Abstract
Entropy is widely employed in many applied sciences to measure the heterogeneity of observations. Recently, many attempts have been made to build entropy measures for spatial data, in order to capture the influence of space over the variable outcomes. The main limit of these developments is that all indices are computed conditional on a single distance and do not cover the whole spatial configuration of the phenomenon under study. Moreover, most of them do not satisfy the desirable additivity property between local and global spatial measures. This work reviews some recent developments, based on univariate distributions, and compares them to a new approach which considers the properties of entropy measures linked to bivariate distributions. This perspective introduces substantial innovations. Firstly, Shannon’s entropy may be decomposed into two terms: spatial mutual information, accounting for the role of space in determining the variable outcome, and spatial global residual entropy, summarizing the remaining heterogeneity carried by the variable itself. Secondly, these terms both satisfy the additivity property, being sums of partial entropies measuring what happens at different distance classes. The proposed indices are used for measuring the spatial entropy of a marked point pattern on rainforest tree species. The new entropy measures are shown to be more informative and to answer a wider set of questions than the current proposals of the literature.
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Acknowledgements
This work is developed under the PRIN2015 supported-project ’Environmental processes and human activities: capturing their interactions via statistical methods (EPHASTAT)’ (Grant Number 20154X8K23) funded by MIUR (Italian Ministry of Education, University and Scientific Research).
Asregards author Linda Altieri, the research work underlying this paper was partially funded by the FIRB 2012 (Grant Number RBFR12URQJ) ’Statistical modelling of environmental phenomena: pollution, meteorology, health and their interactions’ for research projects by MIUR (Italian Ministry of Education, University and Scientific Research).
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Altieri, L., Cocchi, D. & Roli, G. A new approach to spatial entropy measures. Environ Ecol Stat 25, 95–110 (2018). https://doi.org/10.1007/s10651-017-0383-1
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DOI: https://doi.org/10.1007/s10651-017-0383-1