Autocovariance Varieties of Moving Average Random Fields
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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