On the Fundamental Diagram for Freeway Traffic: Exploring the Lower Bound of the Fitting Error and Correcting the Generalized Linear Regression Models
In traffic flow, the relationship between speed and density exhibits decreasing monotonicity and continuity, which is characterized by various models such as the Greenshields and Greenberg models. However, some existing models, i.e., the Underwood and Northwestern models, introduce bias by incorrect...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-08-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/16/3460 |
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| author | Yidan Shangguan Xuecheng Tian Sheng Jin Kun Gao Xiaosong Hu Wen Yi Yu Guo Shuaian Wang |
| author_facet | Yidan Shangguan Xuecheng Tian Sheng Jin Kun Gao Xiaosong Hu Wen Yi Yu Guo Shuaian Wang |
| author_sort | Yidan Shangguan |
| collection | DOAJ |
| description | In traffic flow, the relationship between speed and density exhibits decreasing monotonicity and continuity, which is characterized by various models such as the Greenshields and Greenberg models. However, some existing models, i.e., the Underwood and Northwestern models, introduce bias by incorrectly utilizing linear regression for parameter calibration. Furthermore, the lower bound of the fitting errors for all these models remains unknown. To address above issues, this study first proves the bias associated with using linear regression in handling the Underwood and Northwestern models and corrects it, resulting in a significantly lower mean squared error (MSE). Second, a quadratic programming model is developed to obtain the lower bound of the MSE for these existing models. The relative gaps between the MSEs of existing models and the lower bound indicate that the existing models still have a lot of potential for improvement. |
| first_indexed | 2024-03-10T23:46:40Z |
| format | Article |
| id | doaj.art-906f901d0bf248d884158d6c0577cf85 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T23:46:40Z |
| publishDate | 2023-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-906f901d0bf248d884158d6c0577cf852023-11-19T02:02:18ZengMDPI AGMathematics2227-73902023-08-011116346010.3390/math11163460On the Fundamental Diagram for Freeway Traffic: Exploring the Lower Bound of the Fitting Error and Correcting the Generalized Linear Regression ModelsYidan Shangguan0Xuecheng Tian1Sheng Jin2Kun Gao3Xiaosong Hu4Wen Yi5Yu Guo6Shuaian Wang7Department of Logistics and Maritime Studies, Faculty of Business, The Hong Kong Polytechnic University, Hung Hom, Hong KongDepartment of Logistics and Maritime Studies, Faculty of Business, The Hong Kong Polytechnic University, Hung Hom, Hong KongCollege of Civil Engineering and Architecture, Zhejiang University, Hangzhou 310058, ChinaDepartment of Architecture and Civil Engineering, Chalmers University of Technology, 412 96 Göteborg, SwedenState Key Laboratory of Mechanical Transmission/Automotive Collaborative Innovation Center, Chongqing University, Chongqing 400044, ChinaDepartment of Building and Real Estate, The Hong Kong Polytechnic University, Hung Hom, Hong KongDepartment of Logistics and Maritime Studies, Faculty of Business, The Hong Kong Polytechnic University, Hung Hom, Hong KongDepartment of Logistics and Maritime Studies, Faculty of Business, The Hong Kong Polytechnic University, Hung Hom, Hong KongIn traffic flow, the relationship between speed and density exhibits decreasing monotonicity and continuity, which is characterized by various models such as the Greenshields and Greenberg models. However, some existing models, i.e., the Underwood and Northwestern models, introduce bias by incorrectly utilizing linear regression for parameter calibration. Furthermore, the lower bound of the fitting errors for all these models remains unknown. To address above issues, this study first proves the bias associated with using linear regression in handling the Underwood and Northwestern models and corrects it, resulting in a significantly lower mean squared error (MSE). Second, a quadratic programming model is developed to obtain the lower bound of the MSE for these existing models. The relative gaps between the MSEs of existing models and the lower bound indicate that the existing models still have a lot of potential for improvement.https://www.mdpi.com/2227-7390/11/16/3460speed and density relationshiplinear regressionquadratic programming |
| spellingShingle | Yidan Shangguan Xuecheng Tian Sheng Jin Kun Gao Xiaosong Hu Wen Yi Yu Guo Shuaian Wang On the Fundamental Diagram for Freeway Traffic: Exploring the Lower Bound of the Fitting Error and Correcting the Generalized Linear Regression Models Mathematics speed and density relationship linear regression quadratic programming |
| title | On the Fundamental Diagram for Freeway Traffic: Exploring the Lower Bound of the Fitting Error and Correcting the Generalized Linear Regression Models |
| title_full | On the Fundamental Diagram for Freeway Traffic: Exploring the Lower Bound of the Fitting Error and Correcting the Generalized Linear Regression Models |
| title_fullStr | On the Fundamental Diagram for Freeway Traffic: Exploring the Lower Bound of the Fitting Error and Correcting the Generalized Linear Regression Models |
| title_full_unstemmed | On the Fundamental Diagram for Freeway Traffic: Exploring the Lower Bound of the Fitting Error and Correcting the Generalized Linear Regression Models |
| title_short | On the Fundamental Diagram for Freeway Traffic: Exploring the Lower Bound of the Fitting Error and Correcting the Generalized Linear Regression Models |
| title_sort | on the fundamental diagram for freeway traffic exploring the lower bound of the fitting error and correcting the generalized linear regression models |
| topic | speed and density relationship linear regression quadratic programming |
| url | https://www.mdpi.com/2227-7390/11/16/3460 |
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