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Spatial Entropy, Geo-Information and Spatial Surprise

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Spatial Entropy and Landscape Analysis

Part of the book series: RaumFragen: Stadt – Region – Landschaft ((RFSRL))

Abstract

Landscape entropy can be calculated using Shannon’s formula. Focusing on multicolored square maps, it is shown how and why the spatial information of their square cells relates to spatial entropy. Both spatial information and spatial entropy correspond to the “spatial surprise” that is experienced by the map’s viewer: the former from a single cell, the latter from the image as a whole.

Only entropy comes easy

“Только энтропия даётся легко”

(Anton Chekhov, 1860–1904)

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References

  • Addiscott TM (2010) Entropy, non-linearity and hierarchy in ecosystems. Geoderma 160(1):57–63

    Article  Google Scholar 

  • Altieri L, Cocchi D, Roli, G (2018) Measuring heterogeneity in urban expansion via spatial entropy. arXiv:1805.04274v1 [stat.AP] 11 May 2018

  • Baiesi M, Orlandini E, Stella AL (2010) The entropic cost to tie a knot. J Stat Mech Theory Exp 2010(6):P06012

    Article  Google Scholar 

  • Band G, Boyland P (2007) The Burau estimate for the entropy of a braid. Algebr Geom Topol 7(3):1345–1378

    Article  Google Scholar 

  • Batty M (1974) Spatial Entropy. Geogr Anal 6:1–31

    Article  Google Scholar 

  • Batty M (1976) Entropy in spatial aggregation. Geogr Anal 8:1–21

    Article  Google Scholar 

  • Boltzmann L (1872) Weitere studien uber das warmegleichgewicht unter gasmolekulen [Further studies of the warm equilibrium of gas molecules]. Wiener Berichte 66:275–370

    Google Scholar 

  • Bueso MC, Angulo JM, Alonso FJ (1998) A state-space model approach to optimum spatial sampling design based on entropy. Environ Ecol Stat 5:29–44

    Article  Google Scholar 

  • Cabral P, Augusto G, Tewolde M, Araya Y (2013) Entropy in urban systems. Entropy, 15(12), 5223–5236, 1–14

    Google Scholar 

  • Chakir, R (2009) Spatial downscaling of agricultural land-use data: an econometric approach using cross entropy. Land Econ 85(2) May, 238–251

    Google Scholar 

  • Chapman EJ, Childers DL, Vallino JJ (2016) How the second law of thermodynamics has informed ecosystem ecology through its history. Bioscience 66(1):27–39

    Article  Google Scholar 

  • Chen Y, Huang L (2018) Spatial measures of urban systems: from Entropy to fractal dimension. Entropy 20(991):1–21.

    Article  Google Scholar 

  • Chen Y, Wang J, Feng J (2017) Understanding the fractal dimensions of urban forms through spatial entropy. Entropy 19(600):1–18.

    Article  Google Scholar 

  • Christakos G, Li X (1998) Bayesian maximum entropy analysis and mapping: a farewell to kriging estimators? Math Geol 30(4):435–462

    Article  Google Scholar 

  • Christakos G (1990) A Bayesian/maximum-entropy view to the spatial estimation problem. Math Geol 22(7):763–776

    Article  Google Scholar 

  • Clausius R (1864) Abhandlungen über die mechanische Wärmetheorie I. Vieweg und Sohn, Braunschweig

    Google Scholar 

  • Cushman SA (2016) Calculating the configurational entropy of a landscape mosaic. Landscape Ecol 31(3):481–489

    Article  Google Scholar 

  • Cushman SA (2018) Entropy in Landscape Ecology. Entropy 20(5):314

    Article  Google Scholar 

  • Czyż T, Hauke J (2015) Entropy in regional analysis. Quaestiones Geographicae 34(4):69–78

    Article  Google Scholar 

  • Dourish P (2001) Where the action is. Foundations of embodied interaction. MIT Press, Cambridge

    Book  Google Scholar 

  • Edren SMC, Wisz MS, Teilmann J, Dietz R, Soderkvist J (2010) Modelling spatial patterns in harbour porpoise satellite telemetry data using maximum entropy. Ecography 33:698–708

