# An Entropy-Based Approach for Evaluating Travel Time Predictability Based on Vehicle Trajectory Data

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Entropy of Travel Time Series

#### 2.2. Travel Time Predictability

## 3. Experiments

#### 3.1. Study Area and Data

#### 3.2. Entropy and Predictability

#### 3.3. Analysis and Discussion

#### 3.3.1. Time Scales

#### 3.3.2. Tolerance

#### 3.3.3. Series Length

#### 3.3.4. The Validity of Travel Time Predictability

## 4. Discussion and Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Wu, C.H.; Ho, J.M.; Lee, D.T. Travel-time prediction with support vector regression. IEEE Trans. Intell. Transp. Syst.
**2005**, 5, 276–281. [Google Scholar] [CrossRef] - Mori, U.; Mendiburu, A.; Álvarez, M.; Lozano, J.A. A review of travel time estimation and forecasting for Advanced Traveller Information Systems. Transportmetr. A Transp. Sci.
**2015**, 11, 1–39. [Google Scholar] [CrossRef] - Oh, S.; Byon, Y.J.; Jang, K.; Yeo, H. Short-Term Travel-Time Prediction on Highway: A Review of the Data-Driven Approach. Transp. Rev.
**2015**, 35, 4–32. [Google Scholar] [CrossRef] - Vlahogianni, E.I.; Karlaftis, M.G.; Golias, J.C. Short-term traffic forecasting: Where we are and where we’re going. Transp. Res. Part C Emerg. Technol.
**2014**, 43, 3–19. [Google Scholar] [CrossRef] - Lin, H.E.; Zito, R.; Taylor, M. A review of travel-time prediction in transport and logistics. Proc. East. Asia Soc. Transp. Stud.
**2005**, 5, 1433–1448. [Google Scholar] - Van Lint, J.W.C.; van Hinsbergen, C.P.I. Short-Term Traffic and Travel Time Prediction Models. Artif. Intell. Appl. Crit. Trans. Issues
**2012**, 22, 22–41. [Google Scholar] - Vlahogianni, E.I.; Golias, J.C.; Karlaftis, M.G. Short-term traffic forecasting: Overview of objectives and methods. Transp. Rev.
**2004**, 24, 533–557. [Google Scholar] [CrossRef] - Abrantes, P.A.L.; Wardman, M.R. Meta-analysis of UK values of travel time: An update. Transp. Res. Part A Policy Pract.
**2011**, 45, 1–17. [Google Scholar] [CrossRef] - Jara-Diaz, S.R. Transport Economic Theory; Emerald Group Publishing Limited: Bingley, UK, 2007; pp. 11–49. [Google Scholar]
- Li, Z.; Hensher, D.A.; Rose, J.M. Willingness to pay for travel time reliability in passenger transport: A review and some new empirical evidence. Transp. Res. Part E Log. Transp. Rev.
**2010**, 46, 384–403. [Google Scholar] [CrossRef] - Van Lint, J.W.C.; Zuylen, H.J.V.; Tu, H. Travel time unreliability on freeways: Why measures based on variance tell only half the story. Transp. Res. Part A Policy Pract.
**2008**, 42, 258–277. [Google Scholar] [CrossRef] - Shires, J.D.; de Jong, G.C. An international meta-analysis of values of travel time savings. Eval. Program Plan.
**2009**, 32, 315–325. [Google Scholar] [CrossRef] [PubMed] - Wardman, M.; Batley, R. Travel time reliability: A review of late time valuations, elasticities and demand impacts in the passenger rail market in Great Britain. Transportation
**2014**, 41, 1041–1069. [Google Scholar] [CrossRef] - Noland, R.B.; Polak, J.W. Travel time variability: A review of theoretical and empirical issues. Transp. Rev.
**2002**, 22, 39–54. [Google Scholar] [CrossRef] - Carrion, C.; Levinson, D. Value of travel time reliability: A review of current evidence. Transp. Res. Part A Policy Pract.
**2012**, 46, 720–741. [Google Scholar] [CrossRef] - Rietveld, P.; Bruinsma, F.R.; Vuuren, D.V. Coping with unreliability in public transport chains: A case study for Netherlands. Transp. Res. Part A Policy Pract.
