## 2 Literature Review
### Bibliometric Review
Figure 12: Mean absolute percentage error (MAPE—%) for the multivariate SARIMAX model with different exogenous variables. Figure 13: Mean absolute percentage error (MAPE—%) for the multivariate FB Prophet model with different exogenous variables. For the original time series, the SARIMAX achieved its best MAPE value for a training window of 215 months and considering climatic exogenous variables, while FB Prophet achieved it for 144 months and industrial variables. For the smoothed version, SARIMAX tested the best with 180 months and economic variables, while FB Prophet with 36 months and industrial variables. Both SARIMAX and FB Prophet are compared in these optimal scenarios with different metrics in Table 6. These same scenarios are used to visually demonstrate the validation in Figure 14.

SARIMAX achieved Highly Accurate Prediction in all combinations of exogenous variables, both for the original and the smoothed time series. FB Prophet, contrarily, appears to have reacted poorly to the economic exogenous variables, achieving an Inaccurate Prediction when incorporating all variables in the smoothed version. When using exclusively industrial or climatic variables, it consistently reached Highly Accurate Predictions. ### Models Fit Test

To estimate the adequacy of these models in each scenario, the Ljung-Box test is performed. The test is performed with a maximum lag of four, considering the default parameter of \(max\_lags=floor(number\_of\_observations/5)\). Its hypotheses are as follows:

**Hypothesis 7**. : _There is no autocorrelation in the data up to a specified lag._

**Hypothesis 8**. : _There is significant autocorrelation present in the data._ \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Smoothing** & **Metric** & **SARIMAX** & **FB Prophet** \\ \hline \multirow{4}{*}{No} & MAPE (\%) & 1.28 & 1.62 \\  & MPE (\%) & 0.08 & 0.29 \\  & RMSE (MWh) & \(6.94\times 10^{5}\) & \(9.06\times 10^{5}\) \\  & NRMSE\_minmax & 0.13 & 0.17 \\ \hline \multirow{4}{*}{Yes} & MAPE (\%) & 1.18 & 0.84 \\  & MPE (\%) & \(-0.80\) & 0.10 \\ \cline{1-1}  & RMSE (MWh) & \(5.81\times 10^{5}\) & \(4.75\times 10^{5}\) \\ \cline{1-1}  & NRMSE\_minmax & 0.20 & 0.16 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Comparison of MAPE, MPE, RMSE, and NRMSE_minmax between the multivariate models in their optimal conditions, according to Figures 12 and 13. Figure 14: Comparison between the actual energy consumption and forecasting multivariate models in their optimal training window sizes, according to Figures 12 and 13. The Ljung-Box test is run on the residuals (i.e., \(y-\hat{y}\)) of each model, in their optimal window sizes, according to Section 5.2. Significant autocorrelation in the residuals suggests that the model fails to capture some foundational patterns in the data, indicating potential deficiencies in the estimation process. Its results help ensure the validity of the model's assumptions, particularly regarding the independence and randomness of the residuals. This helps improve the forecasting model's reliability. The fourth lag demonstrated the highest p-values for all models. From Table 7, it is noted that only FB Prophet and SARIMA, when applied to the original time series, were able to reject \(H_{1}\). This means that the forecasting models adequately capture the temporal dependencies present in the time series, and the residuals exhibit randomness and independence. The residuals for forecasting the smoothed version, however, present significant autocorrelation, implying a lack of fit. Holt-Winters and TBATS showed a lack of fit for both scenarios. Upon restricting the accuracy evaluation for SARIMA and FB Prophet applied to the original time series, as these are the only models that demonstrated to be fit, it is concluded that SARIMA is the most accurate fit model for the univariate forecasting.