Figure 4.

The vector-space illustration of global mean regression. (A) The characteristic time series for network A can be described as a single point in a high-dimensional vector space. Relative to 0 (the zero time series, black dot) the orthogonal plane (dotted line in this example) separates the vector space into an area of positive correlation (r_A > 0) and a subspace of time series negatively correlated with A. The correlation between A and any other point is defined by the (cosine of) the inner angle: all points within ± 90° are positively correlated with A, whereas all other points are negatively correlated with A; (B) when regressing out the mean of two network-specific time series A and B, the 0 reference point is moved half-way between the two points and the original time series get projected onto the subspace perpendicular to this mean, thereby inducing perfect anti-correlation between A and B as the new characteristic vectors are now aligned at 180°; (C) in the more general case of multiple networks (grey dots) the range of possible differences in pair-wise correlations is again determined by the maximum range of the inner angles α: if α is small, pair-wise correlations differ by only a small amount and delineation of different networks becomes difficult, in this example all pair-wise correlations are positive; (D) the global mean necessarily lies within the convex hull spanned by all the individual characteristic time series. Global time-series regression moves the 0 reference point somewhere into the convex hull, thereby inevitably inducing spurious negative correlations between the characteristic time series associated with different RSNs. Global mean regression does increase the maximum inner angle between pairs of time courses and therefore facilitates delineation of networks from each other; the resulting correlation scores (and signs thereof), however, are no longer interpretable and reference to these should be avoided.