# Modeling Stochastic Data Using Copulas for Applications in the Validation of Autonomous Driving

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Safety Validation of Automated Vehicles

#### 2.2. Copulas

#### 2.2.1. Definition

**Definition 1**

**Theorem 1**

**Example 1.**

#### 2.2.2. Bivariate Parametric Copula Models

**Definition 2**

**Remark 1.**

#### 2.2.3. Vine Copulas

**Theorem 2.**

#### 2.3. Dependence Measures

**Definition 3**

**Definition 4**

**Definition 5**

**Definition 6**

## 3. Application to Traffic Data

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Measured data (grey) and associated simulated data (+) generated with the underlying copula.

Name | Bivariate Archimedean Copula $\mathit{C}({\mathit{u}}_{1},{\mathit{u}}_{2})$ | $\mathit{\delta}$ | $\mathit{\theta}$ |
---|---|---|---|

Clayton | ${\left(\right)}^{{u}_{1}^{-\delta}}$ | $\delta \in (0,\infty )$ | − |

Gumbel | $exp\left(\right)open="("\; close=")">-{\left(\right)}^{{\left(\right)}^{-}}+{\left(\right)}^{-}\delta 1/\delta $ | $\delta \ge 1$ | − |

Frank | $-\frac{1}{\delta}ln\left(\right)open="("\; close=")">\frac{1}{1-{e}^{-\delta}}\left(\right)open="["\; close="]">(1-{e}^{-\delta})-(1-{e}^{-\delta {u}_{1}})(1-{e}^{-\delta {u}_{2}})$ | $\delta \ne 0$ | − |

Joe | $1-{\left(\right)}^{{(1-{u}_{1})}^{\delta}}1/\delta $ | $\delta \ge 1$ | − |

BB1 | ${\left(\right)}^{1}1/\delta $ | $\delta \ge 1$ | $\theta >0$ |

BB7 | $1-{\left(\right)}^{1}+{\left(\right)}^{1}-1$ | $\delta >0$ | $\theta \ge 1$ |

Parameter | Description | Unit |
---|---|---|

VelCar | velocity of cars per frame | $(\mathrm{m}/\mathrm{s})$ |

TrafficCar | traffic density per car per frame | $(-)$ |

WaitTime | waiting time per car before entering roundabout | $\left(\mathrm{s}\right)$ |

DistCar | minimal distance for one car to the other around him | $\left(\mathrm{m}\right)$ |

${\mathit{\rho}}_{\mathit{S}}$ | TrafficCar | VelCar | WaitTime | DistCar | |
---|---|---|---|---|---|

$\mathit{\tau}$ | |||||

TrafficCar | $1.0$ | $-0.45$ | $0.43$ | $-0.79$ | |

VelCar | $-0.36$ | $1.0$ | $-0.51$ | $0.49$ | |

WaitTime | $0.41$ | $-0.41$ | $1.0$ | $-0.39$ | |

DistCar | $-0.64$ | $0.34$ | $-0.31$ | $1.0$ |

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**MDPI and ACS Style**

Lotto, K.; Nagler, T.; Radic, M.
Modeling Stochastic Data Using Copulas for Applications in the Validation of Autonomous Driving. *Electronics* **2022**, *11*, 4154.
https://doi.org/10.3390/electronics11244154

**AMA Style**

Lotto K, Nagler T, Radic M.
Modeling Stochastic Data Using Copulas for Applications in the Validation of Autonomous Driving. *Electronics*. 2022; 11(24):4154.
https://doi.org/10.3390/electronics11244154

**Chicago/Turabian Style**

Lotto, Katrin, Thomas Nagler, and Mladjan Radic.
2022. "Modeling Stochastic Data Using Copulas for Applications in the Validation of Autonomous Driving" *Electronics* 11, no. 24: 4154.
https://doi.org/10.3390/electronics11244154