发表机构
University of Göttingen（哥廷根大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究将惩罚变换模型扩展为条件形状PTM，通过小批量随机变分推断拟合，在模拟与真实观测中验证其能协调分布回归的形状灵活性与可解释性，预测性能优于固定形状PTM和高斯位置尺度模型。
AI 中文摘要
分布回归的核心挑战在于，如何在允许响应变量的条件分布形状随协变量灵活变化的同时，保留其均值和标准差的直接可解释效应。我们将惩罚变换模型（PTM）家族扩展为条件形状PTM，为条件均值、标准差以及位置和尺度之外的标准化分布形状分配独立的结构化加性预测器。协变量依赖的单调变换将标准化响应映射到固定参考分布，而仿射标准化则强制诱导的标准化分布均值为0、方差为1，因此前两个预测器恰好对应条件均值和标准差。形状预测器可容纳选定的线性、非线性、组特异性、空间及交互效应；正则化将无支撑的偏离收缩至参考族位置尺度模型。我们使用带模型对齐高斯块的小批量随机变分推断与分阶段优化来拟合PTM。在模拟实验中，PTM可恢复平滑的均值和标准差效应，以及从偏度到双峰性的协变量依赖转变，同时抑制不必要的形状效应。在故意设定错误的设计下，其在测试集密度和分布函数准确性上仍与结构化加性狄利克雷过程混合物具有竞争力，尽管所有模型均存在覆盖不足，且两种灵活方法均遗漏了精细特征。对13425条挪威水导率观测值和1182514条德国日气温观测值的应用，展示了选择性的组特异性、季节性和空间形状变化。预测性能指标表明，条件形状PTM优于固定形状PTM和高斯位置尺度模型。
英文摘要
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.
Comments55 pages, 25 figures; Revised Figure 14 to isolate seasonal effects and space-season interactions and updated the accompanying text; corrected city-color mapping. Fitted models and main conclusions unchanged