    Article  Google Scholar 

  • Feng Q, Chai L (2009) Ecosystem evolution dynamics based on generalized entropy principle. Sci Technol Rev 4:36–41

    Google Scholar 

  • Fistola R (2012) Urban entropy vs sustainability: a new town planning perspective. The Sustainable City VII: Urban Regeneration and Sustainability 155:1195

    Google Scholar 

  • Forman RTT (1995) Land mosaics: the ecology of landscapes and regions. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Francois N, Xia H, Punzmann H, Faber B, Shats M (2015) Braid entropy of two-dimensional turbulence. Sci Rep 5(1):1–8

    Google Scholar 

  • Franks J, Williams RF (1985) Entropy and knots. Trans Am Math Soc 291(1):241–253

    Article  Google Scholar 

  • Gatrell AG (1977) Complexity and redundancy in binary maps. Geogr Anal 9:29–41

    Article  Google Scholar 

  • Ghrist RW (1998) Chaotic knots and wild dynamics. Chaos, Solitons Fractals 9(4–5):583–598

    Article  Google Scholar 

  • Haynes KE, Storbecks JE (1978) The entropy paradox and the distribution of urban population. Socioecon Plann Sci 12(1):1–6

    Article  Google Scholar 

  • Heikkila EJ, Hu L (2006) Adjusting spatial-entropy measures for scale and resolution effects. Environ Plann B Plann Des 33:845–861

    Article  Google Scholar 

  • Huynh HN (2019) Spatial point pattern and urban morphology: perspectives from entropy, complexity and networks. arXiv:1904.09787v2 [physics.soc-ph] 14 Aug 2019

  • Ji W, Wu J, Zhang M, Liu Z, Shi G, Xie X (2019) Blind image quality assessment with joint entropy degradation. IEEE Access 7:30925–30936

    Article  Google Scholar 

  • Joshi PK, Lele N, Agarwal SP (2006) Entropy as an indicator of fragmented Landscape. Curr Sci 91(3):276–278

    Google Scholar 

  • Kandziora M, Burkhard B, Müller F (2013) Interactions of ecosystem properties, ecosystem integrity and ecosystem service indicators—A theoretical matrix exercise. Ecol Ind 28:54–78

    Article  Google Scholar 

  • Khinchin A (1957) The entropy concept in probability theory-Mathematical foundations of Information theory. Dover, New York

    Google Scholar 

  • Lang W, Long Y, Chen T (2018) Rediscovering Chinese cities through the lens of land-use patterns. Land Use Policy 79:362–374

    Article  Google Scholar 

  • Leibovici DG (2009) Defining spatial entropy from multivariate distributions of co-occurrences. In: Stewart Hornsby K et al (eds) COSIT 2009, Springer Lecture Notes in Computer Science 5756, 392–404

    Google Scholar 

  • Leopold LB, Langbein WB (1962) The concept of entropy in landscape evolution. U.S.G.S. Prof.Paper 500-A:A1–A20

    Google Scholar 

  • Li X, Claramunt C (2006) A spatial entropy-based decision tree for classification of geographical information. Trans GIS 10(3):451–467

    Article  Google Scholar 

  • Lin H (2015) Thermodynamic entropy fluxes reflect ecosystem characteristics and succession. Ecol Model 298:75–86

    Article  Google Scholar 

  • Lin Z, Xia B (2013) Sustainability analysis of the urban ecosystem in Guangzhou City based on information entropy between 2004 and 2010. J Geog Sci 23(3):417–435

    Article  Google Scholar 

  • Liu L, Liu B, Huang H, Bovik AC (2014) No-reference image quality assessment based on spatial and spectral entropies. Signal Process Image Commun 29(8):856–863

    Google Scholar 

  • Liu L, Peng Z, Wu H, Jiao H, Yu Y, Zhao J (2018) Fast identification of Urban Sprawl based on K-Means clustering with population density and local spatial entropy. Sustainability 10(2683):1–16

    Google Scholar 

  • MacArthur RH (1955) Fluctuation of animal populations and a measure of community stability. Ecology 36:533–536

