A statistical test on the local effects of spatially structured variance

Spatial variance is an important characteristic of spatial random variables. It describes local deviations from average global conditions and is thus a proxy for spatial heterogeneity. Investigating instability in spatial variance is a useful way of detecting spatial boundaries, analysing the internal structure of spatial clusters and revealing simultaneously acting geographic phenomena. Recently, a corresponding test statistic called ‘Local Spatial Heteroscedasticity’ (LOSH) has been proposed. This test allows locally heterogeneous regions to be mapped and investigated by comparing them with the global average mean deviation in a data set. While this test is useful in stationary conditions, its value is limited in a global heterogeneous state. There is a risk that local structures might be overlooked and wrong inferences drawn. In this paper, we introduce a test that takes account of global spatial heterogeneity in assessing local spatial effects. The proposed measure, which we call ‘Local Spatial Dispersion’ (LSD), adapts LOSH to local conditions by omitting global information beyond the range of the local neighbourhood and by keeping the related inferential procedure at a local level. Thereby, the local neighbourhoods might be small and cause small-sample issues. In the view of this, we recommend an empirical Bayesian technique to increase the data that is available for resampling by employing empirical prior knowledge. The usefulness of this approach is demonstrated by applying it to a Light Detection and Ranging-derived data set with height differences and by making a comparison with LOSH. Our results show that LSD is uncorrelated with non-spatial variance as well as local spatial autocorrelation. It thus discloses patterns that would be missed by LOSH or indicators of spatial autocorrelation. Furthermore, the empirical outcomes suggest that interpreting LOSH and LSD together is of greater value than interpreting each of the measures individually. In the given example, local interactions can be statistically detected between variance and spatial patterns in the presence of global structuring, and thus reveal details that might otherwise be overlooked.