A Moran eigenvector spatial filtering specification of entropy measures

Abstract

Regional science investigations of geographical disparities in socio-economic development sometimes utilize entropy, which measures a phenomenon's distributional uniformity across geographical space. Entropy also is widely utilized to measure random phenomenon dispersion, and often used to identify the most probable allocation of a phenomenon in space. Its common formulation is with empirical frequencies, following Shannon. Batty introduces spatial entropy assuming equal probability over space. His specification considers probabilities as fundamentally being spatially independent, which does not hold in most empirical geographical analyses. Hence, an entropy measure can be further modified by controlling extra variation caused by spatial autocorrelation. This paper proposes a Moran eigenvector spatial filtering (MESF) entropy specification that accounts for spatial autocorrelation when modelling georeferenced data. Using eigenvectors from a transformed spatial weights matrix, MESF identifies and isolates spatially autocorrelated components within a georeferenced variable. Coupling it with a non-normal distribution, such as a binomial or beta probability model, which researchers often employ to describe empirical probabilities, expands its utility. The proposed method is examined with an application to regional income inequality in Poland during 2005–2012. This application demonstrates that accounting for spatial autocorrelation further enhances an entropy measure, showing that the MESF specification provides a flexible method for controlling spatial autocorrelation in an entropy formulation.