  A problem of great interest in real world systems, where multiple time series
measurements are available, is the estimation of the intra-system causal
relations. For instance, electric cortical signals are used for studying
functional connectivity between brain areas, their directionality, the direct
or indirect nature of the connections, and the spectral characteristics (e.g.
which oscillations are preferentially transmitted). The earliest spectral
measure of causality was Akaike's (1968) seminal work on the noise contribution
ratio, reflecting direct and indirect connections. Later, a major breakthrough
was the partial directed coherence of Baccala and Sameshima (2001) for direct
connections. The simple aim of this study consists of two parts: (1) To expose
a major problem with the partial directed coherence, where it is shown that it
is affected by irrelevant connections to such an extent that it can
misrepresent the frequency response, thus defeating the main purpose for which
the measure was developed, and (2) To provide a solution to this problem,
namely the "isolated effective coherence", which consists of estimating the
partial coherence under a multivariate auto-regressive model, followed by
setting all irrelevant associations to zero, other than the particular
directional association of interest. Simple, realistic, toy examples illustrate
the severity of the problem with the partial directed coherence, and the
solution achieved by the isolated effective coherence. For the sake of
reproducible research, the software code implementing the methods discussed
here (using lazarus free-pascal "www.lazarus.freepascal.org"), including the
test data as text files, are freely available at:
https://sites.google.com/site/pascualmarqui/home/icoh-isolated-effective-coherence
