# A Statistical Learning View of Simple Kriging

arXiv: https://arxiv.org/abs/2202.07365 · HTML: https://mlsanity.com/paper/2202.07365/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 3 of its 53 references that are in the map.
- First posted on arXiv 15 Feb 2022.

## Builds on (most cited first, 3 of 3)

- [Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences](https://mlsanity.com/paper/1807.02582/) (2018)
- [Learning Whenever Learning is Possible: Universal Learning under General Stochastic Processes](https://mlsanity.com/paper/1706.01418/) (2017)
- [Affine-Invariant Integrated Rank-Weighted Depth: Definition, Properties and Finite Sample Analysis](https://mlsanity.com/paper/2106.11068/) (2021)

## Abstract (arXiv)

In the Big Data era, with the ubiquity of geolocation sensors in particular, massive datasets exhibiting a possibly complex spatial dependence structure are becoming increasingly available. In this context, the standard probabilistic theory of statistical learning does not apply directly and guarantees of the generalization capacity of predictive rules learned from such data are left to establish. We analyze here the simple Kriging task from a statistical learning perspective, i.e. by carrying out a nonparametric finite-sample predictive analysis. Given d≥ 1 values taken by a realization of a square integrable random field X=Xss∈ S, Ssubset ℝ2, with unknown covariance structure, at sites s1,; ldots,; sd in S, the goal is to predict the unknown values it takes at any other location s∈ S with minimum quadratic risk. The prediction rule being derived from a training spatial dataset: a single realization X' of X, independent from those to be predicted, observed at n≥ 1 locations σ1,; ldots,; σn in S. Despite the connection of this minimization problem with kernel ridge regression, establishing the generalization capacity of empirical risk minimizers is far from straightforward, due to the non independent and identically distributed nature of the training data X'σ1,; ldots,; X'σn involved in the learning procedure. In this article, non-asymptotic bounds of order O_mathbbP(1/√n) are proved for the excess risk of a plug-in predictive rule mimicking the true minimizer in the case of isotropic stationary Gaussian processes, observed at locations forming a regular grid in the learning stage. These theoretical results are illustrated by various numerical experiments, on simulated data and on real-world datasets.