{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,24]],"date-time":"2026-02-24T21:28:04Z","timestamp":1771968484095,"version":"3.50.1"},"reference-count":48,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2020,8,26]],"date-time":"2020-08-26T00:00:00Z","timestamp":1598400000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["41901316"],"award-info":[{"award-number":["41901316"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100006132","name":"State Key Laboratory of Earth Surface Processes and Resource Ecology","doi-asserted-by":"publisher","award":["2020-KF-03"],"award-info":[{"award-number":["2020-KF-03"]}],"id":[{"id":"10.13039\/501100006132","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Entropy is a fundamental concept in thermodynamics that is important in many fields, including image processing, neurobiology, urban planning, and sustainability. As of recently, the application of Boltzmann entropy for landscape patterns was mostly limited to the conceptual discussion. However, in the last several years, a number of methods for calculating Boltzmann entropy for landscape mosaics and gradients were proposed. We developed an R package belg as an open source tool for calculating Boltzmann entropy of landscape gradients. The package contains functions to calculate relative and absolute Boltzmann entropy using the hierarchy-based and the aggregation-based methods. It also supports input raster with missing (NA) values, allowing for calculations on real data. In this study, we explain ideas behind implemented methods, describe the core functionality of the software, and present three examples of its use. The examples show the basic functions in this package, how to adjust Boltzmann entropy values for data with missing values, and how to use the belg package in larger workflows. We expect that the belg package will be a useful tool in the discussion of using entropy for a description of landscape patterns and facilitate a thermodynamic understanding of landscape dynamics.<\/jats:p>","DOI":"10.3390\/e22090937","type":"journal-article","created":{"date-parts":[[2020,8,26]],"date-time":"2020-08-26T09:05:37Z","timestamp":1598432737000},"page":"937","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["belg: A Tool for Calculating Boltzmann Entropy of Landscape Gradients"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1057-3721","authenticated-orcid":false,"given":"Jakub","family":"Nowosad","sequence":"first","affiliation":[{"name":"Institute of Geoecology and Geoinformation, Adam Mickiewicz University, Krygowskiego 10, 61-680 Poznan, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1714-779X","authenticated-orcid":false,"given":"Peichao","family":"Gao","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Earth Surface Processes and Resource Ecology, Beijing Normal University, Beijing 100875, China"},{"name":"Faculty of Geographical Science, Beijing Normal University, Beijing 100875, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,8,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"22","DOI":"10.1016\/j.landurbplan.2017.10.002","article-title":"The Low-Entropy City: A Thermodynamic Approach to Reconnect Urban Systems with Nature","volume":"168","author":"Pelorosso","year":"2017","journal-title":"Landsc. Urban Plan."},{"key":"ref_2","unstructured":"Swati, B.K., and Venkanna, B.V. (2010). Basic Thermodynamics, PHI Learning."},{"key":"ref_3","first-page":"275","article-title":"Weitere Studien \u00fcber Das W\u00e4rmegleichgewicht Unter Gasmolek\u00fclen [Further Studies on the Thermal Equilibrium of Gas Molecules]","volume":"66","author":"Boltzmann","year":"1872","journal-title":"Sitzungsberichte Akad. Der Wiss."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Thess, A. (2011). The Entropy Principle: Thermodynamics for the Unsatisfied, Springer.","DOI":"10.1007\/978-3-642-13349-7"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1126\/science.192.4235.101","article-title":"Net Energy Analysis: An Economic Assessment","volume":"192","author":"Huettner","year":"1976","journal-title":"Science"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Gao, P.C., Wang, H.Y., Cushman, S.A., Cheng, C.X., Song, C.Q., and Ye, S.J. (2020). Sustainable Land-Use Optimization Using NSGA-II: Theoretical and Experimental Comparisons of Improved Algorithms. Landsc. Ecol.","DOI":"10.1007\/s10980-020-01051-3"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"3948","DOI":"10.1080\/01431161.2019.1711242","article-title":"Unsupervised Band Selection Based on Weighted Information Entropy and 3D Discrete Cosine Transform for Hyperspectral Image Classification","volume":"41","author":"Sawant","year":"2020","journal-title":"Int. J. Remote Sens."