{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:35:41Z","timestamp":1760240141860,"version":"build-2065373602"},"reference-count":65,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2019,3,28]],"date-time":"2019-03-28T00:00:00Z","timestamp":1553731200000},"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":["41571414"],"award-info":[{"award-number":["41571414"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Tsinghua University Initiative Scientific Research Program","award":["2015THZ01"],"award-info":[{"award-number":["2015THZ01"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Urban scaling laws describe powerful universalities of the scaling relationships between urban attributes and the city size across different countries and times. There are still challenges in precise statistical estimation of the scaling exponent; the properties of variance require further study. In this paper, a statistical regression method based on the maximum likelihood estimation considering the lower bound constraints and the heterogeneous variance of error structure, termed as CHVR, is proposed for urban scaling estimation. In the CHVR method, the heterogeneous properties of variance are explored and modeled in the form of a power-of-the-mean variance model. The maximum likelihood fitting method is supplemented to satisfy the lower bound constraints in empirical data. The CHVR method has been applied to estimating the scaling exponents of six urban attributes covering three scaling regimes in China and compared with two traditional methods. Method evaluations based on three different criteria validate that compared with both classical methods, the CHVR method is more effective and robust. Moreover, a statistical test and long-term variations of the parameter in the variance function demonstrate that the proposed heterogeneous variance function can not only describe the heterogeneity in empirical data adequately but also provide more meaningful urban information. Therefore, the CHVR method shows great potential to provide a valuable tool for effective urban scaling studies across the world and be applied to scaling law estimation in other complex systems in the future.<\/jats:p>","DOI":"10.3390\/e21040337","type":"journal-article","created":{"date-parts":[[2019,3,29]],"date-time":"2019-03-29T03:50:21Z","timestamp":1553831421000},"page":"337","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["An Urban Scaling Estimation Method in a Heterogeneity Variance Perspective"],"prefix":"10.3390","volume":"21","author":[{"given":"Wenjia","family":"Wu","sequence":"first","affiliation":[{"name":"Institute of Geomatics, Department of Civil Engineering, Tsinghua University, Beijing 100084, China"},{"name":"3S Center, Tsinghua University, Beijing 100084, China"},{"name":"National Engineering Laboratory for Green and Safe Construction Technology in Urban Rail Transit, Tsinghua University, Beijing 100084, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hongrui","family":"Zhao","sequence":"additional","affiliation":[{"name":"Institute of Geomatics, Department of Civil Engineering, Tsinghua University, Beijing 100084, China"},{"name":"3S Center, Tsinghua University, Beijing 100084, China"},{"name":"National Engineering Laboratory for Green and Safe Construction Technology in Urban Rail Transit, Tsinghua University, Beijing 100084, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Qifan","family":"Tan","sequence":"additional","affiliation":[{"name":"Institute of Geomatics, Department of Civil Engineering, Tsinghua University, Beijing 100084, China"},{"name":"3S Center, Tsinghua University, Beijing 100084, China"},{"name":"National Engineering Laboratory for Green and Safe Construction Technology in Urban Rail Transit, Tsinghua University, Beijing 100084, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1714-779X","authenticated-orcid":false,"given":"Peichao","family":"Gao","sequence":"additional","affiliation":[{"name":"Faculty of Geographical Science, Beijing Normal University, Beijing100084, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,3,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1418","DOI":"10.1126\/science.1239870","article-title":"A theory of city size","volume":"340","author":"Batty","year":"2013","journal-title":"Science"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"7301","DOI":"10.1073\/pnas.0610172104","article-title":"Growth, innovation, scaling, and the pace of life in cities","volume":"104","author":"Bettencourt","year":"2007","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1438","DOI":"10.1126\/science.1235823","article-title":"The origins of scaling in cities","volume":"340","author":"Bettencourt","year":"2013","journal-title":"Science"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Pumain, D., Paulus, F., Vacchiani-Marcuzzo, C., and Lobo, J. (2006). An evolutionary theory for interpreting urban scaling laws. Cybergeo Eur. J. Geogr.","DOI":"10.4000\/cybergeo.2519"},{"key":"ref_5","first-page":"395","article-title":"and Liu, M.H. Study on fractal dimension of size distribution of cities","volume":"14","author":"Chen","year":"1998","journal-title":"Bull. Sci. Technol."