# Scenario Optimisation and Sensitivity Analysis for Safe Automated Driving Using Gaussian Processes
## 1 Introduction  
With the help of technological advances in vehicle technology, fatal accidents have been steadily declining over the past decades [1]. The rate at which fatal accidents are declining has slowed down, however, suggesting that further improvements can only be achieved by significantly reducing the proportion of accidents which are caused by human error. The latest developments in advanced driver assistance systems and proof-of-concept prototypes of automated vehicles (AVs) are very promising to overcome this challenge [2]. The introduction of higher levels of automation (SAE levels 3+ [3]) into wider public use is currently hindered by the satisfactory assurance of their safety in every possible situation. For instance, to formulate a statistical safety case which supports the superiority of AVs over human drivers, an estimated 440 million km of on-road driving with AVs is necessary [4].  
Current research thus focuses on reducing the necessary testing with physical hardware by utilizing virtual testing in simulation. A common approach focuses on the circumstance that most day-to-day driving is uneventful and thus tries to prioritise dangerous and safety-critical scenarios which happen very rarely [5]. Most methods are thereforeaimed at finding these critical scenarios by pursuing a scenario-based testing approach, where AVs are evaluated with the help of vehicle simulation [6]. When implemented in simulation, scenario-based testing allows for an automatic generation of interesting test cases, which can then be further evaluated in tests involving physical hardware.  
A terminology for describing scenarios in the different phases of development, test and validation of AVs is presented in Reference [7]. Scenarios can be abstracted into three levels, depending on where they are used in the development process. The highest abstraction is the functional scenario, which describes the involved entities and behaviours semantically, mostly in natural language. A logical scenario, as the middle level of abstraction, specifies the parameters which were described linguistically in the functional scenario. The logical scenario also specifies value ranges for each parameter in the scenarios state space, thus giving a formal description of the scenario. On the lowest level of abstraction, a concrete scenario specifies concrete values for each parameter from the respective value ranges. We make use of this terminology, which is visualised in Figure 1.  
A challenging problem in scenario-based testing is the combinatorial explosion of possible concrete scenarios. Defining a concrete scenario to include the interactions of the AV with its surroundings, especially with other traffic participants leads to an indefinite amount of concrete scenarios. There is a need for a novel method that efficiently finds the parameter combinations that lead to critical concrete scenarios. In our previous work [8], we have shown how stochastic machine learning can be used to estimate the performance boundary of an AV in a given logical scenario, by simulating a limited number of concrete scenarios. The performance boundary separates the scenario space into regions according to their criticality.