# Hierarchical Modelling for CO\({}_{2}\) Variation Prediction for HVAC System Operation
## 6 Results and Discussion
### Parameter Selection
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{2}{c}{**History and Future**} & \multirow{2}{*}{**Ensemble Algorithm**} & \multirow{2}{*}{**CNN Model**} & \multirow{2}{*}{**CNN FCL**} & \multirow{2}{*}{**Method**} & \multirow{2}{*}{**MAE**} \\ \cline{3-3} \cline{5-6} \multicolumn{1}{c}{**Time Window (in Minutes)**} & & & & & \\ \hline \multirow{6}{*}{h-5\_f-5} & RR & VGG\_16 & [64] & holistic & 1.61 \\ \cline{2-6}  & DT & Resnet\_152 & [256] & local & 0.65 \\ \cline{2-6}  & RF & VGG\_16 & [512] & holistic & **0.4** \\ \cline{2-6}  & NN & Resenet\_152 & [512; 256] & holistic & 1.3 \\ \hline \multirow{6}{*}{h-10\_f-10} & RR & VGG\_19 & [4096] & local & 3.25 \\ \cline{2-6}  & DT & VGG\_19 & [512] & local & 0.84 \\ \cline{2-6}  & RF & Resnet\_152 & [128; 64] & local & **0.765** \\ \cline{2-6}  & NN & Resnet\_101 & [256; 128] & local & 2.63 \\ \hline \multirow{6}{*}{h-15\_f-15} & RR & VGG\_16 & None & local & 5.54 \\ \cline{2-6}  & DT & Resnet\_152 & [256; 128] & local & **0.98** \\ \cline{2-6}  & RF & Xception & None & local & 1.22 \\ \cline{2-6}  & NN & Resnet\_101 & [256] & local & 4.48 \\ \hline \multirow{6}{*}{h-20\_f-20} & RR & Resnet\_50 & [128] & holistic & 6.07 \\ \cline{2-6}  & DT & Resnet\_101 & [128; 64] & local & 1.18 \\ \cline{1-1} \cline{2-6}  & RF & VGG\_19 & [4096] & holistic & **0.84** \\ \cline{1-1} \cline{2-6}  & NN & Resnet\_50 & [128; 64] & local & 4.91 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameter Selection on the Training Dataset. variation direction with better accuracy when fed with GAF representations that accentuate local differences within a time window. The summarized results in Table 2 have shown some parameter combinations that clearly outperform others. However, a clear limitation of the MAE parameter in this application is that it is skewed toward the values that constitute the majority of the response variable. In order to clarify this caveat, Figure 4 shows the percentage distributions of different absolute values of thresholds defined in the evaluation criteria. These results suggest that more than 60% of the variations are under 5 ppm. Therefore, a model with a low MAE is not fully representative of its performance on drastic changes in data. These radical changes are of higher importance for HVAC systems and building operators, but they are less common in the studied dataset. As a result, the crude MAE is replaced by a metric that averages the MAEs over each of the defined thresholds. This new metric is referred to as **Thresh_MAE**. Figure 5 shows the differences in MAE for each of the defined thresholds for the two best-performing models in Table 2 that use DT or RF for ensemble learning. Figure 5a shows that RF combined with the other parameters included in Table 2 outperforms its DT counterpart when no thresholds are imposed. However, the performances start to diverge with increasing thresholds until a huge gap in performance is prominent. This trend shows that the RF model performs well on small variations in CO2 while DT outperforms RF when more drastic changes are involved. A similar observation can be applied to Figure 5b, whereby DT outperforms RF when thresholds are considered. However, a notable separation exists between the results of the RF for higher thresholds with the expansion of the prediction time horizon. This discrepancy is attributed to the abundance of more extreme variations with the expansion of future time windows. This fact enables the models to better map the relationships between these extreme values and the input variables, manifested by the lower MAE for higher thresholds. Figure 4: Thresholds’ Data Percentages. Following the newly defined metric, **Thresh_MAE**, the best-performing parameters are shown in Table 3. As opposed to pure MAE, DT clearly outperformed all other ensemble learners and showed superior performance in predicting drastic variations. This observation is demonstrated by the decrease in Thresh_MAE value compared to MAE value, when DT is involved, for all h&f combinations.