Reconciling Interpretability with Covariate-Dependent Shape Flexibility in Penalized Transformation Models for Distributional Regression
Abstract
A central challenge in distributional regression is to allow the shape of the conditional distribution of the response variable to vary flexibly with covariates while retaining directly interpretable effects on its mean and standard deviation. We extend the penalized transformation model (PTM) family into a conditional-shape PTM, which assigns separate structured additive predictors to the conditional mean, standard deviation, and standardized distributional shape beyond location and scale. A covariate-dependent monotone transformation maps the standardized response to a fixed reference distribution, while affine standardization enforces mean zero and variance one for the induced standardized distribution. Thus, the first two predictors remain exactly the conditional mean and standard deviation. The shape predictor accommodates selected linear, nonlinear, group-specific, spatial, and interaction effects; regularization shrinks unsupported departures toward a reference-family location-scale model. We fit the PTM using mini-batched stochastic variational inference with model-aligned Gaussian blocks and staged optimization. In simulations, the PTM recovers smooth mean and standard-deviation effects and a covariate-dependent transition from skewness to bimodality while suppressing unnecessary shape effects. Under a deliberately misspecified design, it remains competitive with a structured additive Dirichlet-process mixture in test-set density and distribution-function accuracy, although all models show undercoverage and both flexible methods miss fine features. Applications to 13,425 Norwegian water-conductivity observations and 1,182,514 German daily-temperature observations demonstrate selective group-specific, seasonal, and spatial shape variation. Predictive performance criteria favor the conditional-shape PTM over a fixed-shape PTM and a Gaussian location-scale model.
Related
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BibTeX
@misc{brachem2026reconciling,
title = {{Reconciling Interpretability with Covariate-Dependent Shape Flexibility in Penalized Transformation Models for Distributional Regression}},
author = {Johannes Brachem and Thomas Kneib},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02662},
eprint = {2609.02662},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02662},
}
RIS
TY - UNPB
TI - Reconciling Interpretability with Covariate-Dependent Shape Flexibility in Penalized Transformation Models for Distributional Regression
AU - Johannes Brachem
AU - Thomas Kneib
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.02662
UR - https://arxiv.org/abs/2609.02662
AB - A central challenge in distributional regression is to allow the shape of the conditional distribution of the response variable to vary flexibly with covariates while retaining directly interpretable effects on its mean and standard deviation. We extend the penalized transformation model (PTM) family into a conditional-shape PTM, which assigns separate structured additive predictors to the conditional mean, standard deviation, and standardized distributional shape beyond location and scale. A covariate-dependent monotone transformation maps the standardized response to a fixed reference distribution, while affine standardization enforces mean zero and variance one for the induced standardized distribution. Thus, the first two predictors remain exactly the conditional mean and standard deviation. The shape predictor accommodates selected linear, nonlinear, group-specific, spatial, and interaction effects; regularization shrinks unsupported departures toward a reference-family location-scale model. We fit the PTM using mini-batched stochastic variational inference with model-aligned Gaussian blocks and staged optimization. In simulations, the PTM recovers smooth mean and standard-deviation effects and a covariate-dependent transition from skewness to bimodality while suppressing unnecessary shape effects. Under a deliberately misspecified design, it remains competitive with a structured additive Dirichlet-process mixture in test-set density and distribution-function accuracy, although all models show undercoverage and both flexible methods miss fine features. Applications to 13,425 Norwegian water-conductivity observations and 1,182,514 German daily-temperature observations demonstrate selective group-specific, seasonal, and spatial shape variation. Predictive performance criteria favor the conditional-shape PTM over a fixed-shape PTM and a Gaussian location-scale model.
ER -