# A Three-Phase Fundamental Diagram from Three-Dimensional Traffic Data

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## Abstract

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## 1. Introduction

- Unlike most studies that focus on traffic data from a single source, we use data from multiple geographic locations in Europe and the US and analyze the fundamental relationships among flow, density and speed in the 3D space instead of the commonly adopted two-variable representation of the FD. In addition, we use a set of statistical tools including model-based clustering, hypothesis testing and regression to analyze the traffic data.
- Following the above exercise, we discover three data clusters representing three traffic regimes, two of which are contained in the free-flow phase and the third corresponds to the congested phase. Moreover, we are able to detect a statistically significant gap between the first two regimes and the third one. These findings are validated using multiple data sources, and the main features (regimes and gaps) are consistent across different geographical areas.
- Building on the first two, we propose a new three-phase macroscopic traffic flow model, which exhibits all the characteristics shown by our data analyses and combines the features of the ARZ, CGARZ and phase transition models. A complete characterization of solutions of the Riemann problems is provided.

## 2. Data Analysis

#### 2.1. Experimental Data

- Flux (denoted as f), also known as flow or volume, is the number of vehicles passing through a fixed location per unit of time.
- Velocity (denoted as v) is the average speed of vehicles per unit of time.
- Occupancy (denoted as o) is the percentage of time that a vehicle covers the sensor over the unit time of data collection.

#### 2.2. Statistical Tools

#### 2.2.1. Cluster Analysis

#### 2.2.2. Three Phase Traffic

**“free choice phase”**, which corresponds to the situation of a relatively empty road, whereby drivers choose their speed independently without influence from or interaction with other vehicles.

#### 2.2.3. Gap Analysis

## 3. A Macroscopic Second-Order Model Accounting for the 3 Phases

- Non-negative: $v(\rho ,w)\ge 0$ for all $\rho \in [0,{\rho}_{\mathrm{max}}]$, $w\in [{w}_{\mathrm{min}},{w}_{\mathrm{max}}]$;
- Continuous: ${v}_{FC}({\rho}_{FC},w)={v}_{FP}\left({\rho}_{FC}\right)$ and ${v}_{C}({\rho}_{FP},w)={v}_{FP}\left({\rho}_{FP}\right)$ for all $w\in [{w}_{\mathrm{min}},{w}_{\mathrm{max}}]$
- Vanishing at maximal density: $v({\rho}_{\mathrm{max}},w)={v}_{C}({\rho}_{\mathrm{max}},w)=0$ for all $w\in [{w}_{\mathrm{min}},{w}_{\mathrm{max}}]$
- Non-decreasing with respect to w: $\frac{\partial v}{\partial w}(\rho ,w)\ge 0$ for $\rho \in [0,{\rho}_{\mathrm{max}}]$

#### 3.1. Riemann Solver

**1-rarefaction waves.**Two points $({\rho}_{l},{\rho}_{l}{w}_{l})$ and $({\rho}_{r},{\rho}_{r}{w}_{r})$ are connected by a 1-rarefaction wave if and only if$${w}_{l}={w}_{r}\phantom{\rule{2.em}{0ex}}\mathrm{and}\phantom{\rule{2.em}{0ex}}{\lambda}_{1}({\rho}_{l},{w}_{l})<{\lambda}_{1}({\rho}_{r},{w}_{r}).$$**1-shock waves.**Two points $({\rho}_{l},{\rho}_{l}{w}_{l})$ and $({\rho}_{r},{\rho}_{r}{w}_{r})$ are connected by a 1-shock wave if and only if$${w}_{l}={w}_{r}\phantom{\rule{2.em}{0ex}}\mathrm{and}\phantom{\rule{2.em}{0ex}}{\lambda}_{1}({\rho}_{l},{w}_{l})>{\lambda}_{1}({\rho}_{r},{w}_{r}).$$In this case, the jump discontinuity moves with speed$$\sigma ={\displaystyle \frac{{\rho}_{l}v({\rho}_{l},{w}_{l})-{\rho}_{m}v({\rho}_{m},{w}_{m})}{{\rho}_{l}-{\rho}_{m}}}.$$**2-contact discontinuity.**Two points $({\rho}_{l},{\rho}_{l}{w}_{l})$ and $({\rho}_{r},{\rho}_{r}{w}_{r})$ are connected by a 2-contact wave if and only if$$v({\rho}_{l},{w}_{l})=v({\rho}_{r},{w}_{r}).$$

