Adaptive estimation of stationary Gaussian fields

TL;DR

This paper introduces a new adaptive model selection method for estimating the covariance of stationary Gaussian fields on a lattice, leveraging Gaussian Markov random fields and penalization to achieve optimality and adaptivity.

Contribution

It proposes a novel penalized model selection procedure for spatial Gaussian fields that is adaptive and minimax optimal under certain conditions.

Findings

The method satisfies a nonasymptotic oracle inequality.

It is minimax adaptive when the field is a GMRF.

The procedure adapts to the approximation rate of the true distribution.

Abstract

We study the nonparametric covariance estimation of a stationary Gaussian field X observed on a regular lattice. In the time series setting, some procedures like AIC are proved to achieve optimal model selection among autoregressive models. However, there exists no such equivalent results of adaptivity in a spatial setting. By considering collections of Gaussian Markov random fields (GMRF) as approximation sets for the distribution of X, we introduce a novel model selection procedure for spatial fields. For all neighborhoods m in a given collection M, this procedure first amounts to computing a covariance estimator of X within the GMRFs of neighborhood m. Then, it selects a neighborhood by applying a penalization strategy. The so-defined method satisfies a nonasymptotic oracle type inequality. If X is a GMRF, the procedure is also minimax adaptive to the sparsity of its neighborhood. More generally, the procedure is adaptive to the rate of approximation of the true distribution by GMRFs with growing neighborhoods.