预测人类移动行为的介入机会类模型研究进展

刘二见; 闫小勇

doi:10.7498/aps.69.20201119

摘要

预测地点间人类的移动在人类迁徙、交通预测、疾病传播、商品贸易、社会交往等诸多方面具有重要的意义. 介入机会模型是最早从个体目的地选择行为角度建立的预测人类移动的模型, 它将起终点之间的介入机会作为影响人类移动的关键因素, 启发研究者提出了许多新的介入机会类模型. 介入机会类模型在很多学科领域也获得了广泛的应用. 本文首先对包括介入机会模型、辐射类模型、人口权重机会类模型、探索类介入机会模型和统一机会模型等在内的介入机会类模型的研究进展进行综述, 然后对这些介入机会类模型在空间交互和疾病传播方面的应用进行介绍, 最后对该类模型未来的研究方向进行探讨.

关键词:

Abstract

Predicting human mobility between locations is of great significance for investigating the population migration, traffic forecasting, epidemic spreading, commodity trade, social interaction and other relevant areas. The intervening opportunity (IO) model is the model established earliest from the perspective of individual choice behavior to predict human mobility. The IO model takes the total number of opportunities between the origin location and the destination as a key factor in determining human mobility, which has inspired researchers to propose many new IO class models. In this paper, we first review the research advances in the IO class models, including the IO model, radiation class models, population-weighted opportunity class models, exploratory IO class models and universal opportunity model. Among them, although the IO model has an important theoretical value, it contains parameters and has low prediction accuracy, so it is rarely used in practice. The radiation class models are built on the basis of the IO model on the assumption that the individual will choose the closest destination whose benefit is higher than the best one available in origin location. The radiation class models can better predict the commuting behavior between locations. The population-weighted opportunity class models are established on the assumption that when seeking a destination, the individual will not only consider the nearest locations with relatively large benefits, but also consider all locations in the range of alternative space. The population-weighted opportunity class models can better predict intracity trips and intercity travels. The exploratory IO class models are built on condition that the destination selected by the individual presents a higher benefit than the benefit of the origin and the benefits of the intervening opportunities. The exploratory IO class models can better predict the social interaction between individuals, intracity trips and intercity travels. The universal opportunity model is developed on the assumption that when an individual selects a destination, she/he will comprehensively compare the benefits between the origin and the destination and their intervening opportunity. The universal opportunity model presents a new universal framework for IO class models and can accurately predict the movements on different spatiotemporal scales. The IO class models have also been widely used in many fields, including predicting trip distribution in transportation science, modeling the purchasing behaviors of consumers in economics, detecting complex network communities in network science, measuring spatial interaction in economic geography and predicting infectious disease transmission in epidemiology. This paper focuses on the applications of IO class models in spatial interaction and epidemic spreading, and finally presents the discussion on the possible future research directions of these models.

Keywords:

作者及机构信息

刘二见^1,2,
闫小勇^2,3

1.
北京交通大学, 综合交通运输大数据应用技术交通运输行业重点实验室, 北京　100044

2.
北京交通大学, 交通系统科学与工程研究院, 北京　100044

3.
电子科技大学, 复杂性实验室, 成都　611731

通信作者: 闫小勇, yanxy@bjtu.edu.cn

基金项目: 中央高校基本科研业务费专项资金 (批准号: 2019YJS092)和国家自然科学基金 (批准号: 71822102, 71671015, 61304177) 资助的课题

Authors and contacts

1.
Key Laboratory of Transport Industry of Big Data Application Technologies for Comprehensive Transport, Ministry of Transport, Beijing Jiaotong University, Beijing 100044, China

2.
Institute of Transportation System Science and Engineering, Beijing Jiaotong University, Beijing 100044, China

3.
Complex Laboratory, University of Electronic Science and Technology of China, Chengdu 611731, China

Corresponding author: Yan Xiao-Yong, yanxy@bjtu.edu.cn

Funds: Project supported by the Fundamental Research Fund for the Central Universities, China (Grant No. 2019YJS092) and the National Natural Science Foundation of China (Grant Nos. 71822102, 71671015, 61304177)