- If $({\rho}_{l},{\rho}_{l}{w}_{l})$ and $({\rho}_{r},{\rho}_{r}{w}_{r})$ belongs both to ${\mathrm{\Omega}}_{FP}$ or ${\mathrm{\Omega}}_{C}$, the Riemann solver is defined as above.
- If $({\rho}_{l},{\rho}_{l}{w}_{l})\in {\mathrm{\Omega}}_{C}$ and $({\rho}_{r},{\rho}_{r}{w}_{r})\in {\mathrm{\Omega}}_{FP}$, the intermediate point $({\rho}_{m},{\rho}_{m}{w}_{m})$ belongs to ${\mathrm{\Omega}}_{FP}$. Let $({\rho}_{c},{\rho}_{c}{w}_{c})\in \partial {\mathrm{\Omega}}_{C}$ the point defined by$${w}_{c}={w}_{l},\phantom{\rule{2.em}{0ex}}v({\rho}_{c},{w}_{c})={v}_{C}^{\mathrm{max}}.$$$$\sigma ={\displaystyle \frac{{\rho}_{c}{v}_{C}^{\mathrm{max}}-{\rho}_{FP}{v}_{FP}^{\mathrm{min}}}{{\rho}_{c}-{\rho}_{FP}}},$$
- If $({\rho}_{l},{\rho}_{l}{w}_{l})\in {\mathrm{\Omega}}_{FP}$ and $({\rho}_{r},{\rho}_{r}{w}_{r})\in {\mathrm{\Omega}}_{C}$, the intermediate point $({\rho}_{m},{\rho}_{m}{w}_{m})$ belongs to ${\mathrm{\Omega}}_{C}$. Therefore, the solution always contains a 1-wave (shock phase-transition) from $({\rho}_{l},{\rho}_{l}{w}_{l})$ to $({\rho}_{m},{\rho}_{m}{w}_{m})$, followed by a 2-contact discontinuity. Notice that the solution may also contain an intermediate 1-wave in the congested phase.

## 4. Numerical Scheme and Simulations

#### 4.1. Numerical Scheme

**Step 1:**Evolution in time.This step consists in solving the Riemann problem at each cell interface ${x}_{j+1/2}$ with initial data $({\mathbf{u}}_{j}^{n},{\mathbf{u}}_{j+1}^{n})$, obtaining an exact solution ${\overline{\mathbf{u}}}_{\nu}(x,{t}^{n+1}-)$.**Step 2:**Projection to time ${t}^{n+1}$Once all Riemann problems at interfaces are solved, Chalons and Goatin [25] proposed a new averaging procedure. The idea is that, since the solution can contain states in different phases, the average is not done on the regular mesh cells but on modified non-uniform cells that contain only values belonging to the same phase. We denote this modified cells by ${\overline{\mathcal{C}}}_{j}^{n}=[{\overline{x}}_{j-1/2}^{n},{\overline{x}}_{j+1/2}^{n}[$. Afterwards, a sampling strategy allows us to recover a piecewise constant solution on the initial mesh cells ${C}_{j}$.

#### 4.2. Numerical Simulations

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Data Description

- Rome: Three sensors over one week (June 2006) aggregated every 1 min
- Las Vegas: Fifty sensors over five years (2010–2015) aggregated over every 10 min
- Sophia Antipolis: Four sensors for eight months (January–August 2014) every 6 min

**free phase**) and much larger variation when occupancy is larger than the threshold (known as the

**congestion phase**). Furthermore, both flux vs. speed and flux vs. occupancy plots suggest a possible “gap" between free and congestion phases, which corresponds to phase transition. These are important features that need to be taken into consideration in the mathematical modeling. Such pairwise plots are useful to generate data-driven hypotheses that need to be formally tested statistically and validated across different datasets.

**Figure A1.**Pairwise scatterplots of Rome Data in the road Viale del Muro Torto: flux vs. occupancy (

**left**); flux vs. speed (

**middle**); and occupancy vs. speed (

**right**).

## Appendix B. Results of Cluster Analysis

**Table A1.**Results of model-based cluster analysis using Rome, Las Vegas and Sophia Antipolis data. Phase: estimated phase using cluster analysis. FP, free phase; C, congestion phase. Density, estimated density value at the percentile.

Dataset | Percentile (%) | Phase | Density | Test Statistic | p-Value |
---|---|---|---|---|---|

Rome | 97.5 | FP | 10 | 0 | 1 |

2.50 | C | 10 | |||

97 | FP | 9 | −1.79 | 0.037 | |

3 | C | 10 | |||

95 | FP | 9 | −3.15 | <0.001 | |

5 | C | 11 | |||

Las Vegas | 97.5 | FP | 12 | −0.91 | $0.18$ |

2.50 | C | 13 | |||

97 | FP | 12 | −2.17 | 0.015 | |

3 | C | 14 | |||

95 | FP | 11 | −5.75 | <0.001 | |

5 | C | 16 | |||

Sophia | 97.5 | FP | 5 | −2.82 | 0.002 |

2.50 | C | 10 | |||

97 | FP | 5 | −6.44 | <0.001 | |

3 | C | 12 | |||

95 | FP | 4 | −5.79 | <0.001 | |

5 | C | 12 |

#### Appendix Quantifying the Improved Goodness of Fit Through RSS Comparisons

**Table A2.**RSS analysis with two clusters with flux and occupancy. ${\beta}_{0}$ is the value of the intercept and ${\beta}_{1}$ the occupancy.

