Statistical tests for the local spatial variance

Abstract

The concept of local variance is used in the literature on image processing, and, to a lesser extent, in spatial analyses of local heterogeneity. Ord and Getis’ local statistic of heterogeneity (LOSH) is used to test the null hypothesis that the local variance is no different from the variance for the entire study area. LOSH is a ratio of two variances, and it is shown here to also be closely related to the Geary statistic, which measures spatial autocorrelation. In this article, the Brown-Forsythe statistic is proposed as a way to test a similar null hypothesis—namely, that the local spatial variance is no different from that observed elsewhere in the study area. Rejection of the null in favor of significant homogeneity can be interpreted as an indicator of local positive spatial autocorrelation. The merits of the proposed test are illustrated via simulations of both null and alternative hypotheses, and the statistic is used to find local areas of homogeneous values in the classic spatial dataset on wheat yields in Rothamsted, England.

Keywords:

• Local variance
• Geary statistic
• LOSH statistic
• local homogeneity and heterogeneity

Acknowledgement

I acknowledge the helpful comments made by the referees.

Disclosure statement

No potential conflict of interest was reported by the author.

Data and codes availability statement

The data and code that support this work are available on figshare.com with the DOI identifiers 10.6084/m9.figshare.16441296, 10.6084/m9.figshare.16434927, 10.6084/m9.figshare.16439874, and 10.6084/m9.figshare.16441269

Notes

1 https://kwstat.github.io/agridat/reference/mercer.wheat.uniformity.html (Last accessed January 2022)

Additional information

Notes on contributors

Peter A. Rogerson

Peter Rogerson is a Foundation Professor in the School of Geographical Sciences and Urban Planning, at Arizona State University, and a SUNY Distinguished Emeritus Professor at the University at Buffalo. His interests are in the spatial aspects of demographic change and the methods of spatial analysis and spatial statistics.