Local entropy map: a nonparametric approach to detecting spatially varying multivariate relationships

The relationship between two or more variables may change over the geographic space. The change can be in parameter values (e.g., regression coefficients) or even in relation forms (e.g., linear, quadratic, or exponential). Existing local spatial analysis methods often assume a relationship form (e.g., a linear regression model) for all regions and focus only on the change in parameter values. Therefore, they may not be able to discover local relationships of different forms simultaneously. This research proposes a nonparametric approach, a local entropy map, which does not assume a prior relationship form and can detect the existence of multivariate relationships regardless of their forms. The local entropy map calculates an approximation of the Rényi entropy for the multivariate data in each local region (in the geographic space). Each local entropy value is then converted to a p-value by comparing to a distribution of permutation entropy values for the same region. All p-values (one for each local region) are processed by several statistical tests to control the multiple-testing problem. Finally, the testing results are mapped and allow analysts to locate and interactively examine significant local relationships. The method is evaluated with a series of synthetic data sets and a real data set.