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 I_i, local Geary's c_i, 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 I_i. 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.