not considered, and e_xsis the error when x_r was used.

To quantitatively evaluate the performance of our lsGC method, we compare this approach to a conditional Granger causality method (referred to as cGC here) [4] from the literature, which does not require any low-dimensional embedding and obtains GC coefficients in the original ensemble space. For cGC, Z = X where W is the identity matrix. However, the need for lsGC becomes relevant, when N and T are very similar or when N > T, where N is the size of the ensemble, i.e. the number of time-series, and T is the length of the series. The quantities N and d both increases the number of parameters to be computed, whose estimation is limited by length T. The lsGC method alleviates this problem, as it incorporates a reversible dimension reduction step by decreasing the number of free parameters to be estimated for time-series modelling.