New statistical tests for spatial random fields

空间数据分析中的一个常用假设是用于模拟数据生成过程的空间随机场的（二阶）平稳性。这一假设简化了模型的数学分析，从而大大促进了统计方法的后续发展。另一个经常用来获得更强统计分析的假设是平稳随机场各向同性的假设。然而，如果平稳性或各向同性假设是错误的，那么任何基于这些假设的统计分析和结论都将是误导性的。在本项目中，我们将开发新的复杂的统计工具来验证基于给定数据集的平稳性（或各向同性）假设。我们的方法是基于一个潜在的非平稳（或非各向同性）随机场从平稳（或各向同性）随机场的偏差的措施建设。当且仅当二阶平稳性（或各向同性）的假设成立时，测度为零。我们构造估计的这种措施的非线性函数的本地空间（锥形）周期图在一组特定的频率和空间位置评估的总和，并分析其概率特性，如一致性和弱收敛。特别是，我们将研究不同的抽样方案和几何形状的抽样区域上的渐近行为（收敛速度，标准化，渐近方差）的新估计的影响。此外，我们还将开发测试的假设，从平稳性（或各向同性）的偏差是科学相关的。

A commonly used assumption in the analysis of spatial data is that of (second order) stationarity for the spatial random field used to model the data generating process. This assumption simplifies the mathematical analysis of the model and, as a consequence, substantially facilitates the subsequent development of statistical methodology. Another frequently made assumption to obtain a more powerful statistical analysis is the assumption of isotropy of a stationary random field. However, if stationarity or isotropy are erroneously assumed, then any statistical analysis and conclusion based on these assumptions will be misleading.In this project, we will develop new sophisticated statistical tools to validate the assumption of stationarity (or isotropy) on the basis of a given data set. Our approach is based on the construction of a measure for the deviation of a potentially non-stationary (or non-isotropic) random field from a stationary (or isotropic) random field. The measure vanishes if and if only the assumption of second order stationarity (or isotropy) holds. We construct estimates of this measure as sums of nonlinear functions of local spatial (tapered) periodograms evaluated at a specific set of frequencies and spatial locations and analyze their probabilistic properties such as consistency and weak convergence. In particular, we will study the impact of different sampling schemes and the geometry of the sampling region on the asymptotic behavior (rates of convergence, standardization, asymptotic variances) of the new estimates. Moreover, we will also develop tests for the hypothesis that the deviation from stationarity (or isotropy) is scientifically relevant.