# Sensitivity Analysis for Microscopic Crowd Simulation
## 1 Introduction
### Overview of Sensitivity Analysis Methods  
We start with a brief overview of sensitivity analysis methods. The two main types of sensitivity analyses are local and global sensitivity analysis. Local approaches study the sensitivity of the model output around a fixed base point. These approaches are often deterministic. On the other hand, global sensitivity analysis can be used to study the impact of parameters over a wider range. In contrast to local sensitivity analysis, it typically requires probability distributions for the uncertain input parameters. Often, local methods can be expanded towards a global analysis. Consequently, there is no clear distinction between local and global methods. Instead, they can be seen as general concepts. A typical local method is the one-at-a-time approach. The idea behind it is intuitive: only one parameter is varied around a base point, while all others are held constant. One-at-a-time methods and their generalizations are subsumed under the umbrella term screening methods. Screening methods study models locally, but not necessarily at only one base point. Another approach that is commonly used on local sensitivity analysis is based on differentiation in the neighborhood of a base point. Partial derivatives indicate in which parameter directions the model output changes most. Often, the derivatives are not available analytically, but need to be estimated. We differentiate between five types of global sensitivity analysis methods: first, in regression-based methods, a linear surrogate is defined and investigated instead of the actual model. If the hypothesis of a linear relationship between inputs and quantity of interest does not hold, rank-based methods can be used to circumvent that problem. Based on the linear regression, performance indices as correlation coefficients are evaluated in order to gain insights on the parameter sensitivities. Second, variance-based methods are the most popular approach for global sensitivity analysis. Using an analysis of variance, or ANOVA, decomposition, a non-linear regression is performed. Subsequently, the variance in the quantity of interest is studied based on the regression model. A well-known variance-based metric is Sobol' indices [20]. Third, moment-independent or density-based methods investigate the full probability density function of the quantity of interest instead of focusing on moments, such as the variance [21]. Fourth, screening methods with several base points serve as a generalization of the local one-at-a-time idea. In order to use these local approaches globally, the model is evaluated at a number of locations in the parameter space. A common method of this type is Morris screening [22]. The main use of screening methods is to identify non-influential variables among a large number of input parameters [23]. One-at-a-time methods can also be adapted for global sensitivity analysis, but they are not suited for non-linear models [24]. Lastly, derivative-based approaches can also be applied for global sensitivity analysis. As for screening methods, the derivative-based approach can be extended from a single base point to several base points to approximate global sensitivity. Measures of this type are the derivative-based global sensitivity metrics that rely on randomized samples for the base points [25]. In addition, [PERSON] and [PERSON] proposed activity scores that are calculated from gradients as a new sensitivity metric [26]. Because there is a vast variety of global sensitivity methods available, users must choose the most suitable method for their application. The choice can be supported by decision trees as provided in [27]. In addition, several publications deal with links between different metrics: links between derivative-based and variance-based indices are e.g., shown in [28] for derivative-based global sensitivity measures [29] and Sobol' indices and in [26] for activity scores and Sobol' indices.