Spectral Density Estimation for Random Fields via Periodic Embeddings

TL;DR

This paper presents new methods for estimating the spectral density of random fields on lattices using periodic embeddings, improving computational efficiency and accuracy, with applications to satellite temperature data.

Contribution

It introduces a novel periodic embedding approach for spectral density estimation that is computationally efficient and reduces bias in periodogram smoothing.

Findings

Imputed data periodogram properties are theoretically analyzed.

Proposed methods outperform existing approaches in simulations.

Application demonstrates effectiveness on real satellite temperature data.

Abstract

We introduce methods for estimating the spectral density of a random field on a $d$-dimensional lattice from incomplete gridded data. Data are iteratively imputed onto an expanded lattice according to a model with a periodic covariance function. The imputations are convenient computationally, in that circulant embedding and preconditioned conjugate gradient methods can produce imputations in $O(n\log n)$ time and $O(n)$ memory. However, these so-called periodic imputations are motivated mainly by their ability to produce accurace spectral density estimates. In addition, we introduce a parametric filtering method that is designed to reduce periodogram smoothing bias. The paper contains theoretical results studying properties of the imputed data periodogram and numerical and simulation studies comparing the performance of the proposed methods to existing approaches in a number of scenarios. We present an application to a gridded satellite surface temperature dataset with missing values.