Spatially-Indexed Longitudinal Distributional Outcome Regression for Environmental Monitoring
Abstract
Characterizing longitudinal changes in region-specific distribution of environmental exposures, such as total nitrate (TNO3) concentrations, is critical for understanding localized ecological risks that are otherwise obscured by standard mean-level modeling. However, modeling longitudinal distributional outcomes across spatial regions presents significant methodological challenges. The random objects are spatio-temporally dependent, and mathematical constraints are inherent to distributional representations, such as, monotonicity of quantile functions. To address this, we propose a novel spatially-indexed longitudinal distributional outcome regression model. The distributional coefficients corresponding to the fixed effects of covariates are modeled using Bernstein basis polynomials, while spatio-temporal random effects are flexibly captured via tensor product expansions of splines. We develop a scalable Markov Chain Monte Carlo (MCMC) algorithm to explicitly account for spatial dependencies, and introduce a fast two-stage projected-posterior approach to preserve the monotonicity of the predicted subject-specific quantile functions. Extensive simulation studies demonstrate that the proposed framework achieves superior estimation accuracy and predictive performance compared to standard non-spatial distributional outcome regression. We apply our methodology to predict monthly, site-specific distributions of TNO3 concentrations across the contiguous United States. Accounting for spatial correlation provides substantially lower uncertainty in the estimated distributional effects and improves predictive performance over the non-spatial alternative, offering a robust, interpretable tool for spatio-temporal environmental monitoring.
Related
- On the Calibration of Isotonic Distributional Regression
- Distributional Regression for Arbitrary Response Variables Using ReLU Neural Networks
- Sufficiently Reduced Distributional Regression
- Belted Engression: Sufficient Dimension Reduction for Generative Distributional Regression
- Stochastic Variational Inference for Vine Copula Distributional Regression
BibTeX
@misc{ghosal2026spatiallyindexed,
title = {{Spatially-Indexed Longitudinal Distributional Outcome Regression for Environmental Monitoring}},
author = {Rahul Ghosal and Suman Majumder and Indranil Sahoo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15961},
eprint = {2609.15961},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15961},
}
RIS
TY - UNPB
TI - Spatially-Indexed Longitudinal Distributional Outcome Regression for Environmental Monitoring
AU - Rahul Ghosal
AU - Suman Majumder
AU - Indranil Sahoo
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15961
UR - https://arxiv.org/abs/2609.15961
AB - Characterizing longitudinal changes in region-specific distribution of environmental exposures, such as total nitrate (TNO_3) concentrations, is critical for understanding localized ecological risks that are otherwise obscured by standard mean-level modeling. However, modeling longitudinal distributional outcomes across spatial regions presents significant methodological challenges. The random objects are spatio-temporally dependent, and mathematical constraints are inherent to distributional representations, such as, monotonicity of quantile functions. To address this, we propose a novel spatially-indexed longitudinal distributional outcome regression model. The distributional coefficients corresponding to the fixed effects of covariates are modeled using Bernstein basis polynomials, while spatio-temporal random effects are flexibly captured via tensor product expansions of splines. We develop a scalable Markov Chain Monte Carlo (MCMC) algorithm to explicitly account for spatial dependencies, and introduce a fast two-stage projected-posterior approach to preserve the monotonicity of the predicted subject-specific quantile functions. Extensive simulation studies demonstrate that the proposed framework achieves superior estimation accuracy and predictive performance compared to standard non-spatial distributional outcome regression. We apply our methodology to predict monthly, site-specific distributions of TNO_3 concentrations across the contiguous United States. Accounting for spatial correlation provides substantially lower uncertainty in the estimated distributional effects and improves predictive performance over the non-spatial alternative, offering a robust, interpretable tool for spatio-temporal environmental monitoring.
ER -