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"537","DOI":"10.2495\/SC140451","article-title":"The Sustainable City and the Smart City: Measuring Urban Entropy First","volume":"191","author":"Fistola","year":"2014","journal-title":"WIT Trans. Ecol. Environ."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"158","DOI":"10.1016\/j.pneurobio.2016.03.005","article-title":"The Application of Information Theory for the Research of Aging and Aging-Related Diseases","volume":"157","author":"Blokh","year":"2017","journal-title":"Prog. Neurobiol."},{"key":"ref_10","unstructured":"Forman, R.T.T., and Godron, M. (1986). Landscape Ecology, John Wiley & Sons."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1007\/s10980-011-9689-9","article-title":"Spatial Pattern Formation of Coastal Vegetation in Response to External Gradients and Positive Feedbacks Affecting Soil Porewater Salinity: A Model Study","volume":"27","author":"Jiang","year":"2012","journal-title":"Landsc. Ecol."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1007\/BF00156229","article-title":"Biocybernetic and Thermodynamic Perspectives of Landscape Functions and Land Use Patterns","volume":"1","author":"Naveh","year":"1987","journal-title":"Landsc. Ecol."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1007\/BF00131538","article-title":"A Hierarchical Framework for the Analysis of Scale","volume":"3","author":"Johnson","year":"1989","journal-title":"Landsc. Ecol."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"439","DOI":"10.1086\/419172","article-title":"From Balance of Nature to Hierarchical Patch Dynamics: A Paradigm Shift in Ecology","volume":"70","author":"Wu","year":"1995","journal-title":"Q. Rev. Biol."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1161","DOI":"10.1007\/s10980-012-9763-y","article-title":"Highlighting Order and Disorder in Social\u2013Ecological Landscapes to Foster Adaptive Capacity and Sustainability","volume":"28","author":"Zurlini","year":"2013","journal-title":"Landsc. Ecol."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1007\/s10980-014-0108-x","article-title":"Thermodynamics in Landscape Ecology: The Importance of Integrating Measurement and Modeling of Landscape Entropy","volume":"30","author":"Cushman","year":"2015","journal-title":"Landsc. Ecol."},{"key":"ref_17","unstructured":"Bailey, K.D. (2009). Entropy Systems Theory. Parra-Luna, Eolss Publishers\/UNESCO: Systems Science and Cybernetics."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1007\/s10980-014-0105-0","article-title":"A Review on the Use of Entropy in Landscape Ecology: Heterogeneity, Unpredictability, Scale Dependence and Their Links with Thermodynamics","volume":"30","author":"Vranken","year":"2015","journal-title":"Landsc. Ecol."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1016\/j.ecoinf.2012.04.002","article-title":"Calculating Landscape Diversity with Information-Theory Based Indices: A GRASS GIS Solution","volume":"17","author":"Rocchini","year":"2013","journal-title":"Ecol. Informatics"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1016\/j.landurbplan.2016.05.004","article-title":"Detection of Landscape Heterogeneity at Multiple Scales: Use of the Quadratic Entropy Index","volume":"153","year":"2016","journal-title":"Landsc. Urban Plan."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1002\/j.1538-7305.1948.tb01338.x","article-title":"A Mathematical Theory of Communication","volume":"27","author":"Shannon","year":"1948","journal-title":"Bell Syst. Tech. J."},{"key":"ref_22","unstructured":"Shannon, C.E., and Weaver, W. (1949). The Mathematical Theory of Communication, The University of Illinois Press."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"2091","DOI":"10.1007\/s10980-019-00830-x","article-title":"Information Theory as a Consistent Framework for Quantification and Classification of Landscape Patterns","volume":"34","author":"Nowosad","year":"2019","journal-title":"Landsc. Ecol."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1016\/0375-9601(95)00867-5","article-title":"A Statistical Measure of Complexity","volume":"209","author":"Mancini","year":"1995","journal-title":"Phys. Lett. A"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1016\/j.ecolmodel.2015.03.014","article-title":"Statistical-Thermodynamics Modelling of the Built Environment in Relation to Urban Ecology","volume":"307","author":"Mohajeri","year":"2015","journal-title":"Ecol. Model."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1016\/j.physleta.2015.09.048","article-title":"Application of a Boltzmann-Entropy-like Concept in an Agent-Based Multilane Traffic Model","volume":"380","author":"Sugihakim","year":"2016","journal-title":"Phys. Lett. A"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Li, Z.L. (2007). Algorithmic Foundation of Multi-Scale Spatial Representation, CRC Press.","DOI":"10.1201\/9781420008432"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"481","DOI":"10.1007\/s10980-015-0305-2","article-title":"Calculating the Configurational Entropy of a Landscape Mosaic","volume":"31","author":"Cushman","year":"2016","journal-title":"Landsc. Ecol."