用可变缩放核提升克里金模型对不均匀数据的建模能力
Kriging via variably scaled kernels
- 引入可变缩放核，动态调整数据相关性结构
- 在突变或不连续场景中重建精度显著提升
- 适合需精准不确定性估计的复杂空间建模任务
经典高斯过程与克里金模型通常基于平稳核，其相关性仅依赖于数据点间的相对距离。该假设虽利于解析求解，却限制了对异质相关结构的表达能力。本文研究可变缩放核作为构建非平稳高斯过程的有效工具，通过缩放函数显式改变数据相关性，使模型能刻画具有突变或不连续特性的目标。我们通过可变缩放核的幂函数分析预测不确定性，并厘清其与经典非平稳核的关系。数值实验表明，基于可变缩放核的高斯过程在重建精度上更优，且不确定性估计能准确反映数据内在结构。
原文摘要 · Abstract (English)
Classical Gaussian processes and Kriging models are commonly based on stationary kernels, whereby correlations between observations depend exclusively on the relative distance between scattered data. While this assumption ensures analytical tractability, it limits the ability of Gaussian processes to represent heterogeneous correlation structures. In this work, we investigate variably scaled kernels as an effective tool for constructing non-stationary Gaussian processes by explicitly modifying the correlation structure of the data. Through a scaling function, variably scaled kernels alter the correlations between data and enable the modeling of targets exhibiting abrupt changes or discontinuities. We analyse the resulting predictive uncertainty via the variably scaled kernel power function and clarify the relationship between variably scaled kernels-based constructions and classical non-stationary kernels. Numerical experiments demonstrate that variably scaled kernels-based Gaussian processes yield improved reconstruction accuracy and provide uncertainty estimates that reflect the underlying structure of the data
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