Reconciling Interpretability with Covariate-Dependent Shape Flexibility in Penalized Transformation Models for Distributional Regression
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惩罚变换分布回归：兼顾可解释性与协变量依赖的形状灵活性
- 研究问题
- 如何让条件分布形状随协变量灵活变化，同时使模型中的均值和标准差效应保持直接、准确的解释？
- 方法与思路
- 作者为均值、标准差及标准化分布形状设置独立的结构加性预测器，结合单调变换、正则化和随机变分推断。
- 材料陈述的贡献
- 摘要称，模型可刻画选择性的形状变化并保持位置尺度解释；所报告应用的预测指标优于两种较简单模型。
- 阅读关注点
- 应核对标准化如何保证均值与方差约束、形状正则化的选择，以及错设下欠覆盖和细节遗漏的原因。
- 阅读边界
- 依据 arXiv v2 预印本摘要，不能据此声称已同行评审或接收；摘要已报告错设下欠覆盖，不能推断普遍优越性。
原文摘要
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.