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 <i>G </i>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 <i>p</i>-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 <i>p</i>-values of the local Moran's <i>I<sub>i</sub></i>, local Geary's <i>c<sub>i</sub></i>, and the modified Ord and Getis <i>G </i>statistics based on the distributional theory of quadratic forms in normal variables. Furthermore, an approximate method, called three-moment χ<sup>2</sup> 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 χ<sup>2</sup> method is useful in some situations although it is inappropriate for approximating the null distribution of <i>I<sub>i</sub></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.
