Extracting individual variable information for their decoupling, direct mutual information and multi-feature Granger causality

Working with multiple variables they usually contain difficult to control complex dependencies. This article proposes extraction of their individual information, e.g. overlineX|Y as random variable containing information from X, but with removed information about Y, by using (x,y) leftrightarrow (barx=textrmCDFX|Y=y(x),y) reversible normalization. One application can be decoupling of individual information of variables: reversibly transform (X1,ldots,Xn)leftrightarrow(tildeX1,ldots tildeXn) together containing the same information, but being independent: forallineq j tildeXiperp tildeXj, tildeXiperp Xj. 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).