文章全文 : translate this paragraph

参考文献

[1]	闫小勇 2019 超越引力定律 (北京: 科学出版社) 第1−3页 Yan X Y 2019 Beyond Gravity Law (Beijing: China Science Press) pp1−3 (in Chinese)
[2]	Desart H G 1846 Chemin de fer Direct de Bruxelles vers Gand, par Alost, en Communication avec les Stations Diverses (Bruxelles: Devroye) p16
[3]	Huang Z R, Wang P, Zhang F, Gao J X, Schich M 2018 Transport. Res. Part B 114 147 Google Scholar
[4]	Xia C Y, Wang Z H, Zheng C Y, Guo Q T, Shi Y T, Dehmer M, Chen Z Q 2019 Inf. Sci. 471 185 Google Scholar
[5]	Jia J S, Lu X, Yuan Y, Xu G, Jia J M, Christakis N A 2020 Nature 582 389 Google Scholar
[6]	Giles J R, Erbach-Schoenberg E Z, Tatem A J, et al. 2020 Proc. Natl. Acad. Sci. U.S.A. 117 22572 Google Scholar
[7]	周涛, 韩筱璞, 闫小勇, 杨紫陌, 赵志丹, 汪秉宏 2013 电子科技大学学报 42 481 Google Scholar Zhou T, Han X P, Yan X Y, Yang Z M, Zhao Z D, Wang B H 2013 J. Uiv. Electron. Sci. Technol. China 42 481 Google Scholar
[8]	闫小勇 2020 69 088903 Yan X Y 2020 Acta Phys. Sin. 69 088903
[9]	Stouffer S A 1940 Am. Sociol. Rev. 5 845 Google Scholar
[10]	Newton I 1729 Mathematical Principles of Natural Philosophy (London: Benjamin Motte Publisher) p5
[11]	Wilson A G 1967 Transp. Res. 1 253 Google Scholar
[12]	Pressé S, Ghosh K, Lee J, Dill K A 2013 Rev. Mod. Phys. 85 1115 Google Scholar
[13]	Barbosa H, Barthélemy M, Ghoshal G, et al. 2018 Phys. Rep. 734 1 Google Scholar
[14]	Simini F, González M C, Maritan A, Barabási A L 2012 Nature 484 96 Google Scholar
[15]	Ren Y H, Ercsey-Ravasz M, Wang P, González M C, Toroczkai Z 2014 Nat. Commun. 5 5347 Google Scholar
[16]	Simini F, Maritan A, Néda Z 2013 PLoS One 8 e60069 Google Scholar
[17]	Varga L, Tóth G, Néda Z 2016 Regional Statistics 6 27 Google Scholar
[18]	Varga L, Tóth G, Néda Z 2018 EPJ Data Sci. 7 37 Google Scholar
[19]	Yan X Y, Zhao C, Fan Y, Di Z R, Wang W X 2014 J. R. Soc. Interface 11 20140834 Google Scholar
[20]	Yan X Y, Wang W X, Gao Z Y, Lai Y C 2017 Nat. Commun. 8 1639 Google Scholar
[21]	Sim A, Yaliraki S N, Barahona M, Stumpf M P 2015 J. R. Soc. Interface 12 20150315 Google Scholar
[22]	Liu E J, Yan X Y 2019 Physica A 526 121023 Google Scholar
[23]	Liu E J, Yan X Y 2020 Sci. Rep. 10 4657 Google Scholar
[24]	闫小勇 2017 科技导报 35 15 Google Scholar Yan X Y 2017 Sci. Technol. Rev. 35 15 Google Scholar
[25]	Kang C G, Liu Y, Guo D S, Qin K 2015 PLoS One 10 e0143500 Google Scholar
[26]	Yang Y X, Herrera C, Eagle N, Gonzélez M C 2014 Sci. Rep. 4 5662 Google Scholar
[27]	闫小勇 2014 博士学位论文 (北京: 北京师范大学) 第39−43页 Yan X Y 2014 Ph. D. Dissertation (Beijing: Beijing Normal University) pp39−43 (in Chinese)
