*4.2. Case Study Results*

In this study, we applied both the AVG and the EB method in two di fferent cases: case (A) where scores were calculated for individual forecaster of each one of the methods ANN, hybrid, RCGA-FCM, and SOGA-FCM, and case (B), where scores were calculated for each ensemble forecaster (ANN ensemble, hybrid ensemble, RCGA-FCM ensemble, and SOGA-FCM ensemble).

Considering case (A), Table 1 shows the calculated errors and scores based on the EB method for individual forecaster of the two forecasting methods: ANN and hybrid for the city of Athens. The rest calculated errors and scores, based on the EB method, for individual forecaster for the other two remaining forecasting methods RCGA-FCM and SOGA-FCM for Athens can be found in Appendix A of the paper (Table A3). In Appendix A, parts of the corresponding results for the other two examined cities (Larissa and Thessaloniki) are also presented (Tables A4 and A5).


**Table 1.** Case (A)-Calculated errors and weights for each ensemble forecaster based on scores for EB (error-based) method (Athens).

MSE: Mean Square Error, MAE: Mean Absolute Error.

Considering case (B), Table 2 presents the calculated weights based on scores for each ensemble forecaster (ANN (ensemble, hybrid ensemble, RCGA ensemble, and SOGA ensemble) for all three cities.

The calculated weights, based on scores for the EB method, were computed using Equation (17). According to this equation, the weights of the component forecasts are inversely proportional to their in-sample forecasting errors, concluding that the model with more error is assigned less weight to it and vice versa [80]. In this work, as the values of errors were high for certain ensemble forecasters, the corresponding weights were approximately zero, so they were considered to have a zero value for further predictions.

**Table 2.** Case (B)-Calculated weights for each ensemble forecaster based on scores for the EB method.


ANN: Artificial Neural Network, RCGA-FCM: Real Codded Genetic Algorithm-Fuzzy Cognitive Map, SOGA-FCM: Structure Optimization Genetic Algorithm-Fuzzy Cognitive Map.

The obtained forecasting results of the individual and combination methods are depicted in Tables 3–8, respectively, for the three cities. In each of these tables, the best results (i.e., those associated with the least values of error measures) are presented in bold letters. In Figures 4 and 5, the forecasting results concerning Thessaloniki and Larissa are visually illustrated for both ensemble methods (AVG, EB). Moreover, Figure 6 gathers the forecasting results for all three cities considering the best ensemble method.

**Table 3.** Calculated errors for individual forecaster based on scores for Athens.



**Table 5.** Calculated errors for individual forecaster based on scores for Thessaloniki.



**Table 6.** Calculated errors for each ensemble forecaster based on scores for Thessaloniki.

**Table 7.** Calculated errors for individual forecaster based on scores for Larissa.


**Table 8.** Case (B) -Calculated errors for each ensemble forecaster based on scores for Larissa.


(a) 

(b) **Figure 4.** Forecasting results for Thessaloniki considering the two ensemble methods (AVG, EB) based on scores. (**a**) Validation, (**b**) Testing. AVG, simple average; EB, error-based.

(**a**)

**Figure 5.** Forecasting results for Larissa considering the two ensemble methods (AVG, EB) based on scores. (**a**) Validation, (**b**) Testing.

**Figure 6.** *Cont.*

**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.
