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Constrained Sampling from a Kernel Density Estimator to Generate Scenarios for the Assessment of Automated Vehicles

Erwin de Gelder, Eric Cator, Jan-Pieter Paardekooper, Olaf Op den Camp, Bart De Schutter

TL;DR

This paper proposes a method to sample from a pdf estimated using KDE, such that the samples satisfy a linear equality constraint and presents an algorithm of the method in pseudo-code that can be used to generating scenarios that have, e.g., a predetermined starting speed or to generate different types of scenarios.

Abstract

The safety assessment of automated vehicles (AVs) is an important aspect of the development cycle of AVs. A scenario-based assessment approach is accepted by many players in the field as part of the complete safety assessment. A scenario is a representation of a situation on the road to which the AV needs to respond appropriately. One way to generate the required scenario-based test descriptions is to parameterize the scenarios and to draw these parameters from a probability density function (pdf). Because the shape of the pdf is unknown beforehand, assuming a functional form of the pdf and fitting the parameters to the data may lead to inaccurate fits. As an alternative, Kernel Density Estimation (KDE) is a promising candidate for estimating the underlying pdf, because it is flexible with the underlying distribution of the parameters. Drawing random samples from a pdf estimated with KDE is possible without the need of evaluating the actual pdf, which makes it suitable for drawing random samples for, e.g., Monte Carlo methods. Sampling from a KDE while the samples satisfy a linear equality constraint, however, has not been described in the literature, as far as the authors know. In this paper, we propose a method to sample from a pdf estimated using KDE, such that the samples satisfy a linear equality constraint. We also present an algorithm of our method in pseudo-code. The method can be used to generating scenarios that have, e.g., a predetermined starting speed or to generate different types of scenarios. This paper also shows that the method for sampling scenarios can be used in case a Singular Value Decomposition (SVD) is used to reduce the dimension of the parameter vectors.

Constrained Sampling from a Kernel Density Estimator to Generate Scenarios for the Assessment of Automated Vehicles

TL;DR

This paper proposes a method to sample from a pdf estimated using KDE, such that the samples satisfy a linear equality constraint and presents an algorithm of the method in pseudo-code that can be used to generating scenarios that have, e.g., a predetermined starting speed or to generate different types of scenarios.

Abstract

The safety assessment of automated vehicles (AVs) is an important aspect of the development cycle of AVs. A scenario-based assessment approach is accepted by many players in the field as part of the complete safety assessment. A scenario is a representation of a situation on the road to which the AV needs to respond appropriately. One way to generate the required scenario-based test descriptions is to parameterize the scenarios and to draw these parameters from a probability density function (pdf). Because the shape of the pdf is unknown beforehand, assuming a functional form of the pdf and fitting the parameters to the data may lead to inaccurate fits. As an alternative, Kernel Density Estimation (KDE) is a promising candidate for estimating the underlying pdf, because it is flexible with the underlying distribution of the parameters. Drawing random samples from a pdf estimated with KDE is possible without the need of evaluating the actual pdf, which makes it suitable for drawing random samples for, e.g., Monte Carlo methods. Sampling from a KDE while the samples satisfy a linear equality constraint, however, has not been described in the literature, as far as the authors know. In this paper, we propose a method to sample from a pdf estimated using KDE, such that the samples satisfy a linear equality constraint. We also present an algorithm of our method in pseudo-code. The method can be used to generating scenarios that have, e.g., a predetermined starting speed or to generate different types of scenarios. This paper also shows that the method for sampling scenarios can be used in case a Singular Value Decomposition (SVD) is used to reduce the dimension of the parameter vectors.

Paper Structure

This paper contains 8 sections, 26 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem definition
  3. Method
  4. Example
  5. Sampling with a linear equality constraint
  6. Applying conditional sampling with parameter reduction
  7. Discussion
  8. Conclusions

Figures (3)

  • Figure 1: The histogram shows the result of the conditional sampling according to \ref{['alg:constrained hard']}. The red line represents the true pdf.
  • Figure 2: 50 scenarios sampled from the kde with a constraint on the initial speed and the initial acceleration.
  • Figure 3: 50 scenarios sampled from the kde with a constraint on the initial speed and the end speed.