# On the Fundamental Diagram for Freeway Traffic: Exploring the Lower Bound of the Fitting Error and Correcting the Generalized Linear Regression Models

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## Abstract

**:**

## 1. Introduction

## 2. Correcting Generalized Linear Regression Models

#### 2.1. Analysis

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

#### 2.2. Correction

Algorithm 1: An enumeration algorithm. |

Input: A set of candidate pairs of parameters $\left\{\left(\left({v}_{fi},{k}_{0j}\right)|i=1,2,\dots ,M;j=1,2,\dots ,N\right)\right\}$.Output: The minimum MSE, the optimal values of parameters.$MSE\left({v}_{fi},{k}_{0j}\right)$ denotes the MSE value of the pair of parameters $\left({v}_{fi},{k}_{0j}\right)$; the minimum MSE and its corresponding optimal parameters are denoted as ${MSE}^{\prime}$, ${{v}_{f}}^{\prime}$, ${{k}_{0}}^{\prime}$. Initialize the ${MSE}^{\prime}=\infty $, ${{v}_{f}}^{\prime}=0$, ${{k}_{0}}^{\prime}=0$. For $i=1,2,\dots ,M$ do:For $j=1,2,\dots ,N$ do:Calculate the MSE value $MSE\left({v}_{fi},{k}_{0j}\right)$ for the pair of parameters $\left({v}_{fi},{k}_{0j}\right)$. If $MSE\left({v}_{fi},{k}_{0j}\right)\le {MSE}^{\prime}$ do:${{v}_{f}}^{\prime}={v}_{fi}$, ${{k}_{0}}^{\prime}={k}_{0j}$, ${MSE}^{\prime}=MSE\left({v}_{fi},{k}_{0j}\right).$ End ifEnd forEnd for |

## 3. Lower Bound of the Fitting Error of Existing Models

#### 3.1. MSE Values of Existing Models

#### 3.2. Quadratic Programming Model

#### 3.3. Results

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Unbiased case in the Underwood model. (

**a**) The relationship between $v$ and $k.$ (

**b**) The relationship between $\mathrm{l}\mathrm{n}v$ and $k$.

**Figure 3.**Biased case in the Underwood model. (

**a**) The relationship between $v$ and $k$. (

**b**) The relationship between $\mathrm{l}\mathrm{n}v$ and $k$.

**Figure 4.**Biased case in the Northwestern model. (

**a**) The relationship between $v$ and $k$. (

**b**) The relationship between $\mathrm{l}\mathrm{n}v$ and ${k}^{2}$.

**Figure 10.**Average values of MSE for different density intervals. (

**a**) The Underwood model. (

**b**) The Northwestern model.

**Table 1.**Four speed–density models (Qu et al., 2015) [19].

Models | Function | Parameters |
---|---|---|

Greenshields [1] | $v={v}_{f}\left(1-\frac{k}{{k}_{j}}\right)$ | ${v}_{f}$, ${k}_{j}$ |

Greenberg [3] | $v={v}_{0}ln\left(\frac{{k}_{j}}{k}\right)$ | ${v}_{0}$, ${k}_{j}$ |

Underwood [5] | $v={v}_{f}\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{k}{{k}_{0}}\right)$ | ${v}_{f}$, ${k}_{0}$ |

Northwestern [20] | $v={v}_{f}\mathrm{e}\mathrm{x}\mathrm{p}\left[-\frac{1}{2}{\left(\frac{k}{{k}_{0}}\right)}^{2}\right]$ | ${v}_{f}$, ${k}_{0}$ |

Models | Function | Transformation | Original MSE | Corrected MSE |
---|---|---|---|---|

Greenshields (Greenshields et al., 1935) [1] | $v={v}_{f}\left(1-\frac{k}{{k}_{j}}\right)$ | $v=y$, $k=x$ | 46.727 | 46.727 |

Greenberg (1959) [3] | $v={v}_{0}\mathrm{ln}\left(\frac{{k}_{j}}{k}\right)$ | $v=y$, $\mathrm{l}\mathrm{n}k=x$ | 107.948 | 107.948 |

Underwood (1961) [5] | $v={v}_{f}\mathrm{e}\mathrm{x}\mathrm{p}\left(1-\frac{k}{{k}_{0}}\right)$ | $\mathrm{l}\mathrm{n}v=y$, $k=x$ | 59.4544 | 50.3609 |

Northwestern (Drake et al., 1967) [20] | $v={v}_{f}\mathrm{e}\mathrm{x}\mathrm{p}\left[-\frac{1}{2}{\left(\frac{k}{{k}_{0}}\right)}^{2}\right]$ | $\mathrm{l}\mathrm{n}v=y$, ${k}^{2}=x$ | 44.3233 | 25.9371 |

