为比例数据提供可信赖的预测区间,解决传统方法在复合空间中的几何缺陷。
Conformal Prediction for Compositional Data
- 基于狄利克雷回归,用分位数残差与密度区域逼近构建预测集。
- 网格离散化方法显著缩小预测区域并减少覆盖过度,保持良好覆盖率。
- 适用于睡眠分期与植物生物量分配等真实场景,结果具可解释性。
狄利克雷回归模型适用于响应变量为和为1的比例数据的复合数据。然而,目前尚无成熟方法在该情境下构建有效的预测集,尤其未充分考虑复合空间的几何特性。本文研究基于置信传播的策略,在狄利克雷回归中构造有效预测区域,评估了三种方法:基于分位数残差的方法、最高密度区域(HDR)的近似构造,以及在单纯形上采用网格离散化的近似HDR改进方法。通过不同模型复杂度、响应维数和协变量结构的模拟实验分析性能。结果显示,HDR近似方法在覆盖率上表现稳健;网格离散化方法有效降低过覆盖问题,并减小预测区域面积;分位数法生成区域较大但覆盖充足。在两个真实数据集(睡眠阶段、植物生物量分配)上验证了方法的实际可行性,结果在复合空间内具有合理解释性。最后讨论了未来扩展方向。
原文摘要 · Abstract (English)
Dirichlet regression models are suitable for compositional data, in which the response variable represents proportions that sum to one. However, there are still no well-established methods for constructing valid prediction sets in this context, especially considering the geometry of the compositional space. In this work, we investigate conformal prediction-based strategies for constructing valid predictive regions in Dirichlet regression models. We evaluate three distinct approaches: a method based on quantile residuals, an approximate construction of highest density regions (HDR), and an adaptation of the approximate HDR using grid-based discretization over the simplex. The performance of the methods was analyzed through simulation studies under different scenarios, varying the model complexity, response dimensionality, and covariate structure. The results indicated that the HDR approximation approach exhibits good robustness in terms of coverage, while the grid discretization proved effective in reducing overcoverage and the area of the prediction region compared to the original method. The quantile method provided larger prediction regions compared to the grid method, while maintaining adequate coverage. The methodologies were also applied to two real datasets: one concerning sleep stages and another on biomass allocation in plants. In both cases, the proposed methods demonstrated practical feasibility and produced coherent interpretations within the compositional space. Finally, we discuss possible extensions of this work
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