Time delay multi-feature correlation analysis to extract subtle dependencies from EEG signals

Electroencephalography (EEG) signals are resultants of extremely complex brain activity. Some details of this hidden dynamics might be accessible through e.g. joint distributions $\rho_{\Delta t}$ of signals of pairs of electrodes shifted by various time delays (lag $\Delta t$). A standard approach is monitoring a single evaluation of such joint distributions, like Pearson correlation (or mutual information), which turns out relatively uninteresting - as expected, there is usually a small peak for zero delay and nearly symmetric drop with delay. In contrast, such a complex signal might be composed of multiple types of statistical dependencies - this article proposes approach to automatically decompose and extract them. Specifically, we model such joint distributions as polynomials, estimated separately for all considered lag dependencies, then with PCA dimensionality reduction we find the dominant joint density distortion directions $f_v$. This way we get a few lag dependent features $a_i(\Delta t)$ describing separate dominating statistical dependencies of known contributions: $\rho_{\Delta t}(y,z)\approx \sum_{i=1}^r a_i(\Delta t)\, f_{v_i}(y,z)$. Such features complement Pearson correlation, extracting hidden more complex behavior, e.g. with asymmetry which might be related with direction of information transfer, extrema suggesting characteristic delays, or oscillatory behavior suggesting some periodicity. There is also discussed extension of Granger causality to such multi-feature joint density analysis, suggesting e.g. two separate causality waves. While this early article is initial fundamental research, in future it might help e.g. with understanding of cortex hidden dynamics, diagnosis of pathologies like epilepsy, determination of precise electrode position, or building brain-computer interface.