[28]	Zhao Y M, Zeng A, Yan X Y, Wang W X, Lai Y C 2016 New J. Phys. 18 053025 Google Scholar
[29]	Hu B Y, Ma Y L, Pei Y L, Gao W 2020 Transportmetrica A 16 1062 Google Scholar
[30]	Forghani M, Karimipour F 2018 Trans. GIS 22 1008 Google Scholar
[31]	Bassolas A, Barbosa-Filho H, Dickinson B, et al. 2019 Nat. Commun. 10 1 Google Scholar
[32]	Zhao Y B, Wu G Z, Gong Y X, Yang M Z, Ni H G 2019 Sci. Total Environ. 679 378 Google Scholar
[33]	Cazabet R, Borgnat P, Jensen P 2017 Proceedings of the 8th International Conference on Complex Networks Dubrovnik, Croatia, March 21–24, 2017 p47
[34]	Li F Z, Feng Z M, Li P, You Z 2017 PLoS One 12 e0171107 Google Scholar
[35]	Wang X B, Li F Z, Bu W, Yu X 2017 Proceedings of 4th International Conference on Industrial Economics System and Industrial Security Engineering Kyoto, Japan, July 24–27, 2017 p1
[36]	Tian Y 2020 Land 9 159 Google Scholar
[37]	Zheng W S, Kuang A P, Wang X W, Chen J 2020 Chin. Geogra. Sci. 30 677 Google Scholar
[38]	Kraemer M U G, Golding N, Bisanzio D, et al. 2019 Sci. Rep. 9 1 Google Scholar
[39]	Dalziel B D, Pourbohloul B, Ellner S P 2013 Proc. R. Soc. B 280 20130763 Google Scholar
[40]	Tizzoni M, Bajardi P, Decuyper A, et al. 2014 PLoS Comput. Biol. 10 e1003716 Google Scholar
[41]	Wen T H, Hsu C S, Hu M C 2018 Int. J. Health Geographics 17 1 Google Scholar
[42]	Kramer A M, Pulliam J T, Alexander L W, Park A W, Rohani P 2016 R. Soc. Open Sci. 3 160294 Google Scholar
[43]	Zhang G H, Li J M, Li S J, Wang Y 2018 Int. J. Environ. Res. Public Health 15 1824 Google Scholar
[44]	祖正虎, 许晴, 张斌, 徐展凯, 郑涛 2015 系统工程理论与实践 35 2513 Google Scholar Zu Z H, XU Q, Zhang B, Xu Z K, Zheng T 2015 System Eng. Theor. Prac 35 2513 Google Scholar
[45]	Zhu G H, Xiao J P, Zhang B, et al. 2018 Sci. Total Environ. 622 252 Google Scholar
[46]	Sjödin H, Johansson A F, Brännström Å, et al. 2020 Int. J. Epidemiol. 0 1 Google Scholar
[47]	Akhmetzhanov A R, Mizumoto K, Jung S M, Linton N M, Omori R, Nishiura H 2020 arXiv: 2020.04.24.20077800 [medical]
[48]	Porojan A 2001 Open Econ. Rev. 12 265 Google Scholar
[49]	de Dios Ortuzar J, Willumsen L G 2011 Modelling Transport (West Sussex: John Wiley & Sons) p182
[50]	Sheppard E S 1978 Geogr. Anal. 10 386 Google Scholar
[51]	Hua C I, Porell F 1979 Int. Regional Sci. Rev. 4 97 Google Scholar
[52]	Barthélemy M 2011 Phys. Rep. 499 147 Google Scholar
[53]	Pan R K, Kaski K, Fortunato S 2012 Sci. Rep. 2 902 Google Scholar
[54]	Szell M, Sinatra R, Petri G, Thurner S, Latora V 2012 Sci. Rep. 2 457 Google Scholar