**2001**, 35, 539–559. [Google Scholar] [CrossRef] - Yue, Y.; Yeh, A.G.O.; Zhuang, Y. Prediction time horizon and effectiveness of real-time data on short-term traffic predictability. Proc. Intell. Transp. Syst. Conf.
**2007**, 962–967. [Google Scholar] - Foell, S.; Phithakkitnukoon, S.; Kortuem, G.; Veloso, M. Predictability of Public Transport Usage: A Study of Bus Rides in Lisbon, Portugal. IEEE Trans. Intell. Transp. Syst.
**2015**, 16, 2955–2960. [Google Scholar] [CrossRef] - Siddle, D. Travel Time Predictability. Available online: https://trid.trb.org/view.aspx?id=1323043 (accessed on 11 April 2017).
- Wang, J.; Mao, Y.; Li, J.; Xiong, Z.; Wang, W.X. Predictability of road traffic and congestion in urban areas. PLoS ONE
**2015**, 10, e0121825. [Google Scholar] [CrossRef] [PubMed] - Du, Y.; Chai, Y.W.; Yang, J.W.; Liang, J.H.; Lan, J.H. Predictability of Resident activity in Beijing Based on GPS Data. Geogr. Geo Inf. Sci.
**2015**, 31, 47–51. [Google Scholar] - Song, C.; Qu, Z.; Blumm, N.; Barabási, A.L. Limits of predictability in human mobility. Science
**2010**, 327, 1018–1021. [Google Scholar] [CrossRef] [PubMed] - Shannon, C.E. A Mathematical Theory of Communication: The Bell System Technical Journal. Bell Syst. Tech. J.
**1948**, 27, 3–55. [Google Scholar] [CrossRef] - Fano, R.M.; Hawkins, D. Transmission of Information: A Statistical Theory of Communications. Am. J. Phys.
**1961**, 29, 793–794. [Google Scholar] [CrossRef] - Kontoyiannis, I.; Algoet, P.H.; Suhov, Y.M.; Wyner, A.J. Nonparametric entropy estimation for stationary processes and random fields, with applications to English text. IEEE Trans. Inf. Theory
**2007**, 44, 1319–1327. [Google Scholar] [CrossRef] - Wyner, A.D.; Ziv, J. The sliding-window lempel-ziv algorithm is asymptotically optimal. Proc. IEEE
**1994**, 82, 872–877. [Google Scholar] [CrossRef] - Xie, X.X.; Li, S.; Zhang, C.L.; Li, J.K. Study on the application of Lempel-Ziv complexity in the nonlinear detecting. Complex Syst. Complex. Sci.
**2005**, 2, 61–66. [Google Scholar] - Costa, M.; Goldberger, A.L.; Peng, C.K. Multiscale entropy analysis of complex physiologic time series. Phys. Rev. Lett.
**2002**, 92, 705–708. [Google Scholar] [CrossRef] [PubMed] - Wu, S.D.; Wu, C.W.; Lin, S.G.; Lee, K.Y.; Peng, C.K. Analysis of complex time series using refined composite multiscale entropy. Phys. Lett. A
**2014**, 378, 1369–1374. [Google Scholar] [CrossRef] - González, M.C.; Hidalgo, C.A. Understanding individual human mobility patterns. Nature
**2008**, 453, 779. [Google Scholar] [CrossRef] [PubMed] - Pappalardo, L.; Simini, F.; Rinzivillo, S.; Pedreschi, D.; Giannotti, F.; Barabási, A.L. Returners and explorers dichotomy in human mobility. Nat. Commun.
**2015**, 6. [Google Scholar] [CrossRef] [PubMed] - Pappalardo, L.; Rinzivillo, S.; Qu, Z.; Pedreschi, D.; Giannotti, F. Understanding the patterns of car travel. Eur. Phys. J. Spec. Top.
**2013**, 215, 61–73. [Google Scholar] [CrossRef] - Hu, Y.; Miller, H.J.; Li, X. Detecting and analyzing mobility hotspots using surface networks. Trans. GIS
**2014**, 18, 911–935. [Google Scholar] [CrossRef] - Li, X.J.; Li, X.; Tang, D.; Xu, X. Deriving features of traffic flow around an intersection from trajectories of vehicles. In Proceedings of the 2010 International Conference on Geoinformatics: Giscience in Change, Geoinformatics, Beijing, China, 18–20 June 2010; pp. 1–5. [Google Scholar]
- Box, G.E.P.; Jenkins, G.M. Time series analysis: Forecasting and control. J. Oper. Res. Soc.
**1971**, 22, 199–201. [Google Scholar] - Rumelhart, D.E.; Hinton, G.E.; Williams, R.J. Learning representations by back-propagating errors. Cogn. Model.
**1988**, 5, 1. [Google Scholar] [CrossRef]

