## 3 The Proposed Forecast Combination Methodology  
In the rest of the paper, we explored a new advanced forecasting approach by introducing a different split of dataset in the case of daily, weekly, or monthly forecasting, as well as a combination of forecasts from multiple structurally different models, like ANN and FCM with various efficient learning algorithms and hybrid configurations of them. Also, the two most popular and usually used ensemble methods, the AVG and the EB methods, were applied to the ensemble forecasts to improve the prediction accuracy. In the described ensemble scheme, the selection of the appropriate validation set, i.e., the selection of the parameter \(N_{rd}\) and the group size \(N_{tr}\), is very important. The validation set should reflect the characteristics of the testing dataset that is practically unknown in advance. As such, in this study, we set the following process of data split. The data split takes place by removing 15% of the total dataset \(N\) and saving for later use as testing data. The remaining 85% of the dataset is then split again into an 82/18 ratio, resulting in the following portions: 70% for training and 15% for validation. Also, the group size \(N_{tr}\) (i.e., the training data) should be appropriately selected so that it is neither too small nor too large. Due to the problem nature, as we work with time-series data, the most efficient method for resampling is the boosting/bootstrapping method [85]. In boosting, resampling is strategically geared to provide the most informative training data for each consecutive predictor. Therefore, in this study, an appropriate bootstrapping method was applied, so that the training dataset should have the same size at each resampling set, and the validation and testing sets should keep the same size (after excluding the \(k\)-values from the in-sample dataset). The proposed effective forecast combination methodology for time series forecasting, presented in the paper, includes three main processing steps: data pre-processing to handle missing values, normalize the collected time-series data, and split the dataset; the various forecasting methods of ANNs, RCGA-FCMs, SOGA-FCMs, and hybrid SOGA FCM-ANN with their ensembles; and evaluation of the prediction results, implementing the two most popular and used ensemble methods of simple average (AVG) and error-based (EB). Figure 1 visually illustrates the suggested methodology. In the followed approach, data preprocessing included outlier detection and removal, handling missing data, and data normalization, all of which were in accordance with the principles of Data Science practices described in corresponding literature. For outlier detection, the Z-score was first calculated for each sample on the data set (using the standard deviation value that is presented in the descriptive statistics Tables A1 and A2 in Appendix A). Then, a threshold was specified, and the data points that lied beyond this threshold were classified as outliers and were removed. Mean imputation was performed to handle missing values. Specifically, for numerical features, missing values were replaced by the mean feature value. Each dataset was normalized to [0,1] before the forecasting models were applied. The normalized datasets were taking again their original values, while the testing phase was implemented. The data normalizations were carried out mathematically, as follows:  
\[y_{i}^{(new)}=\frac{y_{i}-y^{(min)}}{y^{(max)}-y^{(min)}},\forall i=1,\;2,\;\ldots,\;N \tag{18}\]  
where \(\mathrm{Y}=\left[y_{1},\;y_{2},y_{3},\;\ldots,y_{\mathrm{N}_{min}}\right]^{ \mathrm{T}}\) is the training dataset, and \(\mathrm{Y}^{(new)}=\left[y_{1}^{(new)},\;y_{1}^{(new)},\;\ldots,y_{N}^{(new)}\right] ^{\mathrm{T}}\) is the normalized dataset, \(y^{(min)}\) and \(y^{(max)}\) are, respectively, the minimum and maximum values of the training dataset \(\mathrm{Y}\).