为度量空间中的回归模型提供可信赖的不确定性量化方法。
Conformal and kNN Predictive Uncertainty Quantification Algorithms in Metric Spaces
- 基于同方差性定义新型置信预测算法，保证有限样本覆盖率。
- 提出kNN方法自适应局部预测半径，无需平滑假设。
- 适用于复杂对象如分布和图拉普拉斯矩阵，适合医学个性化建模。
本文提出一种在度量空间中进行回归模型不确定性量化的框架。通过引入同方差性概念，构建了具有有限样本边际覆盖保证且收敛速度快于理论最优的置信预测算法。针对异方差情形，设计了一种kNN方法，可在一般度量空间中实现局部自适应预测半径，虽无相同有限样本保证，但能改善局部覆盖率校准且不依赖平滑假设。两种方法均兼容多种回归算法并支持大规模数据，便于实践者使用其首选模型并融入领域知识。基于异方差kNN方法，进一步开发了面向度量空间值时间序列的灵活顺序扩展，采用最近邻专家聚合策略。在最弱条件下证明了所提估计器的一致性。最后，通过涉及概率分布与图拉普拉斯等随机对象的个性化医疗应用，展示了该框架的实际效用。
原文摘要 · Abstract (English)
This paper introduces a framework for uncertainty quantification in regression models defined on metric spaces. Using a proposed notion of homoscedasticity, we define a conformal prediction algorithm that provides finite-sample marginal coverage guarantees and fast convergence rates to the oracle prediction region. For heteroscedastic settings, we introduce a kNN procedure that yields locally adaptive prediction radii in general metric spaces. Although this procedure does not provide the same finite-sample guarantees as the conformal algorithm, it is designed to improve local coverage calibration without imposing smoothing assumptions. Both procedures are compatible with a broad range of regression algorithms and scale to large datasets, allowing practitioners to use their preferred models and incorporate domain-specific knowledge. Building on the heteroscedastic $k$NN approach, we also develop a flexible sequential extension for metric-space-valued time series based on nearest-neighbor expert aggregation. We establish the consistency of the proposed estimators under minimal conditions. Finally, we illustrate the practical utility of our framework in personalized medicine applications involving random objects such as probability distributions and graph Laplacians.
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