{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:48:17Z","timestamp":1760233697730,"version":"build-2065373602"},"reference-count":25,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2021,2,7]],"date-time":"2021-02-07T00:00:00Z","timestamp":1612656000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100000002","name":"National Institutes of Health","doi-asserted-by":"publisher","award":["1R01LM012607;  1R01AI130460"],"award-info":[{"award-number":["1R01LM012607;  1R01AI130460"]}],"id":[{"id":"10.13039\/100000002","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001665","name":"Agence Nationale de la Recherche","doi-asserted-by":"publisher","award":["ANR-15- IDEX-02"],"award-info":[{"award-number":["ANR-15- IDEX-02"]}],"id":[{"id":"10.13039\/501100001665","id-type":"DOI","asserted-by":"publisher"}]},{"name":"NSF project KI-Net","award":["DMS # 1107444"],"award-info":[{"award-number":["DMS # 1107444"]}]},{"name":"NSF project","award":["CNS # 1446715"],"award-info":[{"award-number":["CNS # 1446715"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper uses empirical traffic data collected from three locations in Europe and the US to reveal a three-phase fundamental diagram with two phases located in the uncongested regime. Model-based clustering, hypothesis testing and regression analyses are applied to the speed\u2013flow\u2013occupancy relationship represented in the three-dimensional space to rigorously validate the three phases and identify their gaps. The finding is consistent across the aforementioned different geographical locations. Accordingly, we propose a three-phase macroscopic traffic flow model and a characterization of solutions to the Riemann problems. This work identifies critical structures in the fundamental diagram that are typically ignored in first- and higher-order models and could significantly impact travel time estimation on highways.<\/jats:p>","DOI":"10.3390\/axioms10010017","type":"journal-article","created":{"date-parts":[[2021,2,8]],"date-time":"2021-02-08T20:51:51Z","timestamp":1612817511000},"page":"17","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["A Three-Phase Fundamental Diagram from Three-Dimensional Traffic Data"],"prefix":"10.3390","volume":"10","author":[{"given":"Maria","family":"Delle Monache","sequence":"first","affiliation":[{"name":"University Grenoble Alpes, Inria, CNRS, Grenoble INP, GIPSA-Lab, 38000 Grenoble, France"}]},{"given":"Karen","family":"Chi","sequence":"additional","affiliation":[{"name":"Learning Lab, University of Pennsylvania, Philadelphia, PA 19104, USA"}]},{"given":"Yong","family":"Chen","sequence":"additional","affiliation":[{"name":"Department of Biostatistics, Epidemiology and Informatics (DBEI), University of Pennsylvania, Philadelphia, PA 19104, USA"}]},{"given":"Paola","family":"Goatin","sequence":"additional","affiliation":[{"name":"Universit\u00e9 C\u00f4te d\u2019Azur, Inria, CNRS, LJAD, 06902 Sophia Antipolis, France"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3529-6246","authenticated-orcid":false,"given":"Ke","family":"Han","sequence":"additional","affiliation":[{"name":"Institute of System Science and Engineering, Southwest Jiaotong University, Chengdu 611756, China"}]},{"given":"Jing-mei","family":"Qiu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Delaware, Newark, DE 19713, USA"}]},{"given":"Benedetto","family":"Piccoli","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, Rutgers University-Camden, Camden, NJ 08102, USA"}]}],"member":"1968","published-online":{"date-parts":[[2021,2,7]]},"reference":[{"key":"ref_1","first-page":"317","article-title":"On kinematic waves. II. A theory of traffic flow on long crowded roads","volume":"229","author":"Lighthill","year":"1955","journal-title":"Proc. R. Soc. A"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1287\/opre.4.1.42","article-title":"Shock waves on the highway","volume":"4","author":"Richards","year":"1956","journal-title":"Oper. Res."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"916","DOI":"10.1137\/S0036139997332099","article-title":"Resurrection of \u201csecond order\u201d models of traffic flow","volume":"60","author":"Aw","year":"2000","journal-title":"SIAM J. Appl. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1016\/S0191-2615(00)00050-3","article-title":"A non-equilibrium traffic model devoid of gas-like behavior","volume":"36","author":"Zhang","year":"2002","journal-title":"Transp. Res. Part B Methodol."},{"key":"ref_5","unstructured":"Garavello, M., Han, K., and Piccoli, B. (2016). Models for Vehicular Traffic on Networs, American Institute of Mathematical Sciences."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"708","DOI":"10.1137\/S0036139901393184","article-title":"Hyperbolic phase transitions in traffic flow","volume":"63","author":"Colombo","year":"2003","journal-title":"SIAM J. Appl. Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2652","DOI":"10.1137\/090752468","article-title":"A 2-phase traffic model based on a speed bound","volume":"70","author":"Colombo","year":"2010","journal-title":"SIAM J. Appl. Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1142\/S0219891610002025","article-title":"Road networks with phase transitions","volume":"7","author":"Colombo","year":"2010","journal-title":"J. Hyperbolic Differ. Eq."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1016\/j.mcm.2006.01.016","article-title":"The Aw\u2013Rascle vehicular traffic flow model with phase transitions","volume":"44","author":"Goatin","year":"2006","journal-title":"Math. Comput. Model."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1862","DOI":"10.1137\/S0036141004402683","article-title":"Traffic flow on a road network","volume":"36","author":"Coclite","year":"2005","journal-title":"SIAM J. Math. Anal."},{"key":"ref_11","unstructured":"Gerlough, D.L., and Huber, M.J. (1976). Traffic Flow Theory, Transportation Research Board. No. HS-006 783."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1016\/j.physa.2003.10.017","article-title":"Three-phase traffic theory and highway capacity","volume":"333","author":"Kerner","year":"2004","journal-title":"Phys. A"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1016\/j.trb.2013.07.005","article-title":"A variational formulation for higher order macroscopic traffic flow models of the GSOM family","volume":"57","author":"Lebacque","year":"2013","journal-title":"Transp. Res. Part Methodol."},{"key":"ref_14","unstructured":"Fan, S., Sun, Y., Piccoli, B., Seibold, B., and Work, D.B. (2017). A Collapsed Generalized Aw-Rascle-Zhang Model and Its Model Accuracy. arXiv."},{"key":"ref_15","unstructured":"(2017, April 03). ATAC, S.p.A. Available online: http:\/\/www.atac.roma.it\/."},{"key":"ref_16","unstructured":"(2017, April 03). Regional Transportation Commission of Southern Nevada (RTC), Freeway and Arterial System of Transportation (FAST) Division. Available online: http:\/\/bugatti.nvfast.org\/."},{"key":"ref_17","unstructured":"(2017, April 03). D\u00e9partement des Alpes Maritimes. Available online: https:\/\/www.departement06.fr."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"611","DOI":"10.1198\/016214502760047131","article-title":"Model-based clustering, discriminant analysis, and density estimation","volume":"97","author":"Fraley","year":"2002","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_19","unstructured":"Gross, A.J., and Clark, V. (1975). Survival Distributions: Reliability Applications in the Biomedical Sciences, John Wiley & Sons."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"813","DOI":"10.1137\/140977977","article-title":"A heterogeneous multiclass traffic flow model with creeping","volume":"75","author":"Fan","year":"2015","journal-title":"SIAM J. Appl. Math."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"239","DOI":"10.3934\/nhm.2014.9.239","article-title":"Comparative model accuracy of a data-fitted generalized Aw-Rascle-Zhang model","volume":"9","author":"Fan","year":"2014","journal-title":"Netw. Heterog. Media"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"562","DOI":"10.1016\/j.trb.2008.10.001","article-title":"A conserved higher-order anisotropic traffic flow model: Description of equilibrium and non-equilibrium flows","volume":"43","author":"Zhang","year":"2009","journal-title":"Transp. Res. Part B Methodol."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1137\/090754467","article-title":"A general phase transition model for vehicular traffic","volume":"71","author":"Blandin","year":"2011","journal-title":"SIAM J. Appl. Math."},{"key":"ref_24","first-page":"1529","article-title":"The semigroup generated by a Temple class system with non-convex flux function","volume":"13","author":"Bianchini","year":"2000","journal-title":"Differ. Integral Eq."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"197","DOI":"10.4171\/ifb\/186","article-title":"Godunov scheme and sampling technique for computing phase transitions in traffic flow modeling","volume":"10","author":"Chalons","year":"2008","journal-title":"Interfaces Free Boundaries"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/10\/1\/17\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:20:52Z","timestamp":1760160052000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/10\/1\/17"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,2,7]]},"references-count":25,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2021,3]]}},"alternative-id":["axioms10010017"],"URL":"https:\/\/doi.org\/10.3390\/axioms10010017","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2021,2,7]]}}}