Models | Corrected MSE |
---|---|

Greenshields (Greenshields et al., 1935) [1] | 72.0000 |

Greenberg (1959) [3] | 117.3113 |

Underwood (1961) [5] | 95.7534 |

Northwestern (Drake et al., 1967) [20] | 57.0006 |

Models | MSE | Relative Gap |
---|---|---|

Greenshields [1] | 46.7270 | 137.603% |

Greenberg [3] | 107.9480 | 457.583% |

Underwood [5] | 50.3609 | 160.129% |

Northwestern [20] | 25.9371 | 33.973% |

Average value | 57.8053 | 197.322% |

Sample Size | MSE Values for Linear Regression | MSE Values after Correction | MSE Values for Lower Bound | Relative Gap for Linear Regression | Relative Gap for Corrected Results | |
---|---|---|---|---|---|---|

1 | 100 | 79.266 | 67.860 | 24.084 | 229.126% | 181.766% |

2 | 100 | 44.656 | 43.402 | 10.827 | 312.444% | 300.863% |

3 | 500 | 62.533 | 50.326 | 14.262 | 338.467% | 252.871% |

4 | 500 | 68.246 | 58.360 | 18.631 | 266.316% | 213.249% |

5 | 1000 | 57.880 | 48.574 | 16.318 | 254.691% | 197.667% |

6 | 1000 | 53.204 | 44.782 | 12.246 | 334.443% | 265.675% |

7 | 5000 | 61.323 | 51.584 | 19.455 | 215.196% | 165.137% |

8 | 5000 | 58.771 | 50.546 | 18.529 | 217.175% | 172.787% |

9 | 10,000 | 59.292 | 50.461 | 19.010 | 211.899% | 165.446% |

10 | 10,000 | 60.492 | 51.296 | 19.657 | 207.732% | 160.950% |

11 | 30,000 | 59.321 | 49.987 | 18.852 | 214.672% | 165.157% |

12 | 30,000 | 59.220 | 50.062 | 19.505 | 203.620% | 156.668% |

Sample Size | MSE Values for Linear Regression | MSE Values after Correction | MSE Values for Lower Bound | Relative Gap for Linear Regression | Relative Gap for Corrected Results | |
---|---|---|---|---|---|---|

1 | 100 | 26.520 | 26.288 | 15.640 | 69.562% | 68.082% |

2 | 100 | 99.182 | 65.021 | 24.821 | 299.583% | 161.955% |

3 | 500 | 29.822 | 24.064 | 16.680 | 78.787% | 44.271% |

4 | 500 | 43.881 | 29.784 | 17.6181 | 149.064% | 69.054% |

5 | 1000 | 38.354 | 23.915 | 15.793 | 142.859% | 51.433% |

6 | 1000 | 49.473 | 23.244 | 14.825 | 233.723% | 56.790% |

7 | 5000 | 52.530 | 28.479 | 20.810 | 152.427% | 36.852% |

8 | 5000 | 49.058 | 27.132 | 19.101 | 156.829% | 42.045% |

9 | 10,000 | 40.869 | 24.552 | 17.541 | 132.990% | 39.969% |

10 | 10,000 | 46.855 | 25.510 | 18.658 | 151.124% | 36.722% |

11 | 30,000 | 43.821 | 26.410 | 19.684 | 122.624% | 34.172% |

12 | 30,000 | 43.286 | 26.440 | 19.655 | 120.226% | 34.521% |

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## Share and Cite

**MDPI and ACS Style**

Shangguan, Y.; Tian, X.; Jin, S.; Gao, K.; Hu, X.; Yi, W.; Guo, Y.; Wang, S.
On the Fundamental Diagram for Freeway Traffic: Exploring the Lower Bound of the Fitting Error and Correcting the Generalized Linear Regression Models. *Mathematics* **2023**, *11*, 3460.
https://doi.org/10.3390/math11163460

**AMA Style**

Shangguan Y, Tian X, Jin S, Gao K, Hu X, Yi W, Guo Y, Wang S.
On the Fundamental Diagram for Freeway Traffic: Exploring the Lower Bound of the Fitting Error and Correcting the Generalized Linear Regression Models. *Mathematics*. 2023; 11(16):3460.
https://doi.org/10.3390/math11163460

**Chicago/Turabian Style**

Shangguan, Yidan, Xuecheng Tian, Sheng Jin, Kun Gao, Xiaosong Hu, Wen Yi, Yu Guo, and Shuaian Wang.
2023. "On the Fundamental Diagram for Freeway Traffic: Exploring the Lower Bound of the Fitting Error and Correcting the Generalized Linear Regression Models" *Mathematics* 11, no. 16: 3460.
https://doi.org/10.3390/math11163460