    Article  Google Scholar 

  • Marchettini N, Pulselli FM, Tiezzi E (2006) Entropy and the city. WIT Trans Ecol Environ 93:263–272

    Article  Google Scholar 

  • Margalef DR (1958) Information Entropy in Ecology. Yearb Soc Gen Syst Res 3:36–71

    Google Scholar 

  • Margurran A (1988) Ecological diversity and its measurement. Princeton University Press, Princeton

    Google Scholar 

  • O’ Neill RV (1988) Indices of landscape pattern. Landscape Ecol 1(3):153–162

    Google Scholar 

  • O’ Neill RV, Krummel JR, Gardner RH, Sugihara G, Jackson B, De Angelis DL, Milne BT, Turner MG, Zygmunt B, Christensen SW, Dale VH, Graham RL (1988) Indices of Landscape Pattern. Landscape Ecol 1(3):153–162

    Google Scholar 

  • Papadimitriou F (2002) Modelling indices and indicators of landscape complexity: an approach using GIS. Ecol Ind 2:17–25

    Article  Google Scholar 

  • Papadimitriou F (2009) Modelling spatial landscape complexity using the levenshtein algorithm. Eco Inform 4:48–55

    Article  Google Scholar 

  • Papadimitriou F (2010) Conceptual modelling of landscape complexity. Landsc Res 35(5):563–570

    Article  Google Scholar 

  • Papadimitriou F (2012a) The algorithmic complexity of landscapes. Landsc Res 37(5):599–611

    Article  Google Scholar 

  • Papadimitriou F (2012b) Artificial intelligence in modelling the complexity of mediterranean landscape transformations. Comput Electron Agric 81:87–96

    Article  Google Scholar 

  • Papadimitriou F (2012c) Modelling landscape complexity for land use management in Rio de Janeiro. Brazil. Land Use Policy 29(4):855–861

    Article  Google Scholar 

  • Papadimitriou F (2013) Mathematical modelling of land use and landscape complexity with ultrametric topology. J Land Use Sci 8(2):234–254

    Article  Google Scholar 

  • Papadimitriou F (2020a) Modelling and visualization of landscape complexity with braid topology. In: Edler D, Jenal C, Kühne O (eds) Modern approaches to the visualization of landscapes. Springer VS, Wiesbaden, pp 79–101

    Chapter  Google Scholar 

  • Papadimitriou, F. (2020b). Spatial complexity. Theory, mathematical methods and applications. Springer, Cham

    Google Scholar 

  • Papadimitriou F (2020c) The Probabilistic basis of spatial complexity. In: Complexity S (ed) Theory, mathematical methods and applications. Springer, Cham, pp 51–61

    Google Scholar 

  • Papadimitriou F (2020d) The spatial complexity of 3x3 binary maps. In: Complexity S (ed) Theory, mathematical methods and applications. Springer, Cham, pp 163–178

    Google Scholar 

  • Papadimitriou F (2020e) Geophilosophy and epistemology of spatial complexity. In: Spatial complexity. Theory, mathematical methods and applications. Springer, Cham, pp 263–278

    Google Scholar 

  • Papadimitriou F (2020f) Spatial complexity and the future. In: Spatial complexity. Theory, mathematical methods and applications. Springer, Cham, pp 279–292

    Google Scholar 

  • Pelorosso R, Gobattoni F, Leone A (2017) The low-entropy city: a thermodynamic approach to reconnect urban systems with nature. Landsc Urban Plan 168:22–30

    Article  Google Scholar 

  • Piasecki R (2000) Entropic measure of spatial disorder for systems of finite-sized objects. Physica A 277(1–2):157–173

    Article  Google Scholar 

  • Pielou EC (1969) An introduction to mathematical ecology. Wiley, New York

    Google Scholar 

  • Pielou EC (1975) Ecological diversity. Wiley, New York

    Google Scholar 

  • Pournader M, Ahmadi H, Feiznia S, Karimi H, Reza Peirovan H (2018) Spatial prediction of soil erosion susceptibility: an evaluation of the maximum entropy model. Earth Sci Inf 1–18:6

    Google Scholar 

  • Purvis B, Mao Y, Robinson D (2019) Entropy and its application to urban systems. Entropy 21(1):56

