基于CRPS的单变量符合回归最优分箱
CRPS-Optimal Binning for Univariate Conformal Regression
- Centre for Reliable Machine Learning（可靠机器学习中心）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出基于将协变量排序观测分箱并使用箱内经验CDF作为预测分布的非参数条件分布估计方法，通过动态规划选择最优分箱数，实现数据高效且具有有限样本边际覆盖保证的符合预测。
AI中文摘要：
我们提出了一种基于将协变量排序观测划分为连续箱并使用箱内经验CDF作为预测分布的非参数条件分布估计方法。箱边界选择旨在最小化总留一连续排名概率评分（LOO-CRPS），其具有闭式成本函数，预计算复杂度为O(n² log n)，存储复杂度为O(n²)；全局最优K分箱通过动态规划在O(n² K)时间内恢复。最小化样本内LOO-CRPS在选择K时不合适，因为它导致样本内乐观。我们通过K折交叉验证测试CRPS来选择K，从而获得U型准则，具有明确的最小值。选定K*并拟合完整数据分箱后，我们形成两个互补的预测对象：基于CRPS作为非符合分数的Ven prediction band和符合预测集，该预测集在任何指定水平ε下具有有限样本边际覆盖保证。符合预测是转导的且数据高效，因为所有观测用于分箱和p值计算，无需保留保留集。在实际基准测试中，与拆分符合竞争对手（高斯拆分符合、CQR、CQR-QRF和符合等分布回归）相比，该方法产生显著更窄的预测区间，同时保持接近名义覆盖。
英文摘要：
We propose a method for non-parametric conditional distribution estimation based on partitioning covariate-sorted observations into contiguous bins and using the within-bin empirical CDF as the predictive distribution. Bin boundaries are chosen to minimise the total leave-one-out Continuous Ranked Probability Score (LOO-CRPS), which admits a closed-form cost function with $O(n^2 \log n)$ precomputation and $O(n^2)$ storage; the globally optimal $K$-partition is recovered by a dynamic programme in $O(n^2 K)$ time. Minimisation of within-sample LOO-CRPS turns out to be inappropriate for selecting $K$ as it results in in-sample optimism. We instead select $K$ by $K$-fold cross-validation of test CRPS, which yields a U-shaped criterion with a well-defined minimum. Having selected $K^*$ and fitted the full-data partition, we form two complementary predictive objects: the Venn prediction band and a conformal prediction set based on CRPS as the nonconformity score, which carries a finite-sample marginal coverage guarantee at any prescribed level $\varepsilon$. The conformal prediction is transductive and data-efficient, as all observations are used for both partitioning and p-value calculation, with no need to reserve a hold-out set. On real benchmarks against split-conformal competitors (Gaussian split conformal, CQR, CQR-QRF, and conformalized isotonic distributional regression), the method produces substantially narrower prediction intervals while maintaining near-nominal coverage.