施引文献

图 1 介入机会说明　(a) 原始地图, 其中每个圆代表一个地点, 地点i是起点, 其他地点是潜在的目的地; 人口越多的地点, 机会越多, 圆面积越大; (b)介入机会, 按照距离起点的远近排序, 目的地j与起点i之间的所有地点的介入机会为$ s_{ij} = m_a+m_b+m_c+m_d $(不包含起终点)或$ S_{ij} = m_i+s_{ij}+m_j $(包含起终点), 其中$ m_l $是地点l的机会

Fig. 1. Intervening opportunities illustration. (a) Sketch map. Each circle represents a location. Location i is the origin, and the other locations are potential destinations. The larger the population, the more opportunities, the larger the circle. (b) Intervening opportunities. All locations are sorted by their distance from location i. The intervening opportunities are all opportunities of the locations between i and j such that $ s_{ij} = m_a+m_b+m_c+m_d $ (excluding the origin and destination) or $ S_{ij} = m_i+s_{ij}+m_j $ (including the origin and destination), where $ m_l $ is the number of opportunities at location l.

下载: 全尺寸图片幻灯片

图 2 统一机会模型对14个数据集的最优参数取值

Fig. 2. The optimal values of the parameters $ \alpha $ and $ \beta $ for the fourteen data sets

下载: 全尺寸图片幻灯片

表 1 几种介入机会类模型的属性特征比较

Table 1. Comparison of attributes of several intervening opportunity models.

模型名称	适用类型	有无参数
介入机会模型	无	有
辐射模型	通勤	无
人口权重机会模型	城市内、城市间	无
审慎社交模型	个体间社会交往	无
机会优先选择模型	城市内、城市间	无
统一机会模型	求职、迁移、货运、通勤、城市内、城市间	有

