Computation-free Nonparametric testing for Local and Global Spatial Autocorrelation with application to the Canadian Electorate

Abstract: 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 $G^*$ 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 on the results of the 2019 federal Canadian election in the province of Alberta. We recorded the percentage of the vote gained by the conservative candidate in each riding. This data is not normal, and the sample size is fixed at $n=34$ ridings making the parametric approach invalid. In contrast, running a classic permutation test for every riding, for multiple test statistics, with various neighbourhood structures, and multiple testing correction would require the simulation of millions of permutations. We are able to achieve similar statistical power on this dataset to the permutation test without the need for tedious simulation. We also consider data simulated across the entire electoral map of Canada.