    Article  Google Scholar 

  • Rahimipoor S, Edoyan H, Hashemi M (2011) The ambiguity of self and entropy in Samuel Beckett’s plays. Procedia Soc Behav Sci 28:914–918

    Article  Google Scholar 

  • Razlighi QR, Rahman MT, Kehtarnavaz N (2011) Fast computation methods for estimation of image spatial entropy. J Real-Time Image Proc 6(2):137–142

    Article  Google Scholar 

  • Riitters KH, O’Neill RV, Hunsaker CT, Wickham JD, Yankee DH, Timmins SP, Jones KB, Jackson BL (1995) A factor analysis of vegetation pattern and structure metrics. Landscape Ecol 10:23–39

    Article  Google Scholar 

  • Rosenzweig ML (1995) Species diversity in space and time. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Scheidegger AE (1964) Some implications of statistical mechanics in geomorphology. Hydrol Sci J 9(1):12–16

    Google Scholar 

  • Shannon CE (1949) The mathematical theory of communication. University of Illinois press, Urbana

    Google Scholar 

  • Sherwin WB, Prat i Fornells, N. (2019) The introduction of entropy and information methods to ecology by Ramon Margalef. Entropy 21(8):794

    Article  Google Scholar 

  • Su H-T, You J-Y (2014) Developing an entropy-based model of spatial information estimation and its application in the design of precipitation gauge networks. J Hydrol 519:3316–3327

    Article  Google Scholar 

  • Sukhov VI (1967) Information capacity of map entropy. Geodesy Aerophotogr X:212–215

    Google Scholar 

  • Theil H (1967) Economics and information theory. North-Holland, Amsterdam

    Google Scholar 

  • Turner MG (1989) Effects of changing spatial scale on the analysis of spatial pattern. Landscape Ecol 3(3/4):153–162

    Article  Google Scholar 

  • Turner MG (1990) Spatial and temporal analysis of landscape patterns. Landscape Ecol 4(1):21–30

    Article  Google Scholar 

  • Virgo N, Harvey I (2007) Entropy production in ecosystems. European Conference on Artificial Life. Springer, Berlin, pp 123–132

    Google Scholar 

  • 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 30:51–65

    Article  Google Scholar 

  • Wang B, Wang X, Chen Z (2012) Spatial Entropy based mutual information in hyperspectral band selection for supervised classification. Int J Numer Anal Model 9(2):181–192

    Google Scholar 

  • You L, Wood S (2005) Assessing the spatial distribution of crop areas using a cross-entropy method. Int J Appl Earth Obs Geoinf 7(4):310–323

    Google Scholar 

  • You L, Wood S (2006) An entropy approach to spatial disaggregation of agricultural production. Agric Syst 90:329–347

    Article  Google Scholar 

  • Zaccarelli N, Li BL, Petrosillo I, Zurlini G (2013) Order and disorder in ecological time-series: introducing normalized spectral entropy. Ecol Ind 28:22–30

    Article  Google Scholar 

  • Zdenkovic M, Scheidegger AE (1989) Entropy of landscapes. Z Geomorphol 33(3):361–371

    Article  Google Scholar 

  • Zhang Y, Yang Z, Li W (2006) Analyses of urban ecosystem based on information entropy. Ecol Model 197(1–2):1–12

    Article  Google Scholar 

  • Zhang T, Cheng C, Gao P (2019) Permutation entropy-based analysis of temperature complexity spatial-temporal variation and its driving factors in China. Entropy 21:1001

    Article  Google Scholar 

  • Zhao S, Chai L (2015) A new assessment approach for urban ecosystem health basing on maximum information entropy method. Stoch Env Res Risk Assess 29(6):1601–1613

    Article  Google Scholar 

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Papadimitriou, F. (2022). Spatial Entropy, Geo-Information and Spatial Surprise. In: Spatial Entropy and Landscape Analysis. RaumFragen: Stadt – Region – Landschaft. Springer VS, Wiesbaden. https://doi.org/10.1007/978-3-658-35596-8_1

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  • DOI: https://doi.org/10.1007/978-3-658-35596-8_1

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