Simulation models for pedestrian crowds are a ubiquitous tool in research and industry. It is crucial that the parameters of these models are calibrated carefully and ultimately it will be of interest to compare competing models to decide which model is best suited for a particular purpose. In this contribution, I demonstrate how Approximate Bayesian Computation (ABC), which is already a popular tool in other areas of science, can be used for model fitting and model selection in a pedestrian dynamics context. I fit two different models for pedestrian dynamics to data on a crowd passing in one direction through a bottleneck. One model describes movement in continuous-space, the other model is a cellular automaton and thus describes movement in discrete-space. In addition, I compare models to data using two metrics. The first is based on egress times and the second on the velocity of pedestrians in front of the bottleneck. My results show that while model fitting is successful, a substantial degree of uncertainty about the value of some model parameters remains after model fitting. Importantly, the choice of metric in model fitting can influence parameter estimates. Model selection is inconclusive for the egress time metric but supports the continuous-space model for the velocity-based metric. These findings show that ABC is a flexible approach and highlight the difficulties associated with model fitting and model selection for pedestrian dynamics. ABC requires many simulation runs and choosing appropriate metrics for comparing data to simulations requires careful attention. Despite this, I suggest ABC is a promising tool, because it is versatile and easily implemented for the growing number of openly available crowd simulators and data sets.
Simulation models for pedestrian crowds are a widely used tool [1]. The dynamics these models produce are controlled by parameters that capture the preferred speed of pedestrians or the strength of interactions between pedestrians, for example [1]. As many models are intended to be used or are already used to investigate real world scenarios, a key challenge is to calibrate model parameters, such that simulations produce realistic behaviour [2]. In addition, robust approaches for calibrating parameters facilitate a fair comparison of the predictive potential or goodness of fit across different models [3].
A range of approaches for calibrating model parameters have been suggested [2,[4][5][6][7][8]. They compare models to empirical data at a microscopic level (e.g. trajectories [5]) or at a macroscopic level, where summary statistics for simulations and data are compared (e.g. pedestrian flows [2]). For specific models, it is sometimes possible to formulate a likelihood function linking model and data via probability distributions [6]. However, for most simulation models it is not practical to find explicit, closed-form probability distributions for simulation outcomes. Thus, parameter calibration typically uses an objective function that measures the difference between data and simulations via measures derived from microscopic data [5,7,8]. Parameter estimates are found by optimising the objective function. While this approach is valid, it has three major shortcomings. First, this approach yields point-estimates for parameters and provides no information on the uncertainty associated with estimates. Second, this approach is not suited for model comparison, where the relative quality of different models in describing data is established. The objective function does provide a goodness of fit measure, but it is not clear how differences in model complexity (i.e. number of model parameters) should be accounted for when comparing this measure across models. Third, all numerical optimization procedures are liable to getting stuck in local optima, meaning that the true optimal solution may not be found.
In this contribution, I propose an alternative approach for pedestrian model calibration using Approximate Bayesian Computation (ABC), that is already widely used in other fields of science (e.g. [9]). I show how this flexible framework avoids the issues mentioned above and I demonstrate for representative simulation models that even for simple scenarios (e.g. unidirectional flow through one bottleneck), parameter estimates are associated with substantial uncertainty.
To demonstrate the parameter calibration and model selection approach based on Approximate Bayesian Computation (ABC), I consider two different simulation models and two different approaches for comparing simulations to data. In the following, I first describe the two models, then I outline the ABC approach and finally I provide details on model fitting and selection.
I consider two microscopic models that are derived from popular models in the literature. As this work is intended as a demonstration of principle only, I do not wish to make any claims about the overall quality of either model or similar models.
The first model is a derivative of a popular simulation model that describes the movement of pedestrians in continuous space [10]. To save space, I do not provide a detailed description of this model, but all details can be found in precious work [11]. Briefly, pedestrian-pedestrian and pedestrian-wall interactions are captured in force vectors acting on simulated pedestrians. The movement preferences of simulated pedestrians, e.g. towards a target and away from walls, is encoded in a discrete floor field [11]. Models of this type are commonly referred to as “Social Force models” and I thus refer to this model as “SF model” here. I fit the values of five parameters of this model, keeping the remaining parameters fixed at default levels that have been specified previously [11]. The fitted parameters are the preferred speed of pedestrians, 𝑣 0 , coefficients determining the strength of psychological interactions between pedestrians ( 𝐴, 𝐵 ) and coefficients regulating the strength of physical pedestrian-pedestrian and pedestrian-wall interactions (𝑘, 𝜅). The nomenclature for these parameters is identical to the one used in previous work [11].
The second model is a cellular automaton model that describes the movement of pedestrians in discrete space on a lattice grid with square cells of side-length 0.4m. It is very loosely based on previous work [12]. The model is deliberately kept very simple to provide a contrast to the complex SF model. Let 𝐹 𝑖𝑗 denote the value of cells of a static floor field that encodes the movement preferences of pedestrians in the same way as the floor field used for the SF model. The same algorithm as for the SF model is used to construct 𝐹 𝑖𝑗 (see [11]). The indices 𝑖 and 𝑗 denote the rows and columns of the latti
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