# Predicting the Aggregate Mobility of a Vehicle Fleet within a City Graph
## 6 Evaluation
### Results  
The results of the evaluation are summarized in Table 1. A more illustrative representation is included in Figure 1, where the distribution of the predictive error for each model and each set of evaluation parameters can be seen. Baseline predictor \(1\), denoted as \(I_{N}\), and baseline predictor \(2\), denoted \(\overline{\Pi}\), are defined, respectively, in Equations (15) and (16). Each column corresponds to a different type of model employed to generate the evaluation time series; as explained above, each one of those time series is constructed using a corresponding transition matrix: generator (a) with \(\overline{\Pi}\) and generator (a) with \(\Pi_{r}\).  
It is important to note that the time series generation schemes (a) and (b) proposed in Section 6.2 make use of an MC-type formulation. Hence, predictor 4 in Section 6.1, based on an MC assumption, performs especially well in such favorable scenarios. The MLP-based predictor \(5\), even though it can capture Markovian behaviors, cannot compete with the MC optimal predictor in these scenarios since it does not exploit the available graph knowledge. The GNN-based predictor 6, when incorporating the node id and an attention mechanism, performs better than the proposed baseline schemes 1 and 2, and its performance is very close to that of the MC-assumption-based predictor 4.  
In all scenarios, GNN models outperform their NoGNN counterparts, which demonstrates the effectiveness of the GNN approach. The simpler MLP variants show higher prediction errors than any of the GNN models. This is especially true of the MLP with more hidden neurons. These results can be explained because MLP models do not incorporate any a priori knowledge about graph structure. Although nodes that are not connected in the graph should not affect each other, the layers in the MLP are fully connected. All weights, including those that are zero, need to be learned from training data However, the chosen values for \(T_{S}\) may not be enough, especially when the number of tunable weights is very high.  
As for other factors to take into account, it is worth mentioning that low asset counts, which imply low node occupancy (i.e., the average occupancy of each node close to zero), lead to higher errors in general; therefore, in that scenario, all predictors behave almost indistinguishable, due to the stochasticity of time series generative models. More specifically, it is worth noting the relatively high error of the uniform predictor in datasets generated using a column-wise uniform transition matrix (\(\overline{\Pi}\)). This is due to the stochastic nature of the generative models explained above (see Section 6.2).  
\begin{table}
\begin{tabular}{r r r r r r r r r r r r r r r r r} \hline \hline \multicolumn{1}{c}{**\#assets**} & \multicolumn{1}{c}{**Tans.**} & \multicolumn{1}{c}{**Model**} & \multicolumn{1}{c}{**\(I_{N}\)**} & \multicolumn{1}{c}{**\(\overline{\Pi}\)**} & \multicolumn{1}{c}{**MC**} & \multicolumn{1}{c}{**NoGNN**} & \multicolumn{1}{c}{**NoGNNNid**} & \multicolumn{1}{c}{**GNNN1**} & \multicolumn{1}{c}{**GNN3**} & \multic