  While the linear Pearson correlation coefficient represents a
well-established normalized measure to quantify the interrelation of two
stochastic variables $X$ and $Y$, it fails for multidimensional variables such
as Cartesian coordinates. Avoiding any assumption about the underlying data,
the mutual information $I(X, Y)$ does account for multidimensional
correlations. However, unlike the normalized Pearson correlation, it has no
upper bound ($I \in [0, \infty)$), i.e., it is not clear if say, $I = 0.4$
corresponds to a low or a high correlation. Moreover, the mutual information
(MI) involves the estimation of high-dimensional probability densities (e.g.,
six-dimensional for Cartesian coordinates), which requires a k-nearest neighbor
algorithm, such as the estimator by Kraskov et al. [Phys. Rev. E 69, 066138
(2004)]. As existing methods to normalize the MI cannot be used in connection
with this estimator, a new approach is presented, which uses an entropy
estimation method that is invariant under variable transformations. The
algorithm is numerically efficient and does not require more effort than the
calculation of the (un-normalized) MI. After validating the method by applying
it to various toy models, the normalized MI between the $C_{\alpha}$
-coordinates of T4 lysozyme is considered and compared to a correlation
analysis of inter-residue contacts.
