Valid model-free spatial prediction

Predicting the response at an unobserved location is a fundamental problem in spatial statistics. Given the difficulty in modeling spatial dependence, especially in non-stationary cases, model-based prediction intervals are at risk of misspecification bias that can negatively affect their validity. Here we present a new approach for model-free nonparametric spatial prediction based on the conformal prediction machinery. Our key observation is that spatial data can be treated as exactly or approximately exchangeable in a wide range of settings. In particular, under an infill asymptotic regime, we prove that the response values are, in a certain sense, locally approximately exchangeable for a broad class of spatial processes, and we develop a local spatial conformal prediction algorithm that yields valid prediction intervals without strong model assumptions like stationarity. Numerical examples with both real and simulated data confirm that the proposed conformal prediction intervals are valid and generally more efficient than existing model-based procedures for large datasets across a range of non-stationary and non-Gaussian settings.

空间统计学（spatial statistics）中，预测未观测位点的响应值是一项基础研究问题。由于空间依赖性（spatial dependence）建模存在固有难点，尤其在非平稳（non-stationary）场景下，基于模型的预测区间易遭受模型误设偏差的影响，该偏差会损害预测区间的有效性。为此，本文提出一种基于保形预测（conformal prediction）框架的无模型非参数空间预测新方法。我们的核心发现为：在诸多实际场景中，空间数据可被视作严格或近似可交换的。具体而言，在填充渐近（infill asymptotic）框架下，针对广泛的空间过程类别，我们证明了响应值在特定意义下满足局部近似可交换性；据此我们设计了局部空间保形预测算法，该算法无需平稳性（stationarity）等强模型假设，即可生成有效的预测区间。基于真实数据与模拟数据的数值实验验证表明：在各类非平稳、非高斯（non-Gaussian）场景下，针对大规模数据集，本文提出的保形预测区间不仅具备有效性，且整体效率普遍优于现有基于模型的预测流程。