Extracting individual variable information for their decoupling, direct
  mutual information and multi-feature Granger causality
Jarek Duda
Working with multiple variables they usually contain difficult to control
complex dependencies. This article proposes extraction of their individual
information, e.g. $\overline{X|Y}$ as random variable containing information
from $X$, but with removed information about $Y$, by using $(x,y)
\leftrightarrow (\bar{x}=\textrm{CDF}_{X|Y=y}(x),y)$ reversible normalization.
One application can be decoupling of individual information of variables:
reversibly transform $(X_1,\ldots,X_n)\leftrightarrow(\tilde{X}_1,\ldots
\tilde{X}_n)$ together containing the same information, but being independent:
$\forall_{i\neq j} \tilde{X}_i\perp \tilde{X}_j, \tilde{X}_i\perp X_j$. It
requires detailed models of complex conditional probability distributions - it
is generally a difficult task, but here can be done through multiple dependency
reducing iterations, using imperfect methods (here HCR: Hierarchical
Correlation Reconstruction). It could be also used for direct mutual
information - evaluating direct information transfer: without use of
intermediate variables. For causality direction there is discussed
multi-feature Granger causality, e.g. to trace various types of individual
information transfers between such decoupled variables, including propagation
time (delay).