Statistical Test for Local Patterns of Spatial Association

In recent years, there has been a growing interest in the use of local measures such as Anselin's LISAs and Ord and Getis G statistics to identify local patterns of spatial association. The statistical significance test based on local statistics is one of the most important aspects in performing this kind of analysis, and a randomized permutation approach and normal approximation are commonly used to derive the p-values of the statistics. To circumvent some of the shortcomings of these existing methods and to offer a more formal approach in line with classical statistical framework, we develop in this paper an exact method for computing the p-values of the local Moran's Ii, local Geary's ci, and the modified Ord and Getis G statistics based on the distributional theory of quadratic forms in normal variables. Furthermore, an approximate method, called three-moment χ2 approximation, with explicit calculation formulae is also proposed to achieve a computational cost lower than the exact method. Numerical evaluation on the accuracy of the approximate null distributions of the local statistics demonstrates that the proposed three-moment χ2 method is useful in some situations although it is inappropriate for approximating the null distribution of Ii. The study not only provides an exact test for local patterns of spatial association, but also put the tests of several local statistics within a unified statistical framework.