# Observations on the Relationship between Crash Frequency and Traffic Flow

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## Abstract

**:**

## 1. Motivation

## 2. The Data

#### 2.1. The Crash Database

#### 2.2. The Distribution of the Crash Frequency

#### 2.3. The Traffic Flow Data

#### 2.4. Supplemental Traffic Flow Data

## 3. On Crashes and Cars

#### 3.1. The 2018 Data

#### 3.1.1. Weekly Curves of the Crash Rate

#### 3.1.2. A Remaining Pattern

## 4. Summary and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Distribution of the number of road users involved (

**left**), and the traffic shares (

**right**) in the Berlin crash data set. The Misc traffic mode is being used by the police to denote any traffic mode that cannot be assigned. The y-axis is logarithmic, the numbers on top of the bars are the percentages of the respective shares.

**Figure 2.**Box plot of the crash frequency per hour of the week. The blue bar is the median, the boxes are the 25- and 75-percentiles, the whiskers display the minimum and the maximum of the data.

**Figure 3.**Distribution of the hourly crash counts as a function of the hour of the day on Mondays, displayed as a violin plot. The orange violins are for all crashes, the red ones for the severe crashes (which are shifted left by half an hour). The white circle is the median of the values.

**Figure 4.**Variance versus mean for all crashes (

**left**) and the severe crashes (

**right**). For comparison, a linear relationship is fitted as well, and the theoretical Poisson result is also included (broken red lines).

**Figure 5.**The parameter $\gamma $ as a function of hour of the week, for all crashes (dark-red) as well as for the severe crashes (dark-green). Whenever the fit has failed, $\gamma $ has been computed from $\gamma =({\sigma}^{2}-\mu )/{\mu}^{2}$. These curves are not smoothed.

**Figure 7.**Zoom of the boxplot (

**left**), aggregation is now in 10 min intervals. The speeds have been added to the flow. In the (

**right**) panel, speed and flow are combined to a density plot, which is just the macroscopic fundamental diagram for the flow data (now in the data set’s native 5 min intervals).

**Figure 8.**Comparison of the weekly flow pattern $\widehat{Q}\left(h\right)$ obtained from the T4C, the LOOP, and the MiD-Q data.

**Figure 9.**Pairwise comparison of five possible exposure values. The density plots display the measure on the right of each row on the y-axis versus the measure on the top of each column on the x-axis, giving more detailed information than just the correlation itself.

**Figure 10.**N versus $\widehat{Q}$ data, together with a line that connects the means computed from the data, and three different models: a gam (dark blue), a second-order polynomial model (green), and a power-law model ${Q}^{{\beta}_{1}}$ (red).

**Figure 12.**$\rho $ versus $\widehat{Q}$ for the severe crashes only, again with the mean values of the empirical data (orange) and the bi-linear model (blue) of Equation (9).

**Figure 13.**N versus Q for the severe crashes only, again with the empirical data and the four different models.

**Figure 14.**Risk index $\rho $ as a function of the hour of the week for the five different exposure variables. The blue rectangles indicate the hours between 0 a.m. and 6 a.m.

**Figure 15.**Risk index $\rho $ as a function of the hour of the week for all crashes (

**top**, black) and the severe crashes (

**bottom**, red). The blue rectangles indicate the hours between 0 a.m. and 6 a.m., the area around the curves is the 99% confidence interval of the mean. They have been computed by bootstrapping.

**Figure 16.**The prediction index ${\Lambda}$ (green) as a function of the hour of the week, together with the safety index $\widehat{\rho}$ (red). The shaded areas are the 99% confidence intervals of the mean values which have been computed by bootstrapping.

**Table 1.**Results of the fits of the three models displayed in Figure 10.

Model | Link | R Function | AIC | Parameters (All $\mathit{p}<2\times {10}^{-16}$) |
---|---|---|---|---|

power | log | gam | 35,129 | $23.57\phantom{\rule{0.166667em}{0ex}}{Q}^{1.50}$ |

polynomial | id | glm.nb | 35,186 | $7.43\phantom{\rule{0.166667em}{0ex}}Q+16.65\phantom{\rule{0.166667em}{0ex}}{Q}^{2}$ |

gam | id | gam | 34,821 | $12.25+2.99\phantom{\rule{0.166667em}{0ex}}s\left(Q\right)$ |

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Wagner, P.; Hoffmann, R.; Leich, A.
Observations on the Relationship between Crash Frequency and Traffic Flow. *Safety* **2021**, *7*, 3.
https://doi.org/10.3390/safety7010003

**AMA Style**

Wagner P, Hoffmann R, Leich A.
Observations on the Relationship between Crash Frequency and Traffic Flow. *Safety*. 2021; 7(1):3.
https://doi.org/10.3390/safety7010003

**Chicago/Turabian Style**

Wagner, Peter, Ragna Hoffmann, and Andreas Leich.
2021. "Observations on the Relationship between Crash Frequency and Traffic Flow" *Safety* 7, no. 1: 3.
https://doi.org/10.3390/safety7010003