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"54","DOI":"10.1080\/04353684.1971.11879355","article-title":"Urban allometric growth","volume":"53","author":"Nordbeck","year":"1971","journal-title":"Geogr. Ann. Ser. B Hum. Geogr."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"315","DOI":"10.3733\/hilg.v06n11p315","article-title":"Body size and metabolism","volume":"6","author":"Kleiber","year":"1932","journal-title":"Hilgardia"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"2473","DOI":"10.1073\/pnas.012579799","article-title":"Allometric scaling of metabolic rate from molecules and mitochondria to cells and mammals","volume":"99","author":"West","year":"2002","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Batty, M. (2013). The New Science of Cities, The MIT Press.","DOI":"10.7551\/mitpress\/9399.001.0001"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"912","DOI":"10.1038\/467912a","article-title":"A unified theory of urban living","volume":"467","author":"Bettencourt","year":"2010","journal-title":"Nature"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Bettencourt, L.M., Lobo, J., Strumsky, D., and West, G.B. (2010). Urban scaling and its deviations: Revealing the structure of wealth, innovation and crime across cities. PLoS ONE, 5.","DOI":"10.1371\/journal.pone.0013541"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"5561","DOI":"10.1038\/srep05561","article-title":"How congestion shapes cities: From mobility patterns to scaling","volume":"4","author":"Louf","year":"2014","journal-title":"Sci. Rep."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Arcaute, E., Hatna, E., Ferguson, P., Youn, H., Johansson, A., and Batty, M. (2015). Constructing cities, deconstructing scaling laws. J. R. Soc. Interface, 12.","DOI":"10.1098\/rsif.2014.0745"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Alves, L.G., Ribeiro, H.V., Lenzi, E.K., and Mendes, R.S. (2013). Distance to the scaling law: A useful approach for unveiling relationships between crime and urban metrics. PLoS ONE, 8.","DOI":"10.1371\/annotation\/7471cfa3-9b0f-4cc8-8bc8-fd9fb3ce5844"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"194715","DOI":"10.1155\/2010\/194715","article-title":"Characterizing growth and form of fractal cities with allometric scaling exponents","volume":"2010","author":"Chen","year":"2010","journal-title":"Discret. Dyn. Nat. Soc."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1841","DOI":"10.1038\/s41467-017-01882-w","article-title":"Simple spatial scaling rules behind complex cities","volume":"8","author":"Li","year":"2017","journal-title":"Nat. Commun."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Pumain, D., and Rozenblat, C. (2018). Two metropolisation gradients in the European system of cities revealed by scaling laws. Environ. Plan. B Urban Anal. City Sci.","DOI":"10.1177\/2399808318785633"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Arbabi, H., Mayfield, M., and Dabinett, G. (2018). Urban performance at different boundaries in England and wales through the settlement scaling theory. Reg. Stud.","DOI":"10.1080\/00343404.2018.1490501"},{"key":"ref_19","unstructured":"Louf, R. (arXiv, 2015). Wandering in cities: A statistical physics approach to urban theory, arXiv."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"4235","DOI":"10.1038\/srep04235","article-title":"Large cities are less green","volume":"4","author":"Oliveira","year":"2014","journal-title":"Sci. Rep."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"80","DOI":"10.1016\/j.compenvurbsys.2016.04.006","article-title":"Diverse cities or the systematic paradox of urban scaling laws","volume":"63","author":"Cottineau","year":"2017","journal-title":"Comput. Environ. Urban Syst."},{"key":"ref_22","first-page":"213","article-title":"The definition of city boundary and scaling law","volume":"72","author":"Dong","year":"2017","journal-title":"Acta Geogr. Sin."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Finance, O., and Cottineau, C. (2018). Are the absent always wrong? Dealing with zero values in urban scaling. Environ. Plan. B Urban Anal. City Sci.","DOI":"10.1177\/2399808318785634"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"665","DOI":"10.1126\/science.1216142","article-title":"Critical truths about power laws","volume":"335","author":"Stumpf","year":"2012","journal-title":"Science"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"661","DOI":"10.1137\/070710111","article-title":"Power-law distributions in empirical data","volume":"51","author":"Clauset","year":"2009","journal-title":"Siam Rev."