Abstract
A very recent proposal of a set of entropy measures for spatial data, based on building pairs of realizations, allows to split the data heterogeneity that is usually assessed via Shannon’s entropy into two components: spatial mutual information, identifying the role of space, and spatial residual entropy, measuring heterogeneity due to other sources. A further decomposition into partial terms deeply investigates the role of space at specific distance ranges. The present work proposes improvements to the method and adds relevant results proving that the new set of spatial entropies satisfies a list of desirable properties. We extend the methodology to sets of realizations greater than pairs. We also show that the approach is more general, better performing and more interpretable than the most popular proposals in the literature, thanks to the property of additivity and a new way of computing entropy that explicitly discards the order within sets. A novel procedure for building the necessary quantities for computations is also provided. A comparative study illustrates the superior performance of the new set of measures over representative spatial configurations. Practical questions are answered by means of a case study on land use data.
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References
Altieri L, Cocchi D, Roli G (2017) A new approach to spatial entropy measures. Environ Ecol Stat 25(1):95–110. https://doi.org/10.1007/s10651-017-0383-1
Anselin L (1995) Local indicators of spatial association—LISA. Geogr Anal 27(2):94–115
Batty M (1974) Spatial entropy. Geogr Anal 6:1–31
Batty M (1976) Entropy in spatial aggregation. Geogr Anal 8:1–21
Batty M (2010) Space, scale, and scaling in entropy maximizing. Geogr Anal 42:395–421
Bondy JA, Murty USR (2008) Graph theory. Springer, New York
Butera I, Vallivero L, Ridolfi L (2018) Mutual information analysis to approach nonlinearity in groundwater stochastic fields. Stoch Environ Res Risk Assess 32:2933–2942. https://doi.org/10.1007/s00477-018-1591-4
Claramunt C (2005) A spatial form of diversity. In: Cohn AG, Mark DM (eds) COSIT 2005, vol 3693. LNCS, New York, pp 218–231
Cover TM, Thomas JA (2006) Elements of information theory, 2nd edn. Wiley, Hoboken
Cressie N (1993) Statistics for spatial data, Revised edn. Wiley, Hoboken
Frosini BV (2004) Descriptive measures of ecological diversity. In Jureckova J, El-Shaarawi AH (eds.), Environmetrics. Encyclopedia of life support systems (EOLSS). Revised edn. 2006. Developed under the Auspices of the UNESCO, Eolss Publishers, Paris, France. http://www.eolss.net. Accessed 25 Feb 2015
He J, Kolovos A (2018) Bayesian maximum entropy approach and its applications: a review. Stoch Environ Res Risk Assess 32:859–877
Hoeting JA, Leecaster M, Bowden D (2000) An improved model for spatially correlated binary responses. J Agric Biol Environ Stat 5:102–114
Karlström A, Ceccato V (2002) A new information theoretical measure of global and local spatial association. Jaharb Regionalwissensc 22:13–40
Leibovici DG (2009) Defining spatial entropy from multivariate distributions of co-occurrences. In: Hornsby KS, Claramunt C, Denis M, Ligozat G (eds) Spatial information theory. COSIT 2009. Lecture notes in computer science, vol 5756. Springer, Berlin, Heidelberg, pp 392–404
Leibovici DG, Birkin MH (2015) On geocomputational determinants of entropic variations for urban dynamics studies. Geogr Anal 47:193–218
Leibovici DG, Claramunt C, LeGuyader D, Brosset D (2014) Local and global spatio-temporal entropy indices based on distance ratios and co-occurrences distributions. Int J Geogr Inf Sci 28(5):1061–1084
Leinster T, Cobbold CA (2012) Measuring diversity: the importance of species similarity. Ecology 93(3):477–489
Li H, Reynolds JF (1993) A new contagion index to quantify spatial patterns of landscapes. Landsc Ecol 8(3):155–162
O’Neill RV, Krummel JR, Gardner RH, Sugihara G, Jackson B, DeAngelis DL, Milne BT, Turner MG, Zygmunt B, Christensen SW, Dale VH, Graham RL (1988) Indices of landscape pattern. Landsc Ecol 1(3):153–162
Paninski L (2003) Estimation of entropy and mutual information. J Neural Comput 15:1191–1253
Parresol BR, Edwards LA (2014) An entropy-based contagion index and its sampling properties for landscape analysis. Entropy 16(4):1842–1859
Patil GP, Taillie C (1982) Diversity as a concept and its measurement. J Am Stat Assoc 77:548–561
Riitters KH, O’Neill RV, Wickham JD, Jones KB (1996) A note on contagion indices for landscape analysis. Landsc Ecol 11(4):197–202
Shannon C (1948) A mathematical theory of communication. Bell Syst Tech J 27(3):379–423
Shannon C (1948) A mathematical theory of communication. Bell Syst Tech J 27(4):623–656
Theil H (1972) Statistical decomposition analysis. North Holland, Amsterdam
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).
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Altieri, L., Cocchi, D. & Roli, G. Advances in spatial entropy measures. Stoch Environ Res Risk Assess 33, 1223–1240 (2019). https://doi.org/10.1007/s00477-019-01686-y
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DOI: https://doi.org/10.1007/s00477-019-01686-y