# Novel Data-Driven Models Applied to Short-Term Electric Load Forecasting
## 4 Results and Discussion
Figure 12: Training (blue line) and validation (green line) MSE loss evolution for successive epochs while training (model: DL architecture (1 LSTM + 1 FC) (\(f\) = 1, \(p\) = 168, \(k\) = 24)). begins to drift. The drift point is different for different samples, with some signals perfectly followed almost all the time and others beginning to separate in the initial forecasts. In all cases, the forecast signal is a smoothed version of the real one. Figures 14-17 present the ability of the DMD algorithm to identify the structure of the time-series (load values). Figure 14 shows the evolution of the signal energy (sum of squares of principal eigenvalues) in relation to the total signal energy (sum squares of all eigenvalues). We observe that to achieve a 99% total signal energy, generally a few eigenvalues are sufficient. In particular, for our dataset, 10 eigenvalues (modes) are sufficient to capture the 99% of total signal energy for a rolling window of 24 and 168 h, and 16 eigenvalues for a rolling window of 720 h.  
\[RRMSE=\frac{\sqrt{\sum_{i=0}^{N}\big{(}Y_{i}-\hat{Y}_{i}\big{)}^{2}}}{\sqrt{ \sum_{i=0}^{N}\big{(}Y_{i}\big{)}^{2}}} \tag{10}\]0 Graph showing the total energy of the signal captured by the linear DMD approximation versus the number of eigenvectors (modes) used to implement the transformation. This graph was obtained with a rolling window of 24 h, and is very similar to the graph for a rolling window of 168 h.  
\[RRMSE=\frac{\sqrt{\sum_{i=0}^{N}\big{(}Y_{i}-\hat{Y}_{i}\big{)}^{2}}}{\sqrt{ \sum_{i=0}^{N}\big{(}Y_{i}\big{)}^{2}}} \tag{10}\]1 Comparison between real 24 h load signals (blue lines) and their forecasts (orange lines) for data in the test set (model: ensemble Type II ([1 LSTM + 2 FC] * 2 + (4 FC)*2) (\(f=1\), \(p=168\), \(k=24\))). \[RRMSE=\frac{\sqrt{\sum_{i=0}^{N}\big{(}Y_{i}-\hat{Y}_{i}\big{)}^{2}}}{\sqrt{ \sum_{i=0}^{N}\big{(}Y_{i}\big{)}^{2}}} \tag{10}\]2 Principal eigenvectors (\(\Phi\)) of the DMD algorithm (best linear mapping) for the temporal evolution of the load values. The eigenvectors are in order of importance (according to their corresponding eigenvalues) and represent the main modes of temporal evolution of the time-series. They clearly show the strong daily periodicity of the load values. The values shown correspond to a rolling window of 168 h (1 week). \[RRMSE=\frac{\sqrt{\sum_{i=0}^{N}\big{(}Y_{i}-\hat{Y}_{i}\big{)}^{2}}}{\sqrt{ \sum_{i=0}^{N}\big{(}Y_{i}\big{)}^{2}}} \tag{10}\]3 Principal eigenvectors (\(\Phi\)) of the DMD algorithm (best linear mapping) for the temporal evolution of the load values. The eigenvectors are in order of importance (according to their corresponding eigenvalues) and represent the main modes of temporal evolution of the time-series. They clearly show the strong intra-day periodicity of the load values. The values shown correspond to a rolling window of 24 h (1 day). Figures 15-17 show the principal eigenvectors (\(\Phi\)) of the DMD algorithm for the temporal evolution of the load values and for different rolling window lengths (24, 168, and 720 h). The eigenvectors are in order of importance (according to their corresponding eigenvalues) and represent the main modes of temporal evolution of the time-series. They show the strong periodicity of the load values at different levels of granularity: intra-day, daily, and weekly. We implemented all the neural network models (deep learning, Seq2 seq, attention, and gaNet) in python using Tensorflow/Keras [59]. For all other models, we used the scikit-learn python package [60].