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