论文标题
简单Kriging的统计学习观点
A Statistical Learning View of Simple Kriging
论文作者
论文摘要
在大数据时代，尤其是地理位置传感器的普遍存在，显示出可能复杂的空间依赖性结构的大量数据集变得越来越多。在这种情况下，统计学习的标准概率理论不直接应用，并保证从此类数据中学到的预测规则的概括能力来确定。我们在这里从统计学习的角度分析了简单的Kriging任务，即通过进行非参数有限样本的预测分析。给定的$ d \ geq 1 $值通过实现平方集成随机字段$ x = \ {x_s \} _ {s \ in s} $，$ s \ subset \ subset \ mathbb {r}^2 $，带有未知的covariance结构，在网站$ s_1，\; \ ldots，\; $ s $中的s_d $，目标是预测其在S $中的任何其他位置$ s \ in S $中的未知值，并具有最低二次风险。预测规则来自培训空间数据集：单个实现$ x'的$ x $，独立于预测的$ x $，在$ n \ geq 1 $位置$σ_1，\; \ ldots，\; $ s $中的σ_n$。尽管这种最小化问题与内核脊回归有联系，但由于训练数据的非独立和相同分布的性质$ x'_ _ {σ_1}，\; \ ldots，\; x'_ {σ_n} $涉及学习过程。在本文中，证明了$ o _ {\ mathbb {p}}}（1/\ sqrt {n}）$的非扰动范围，是因为有多余的风险模仿插件预测规则，模仿了在同位素平稳的豪斯剂过程中进行真正的最小化的插件预测规则，以定期的级别的阶段进行了定期的级别，该阶段是在定期进行的，该级别的阶段是一个定期的级别，以一定的级别的级别进行级别的级别。这些理论结果通过各种数值实验，模拟数据和现实世界数据集进行了说明。
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.
