## 6 Results
### Adjustment  
A total of 190 models were analyzed, varying the parameters p, q, P, and Q. Of the 20 ARIMA models that were processed, the most significant and with the most accurate predictions were chosen. Some of these models could not be analyzed because their calculation iterations that were carried out sometimes generated values classified as infinite. In the analyzed SARIMA models, those with greater precision in behavior and prediction were taken with respect to the original series, for a total of 32 more representative models. The GRETL software calculated the AIC index when generating each model. Table 3 summarizes the main results obtained for the different models. Models with a lower AIC presented a better goodness of fit and, therefore, a better representation of the data. The ARIMA model that presented the best fit by AIC was the ARIMA (1,1,1), whose mathematical description is reflected in Equation (6). It should be noted that these equations follow the notation described in the theoretical framework:  
\[(1-0.465L)(1-L)(Y_{t})=(1-0.446L)a_{t}. \tag{6}\]  
The prediction made by GRETL for this model is shown in blue in Figure 7 together with the original series (in red). \begin{table}
\begin{tabular}{c c} \hline \hline
**Modelo** & **AIC** \\ \hline ARIMA(1,1,1) & 2688.745 \\ \hline ARIMA(1,1,31) & 2737.489 \\ \hline ARIMA(1,1,32) & 2696.750 \\ \hline ARIMA(2,1,24) & 2719.217 \\ \hline ARIMA(2,1,31) & 2722.380 \\ \hline ARIMA(2,1,32) & 2692.427 \\ \hline ARIMA(4,1,31) & 2774.920 \\ \hline ARIMA(4,1,32) & 2744.881 \\ \hline ARIMA(31,1,24) & 2770.154 \\ \hline ARIMA(31,1,31) & 2776.326 \\ \hline ARIMA(31,1,32) & 2742.363 \\ \hline SARIMA(1,1,1)(0,1,2)(32) & 2291.074 \\ \hline SARIMA(1,1,1)(1,1,2)(32) & 2330.456 \\ \hline SARIMA(1,1,32)(0,1,2)(32) & 2326.070 \\ \hline SARIMA(2,1,32)(0,1,2)(32) & 2326.440 \\ \hline SARIMA(2,1,32)(1,1,2)(32) & 2325.494 \\ \hline SARIMA(32,1,32)(1,1,2)(32) & 2340.973 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: AIC (Akaike information criteria) coefficient for various models. The adjustment was made on a part of the original series (first 100 samples) considering that the remaining part was used to make the prediction. On the other hand, the SARIMA model with the best AIC was the SARIMA (1,1,1) (0,1,2) (32), which is described by Equation (7). \[(1-0.845L)\Big{(}1-L^{32}\Big{)}(1-L)(Y_{t})=(1\ -0.845L)\Big{(}1\ -0.09L-0.904L^{2} \Big{)}\Big{(}1\ -0.99L^{32}\Big{)}u_{t} \tag{7}\]  
Figure 8 shows the prediction (in blue) alongside the original series (in red). Although graphically the two models show similarities, mathematically the SARIMA model presented a better fit, since its AIC was 2291.074 compared to the value of 2688.427 obtained for the ARIMA model. In general, all the SARIMA models presented better AIC indices, with an average value of 2329.9 compared to the ARIMA models, which presented an average AIC of 2695.5. In the next stage, a prediction was generated from a data set and the ability of the models to estimate the actual data was compared. As mentioned above, the models were adjusted on the first 146 samples, after which a prediction of the next 54 samples was made. Table 4 shows some of the models and the mean square error that presented their prediction against the actual values of the original series. Figure 8: Prediction of the SARIMA model (1,1,1) (0,1,2) (32). Figure 7: Prediction of the ARIMA model (1,1,1). As can be seen, the predictions made with SARIMA models presented lower mean square errors than those obtained with higher-order ARIMA. This translates into better predictions of the real data, since the estimated values are, on average, closer to the original series. Equation (8) describes the ARIMA model (2,1,32), which presented the best RMSE (173.62) among the ARIMA models. \[\Big{(}1-0.597L-0.281L^{2}\Big{)}(1-L)(Y_{t})=\Big{(}1-0.484L+0.417L^{32}\Big{)}a _{t} \tag{8}\]  
The prediction made by this model is shown in Figure 9 (in blue) together with the original series (in red).