## 4 Model Considerations
### Stochastic Error Minimization  
Before applying the FEATHERS framework for this case study, another model issue has to be properly addressed, which is minimizing the effect of stochastic errors in FEATHERS. As highlighted before, activity-based models use in most cases a micro-simulation approach; stochastic errors are therefore inherently included in such models [17]. As a result, running a transport micro-simulation model, such as FEATHERS, several times with the same input will generate different outputs due to the random number seed used in each run. Consequently, it is natural to run FEATHERS multiple times, estimate the effects of stochastic errors by analyzing the variation of the outputs between the runs, and use the average value of these outputs for further analysis. The question then becomes: what is the minimum number of runs required to ensure that at least a certain percentile of zones in the study area reach a stable result? To answer this question, 100 successive model runs are performed based on a 50% fraction of the whole Flanders population. After each model run, the prediction file, containing the whole activity \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Travel Mode**} & \multirow{2}{*}{**Study Area (SA)**} & \multirow{2}{*}{\begin{tabular}{c} **No. of Persons** \\ **within the SA (50\%)** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **No. of Households** \\ **within the SA (50\%)** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Running** \\ **Time (Hour)** \\ \end{tabular} } & \multirow{2}{*}{**Accuracy Level**} & \multirow{2}{*}{**Time Saving**} \\ \cline{2-2} \cline{6-6}  & Leuven & & 1,021,660 & 633,216 & 28.8 \\ \cline{2-6}  & Flanders & & 2,395,514 & 1,449,213 & 64.5 & 89.6\% & 55\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison between the Leuven study area and the whole of Flanders with regard to public transport mode. Figure 4: The study area for the city of Leuven with regard to public transport mode. travel pattern or schedule information for each individual, is generated, based on which the average daily number of trips per person (\(X\)) for each of the 327 Superzones can be computed. Next, the concept of confidence interval (\(CI\)) is adopted [18; 19], using the following equation:  
\[CI_{(1-\alpha)\%}=2\times t_{(1-\alpha/2),N-1}\frac{s}{\sqrt{N}}, \tag{3}\]  
where \(CI_{(1-\alpha)\%}\) represents a \((1-\alpha)\%\) confidence interval for the true average value; \(\alpha\) is the probability of the true average value not lying within the confidence interval; \(t_{(1-\alpha/2),N-1}\) is the Student's \(t\)-statistic for the probability of a two-sided error adding up to \(\alpha\) with \(N-1\) degrees of freedom; \(N\) is the required number of model runs; and \(s\) denotes the estimated standard deviation of the results. In this study, a 95% level of confidence is selected and the desired confidence interval, which acts as the predefined stable condition, is set as a 10% fraction of the final average value (after 100 runs) of the corresponding index under study (\(X\)), i.e., \(CI\leq 0.1\times\overline{X}_{100}\), where \(\overline{X}_{100}=\sum_{i=1}^{100}X/100\). Also, the standard deviation of the results among 100 runs is used as the estimation of \(s\). By using Equation (3), an iterative process is applied for each Superzone to estimate the minimum number of model runs required in terms of the average daily number of trips per person (\(X\)). The iteration continues until the estimated number of model runs \(N\) matches, the number of repetitions assumed when looking up the Student's \(t\)-statistic. In this way, the minimum number of FEATHERS runs needed to ensure each Superzone to achieve stable results with respect to the corresponding index can be derived. As indicated in Section 4.2, a total of 135 municipalities (i.e., Superzones) have been identified for the Leuven case study.