Conditional Transformation Models

TL;DR

This paper introduces conditional transformation models that relax traditional assumptions in regression, allowing the entire conditional distribution to depend on explanatory variables, thus enabling more flexible probabilistic modeling.

Contribution

It proposes a novel semiparametric framework for regression that models the full conditional distribution with transformation functions depending on covariates, estimated via regularised scoring rules.

Findings

More effective than kernel-based methods in heteroscedastic settings

Allows modeling of heteroscedasticity and distributional heterogeneity

Consistent estimation of conditional distribution functions

Abstract

The ultimate goal of regression analysis is to obtain information about the conditional distribution of a response given a set of explanatory variables. This goal is, however, seldom achieved because most established regression models only estimate the conditional mean as a function of the explanatory variables and assume that higher moments are not affected by the regressors. The underlying reason for such a restriction is the assumption of additivity of signal and noise. We propose to relax this common assumption in the framework of transformation models. The novel class of semiparametric regression models proposed herein allows transformation functions to depend on explanatory variables. These transformation functions are estimated by regularised optimisation of scoring rules for probabilistic forecasts, e.g. the continuous ranked probability score. The corresponding estimated conditional distribution functions are consistent. Conditional transformation models are potentially useful for describing possible heteroscedasticity, comparing spatially varying distributions, identifying extreme events, deriving prediction intervals and selecting variables beyond mean regression effects. An empirical investigation based on a heteroscedastic varying coefficient simulation model demonstrates that semiparametric estimation of conditional distribution functions can be more beneficial than kernel-based non-parametric approaches or parametric generalised additive models for location, scale and shape.