# Black-Spot Analysis in Hungary Based on Kernel Density Estimation

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## Abstract

**:**

## 1. Introduction

- the problem of examining fixed-length sections;
- ignoring surrounding roads;
- due to the definition applied for black spots, the most affected sections included at least four accidents;
- ignoring the annual average daily traffic (AADT);
- the high procedure time.

## 2. Materials and Methods

#### 2.1. Theoretical Considerations

_{1}, x

_{2}, …, x

_{n}. Let h ∈ N+ be called bandwidth. The kernel function estimation, that is, the shape of the Parzen–Rosenblatt estimate is as follows:

_{i}sample point. In a given point x, the value of the kernel function estimate is the sum of the y coordinates (ordinates) of the n kernels, which are located there. As a result, in x with many sample points, the kernel estimate will be relatively high, and accordingly, fewer sample points will imply a lower value [29,30,31,33].

#### 2.2. Practical Considerations

#### 2.3. Process

## 3. Results

#### 3.1. Weighting with Annual Average Daily Traffic

#### 3.2. Determining the Starting Points and Endpoints of the Accident Black-Spots

#### 3.3. Accident Concentration Sites on the Main Road No. 1

## 4. Discussion

#### 4.1. Applicability of the Kernel Density Estimation Method

#### 4.2. Sensitivity Analysis—The Selection of the Kernel

#### 4.3. Sensitivity Analysis—Determination of Bandwidth

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**The resulting density function of pedestrian and cyclist accidents of the main road No. 1 in Hungary (bandwidth = 300 m).

**Figure 4.**Detail of the resulting density function of pedestrian and cyclist accidents of the main road No. 1 in Hungary (bandwidth = 300 m).

**Figure 5.**Comparison of the three methods (Moving-window, distance matrix based, kernel density estimation).

**Figure 6.**Comparison of kernel functions (the resulting density function of accidents on the main road No. 1 in Hungary).

**Figure 7.**Under-smoothed resultant kernel (bandwidth = 50 m, sample road: Main road No. 1 in Hungary).

**Figure 8.**Over-smoothed resultant kernel (bandwidth = 5000 m, sample road: Main road No. 1 in Hungary).

Author and Publication Year | Road Traffic Accident Investigation | Kernel Functions | Taking into Account the AADT | Bandwidth Rate [m] | Study Area |
---|---|---|---|---|---|

Banos A. (2000) [7] | Yes | Not defined | No/Not defined | Not defined | Urban |

Matthias J. K. (2006) [19] | Yes | Normal | Was available but not used with the kernel | 2000 | Mix |

Pulugurtha S. S. (2007) [9] | Yes | Not defined | No/Not defined | Not defined | Urban |

Blazquez C. A. (2013) [10] | Yes | Normal | No/Not defined | 1000 | Urban |

Sedoník J. (2015) [8] | Yes | Epanechnikov | No/Not defined | 100 and 150 | Mix |

Andrásik R. (2015) [15] | Yes | Epanechnikov | No/Not defined | 100 | Mix |

Michal B. (2013) [16] | Yes | Epanechnikov | No/Not defined | 100 | Urban |

Álvaro B. (2019) [17] | Yes | Normal | Yes, but at intervals | 50 | Urban |

Yunxuan L. (2020) [12] | No | Normal | No/Not defined | 50 | Urban |

Anderson T. K. (2009) [13] | Yes | Not defined | No/Not defined | 200 | Urban |

Saffet E. (2008) [18] | Yes | Not defined | No/Not defined | 500 | Highway |

Seiji H. (2016) [14] | Yes | Not defined | No/Not defined | 250 | Urban |

V. Prasannakumar (2011) [20] | Yes | Not defined | No/Not defined | Not defined | Urban |

Mamoudou S. (2020) [21] | Yes | Normal | No/Not defined | Not defined | Urban |

Zhixiao X. (2008) [22] | Yes | Normal | No/Not defined | Variable | Urban |

Zhixiao X. (2013) [23] | Yes | Normal | No/Not defined | 100 | Urban |

T. Steenberghen (2004) [24] | Yes | Normal | No/Not defined | Not defined | Urban |

Utoyo B. (2012) [25] | Yes | Not defined | No/Not defined | Not defined | Mix |

Liljana Ç. (2013) [26] | Yes | Not defined | No/Not defined | 1000 | Urban |

Guler Y. (2015) [11] | Yes | Not defined | No/Not defined | Not defined | Urban |

Name | Form | Efficiency |
---|---|---|

Normal | $K\left(x\right)=\frac{1}{\mathsf{\sigma}\sqrt{2\mathsf{\pi}}}{\ast \mathrm{e}}^{-\frac{{\left(x-m\right)}^{2}}{2{\mathsf{\sigma}}^{2}}}$ | 95.1% |

Epanechnikov | $K\left(x\right)=\frac{3}{4}\left(1-{x}^{2}\right);\left|x\right|\le 1$ | 100% |

Box | $K\left(x\right)=\frac{1}{2};\left|x\right|\le 1$ | 92.9% |

Triangle | $K\left(x\right)=\left(1-\left|x\right|\right);\left|x\right|\le 1$ | 98.6% |

KDE with AADT | |

Starting Pointskm + m | Endpointskm + m |

3 + 795 | 5 + 649 |

30 + 298 | 31 + 152 |

49 + 550 | 50 + 849 |

63 + 921 | 66 + 717 |

74 + 080 | 74 + 414 |

75 + 581 | 76 + 498 |

84 + 551 | 87 + 583 |

95 + 159 | 96 + 721 |

124 + 027 | 125 + 465 |

133 + 857 | 134 + 196 |

160 + 667 | 164 + 665 |

Distance Matrix Method | |

Starting Pointskm + m | Endpointskm + m |

4 + 451 | 4 + 974 |

64 + 779 | 64 + 792 |

161 + 414 | 161 + 750 |

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Baranyai, D.; Sipos, T.
Black-Spot Analysis in Hungary Based on Kernel Density Estimation. *Sustainability* **2022**, *14*, 8335.
https://doi.org/10.3390/su14148335

**AMA Style**

Baranyai D, Sipos T.
Black-Spot Analysis in Hungary Based on Kernel Density Estimation. *Sustainability*. 2022; 14(14):8335.
https://doi.org/10.3390/su14148335

**Chicago/Turabian Style**

Baranyai, Dávid, and Tibor Sipos.
2022. "Black-Spot Analysis in Hungary Based on Kernel Density Estimation" *Sustainability* 14, no. 14: 8335.
https://doi.org/10.3390/su14148335