Asymptotic theory of cepstral random fields

TL;DR

This paper develops an asymptotic theory for cepstral random fields, providing formulas for parameter estimation and model building, with applications demonstrated through simulations and agricultural data analysis.

Contribution

It introduces recursive formulas linking cepstral coefficients to moving-average fields and establishes asymptotic properties for various estimators in 2D random field models.

Findings

Recursive formulas for cepstral and autocovariance connection

Asymptotic results for Bayesian and likelihood estimators

Simplified model building with guaranteed positive definite covariance

Abstract

Random fields play a central role in the analysis of spatially correlated data and, as a result, have a significant impact on a broad array of scientific applications. This paper studies the cepstral random field model, providing recursive formulas that connect the spatial cepstral coefficients to an equivalent moving-average random field, which facilitates easy computation of the autocovariance matrix. We also provide a comprehensive treatment of the asymptotic theory for two-dimensional random field models: we establish asymptotic results for Bayesian, maximum likelihood and quasi-maximum likelihood estimation of random field parameters and regression parameters. The theoretical results are presented generally and are of independent interest, pertaining to a wide class of random field models. The results for the cepstral model facilitate model-building: because the cepstral coefficients are unconstrained in practice, numerical optimization is greatly simplified, and we are always guaranteed a positive definite covariance matrix. We show that inference for individual coefficients is possible, and one can refine models in a disciplined manner. Our results are illustrated through simulation and the analysis of straw yield data in an agricultural field experiment.