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.

我们研究了整数格上移动平均随机场的自协方差函数 ${\mathbb{Z}}^d$从代数的角度来看。这些自协方差由移动平均系数多项式参数化，从而追踪出代数变异。我们推导出这些变体的维数和度数，并使用它们的代数性质来获得统计结果，例如模型参数的可识别性。我们将参数估计问题与称为欧几里得距离度和最大似然度的代数不变量联系起来。在整个过程中，我们用具体的例子来说明结果。在我们的计算中，我们使用来自交换代数和数值代数几何的工具。