  Quantification of relations between measured variables of interest by
statistical measures of dependence is a common step in analysis of climate
data. The term "connectivity" is used in the network context including the
study of complex coupled dynamical systems. The choice of dependence measure is
key for the results of the subsequent analysis and interpretation. The use of
linear Pearson's correlation coefficient is widespread and convenient. On the
other side, as the climate is widely acknowledged to be a nonlinear system,
nonlinear connectivity quantification methods, such as those based on
information-theoretical concepts, are increasingly used for this purpose.
  In this paper we outline an approach that enables well informed choice of
connectivity method for a given type of data, improving the subsequent
interpretation of the results. The presented multi-step approach includes
statistical testing, quantification of the specific non-linear contribution to
the interaction information, localization of nodes with strongest nonlinear
contribution and assessment of the role of specific temporal patterns,
including signal nonstationarities. In detail we study the consequences of the
choice of a general nonlinear connectivity measure, namely mutual information,
focusing on its relevance and potential alterations in the discovered
dependence structure.
  We document the method by applying it on monthly mean temperature data from
the NCEP/NCAR reanalysis dataset as well as the ERA dataset. We have been able
to identify main sources of observed non-linearity in inter-node couplings.
Detailed analysis suggested an important role of several sources of
nonstationarity within the climate data. The quantitative role of genuine
nonlinear coupling at this scale has proven to be almost negligible, providing
quantitative support for the use of linear methods for this type of data.
