Autocovariance Varieties of Moving Average Random Fields

TL;DR

This paper explores the algebraic structure of autocovariance functions in moving average random fields, revealing their geometric properties and implications for parameter estimation.

Contribution

It introduces a novel algebraic perspective on autocovariance varieties, deriving their dimension, degree, and linking algebraic invariants to statistical identifiability.

Findings

Autocovariance functions form algebraic varieties with computable dimension and degree.

Algebraic properties inform parameter identifiability in moving average models.

Connections established between algebraic invariants and statistical estimation methods.

Abstract

We study the autocovariance functions of moving average random fields over the integer lattice $\mathbb{Z}^d$ from an algebraic perspective. These autocovariances are parametrized polynomially by the moving average coefficients, hence tracing out algebraic varieties. We derive dimension and degree of these varieties and we use their algebraic properties to obtain statistical consequences such as identifiability of model parameters. We connect the problem of parameter estimation to the algebraic invariants known as euclidean distance degree and maximum likelihood degree. Throughout, we illustrate the results with concrete examples. In our computations we use tools from commutative algebra and numerical algebraic geometry.

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