On decompositions of multivariate functions

We present formulas that allow us to decompose a function f of d variables into a sum of 2d terms fu indexed by subsets u of {1, . . . , d}, where each term fu depends only on the variables with indices in u. The decomposition depends on the choice of d commuting projections {Pj}j=1, where Pj(f) does not depend on the variable xj . We present an explicit formula for fu, which is new even for the anova and anchored decompositions; both are special cases of the general decomposition. We show that the decomposition is minimal in the following sense: if f is expressible as a sum in which there is no term that depends on all of the variables indexed by the subset z then, for every choice of {Pj}j=1, the terms fu = 0 for all subsets u containing z. Furthermore, in a reproducing kernel Hilbert space setting, we give sufficient conditions for the terms fu to be mutually orthogonal.