Information Consistency-Based Measures for Spatial Stratified Heterogeneity

As a typical form of spatial heterogeneity, spatial stratified heterogeneity is widely observed in geographical phenomena. Although the q statistic provides a measure of spatial stratified heterogeneity using variance differences, it is not suitable for nominal target variables and neglects information differences between strata at higher order moments. Based on the mutual information and relative entropy between variables, two spatial stratified heterogeneity measures are proposed for nominal and continuous target variables, respectively. Permutation tests are then used to determine their statistical significance. The proposed measures are suitable for either nominal or continuous target variables. They make no assumptions regarding the distribution of target variables, and return a value of zero only when the distribution of the target variable is independent of the explanatory variable. Experiments on five illustrative data sets and three publicly accessible data sets show that the proposed measures are consistent with the q statistic and can detect the existence of spatial stratified heterogeneity when the q statistic fails, so long as there are significant differences between the distributions in different strata.