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Bettencourt, L.M., and Lobo, J. (2016). Urban scaling in Europe. J. R. Soc. Interface, 13.","DOI":"10.1098\/rsif.2016.0005"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"515","DOI":"10.1016\/j.jtbi.2008.10.016","article-title":"On the use of logarithmic transformations in allometric analyses","volume":"257","author":"Packard","year":"2009","journal-title":"J. Theor. Biol."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"549","DOI":"10.1111\/j.1469-185X.2010.00160.x","article-title":"Fitting statistical models in bivariate allometry","volume":"86","author":"Packard","year":"2011","journal-title":"Biol. Rev."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1887","DOI":"10.1890\/11-0538.1","article-title":"On the use of log-transformation vs. Nonlinear regression for analyzing biological power laws","volume":"92","author":"Xiao","year":"2011","journal-title":"Ecology"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"3581","DOI":"10.1242\/jeb.023317","article-title":"Traditional allometric analysis fails to provide a valid predictive model for mammalian metabolic rates","volume":"211","author":"Packard","year":"2008","journal-title":"J. Exp. Biol."},{"key":"ref_31","first-page":"399","article-title":"Urban allometric growth in China: Theory and facts","volume":"64","author":"Li","year":"2009","journal-title":"Acta Geogr. Sin."},{"key":"ref_32","unstructured":"Alain, F., and Zuur, E.N.L. (2009). Mixed Effects Models and Extensions in Ecology with R, Springer."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"150649","DOI":"10.1098\/rsos.150649","article-title":"Is this scaling nonlinear?","volume":"3","author":"Miotto","year":"2016","journal-title":"R. Soc. Open Sci."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1080\/00018730801893043","article-title":"Fluctuation scaling in complex systems: Taylor\u2019s law and beyond","volume":"57","author":"Eisler","year":"2008","journal-title":"Adv. Phys."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Gao, P., Liu, Z., Tian, K., and Liu, G. (2016). Characterizing traffic conditions from the perspective of spatial-temporal heterogeneity. ISPRS Int. J. Geo-Inf., 5.","DOI":"10.3390\/ijgi5030034"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1111\/gean.12046","article-title":"An application of Taylor\u2019s Power Law to measure overdispersion of the unemployed in English labor markets","volume":"47","author":"Greig","year":"2015","journal-title":"Geogr. Anal."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Hanley, Q.S., Khatun, S., Yosef, A., and Dyer, R.-M. (2014). Fluctuation scaling, Taylor\u2019s law, and crime. PLoS ONE, 9.","DOI":"10.1371\/journal.pone.0109004"},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Nomaler, \u00d6., Frenken, K., and Heimeriks, G. (2014). On scaling of scientific knowledge production in US metropolitan areas. PLoS ONE, 9.","DOI":"10.1371\/journal.pone.0110805"},{"key":"ref_39","unstructured":"UN (2014). World Urbanization Prospects: The 2014 Revision, UN."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"96","DOI":"10.1016\/j.jue.2006.12.006","article-title":"Growth, population and industrialization, and urban land expansion of China","volume":"63","author":"Deng","year":"2008","journal-title":"J. Urban Econ."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1007\/s00168-011-0442-8","article-title":"Re-examination of Zipf\u2019s law and urban dynamic in China: A regional approach","volume":"49","author":"Ye","year":"2012","journal-title":"Ann. Reg. Sci."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1016\/j.chaos.2006.05.018","article-title":"Scaling laws and indications of self-organized criticality in urban systems","volume":"35","author":"Chen","year":"2008","journal-title":"Chaos Solitons Fractals"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"392","DOI":"10.1111\/jiec.12563","article-title":"Impact of the economic structure of cities on urban scaling factors: Implications for urban material and energy flows in China","volume":"22","author":"Ramaswami","year":"2018","journal-title":"J. Ind. Ecol."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"122","DOI":"10.1126\/science.276.5309.122","article-title":"A general model for the origin of allometric scaling laws in biology","volume":"276","author":"West","year":"1997","journal-title":"Science"},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Pumain, D., Paulus, F., and Vacchiani-Marcuzzo, C. (2009). Innovation cycles and urban dynamics. Complexity Perspectives in Innovation and Social Change, Springer.","DOI":"10.1007\/978-1-4020-9663-1_9"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"812","DOI":"10.1038\/srep00812","article-title":"There is more than a power law in Zipf","volume":"2","author":"Cristelli","year":"2012","journal-title":"Sci. Rep."