AI 中文总结
针对序数成分数据，提出一种基于Wasserstein距离的线性回归框架，通过线性规划实现全局最优，并引入诊断工具，提升可解释性和预测性能。
AI 中文摘要
我们提出了一个统一的、无需变换的线性回归框架，专门针对序数成分数据，例如教育水平分布或聚合的Likert量表调查响应。传统的对数比率方法常常模糊可解释性，忽略类别的固有顺序，并难以处理边界零值。为克服这些局限性，我们提出一个确定性模型，约束在列随机变换矩阵的空间内，自然保留单纯形的几何结构。通过采用加权1-Wasserstein距离作为损失函数，我们的方法直接将数据的序数性质嵌入估计过程。我们提供了一个计算高效且全局最优的解决方案，明确表述为线性规划（LP）问题。该框架系统地处理四种不同的回归场景，引入一种新颖的序数张量积来严格处理多个成分预测变量之间的交互。此外，我们为模型配备了合适的正则化策略和新颖的诊断工具，包括基于Wasserstein的决定系数（$R^2_W$）和顺序保持指数（OPI）。广泛的模拟研究和两个实证应用证明了所提出方法的稳健预测性能和可解释性。
英文摘要
We present a unified, transformation-free linear regression framework tailored for ordinal compositional data, such as distributions across educational levels or aggregate Likert-scale survey responses. Traditional log-ratio approaches often obscure interpretability, ignore the inherent ordering of categories, and struggle with boundary zero values. To overcome these limitations, we propose a deterministic model constrained to the space of column-stochastic transformation matrices, naturally preserving the fundamental geometry of the simplex. By adopting the weighted 1-Wasserstein distance as the loss function, our method directly embeds the ordinal nature of the data into the estimation process. We provide a computationally efficient and globally optimal solution explicitly formulated as a Linear Programming (LP) problem. The framework systematically addresses four distinct regression scenarios, introducing a novel ordinal tensor product to rigorously handle interactions across multiple compositional predictors. Furthermore, we equip the model with a suitable regularization strategy and novel diagnostic tools, including a Wasserstein-based coefficient of determination ($R^2_W$) and an Order Preservation Index (OPI). Extensive simulation studies and two empirical applications demonstrate the robust predictive performance and interpretability of the proposed methodology.
