  For the evaluation of information flow in bivariate time series, information
measures have been employed, such as the transfer entropy (TE), the symbolic
transfer entropy (STE), defined similarly to TE but on the ranks of the
components of the reconstructed vectors, and the transfer entropy on rank
vectors (TERV), similar to STE but forming the ranks for the future samples of
the response system with regard to the current reconstructed vector. Here we
extend TERV for multivariate time series, and account for the presence of
confounding variables, called partial transfer entropy on ranks (PTERV). We
investigate the asymptotic properties of PTERV, and also partial STE (PSTE),
construct parametric significance tests under approximations with Gaussian and
gamma null distributions, and show that the parametric tests cannot achieve the
power of the randomization test using time-shifted surrogates. Using
simulations on known coupled dynamical systems and applying parametric and
randomization significance tests, we show that PTERV performs better than PSTE
but worse than the partial transfer entropy (PTE). However, PTERV, unlike PTE,
is robust to the presence of drifts in the time series and it is also not
affected by the level of detrending.
