Scaling Geospatial Data from the Perspective of Complexity: Exploring the Scaling Behavior of the Entropogram

A fundamental challenge in geospatial data science is to determine how a property, or its characterization, changes with a change in the scale of measurement. Except for geostatistical regularization of the variogram, which is theoretically well established, the scaling behaviors of a wide range of alternative measures of spatial association remain unclear. This limits the ability to make inferences at scales beyond the scale of measurement. The scaling behavior of the recently introduced entropogram function also remains unclear. Because the entropogram is essentially the generalization of the variogram to categorical spatial variables, the possibility to derive a scaling model for the entropogram exists. Here, the scaling behavior of the entropogram based on the scale effect of Shannon entropy is derived, providing a theoretical basis for the regularization of the entropogram. To validate the developed regularization model for the entropogram, a series of multiscale data was generated. Both theoretical derivation and experimental results showed that the entropogram is scale-invariant, under certain conditions for the generation of the categorical data. This research, thus, generalizes the entropogram to changes in measurement scale, thereby increasing our ability to characterize spatial data and make inferences about the underlying dynamic process. It also provides a reference for the interactions between patterns and processes at different scales.