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Measuring frequency domain Granger causality for multiple blocks of interacting time series. (English) Zbl 1267.92022

Summary: In the past years, several frequency-domain causality measures based on vector autoregressive time series modeling have been suggested to assess directional connectivity in neural systems. The most followed approaches are based on representing the considered set of multiple time series as a realization of two or three vector-valued processes, yielding the so-called J. Geweke linear feedback measures [J. Am. Stat. Assoc. 79, 907–915 (1984; Zbl 0553.62083)] or as a realization of multiple scalar-valued processes, yielding popular measures like the directed coherence (DC) and the partial DC (PDC). In the present study, these two approaches are unified and generalized by proposing novel frequency-domain causality measures which extend the existing measures to the analysis of multiple blocks of time series. Specifically, the block DC (bDC) and block PDC (bPDC) extend DC and PDC to vector-valued processes, while their logarithmic counterparts, denoted as multivariate total feedback \(f^{\mathrm{m}}\) and direct feedback \(g^{\mathrm{m}}\), represent into a full multivariate framework the Geweke measures. Theoretical analysis of the proposed measures shows that they: (i) possess desirable properties of causality measures; (ii) are able to reflect either direct causality (bPDC, \(g^{\mathrm{m}}\)) or total (direct + indirect) causality (bDC, \(f^{\mathrm{m}}\)) between time series blocks; (iii) reduce to the DC and PDC measures for scalar-valued processes, and to the Geweke’s measures for pairs of processes; (iv) are able to capture internal dependencies between the scalar constituents of the analyzed vector processes. Numerical analysis showed that the proposed measures can be efficiently estimated from short time series, allow to represent in an objective, compact way the information derived from the causal analysis of several pairs of time series, and may detect frequency domain causality more accurately than existing measures. The proposed measures find their natural application in the evaluation of directional interactions in neurophysiological settings where several brain activity signals are simultaneously recorded from multiple regions of interest.

MSC:

92C20 Neural biology
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
92-08 Computational methods for problems pertaining to biology

Citations:

Zbl 0553.62083

Software:

TETRAD
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References:

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