  This article advocates factorized and hybrid dimensional decompositions
(FDD/HDD), as alternatives to analysis-of-variance dimensional decomposition
(ADD), for second-moment statistical analysis of multivariate functions. New
formulae revealing the relationships between component functions of FDD and ADD
are proposed. While ADD or FDD is relevant when a function is strongly additive
or strongly multiplicative, HDD, whether formed linearly or nonlinearly,
requires no specific dimensional hierarchies. Furthermore, FDD and HDD lead to
alternative definitions of effective dimension, reported in the current
literature only for ADD. New closed-form or analytical expressions are derived
for univariate truncations of all three decompositions, followed by
mean-squared error analysis of univariate ADD, FDD, and HDD approximations. The
analysis finds appropriate conditions when one approximation is better than the
other. Numerical results affirm the theoretical finding that HDD is ideally
suited to a general function approximation that may otherwise require
higher-variate ADD or FDD truncations for rendering acceptable accuracy in
stochastic solutions.