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Cushman, S.A. (2018). Calculation of Configurational Entropy in Complex Landscapes. Entropy, 20.","DOI":"10.3390\/e20040298"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1849","DOI":"10.1007\/s10980-019-00876-x","article-title":"Calculating Spatial Configurational Entropy of a Landscape Mosaic Based on the Wasserstein Metric","volume":"34","author":"Zhao","year":"2019","journal-title":"Landsc. Ecol."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Zhang, H., Wu, Z.W., Lan, T., Chen, Y.Y., and Gao, P.C. (2020). Calculating the Wasserstein Metric-Based Boltzmann Entropy of a Landscape Mosaic. Entropy, 22.","DOI":"10.3390\/e22040381"},{"key":"ref_32","first-page":"1133","article-title":"A Hierarchy-Based Solution to Calculate the Configurational Entropy of Landscape Gradients","volume":"32","author":"Gao","year":"2017","journal-title":"Landsc. Ecol."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1837","DOI":"10.1007\/s10980-019-00854-3","article-title":"Aggregation-Based Method for Computing Absolute Boltzmann Entropy of Landscape Gradient with Full Thermodynamic Consistency","volume":"34","author":"Gao","year":"2019","journal-title":"Landsc. Ecol."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"2183","DOI":"10.1007\/s10980-019-00814-x","article-title":"Computation of the Boltzmann Entropy of a Landscape: A Review and a Generalization","volume":"34","author":"Gao","year":"2019","journal-title":"Landsc. Ecol."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Zhang, H., and Wu, Z.W. (2020). A Head\/Tail Breaks-Based Method for Efficiently Estimating the Absolute Boltzmann Entropy of Numerical Raster Data. ISPRS Int. J. Geo-Inf., 9.","DOI":"10.3390\/ijgi9020103"},{"key":"ref_36","unstructured":"R Core Team (2020). R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"673","DOI":"10.1007\/s10980-009-9341-0","article-title":"Gradient Modeling of Conifer Species Using Random Forests","volume":"24","author":"Evans","year":"2009","journal-title":"Landsc. Ecol."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Wiens, J.A., and Moss, M.R. (2005). The Gradient Concept of Landscape Structure. Issues and Perspectives in Landscape Ecology, Cambridge University Press.","DOI":"10.1017\/CBO9780511614415"},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Dalarsson, N., Dalarsson, M., and Golubovic, L. (2011). Introductory Statistical Thermodynamics, Academic Press.","DOI":"10.1016\/B978-0-12-384956-4.00009-4"},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Hobbie, R.K., and Roth, B.J. (2015). Intermediate Physics for Medicine and Biology, Springer.","DOI":"10.1007\/978-3-319-12682-1"},{"key":"ref_41","unstructured":"Serway, R.A., and Jewett, J.W. (2009). Physics for Scientists and Engineers with Modern Physics, Brooks\/Cole Publishing Company. [8th ed.]."},{"key":"ref_42","first-page":"e3188v1","article-title":"Extending extitR with extitC++: A Brief Introduction to extitRcpp","volume":"5","author":"Eddelbuettel","year":"2017","journal-title":"PeerJ Prepr."},{"key":"ref_43","unstructured":"Hijmans, R.J. (2020). Raster: Geographic Data Analysis and Modeling, R Foundation for Statistical Computing."},{"key":"ref_44","unstructured":"Pebesma, E. (2020). Stars: Spatiotemporal Arrays, Raster and Vector Data Cubes, R Foundation for Statistical Computing."},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Lovelace, R., Nowosad, J., and Muenchow, J. (2019). Geocomputation with R, CRC Press.","DOI":"10.1201\/9780203730058"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"439","DOI":"10.32614\/RJ-2018-009","article-title":"Simple Features for r: Standardized Support for Spatial Vector Data","volume":"10","author":"Pebesma","year":"2018","journal-title":"R J."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"1","DOI":"10.18637\/jss.v084.i06","article-title":"tmap: Thematic Maps in R","volume":"84","author":"Tennekes","year":"2018","journal-title":"J. Stat. Softw."},{"key":"ref_48","unstructured":"Bengtsson, H. (2020). Future: Unified Parallel and Distributed Processing in R for Everyone, R Foundation for Statistical Computing. R Package Version 1.18.0."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/9\/937\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:06:54Z","timestamp":1760177214000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/9\/937"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,8,26]]},"references-count":48,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2020,9]]}},"alternative-id":["e22090937"],"URL":"https:\/\/doi.org\/10.3390\/e22090937","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,8,26]]}}}