A Statistical Learning View of Simple Kriging

TL;DR

This work reframes simple Kriging as a nonparametric, finite-sample predictive problem for spatial data, where standard i.i.d. generalization theory does not apply. By linking simple Kriging to kernel ridge regression and employing a plug-in approach with a nonparametric covariance estimator derived from a single training realization on a regular grid, it derives non-asymptotic excess risk bounds under Gaussian isotropy in an in-fill setting. The main theoretical results provide rates of convergence of order $O_{\mathbb{P}}(1/\sqrt{n})$ for the excess risk and $\sup_s\|\widehat{\Lambda}_d(s)-\Lambda_d^*(s)\|$ bounds, demonstrating the generalization capacity of the empirical Kriging predictor. Empirical investigations on simulated and real spatial data corroborate the theory, showing that the nonparametric plug-in Kriging predictor closely matches the oracle predictor as the training size $n$ grows and remains robust across several covariance models.

Abstract

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 $d\geq 1$ values taken by a realization of a square integrable random field $X=\{X_s\}_{s\in S}$, $S\subset \mathbb{R}^2$, with unknown covariance structure, at sites $s_1,\; \ldots,\; s_d$ in $S$, the goal is to predict the unknown values it takes at any other location $s\in S$ 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 $n\geq 1$ locations $σ_1,\; \ldots,\; σ_n$ 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 $X'_{σ_1},\; \ldots,\; X'_{σ_n}$ involved in the learning procedure. In this article, non-asymptotic bounds of order $O_{\mathbb{P}}(1/\sqrt{n})$ 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.

Figures (18)

• Figure 1: Dyadic grid at scale $J = 3$ ($n = 81$). Depicted lags: $h_1 = 2^{-J}$ (in red), $h_2 = 2/2^J$ (in blue) and $h_3 = \sqrt{2}/2^J$ (in green).
• Figure 2: Changes in the value of the number $n_h$ of pairs of sites that are at distance $h$ from one another, for different values of the lag $h$, on a dyadic grid at scale $J = 2$ ($n = 25$).
• Figure 3: Estimation of the truncated power law (left) and the Gaussian (right) covariance functions, on a dyadic grid at scale $J = 3$ ($n = 81$), with $\theta = 5$. For each model, the red line corresponds to the true covariance function and the green line to the mean of the estimated one, together with the corresponding mean standard deviation (in green shaded bands), over $100$ replications.
• Figure 4: MSE maps over $100$ realizations of a Gaussian process with truncated power law covariance function for the empirical Kriging predictor with different values of $\theta$ ($J = 3$, $N = 1681$, and $d = 10$).
• Figure 5: MSE maps of over $100$ realizations of a Gaussian process with Gaussian covariance function for the empirical Kriging predictor with different values of $\theta$ ($J = 3$, $N = 1681$, and $d = 10$).

Theorems & Definitions (33)

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