${\mathit{\beta}}_{0}$ | ${\mathit{\beta}}_{1}$ | ${\mathit{R}}^{2}$ | Adj. ${\mathit{R}}^{2}$ | RSS | |
---|---|---|---|---|---|

Free Phase | 77.71 | 249.18 | 0.952 | 0.952 | 157,765,974 |

Congestion | 2184.09 | −11.00 | 0.09746 | 0.09676 | 283,255,197 |

**Table A3.**RSS analysis with two clusters with flux, occupancy and speed. ${\beta}_{0}$ is the value of the intercept, ${\beta}_{1}$ the occupancy, and ${\beta}_{2}$ the speed.

${\mathit{\beta}}_{0}$ | ${\mathit{\beta}}_{1}$ | ${\mathit{\beta}}_{2}$ | ${\mathit{R}}^{2}$ | Adj. ${\mathit{R}}^{2}$ | RSS | |
---|---|---|---|---|---|---|

Free Phase | −248.7 | 253.88 | 5.48 | 0.953 | 0.953 | 154,308,365 |

Congestion | 1941.54 | −8.11 | 6.64 | 0.1032 | 0.1018 | 281,448,225 |

**Table A4.**RSS analysis with three clusters with flux and occupancy. ${\beta}_{0}$ is the value of the intercept and ${\beta}_{1}$ the occupancy.

${\mathit{\beta}}_{0}$ | ${\mathit{\beta}}_{1}$ | ${\mathit{R}}^{2}$ | Adj. ${\mathit{R}}^{2}$ | RSS | |
---|---|---|---|---|---|

Free Choice | 79.56 | 201.3 | 0.563 | 0.5628 | 14,269,933 |

Free Flow | 187.68 | 231.36 | 0.9175 | 0.9175 | 126,714,084 |

Congestion | 2302.55 | −13.87 | 0.1618 | 0.1611 | 240,324,480 |

**Table A5.**Residual sum squared analysis with three clusters with flux, occupancy and speed. ${\beta}_{0}$ is the value of the intercept and ${\beta}_{1}$ the occupancy, ${\beta}_{2}$ the speed.

${\mathit{\beta}}_{0}$ | ${\mathit{\beta}}_{1}$ | ${\mathit{\beta}}_{2}$ | ${\mathit{R}}^{2}$ | Adj. ${\mathit{R}}^{2}$ | RSS | |
---|---|---|---|---|---|---|

Free Choice | −153.88 | 200.36 | 4.03 | 0.6033 | 0.603 | 12,951,197 |

Free Flow | −316.91 | 239.01 | 8.43 | 0.9193 | 0.9192 | 123,984,911 |

Congestion | 2065.61 | −11.05 | 6.5 | 0.1676 | 0.1663 | 283,641,319 |

% RSS Improvement | ||
---|---|---|

2 Clusters | 3 Clusters | |

2D | FP: $13.2\%$ | FC: $9.2\%$ |

FF: $2.2\%$ | ||

C: $15.8\%$ | C: $0.7\%$ | |

3D | FP: $11.3\%$ | FC: - |

FF: - | ||

C: $15.2\%$ | C: - |

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**Figure 1.**3D visualization and cluster analysis results of Rome data suggest the existence of third phase (red) in addition to the free flow (blue) and congestion (green) phases.

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Delle Monache, M.L.; Chi, K.; Chen, Y.; Goatin, P.; Han, K.; Qiu, J.-m.; Piccoli, B. A Three-Phase Fundamental Diagram from Three-Dimensional Traffic Data. *Axioms* **2021**, *10*, 17.
https://doi.org/10.3390/axioms10010017

**AMA Style**

Delle Monache ML, Chi K, Chen Y, Goatin P, Han K, Qiu J-m, Piccoli B. A Three-Phase Fundamental Diagram from Three-Dimensional Traffic Data. *Axioms*. 2021; 10(1):17.
https://doi.org/10.3390/axioms10010017

**Chicago/Turabian Style**

Delle Monache, Maria Laura, Karen Chi, Yong Chen, Paola Goatin, Ke Han, Jing-mei Qiu, and Benedetto Piccoli. 2021. "A Three-Phase Fundamental Diagram from Three-Dimensional Traffic Data" *Axioms* 10, no. 1: 17.
https://doi.org/10.3390/axioms10010017