## 6 Results and Discussion
### Cluster Analysis of the Prediction Errors  
To inform policymaking, it is necessary to have an in-depth understanding of how Uber fares react concerning demand or supply shortages. To this end, in this subsection \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Performance Metrics** & **Linear Regression** & **Decision Tree** & **Random Forest** \\ \hline RMSE (EUR) & 3.41 & 3.85 & 3.40 \\ MAPE (\%) & 8.01 & 6.60 & 6.32 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of performance metrics of the models. Figure 6: Scatterplots of real fare values versus predicted values. cluster analysis is applied using the prediction errors of the models as variables (namely, the distribution of MAPE prediction errors, which are mostly used for comparison purposes), identifying, for example, cases in which models overpredict or underpredict the observed fares. Cluster analysis is a powerful technique for examining group features. Many studies based on clustering approaches have been conducted on feature recognition and analysis. Each observation belonging to one cluster is like the other ones belonging to it and different from all the other ones belonging to other clusters. The \(k\)-means algorithm is one of the most frequently employed techniques in group division and feature analysis [40]. This study applied the \(k\)-means clustering method for which the number of clusters is one of the most critical decisions. In this analysis, the \(k\) cluster number was set to three, as was conducted by other authors, such as [PERSON] et al. (2016) [41]. This study considers the number of clusters based on the distribution of MAPE prediction errors of the three models (i.e., Linear Regression, Decision Tree, and Random Forest), namely: (i) Low error; (ii) Medium error; and (iii) High error. Fifty-nine iterations were necessary to achieve stability in the cluster centers. Table 5 shows the average distance of each variable (i.e., the MAPE prediction errors of the three machine learning models) to every cluster center and Figure 7 shows the number and percentage of observations per cluster. The characteristics of the clusters are described below:  
* Cluster #1 (Low error): This cluster has the smallest average distance values across all MAPE prediction error variables. The two smallest distances were found in the Decision Tree and Random Forest models, indicating that these models predict observations with smaller MAPE prediction errors than Linear Regression. * Cluster #2 (Medium error): This cluster has intermediate average distance values on all MAPE prediction error variables. The smallest distance was found in the Linear Regression model. * Cluster #3 (High error): This cluster has the highest average distance values across all MAPE prediction error variables. The smallest distance was found in the Linear \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**MAPE Prediction Errors**} & \multicolumn{3}{c}{**Cluster**} \\ \cline{2-4}  & **\#1 (Low Error)** & **\#2 (Medium Error)** & **\#3 (High Error)** \\ \hline Linear Regression & 6.030 & 21.458 & 22.544 \\ Decision Tree & 2.555 & 26.611 & 76.960 \\ Random Forest & 2.917 & 25.045 & 55.860 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Final cluster centers. Figure 7: Number and percentage of observations per cluster. Regression model, indicating that this model predicts the observations in the group with larger MAPE prediction errors better. The number of observations in each cluster also shows that cluster #1 (Low error) contains 87.32% of the observations, followed by cluster #2 (Medium error) with 10.72% and cluster #3 (High error) with 1.96%. All valid observations are included in the clusters.