AI 中文总结
针对环境监测中区域分布结果的纵向建模难题，提出空间索引分布回归模型，结合Bernstein基和样条张量积捕捉时空效应，经模拟验证精度更优，并成功应用于美国TNO3浓度预测。
AI 中文摘要
刻画环境暴露（如总硝酸盐（TNO$_3$）浓度）的区域特定分布的纵向变化，对于理解局地生态风险至关重要，而这些风险在标准均值建模中往往被掩盖。然而，跨空间区域对纵向分布结果进行建模面临显著的方法论挑战。这些随机对象具有时空依赖性，且分布表示本身蕴含数学约束，例如分位数函数的单调性。为此，我们提出了一种新颖的空间索引纵向分布结果回归模型。对应于协变量固定效应的分布系数采用Bernstein基多项式建模，而时空随机效应则通过样条的张量积展开灵活捕捉。我们开发了一种可扩展的马尔可夫链蒙特卡洛（MCMC）算法以显式考虑空间依赖性，并引入了一种快速的两阶段投影后验方法，以保持预测的个体特定分位数函数的单调性。大量模拟研究表明，与标准的非空间分布结果回归相比，所提出的框架实现了更优的估计精度和预测性能。我们将该方法应用于预测美国本土各站点每月的TNO$_3$浓度分布。考虑空间相关性可显著降低估计分布效应中的不确定性，并相较于非空间替代方法提升预测性能，为时空环境监测提供了一种稳健、可解释的工具。
英文摘要
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.
