New statistical tests for spatial random fields

空间随机场的新统计检验

• 批准号: 457238972
• 负责人: Professor Dr. Holger Dette
• 金额: --
• 依托单位: Lehrstuhl Stochastik, insbesondere Statistik
• 依托单位国家: 德国
• 项目类别: Research Grants
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/457238972?language=en
• 关键词: New; statistical; tests; spatial; random

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.

空间数据分析中常用的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，因此大大便利了统计方法的后续发展。为了获得更强大的统计分析，另一个经常提出的假设是平稳随机场的各向同性假设。然而，如果假设平稳性或各向同性是错误的，那么任何基于这些假设的统计分析和结论都将具有误导性。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。我们的方法是基于构造潜在的非平稳(或非各向同性)随机场与平稳(或各向同性)随机场的偏差的度量。如果且仅当二阶平稳性(或各向同性)的假设成立，则该度量为零。我们将这一度量的估计构造为局部空间(锥化)周期图在特定频率和空间位置上估计的非线性函数和，并分析了它们的相合性和弱收敛等概率性质。特别是，我们将研究不同的抽样方案和抽样区域的几何形状对新估计的渐近行为(收敛速度、标准化、渐近方差)的影响。此外，我们还将对与平稳性(或各向同性)的偏差具有科学相关性的假设进行测试。