下载: 导出CSV

map

[1]	闫小勇 2019 超越引力定律 (北京: 科学出版社) 第1−3页 Yan X Y 2019 Beyond Gravity Law (Beijing: China Science Press) pp1−3 (in Chinese)
[2]	Desart H G 1846 Chemin de fer Direct de Bruxelles vers Gand, par Alost, en Communication avec les Stations Diverses (Bruxelles: Devroye) p16
[3]	Huang Z R, Wang P, Zhang F, Gao J X, Schich M 2018 Transport. Res. Part B 114 147 Google Scholar
[4]	Xia C Y, Wang Z H, Zheng C Y, Guo Q T, Shi Y T, Dehmer M, Chen Z Q 2019 Inf. Sci. 471 185 Google Scholar
[5]	Jia J S, Lu X, Yuan Y, Xu G, Jia J M, Christakis N A 2020 Nature 582 389 Google Scholar
[6]	Giles J R, Erbach-Schoenberg E Z, Tatem A J, et al. 2020 Proc. Natl. Acad. Sci. U.S.A. 117 22572 Google Scholar
[7]	周涛, 韩筱璞, 闫小勇, 杨紫陌, 赵志丹, 汪秉宏 2013 电子科技大学学报 42 481 Google Scholar Zhou T, Han X P, Yan X Y, Yang Z M, Zhao Z D, Wang B H 2013 J. Uiv. Electron. Sci. Technol. China 42 481 Google Scholar
[8]	闫小勇 2020 69 088903 Yan X Y 2020 Acta Phys. Sin. 69 088903
[9]	Stouffer S A 1940 Am. Sociol. Rev. 5 845 Google Scholar
[10]	Newton I 1729 Mathematical Principles of Natural Philosophy (London: Benjamin Motte Publisher) p5
[11]	Wilson A G 1967 Transp. Res. 1 253 Google Scholar
[12]	Pressé S, Ghosh K, Lee J, Dill K A 2013 Rev. Mod. Phys. 85 1115 Google Scholar
[13]	Barbosa H, Barthélemy M, Ghoshal G, et al. 2018 Phys. Rep. 734 1 Google Scholar
[14]	Simini F, González M C, Maritan A, Barabási A L 2012 Nature 484 96 Google Scholar
[15]	Ren Y H, Ercsey-Ravasz M, Wang P, González M C, Toroczkai Z 2014 Nat. Commun. 5 5347 Google Scholar
[16]	Simini F, Maritan A, Néda Z 2013 PLoS One 8 e60069 Google Scholar
[17]	Varga L, Tóth G, Néda Z 2016 Regional Statistics 6 27 Google Scholar
[18]	Varga L, Tóth G, Néda Z 2018 EPJ Data Sci. 7 37 Google Scholar
[19]	Yan X Y, Zhao C, Fan Y, Di Z R, Wang W X 2014 J. R. Soc. Interface 11 20140834 Google Scholar
[20]	Yan X Y, Wang W X, Gao Z Y, Lai Y C 2017 Nat. Commun. 8 1639 Google Scholar
[21]	Sim A, Yaliraki S N, Barahona M, Stumpf M P 2015 J. R. Soc. Interface 12 20150315 Google Scholar
[22]	Liu E J, Yan X Y 2019 Physica A 526 121023 Google Scholar
[23]	Liu E J, Yan X Y 2020 Sci. Rep. 10 4657 Google Scholar
[24]	闫小勇 2017 科技导报 35 15 Google Scholar Yan X Y 2017 Sci. Technol. Rev. 35 15 Google Scholar
[25]	Kang C G, Liu Y, Guo D S, Qin K 2015 PLoS One 10 e0143500 Google Scholar
[26]	Yang Y X, Herrera C, Eagle N, Gonzélez M C 2014 Sci. Rep. 4 5662 Google Scholar
[27]	闫小勇 2014 博士学位论文 (北京: 北京师范大学) 第39−43页 Yan X Y 2014 Ph. D. Dissertation (Beijing: Beijing Normal University) pp39−43 (in Chinese)
[28]	Zhao Y M, Zeng A, Yan X Y, Wang W X, Lai Y C 2016 New J. Phys. 18 053025 Google Scholar
[29]	Hu B Y, Ma Y L, Pei Y L, Gao W 2020 Transportmetrica A 16 1062 Google Scholar
[30]	Forghani M, Karimipour F 2018 Trans. GIS 22 1008 Google Scholar
[31]	Bassolas A, Barbosa-Filho H, Dickinson B, et al. 2019 Nat. Commun. 10 1 Google Scholar
[32]	Zhao Y B, Wu G Z, Gong Y X, Yang M Z, Ni H G 2019 Sci. Total Environ. 679 378 Google Scholar
[33]	Cazabet R, Borgnat P, Jensen P 2017 Proceedings of the 8th International Conference on Complex Networks Dubrovnik, Croatia, March 21–24, 2017 p47
[34]	Li F Z, Feng Z M, Li P, You Z 2017 PLoS One 12 e0171107 Google Scholar
[35]	Wang X B, Li F Z, Bu W, Yu X 2017 Proceedings of 4th International Conference on Industrial Economics System and Industrial Security Engineering Kyoto, Japan, July 24–27, 2017 p1
[36]	Tian Y 2020 Land 9 159 Google Scholar
[37]	Zheng W S, Kuang A P, Wang X W, Chen J 2020 Chin. Geogra. Sci. 30 677 Google Scholar
[38]	Kraemer M U G, Golding N, Bisanzio D, et al. 2019 Sci. Rep. 9 1 Google Scholar
[39]	Dalziel B D, Pourbohloul B, Ellner S P 2013 Proc. R. Soc. B 280 20130763 Google Scholar
[40]	Tizzoni M, Bajardi P, Decuyper A, et al. 2014 PLoS Comput. Biol. 10 e1003716 Google Scholar
[41]	Wen T H, Hsu C S, Hu M C 2018 Int. J. Health Geographics 17 1 Google Scholar
[42]	Kramer A M, Pulliam J T, Alexander L W, Park A W, Rohani P 2016 R. Soc. Open Sci. 3 160294 Google Scholar
[43]	Zhang G H, Li J M, Li S J, Wang Y 2018 Int. J. Environ. Res. Public Health 15 1824 Google Scholar
[44]	祖正虎, 许晴, 张斌, 徐展凯, 郑涛 2015 系统工程理论与实践 35 2513 Google Scholar Zu Z H, XU Q, Zhang B, Xu Z K, Zheng T 2015 System Eng. Theor. Prac 35 2513 Google Scholar
[45]	Zhu G H, Xiao J P, Zhang B, et al. 2018 Sci. Total Environ. 622 252 Google Scholar
[46]	Sjödin H, Johansson A F, Brännström Å, et al. 2020 Int. J. Epidemiol. 0 1 Google Scholar
[47]	Akhmetzhanov A R, Mizumoto K, Jung S M, Linton N M, Omori R, Nishiura H 2020 arXiv: 2020.04.24.20077800 [medical]
[48]	Porojan A 2001 Open Econ. Rev. 12 265 Google Scholar
[49]	de Dios Ortuzar J, Willumsen L G 2011 Modelling Transport (West Sussex: John Wiley & Sons) p182
[50]	Sheppard E S 1978 Geogr. Anal. 10 386 Google Scholar
[51]	Hua C I, Porell F 1979 Int. Regional Sci. Rev. 4 97 Google Scholar
[52]	Barthélemy M 2011 Phys. Rep. 499 147 Google Scholar
[53]	Pan R K, Kaski K, Fortunato S 2012 Sci. Rep. 2 902 Google Scholar
[54]	Szell M, Sinatra R, Petri G, Thurner S, Latora V 2012 Sci. Rep. 2 457 Google Scholar

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