This work addresses the interpolation of probability measures within a spatial statistics framework. We develop a Kriging approach in the Wasserstein space, leveraging the quantile function representation of the one-dimensional Wasserstein distance. To mitigate the inaccuracies in semivariogram estimation that arise from sparse datasets, we combine this formulation with cross-validation techniques. In particular, we introduce a variant of the virtual cross-validation formulas tailored to quantile functions. The effectiveness of the proposed method is demonstrated on a controlled toy problem as well as on a real-world application from nuclear safety.

翻译：本研究致力于在空间统计框架内对概率测度进行插值。我们基于一维Wasserstein距离的分位数函数表示，在Wasserstein空间中构建了一种克里金方法。为缓解稀疏数据集导致的半变异函数估计误差，我们将该模型与交叉验证技术相结合，特别针对分位数函数设计了虚拟交叉验证公式的变体。通过受控仿真案例与核安全实际应用，验证了所提方法的有效性。