# Conditional Temporal Aggregation for Time Series Forecasting Using Feature-Based Meta-Learning
## 5 Results and Discussion  
The forecasting performance of the meta-learner in the M4 and M5 data sets is summarized in Table 3. CTA stands for conditional temporal aggregation, where a single \"best\" level is selected by the meta-learner, CMTA-PW for conditional multiple temporal aggregation with probability-based weights, and CMTA-PSTR for conditional multiple temporal aggregation with probability-based structural scaling. In all cases, ETS is used for producing the base forecasts. To enable comparisons, we also consider several benchmarks. The first benchmark, CON, refers to conventional forecasting, i.e., producing forecasts only at the original \begin{table}
\begin{tabular}{l c c} \hline \hline
**Hyperparameter** & **Fast-Moving (M4)** & **Slow-Moving (M5)** \\ \hline Number of iterations & 12 & 135 \\ Number of trees & 720 & 540 \\ Learning rate & 0.89 & 0.10 \\ Number of leaves per tree & 86 & 23 \\ Feature fraction & 95.5\% & 94.5\% \\ Bagging fraction & 80.0\% & 88.0\% \\ Bagging frequency & 22 & 25 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Selected hyperparameter values of the meta-learner for the fast-moving (M4) and slow-moving (M5) data sets. \begin{table}
\begin{tabular}{l c c c c} \hline \hline  & \multicolumn{2}{c}{**Fast-Moving (M4)**} & \multicolumn{2}{c}{**Slow-Moving (M5)**} \\
**Method** & **MASE** & **Rank** & **RMSSE** & **Rank** \\ \hline \multicolumn{5}{c}{Benchmarks} \\ CON & 0.9478 & 5 & 0.7761 & 6 \\ CON(Comb/Comb) & 0.9662 & 7 & 0.7912 & 8 \\ CON(Theta/SBA) & 0.9695 & 8 & 0.7864 & 7 \\ MTA-EW & 0.9623 & 6 & 0.7716 & 4 \\ MTA-STR & 0.9199 & 3 & 0.7712 & 3 \\ \hline \multicolumn{5}{c}{Conditional (multiple) temporal aggregation} \\ CTA & 0.9444 & 4 & 0.7746 & 5 \\ CMTA-PW & 0.9136 & 1 & 0.7709 & 1 \\ CMTA-PSTR & 0.9186 & 2 & 0.7711 & 2 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Forecasting performance of the proposed method in the M4 (fast-moving data) and M5 (slow-moving data) series. MASE and RMSSE are used to measure accuracy in each set, respectively. The accuracy of the examined benchmarks is also reported to enable comparisons. frequency of the series using ETS. The second, MTA-EW, is a simple implementation of multiple temporal aggregation where the base ETS forecasts are combined using equal weights. The third benchmark, MTA-STR, involves forecasting with temporal hierarchies, using the ETS framework and the structural scaling matrix as estimator. In addition, to allow comparisons with popular time series forecasting methods, we include two more benchmarks per data set, namely the top performing approaches from the benchmarks considered by the organizers of the M4 and M5 competitions. For M4, this is the simple arithmetic average of SES, Holt, and Damped exponential smoothing (Comb) and the Theta method [62], while for M5, this is the simple arithmetic mean of ETS and ARIMA (\(\text{Com}_{\text{b}}\)) and the Syntetos-Boylan approximation (SBA; [14]). Note that Comb, \(\text{Com}_{\text{b}}\), Theta, and SBA represent conventional forecasting approaches. The results of Table 3 suggest that, in both data sets, the meta-learner outperforms the rest of the forecasting approaches when used for applying multiple temporal aggregation. CMTA-PW is slightly more accurate than CMTA-PSTR, but the differences between the two combination methods are minor. Despite the uncertainty involved in the selection process, CTA also results in slightly more accurate forecasts than CON, being at the same time better than MTA-EW in the M4 data set. Moreover, we observe that the proposed meta-learners always outperform the Comb/ \(\text{Com}_{\text{b}}\) and Theta/SBA methods. This finding is encouraging, highlighting the potential benefits of the proposed method over traditional approaches where temporal aggregation is either neglected or implemented without investigating the contribution of each temporal aggregation level to the final forecasts.