Grouping time series by pairwise measures of redundancy

A novel approach is proposed to group redundant time series in the frame of causality. It assumes that (i) the dynamics of the system can be described using just a small number of characteristic modes, and that (ii) a pairwise measure of redundancy is sufficient to elicit the presence of correlated degrees of freedom. We show the application of the proposed approach on fMRI data from a resting human brain and gene expression profiles from HeLa cell culture.

💡 Research Summary

The paper introduces a straightforward yet powerful methodology for clustering redundant time series based on a pairwise redundancy measure derived from a low‑dimensional representation of the system’s dynamics. The authors begin by assuming that the essential behavior of a complex system can be captured by a small number of characteristic modes, which they obtain via principal component analysis (PCA) of the full multivariate data matrix. After normalizing and lag‑shifting the original series (Xᵢ(t)=xᵢ(t‑1)), they compute the eigenvectors (principal components) of the covariance matrix xᵀx and retain the top nλ components, denoted uα(t).

Only those lagged series that show a statistically significant correlation (after Bonferroni correction) with at least one retained component are kept; these constitute the reduced set {Yᵢ}, of size N. For each pair (Yi, Yj) the authors define projection operators Pi, Pj (onto the one‑dimensional subspaces spanned by Yi and Yj) and Pij (onto the two‑dimensional subspace spanned by both). The pairwise redundancy index is then calculated as

cᵢⱼ = Σα=1ⁿλ (‖Pi uα‖² + ‖Pj uα‖² – ‖Pij uα‖²).

If cᵢⱼ > 0 the two series are deemed redundant (they convey overlapping information about the future modes); if cᵢⱼ < 0 they are synergistic (they provide complementary information). The symmetric matrix C =