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

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