**Figure 4.**The upper bound of travel time predictability in the weekly travel time series and the daily travel time series.

**Figure 12.**The comparisons between travel time predictability and prediction results in the 5 min travel time series with different series lengths.

**Figure 13.**Relationships between travel time predictability and prediction results in travel time series with different time scales.

$\mathit{\tau}$ | $\overline{\mathit{S}\left(\mathit{X}\right)}$ | $\mathit{s}{\mathit{d}}_{\mathit{s}}$ | $\overline{{\mathit{\Pi}}^{\mathit{m}\mathit{a}\mathit{x}}}$ | $\mathit{s}{\mathit{d}}_{{\mathit{\Pi}}^{\mathit{m}\mathit{a}\mathit{x}}}$ | ||
---|---|---|---|---|---|---|

$<14\text{}\mathbf{Days}$ | $\ge 14\text{}\mathbf{Days}$ | $<14\text{}\mathbf{Days}$ | $\ge 14\text{}\mathbf{Days}$ | |||

2 | 1.1752 | 0.0896 | 0.0126 | 0.9896 | 0.0045 | 0.0008 |

4 | 1.1966 | 0.0755 | 0.0125 | 0.9885 | 0.0040 | 0.0007 |

6 | 1.2334 | 0.0623 | 0.0148 | 0.9863 | 0.0040 | 0.0011 |

8 | 1.2415 | 0.1108 | 0.0167 | 0.9856 | 0.0058 | 0.0011 |

10 | 1.3261 | 0.1311 | 0.0193 | 0.9805 | 0.0089 | 0.0013 |

12 | 1.3571 | 0.0953 | 0.0242 | 0.9786 | 0.0066 | 0.0016 |

Prediction Model | Number of Prediction | Number of Success | Success Rate | Average Error (s) | $\overline{{\mathit{\Pi}}^{\mathit{m}\mathit{a}\mathit{x}}}$ |
---|---|---|---|---|---|

ARIMA | 50 | 38 | 90% | 13.46 | 0.952 |

BPNN | 50 | 40 | 91% | 12.42 |

**Table 3.**The standard deviation ($sd$) of travel time predictability and the predicted results with different series lengths.

Series Length (Days) | $\overline{{\mathit{\Pi}}^{\mathit{m}\mathit{a}\mathit{x}}}$ | $\mathit{S}{\mathit{R}}_{\mathit{A}\mathit{R}\mathit{I}\mathit{M}\mathit{A}}$ | $\mathit{S}{\mathit{R}}_{\mathit{B}\mathit{P}\mathit{N}\mathit{N}}$ |
---|---|---|---|

$<14\text{}\mathrm{days}$ | 0.0052 | 0.0339 | 0.0378 |

$\ge 14\text{}\mathrm{days}$ | 0.0009 | 0.0142 | 0.0137 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Xu, T.; Xu, X.; Hu, Y.; Li, X. An Entropy-Based Approach for Evaluating Travel Time Predictability Based on Vehicle Trajectory Data. *Entropy* **2017**, *19*, 165.
https://doi.org/10.3390/e19040165

**AMA Style**

Xu T, Xu X, Hu Y, Li X. An Entropy-Based Approach for Evaluating Travel Time Predictability Based on Vehicle Trajectory Data. *Entropy*. 2017; 19(4):165.
https://doi.org/10.3390/e19040165

**Chicago/Turabian Style**

Xu, Tao, Xianrui Xu, Yujie Hu, and Xiang Li. 2017. "An Entropy-Based Approach for Evaluating Travel Time Predictability Based on Vehicle Trajectory Data" *Entropy* 19, no. 4: 165.
https://doi.org/10.3390/e19040165