# Comparison of Traffic Flow Models with Real Traffic Data Based on a Quantitative Assessment
## 2 Theoretical Background
### Criteria and Methods for the Comparison of Traffic Flow Models  
The need to compare models becomes evident when scientists or practitioners must decide which model will work best for their data and select the best from several models. There are a number of criteria that may be used for assessment and comparison of computational models [24]: goodness of fit (which is the empirical accuracy of the model), simplicity or complexity (regarding models' functional form and number of parameters), generalizability (ability to use the model for forecasting future observations), faithfulness (whether the model represents the regularities that are the basis of the phenomenon being modeled), interpretability (whether the model's parameters are clear and linked to the phenomena being modeled), explanatory adequacy (whether the model's theoretical representation is sufficient to understand the data being observed), and falsifiability (whether potential observations exist that are inconsistent with the model). As pointed out by [PERSON] and [PERSON] [24], among these, the most important and the most widely used are criteria lending to quantification (i.e., goodness of fit, simplicity). However, qualitative criteria may provide a relevant and valuable complement to model assessment. It is the researcher's decision to select criteria to match those features of the model that they consider most important. Most of the above criteria can also apply to models of the fundamental relationship. What makes a good traffic flow model? According to [PERSON] et al. [4], it should have a simple functional form (be \"mathematically elegant\"), fit the data well, have interpretable parameters, and allow the entire range of traffic conditions to be modeled. The latter suggests that a single-regime model is preferred over a multi-regime model, which is confirmed in the works of [PERSON] and [PERSON] [9; 18]. In principle, the criteria given by  
Figure 2: Relationships v(k), v(q), and q(k) in selected multi-regime models (assuming model continuity). [PERSON] and [PERSON] are consistent with those identified by [PERSON]. [PERSON] mentions an additional aspect of having to meet boundary conditions. [PERSON], on the other hand, points out that the model should easily lend itself to mathematical analysis and that its limitations should be known in advance. A review of the above works suggests that researchers are in agreement as to what criteria should be met by the fundamental relationship model. As a result, the criteria they identified may be used to assess and compare models of the fundamental relationship. This type of comparison in the literature usually has an expert basis rather than concrete criteria and measures to assess the criteria. The criteria researchers refer to most often include goodness of fit, which is a quantitative measure. In addition, faithfulness of the model is often assessed by checking how the model fits in with the empirical data and how the phenomena are represented on the fundamental diagram. An example is the work of [PERSON] and [PERSON] [25], who assess models matching data from two sections of an urban arterial in Delhi with goodness of fit, using root mean square error, average relative error, and cumulative residual plots for the assessment. To complement the criterion assessment, expert judgement on the properties of the models on the speed-density diagrams is applied. To assess existing models for a 55 mph motorway section, [PERSON] [26] used two quantitative measures: goodness of fit, measured by mean deviation, and parameter value, which is assessed by checking if they are consistent with the ranges established from empirical data. It is not clear from the publication how these ranges were identified. The same criterion was used by [PERSON] [27] for the same data and the same ranges of parameters as [PERSON] [26], comparing known single and multi-regime models with [PERSON]'s model. [PERSON] et al.