Orthogonal decompositions of multivariate statistical dependence measures

We describe two multivariate statistical dependence measures which can be orthogonally decomposed to separate the effects of pairwise, triplewise, and higher order interactions between the random variables. These decompositions provide a convenient method of analyzing statistical dependencies between large groups of random variables, within which smaller "sub-groups" may exhibit dependencies separately from the rest of the variables. The first dependence measure is a generalization of Pearson\´s φ², and we decompose it using an orthonormal series expansion of joint probability density functions. The second measure is based on the Kullback-Leibler distance, and we decompose it using information geometry. Applications of these techniques include analysis of neural population recordings and multimodal sensor fusion. We discuss in detail the simple example of three jointly defined binary random variables.