In modelling spatial data, it is a crucial aspect to specify the covariance function of the random field appropriately. For the sake of simplicity, the spatial isotropy is often assumed. By approximating the isotropy by a composite hypothesis containing the rotational invariance and axial symmetry of the covariance function, a maximum statistic is proposed to test the assumption of isotropy. The proposed test statistic is constructed by maximizing two Anderson-Darling (A-D) statistics, in which one is built up based on spatial periodogram-ratios of the random field at one sampling location set and its rotated version, and the other is based on spatial periodogram-ratios at the sampling location set and its axial symmetric one. Under the null, the probability distribution of the proposed maximum statistic can be approximated by simulation. The proposed nonparametric test is independent of any smoothing parameters, and is applicable for analyzing irregularly spaced spatial data.

中文翻译：

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不规则空间数据各向同性特性的谱域检验

在对空间数据进行建模时，适当地指定随机场的协方差函数是一个至关重要的方面。为简单起见，通常假设空间各向同性。通过包含协方差函数的旋转不变性和轴对称性的复合假设逼近各向同性，提出了一个最大统计量来检验各向同性的假设。建议的检验统计量是通过最大化两个 Anderson-Darling (AD) 统计量构建的，其中一个基于一个采样位置集及其旋转版本的随机场的空间周期图比率建立，另一个基于空间采样位置集的周期图比率及其轴对称。在零下，建议的最大统计量的概率分布可以通过模拟来近似。