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Calculating spatial configurational entropy of a landscape mosaic based on the Wasserstein metric

  • Yuan Zhao
  • Xinchang ZhangEmail author
Research Article
  • 50 Downloads

Abstract

Context

Entropy is an important concept traditionally associated with thermodynamics and is widely used to describe the degree of disorder in a substance, system, or process. Configurational entropy has received more attention because it better reflects the thermodynamic properties of physical and biological processes. However, as the number of configuration combinations increases, configurational entropy becomes too complex to calculate, and its value is too large to be accurately represented in practical applications.

Objectives

To calculate the spatial configurational entropy of a landscape mosaic based on a statistical metric.

Methods

We proposed a relative entropy using histograms to compare two ecosystems with the Wasserstein metric, and used six digital elevation models and five simulated data to calculate the entropy of the complex ecosystems.

Results

The calculation and simulation showed that the purposed metric captured disorder in the spatial landscape, and the result was consistent with the general configurational entropy. By calculating several spatial scale landscapes, we found that relative entropy can be a trade-off between the rationality of results and the cost of calculation.

Conclusions

Our results show that the Wasserstein metric is suitable to capture the discrepancy using complex landscape mosaic data sets, which provides a numerically efficient approximation for the similarity in the histograms, reducing excessive expansion of the calculated result.

Keywords

Boltzmann entropy Configurational entropy Landscape mosaic Shannon entropy Wasserstein metric 

Notes

Acknowledgements

We thank anonymous reviewers for their constructive comments. This research was supported by the National Natural Science Foundation of China (Grant No. 41431178), the Natural Science Foundation of Guangdong Province in China (Grant No. 2016A030311016), the National Administration of Surveying, Mapping and Geoinformation of China (Grant No. GZIT2016-A5-147) and the Research Institute of Henan Spatio-Temporal Big Data Industrial Technology (Grant No. 2017DJA001).

References

  1. Batty M (1976) Entropy in spatial aggregation. Geogr Anal 8(1):1–21CrossRefGoogle Scholar
  2. Bjoke JT (1996) Framework for entropy-based map evaluation. Am Cartogr 23(2):78–95Google Scholar
  3. Bogaert J, Farina A, Ceulemans R (2005) Entropy increase of fragmented habitats: A sign of human impact? Ecol Indic 5(3):207–212CrossRefGoogle Scholar
  4. Brillouin L (1956) Science and information theory. Academic Press, New YorkCrossRefGoogle Scholar
  5. Cushman SA (2015) Thermodynamics in landscape ecology: the importance of integrating measurement and modeling of landscape entropy. Landscape Ecol 30(1):7–10CrossRefGoogle Scholar
  6. Cushman SA (2016) Calculating the configurational entropy of a landscape mosaic. Landscape Ecol 31(3):481–489CrossRefGoogle Scholar
  7. Cushman SA (2018) Calculation of configurational entropy in complex landscapes. Entropy 20(2984):298.  https://doi.org/10.3390/e20040298 CrossRefGoogle Scholar
  8. Díaz-Varela E, Roces-Díaz JV, Álvarez álvarez P (2016) Detection of landscape heterogeneity at multiple scales: use of the quadratic entropy index. Landscape Urban Plan 153:149–159CrossRefGoogle Scholar
  9. Deza E, Deza MM (2016) Encyclopedia of distances. Springer, New YorkCrossRefGoogle Scholar
  10. Feldman DP, Crutchfield JP (2003) Structural information in two-dimensional patterns: entropy convergence and excess entropy. Phys Rev E 67(5):051104CrossRefGoogle Scholar
  11. Foody GM (1995) Cross-entropy for the evaluation of the accuracy of a fuzzy land cover classification with fuzzy ground data. ISPRS J Photogramm Remote Sens 50(5):2–12CrossRefGoogle Scholar
  12. Gao P, Zhang H, Li Z (2017) A hierarchy-based solution to calculate the configurational entropy of landscape gradients. Landscape Ecol 32(6):1–14Google Scholar
  13. Gatrell AC (1977) Complexity and redundancy in binary maps. Geogr Anal 9(1):29–41CrossRefGoogle Scholar
  14. Goodchild MF (2003) The nature and value of geographic information. Foundations of geographic information science. Taylor and Francis Group, LondonGoogle Scholar
  15. Leibovici DG (2009) Defining spatial entropy from multivariate distributions of co-occurrences. Springer, Berlin, pp 392–404Google Scholar
  16. Leibovici DG, Birkin MH (2015) On geocomputational determinants of entropic variations for urban dynamics studies. Geogr Anal 47(3):193–218CrossRefGoogle Scholar
  17. Leibovici DG, Claramunt C, Le Guyader 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):1061–1084CrossRefGoogle Scholar
  18. Li H, Reynolds JF (1993) A new contagion index to quantify spatial patterns of landscapes. Landscape Ecol 8(3):155–162CrossRefGoogle Scholar
  19. Li Z, Liu Q, Gao P (2016) Entropy-based cartographic communication models:evolution from special to general cartographic information theory. Acta Geod Et Cartogr Sinica 45(7):757–767Google Scholar
  20. Li ZL, Huang PZ (2002) Quantitative measures for spatial information of maps. Int J Geogr Inf Sci 16(7):699–709.  https://doi.org/10.1080/13658810210149416 CrossRefGoogle Scholar
  21. Neumann J (1994) The topological information content of a map an attempt at a rehabilitation of information theory in cartography. Cartogr Int J Geogr Inf Geovisualization 31(1):26–34CrossRefGoogle Scholar
  22. Simpson EH (1949) Measurement of diversity. Nature 163:688CrossRefGoogle Scholar
  23. Snickars F, Weibull JW (1977) A minimum information principle: theory and practice. Reg Sci Urban Econ 7(1–2):137–168CrossRefGoogle Scholar
  24. Tobler W (1997) Introductory comments on information theory and cartography. Cartogr Perspect 26(27):5341–5357Google Scholar
  25. Vajda S, Shannon CE, Weaver W (1949) The mathematical theory of communication. Bell Syst Techn J 27(4):379–423Google Scholar
  26. Villani C (2008) Optimal transport: old and new, vol 338. Springer Science & Business Media, BerlinGoogle Scholar
  27. Vranken I, Baudry J, Aubinet M, Visser M, Bogaert J (2015) A review on the use of entropy in landscape ecology: heterogeneity, unpredictability, scale dependence and their links with thermodynamics. Landscape Ecol 30(1):51–65CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.School of Geography and PlanningSun Yat-Sen UniversityGuangzhouChina
  2. 2.School of Geographical SciencesGuangzhou UniversityGuangzhouChina
  3. 3.College of Environment and PlanningHenan UniversityKaifengChina

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