# Comparison of Traffic Flow Models with Real Traffic Data Based on a Quantitative Assessment
## 3 Materials and Methods
### Method
#### 3.2.1 Preparation of the Data and Determination of Expected Parameter Values  
The scatter on the empirical fundamental diagram may make model assessment and the identification of real boundary traffic parameters more difficult. The classical definition of the fundamental relationship assumes that traffic is stationary and homogeneous, which  
Figure 3: Empirical relations between speed, density, and flow. means that vehicles behave the same way in similar traffic circumstances [22]. This suggests that for the same traffic density, vehicles in a stream will move with the same speed. Building on this, an averaged representation of traffic conditions was determined from the data. This helps to significantly reduce the number of points in the diagram, making a visual assessment easier. It will be possible to state whether the actual traffic conditions are correctly represented in the model. In the proposed approach, observed densities are divided into narrow ranges e.g., 0.1 pc/km wide. The empirical mean of the other parameters, i.e., speed and flow rate, are computed. A similar approach can be found in the works of [PERSON] and [PERSON] and [PERSON] et al. [36,37]. Figure 4 presents the data averaged against density. As we can see, the scatter has been significantly reduced with only the congested traffic still represented by a cloud of points. This may be partly due to a much smaller sample of observed congested traffic--from nearly 110,000 of analyzed time intervals, only 3000 represented densities above 20 pc/h/lane (as a result for high traffic densities, the parameters are averaged on the basis of just a few time intervals). Once aggregated, the data may also be used to determine expected ranges of boundary parameters. However, to determine parameters within the maximal flow area (\(q_{\max},k_{\text{opt}},v_{\text{opt}}\)), the data must represent both free-flow and congested traffic. The expected ranges of boundary parameters are determined as follows (Table 3):  
* \(v_{\text{sw}}\), by determining the 5 th and 95 th percentiles of speed (\(v_{5},v_{95}\)) under conditions of low traffic volume, i.e., for corresponding volumes of \(q\leq 500\) pc/h/lane;
* \(q_{\max}\), by determining the 95 th and 99 th percentiles of traffic volume (\(q_{95},q_{99}\));
* \(v_{\text{opt}}\), by determining the 5 th and 95 th percentiles of speed (\(v_{5},v_{95}\)) under conditions of high traffic volume, where \(q_{\max_{1}}\leq q\leq q_{\max_{2}}\) and in free-flow traffic, i.e., assuming that \(v\geq 60\) km/h (to exclude the effect of vehicle stream speed in congested traffic);
* \(k_{\text{opt}}\), by determining the 5 th and 95 th percentiles of density (\(k_{5},k_{95}\)) occurring in the expected range of optimal speeds (\(v_{\text{opt}_{1}}\leq v\leq v_{\text{opt}_{2}}\)). With the 5 th and 95 th percentile as the boundary values of the range \(v_{\text{sw}},v_{opt}\), \(k_{opt}\) it is possible to determine areas that include the value of the given parameter with a 90% certainty. This approach helps to exclude the effects of extreme values and outliers on the results (meaning parameter ranges). In the case of \(q_{max}\), using the 95 th percentile helps to identify the lower boundary of the expected range, which determines 5% of the highest traffic volumes; the 99 th percentile in the upper boundary of the range helps to exclude the effects of 1% of the highest values on the expected parameter ranges. In the case of maximal density, which is difficult to spot in empirical data, results from a Polish study [38] were used to identify the expected range \(k_{\max}\). Aerial observation of traffic flows (an aeromobile platform about 400 m high covering a section of about 1.5 km)  
Figure 4: Curves v(q) and v(k) that determine the averaged representation of traffic conditions. helped to register jams on a selected dual carriageway and four lane sections of Polish motorways.