Conditional expectation estimation through attributable components

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.

Characterizing the Role of a Single Coupling Layer in Affine Normalizing Flows

“…When the function space of an Optimal Transport (OT) problem with quadratic ground cost is reduced to affine maps, the best possible transport matches mean and covariance of the involved distributions [ 17 ]. In the case of conditional distributions, affine maps become conditional affine maps [ 16 ]. We show such maps to have the same minimizer under maximum likelihood loss (KL divergence) as under OT costs.…”

Enabling personalized decision support with patient-generated data and attributable components

“…Blood pressure, for instance, may be related to many factors like age, exercise, diet, sex, prescribed drugs, and the device used to take the measurement. Here we adapt a optimal transport-based method invented by Tabak and Trigila [22] termed attributable components analysis (ACA). This method was created to explain variability in a quantity of interest based on a set of related or potentially confounding covariates, or "attributable components".…”

Conditional density estimation and simulation through optimal transport

“…3. We use a low-rank tensor factorization, variable separation procedure developed in Tabak and Trigila (2018a) to reduce multivariate functions to sums of products of functions of a single variable. These in turn are approximated as convex combinations of their values on prototypes (Chenyue and Tabak 2018).…”