Conditional expectation estimation through attributable components

Abstract

Abstract A general methodology is proposed for the explanation of variability in a quantity of interest x in terms of covariates z = (z1, …, zL). It provides the conditional mean $\bar{x}(z)$ as a sum of components, where each component is represented as a product of non-parametric one-dimensional functions of each covariate zl that are computed through an alternating projection procedure. Both x and the zl can be real or categorical variables; in addition, some or all values of each zl can be unknown, providing a general framework for multi-clustering, classification and covariate imputation in the presence of confounding factors. The procedure can be considered as a preconditioning step for the more general determination of the full conditional distribution $\boldsymbol{\rho}(x|z) $ through a data-driven optimal-transport barycenter problem. In particular, just iterating the procedure once yields the second order structure (i.e. the covariance) of $\boldsymbol{\rho}(x|z) $. The methodology is illustrated through examples that include the explanation of variability of ground temperature across the continental United States and the prediction of book preference among potential readers.