## 4 Results and Discussion
### Discussion of Results  
The following important observations were noticed after careful analysis of Tables and Figures above. 1. After a thorough analysis of the Tables 3-8, on the basis of examining the MAE and MSE errors, it could be clearly stated that the EB method presented lower errors concerning the individual forecasters (ANN, hybrid, RCGA-FCM, and SOGA-FCM) for all three cities (Athens, Thessaloniki, and Larisa). EB seemed to outperform the AVG method in terms of achieving overall better forecasting results when applied to individual forecasters (see Figure 6). 2. Considering the ensemble forecasters, it could be seen from the obtained results that none of the two forecast combination methods had attained consistently better accuracies compared to each other, as far as the cities of Athens and Thessaloniki were concerned. Specifically, from Tables 3-6, it was observed that the MAE and MSE values across the two combination methods were similar for the two cities; however, their errors were lower than those produced by each separate ensemble forecaster. Figure 6: Forecasting results for the three cities considering the best ensemble method. (**a**) Testing all cities, (**b**) Testing Athens, (**c**) Testing Thessaloniki, (**d**) Testing Larissa. Although the AVG and the EB methods performed similarly for Athens and Thessaloniki datasets, the EB forecast combination technique presented lower MAE and MSE errors than the AVG for the examined dataset of Larissa (see Figure 5). The fact that when a forecasting method presented lower MAE and MSE errors than another means that the accuracy of the results produced with the first method, in terms of predicting consumption, was higher than the latter forecasting methods examined and compared to, so as the overall performance of the ensemble method was. Regarding the amount of improvement that was presented when a forecasting method was applied, slightly better performance of both ensemble forecasting methods could be noticed, and that constituted strong evidence for the efficiency of the examined method in the domain of natural gas demand forecasting. In order to examine the efficiency of the proposed algorithm, a statistical test was conducted to reveal no statistical significance. Concerning the individual methods, a \(t\)-test paired two samples of mean was previously conducted in [60] for the cities of Thessaly (Larissa, Volos, Trikala, and Karditsa), for the year 2016, showing that there was no statistical significance among these techniques. In current work, a \(t\)-test paired two samples of mean was also performed, regarding the ensemble methods (average and error-based) for the examined cities (Athens, Thessaloniki, and Larissa), regarding the dataset of the same year. The results of the hypothesis tests (Tables A6-A8 in Appendix A) revealed no statistical significance between these techniques. In all cases, the calculated \(p\)-value exceeded 0.05, so no statistical significance was noticed from the obtained statistical analysis. Therefore, there was no particular need to conduct a post hoc statistical test, since a post hoc test should only be run when you have an overall statistically significant difference in group means, according to the relevant literature [89; 90]. Furthermore, for comparison purposes, to show the effectiveness of the proposed forecasting combination approach of multivariate time series, the experimental analysis was conducted with a new and well-known effective machine learning technique for time series forecasting, the LSTM (long short-term memory). LSTM algorithm encloses the characteristics of the advanced recurrent neural network methods and is mainly applied for time series prediction problems in diverse domains [91]. LSTM was applied in one day-ahead natural gas consumption prediction concerning the same dataset of the three Greek cities (Athens, Thessaloniki, and Larissa) in [70]. For the LSTM implementation, one feature of the dataset as a time series was selected.