Spatial variables can be observed in many different forms, such as regularly sampled random fields (lattice data), point processes, and randomly sampled spatial processes. Joint analysis of such collections of observations is clearly desirable, but complicated by the lack of an easily implementable analysis framework. It is well known that Fourier transforms provide such a framework, but its form has eluded data analysts. We formalize it by providing a multitaper analysis framework using coupled discrete and continuous data tapers, combined with the discrete Fourier transform for inference. Using this set of tools is important, as it forms the backbone for practical spectral analysis. In higher dimensions it is important not to be constrained to Cartesian product domains, and so we develop the methodology for spectral analysis using irregular domain data tapers, and the tapered discrete Fourier transform. We discuss its fast implementation, and the asymptotic as well as large finite domain properties. Estimators of partial association between different spatial processes are provided as are principled methods to determine their significance, and we demonstrate their practical utility on a large-scale ecological dataset.

翻译：空间变量可以以多种不同形式被观测，例如规则采样的随机场（格点数据）、点过程以及随机采样的空间过程。对此类观测集合进行联合分析显然是必要的，但由于缺乏易于实现的分析框架而变得复杂。众所周知，傅里叶变换提供了这样的框架，但其具体形式一直未能被数据分析者有效掌握。我们通过构建多窗谱分析框架将其形式化，该框架结合了离散与连续数据窗的耦合设计，并采用离散傅里叶变换进行统计推断。使用这套工具至关重要，因为它构成了实用谱分析的支柱。在高维情形中，避免受限于笛卡尔积型区域尤为重要，因此我们开发了适用于不规则区域数据窗的谱分析方法，并引入加窗离散傅里叶变换。我们讨论了其快速实现算法、渐近性质以及大有限域特性。本文提供了不同空间过程间偏关联的估计量，并建立了确定其显著性的原理性方法，最后通过大规模生态数据集展示了这些方法的实际效用。