Measures of local and global spatial association are key tools for exploratory spatial data analysis. Many such measures exist including Moran’s I, Geary’s C, and the Getis–Ord $G$ and $G^{\ast }$ statistics. A parametric approach to testing for significance relies on strong assumptions, which are often not met by real world data. Alternatively, the most popular nonparametric approach, the permutation test, imposes a large computational burden especially for massive graphical networks. Hence, we propose a computation-free approach to nonparametric permutation testing for local and global measures of spatial autocorrelation stemming from generalizations of the Khintchine inequality from functional analysis and the theory of $L^p$-spaces. Our methodology is demonstrated both on the conservative party’s performance in the 2019 federal Canadian election in the province of Alberta—a small network with $n=34$—and Donald Trump’s countywise performance in the 2016 US presidential election—a large network with $n=3105$. Neither dataset is normal, and running a classic permutation test locally for every node, considering various test statistics and neighbourhood structures, and including multiple testing correction would require the simulation of millions of permutations. We achieve similar statistical power on these datasets to the permutation test without the need for tedious simulation.

局部和全球空间关联的测量是探索性空间数据分析的关键工具。存在许多这样的度量，包括 Moran's I、Geary's C 和 Getis-Ord$G$和$G^{*}$统计数据。测试显着性的参数方法依赖于强大的假设，而现实世界的数据通常无法满足这些假设。或者，最流行的非参数方法，即置换测试，会带来很大的计算负担，尤其是对于大规模图形网络。因此，我们提出了一种无需计算的方法来对空间自相关的局部和全局测量进行非参数置换测试，该方法源于泛函分析和理论的 Khintchine 不等式的推广。${\mathrm{大号}}^p$-空格。我们的方法论在保守党在 2019 年阿尔伯塔省联邦加拿大大选中的表现都得到了证明——一个小型网络$n=34$——以及唐纳德·特朗普在 2016 年美国总统大选中的县级表现——一个与$n=3105$. 两个数据集都不是正常的，并且在本地为每个节点运行经典的置换测试，考虑到各种测试统计和邻域结构，并且包括多个测试校正，需要模拟数百万个排列。我们在这些数据集上获得了与置换测试相似的统计能力，而无需进行繁琐的模拟。