},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Ritz, C., and Streibig, J.C. (2008). Nonlinear Regression with R, Springer.","DOI":"10.1007\/978-0-387-09616-2"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1214\/09-STS301","article-title":"The impact of Levene\u2019s test of equality of variances on statistical theory and practice","volume":"24","author":"Gastwirth","year":"2009","journal-title":"Stat. Sci."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"1287","DOI":"10.2307\/1911963","article-title":"A simple test for heteroscedasticity and random coefficient variation","volume":"47","author":"Breusch","year":"1979","journal-title":"Econometrica"},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Pinheiro, J.C., and Bates, D.M. (2000). Mixed-Effects Models in S and S-Plus, corrected third printing, Springer.","DOI":"10.1007\/978-1-4419-0318-1"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"732","DOI":"10.1038\/189732a0","article-title":"Aggregation, variance and the mean","volume":"189","author":"Taylor","year":"1961","journal-title":"Nature"},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Chen, Y., and Jiang, B. (2018). Hierarchical scaling in systems of natural cities. Entropy, 20.","DOI":"10.3390\/e20060432"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"3285","DOI":"10.1016\/j.physa.2011.12.031","article-title":"The mathematical relationship between Zipf\u2019s law and the hierarchical scaling law","volume":"391","author":"Chen","year":"2012","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_54","unstructured":"Wen, H., and Wei, D. (2008). Urban Economics, Tsinghua University Press."},{"key":"ref_55","unstructured":"(2018, November 17). See e.G., Available online: https:\/\/obamawhitehouse.Archives.Gov\/sites\/default\/files\/omb\/assets\/fedreg_|2010\/06282010_metro_standards-complete.Pdf."},{"key":"ref_56","unstructured":"(2018, November 17). See e.G., Available online: http:\/\/www.Census.Gov\/population\/metro\/."},{"key":"ref_57","unstructured":"Kohavi, R. (1995, January 20\u201325). In A study of cross-validation and bootstrap for accuracy estimation and model selection. Proceedings of the IJCAI, Montreal, QC, Canada."},{"key":"ref_58","first-page":"2079","article-title":"On over-fitting in model selection and subsequent selection bias in performance evaluation","volume":"11","author":"Cawley","year":"2010","journal-title":"J. Mach. Learn. Res."},{"key":"ref_59","unstructured":"Greene, W.H. (2003). Econometric Analysis, Pearson Education India."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"223","DOI":"10.2307\/2409177","article-title":"Analyzing tables of statistical tests","volume":"43","author":"Rice","year":"1989","journal-title":"Evolution"},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"0012","DOI":"10.1038\/s41562-016-0012","article-title":"Explaining the prevalence, scaling and variance of urban phenomena","volume":"1","author":"Hausmann","year":"2016","journal-title":"Nat. Hum. Behav."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"440","DOI":"10.1038\/30918","article-title":"Collective dynamics of \u2018small-world\u2019 networks","volume":"393","author":"Watts","year":"1998","journal-title":"Nature"},{"key":"ref_63","first-page":"1063","article-title":"Derivation of relations between urbanization level and velocity from logistic growth model","volume":"25","author":"Yan","year":"2006","journal-title":"Geogr. Res."},{"key":"ref_64","unstructured":"West, G. (2017). Scale: The Universal Laws of Growth, Innovation, Sustainability, and the Pace of Life in Organisms, Cities, Economies, and Companies, Penguin."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"1798","DOI":"10.1038\/srep01798","article-title":"Entangled communities and spatial synchronization lead to criticality in urban traffic","volume":"3","author":"Petri","year":"2013","journal-title":"Sci. Rep."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/21\/4\/337\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:41:12Z","timestamp":1760186472000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/21\/4\/337"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,3,28]]},"references-count":65,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2019,4]]}},"alternative-id":["e21040337"],"URL":"https:\/\/doi.org\/10.3390\/e21040337","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2019,3,28]]}}}