# Predictive Modeling of Photovoltaic Energy Yield Using an ARIMA Approach
## 3 Results
### Specific PV System Yield
\begin{table}
\begin{tabular}{l c c c c c c c c c c} \hline \hline  & & & & & & & & & & & \\ \hline \hline \multirow{2}{*}{**Fit Statistic**} & \multirow{2}{*}{**Mean**} & \multirow{2}{*}{**SE**} & \multirow{2}{*}{**Minimum**} & \multirow{2}{*}{**Maximum**} & \multicolumn{6}{c}{**Percentle**} \\ \cline{5-10}  & & & & & & **5** & **10** & **25** & **50** & **75** & **90** & **95** \\ \hline
**Stationary** & & & & & & & & & & & \\
**R-Squared** & 0.802 & - & 0.802 & 0.802 & 0.802 & 0.802 & 0.802 & 0.802 & 0.802 & 0.802 & 0.802 \\
**RMSE** & 18.929 & - & 18.929 & 18.929 & 18.929 & 18.929 & 18.929 & 18.929 & 18.929 & 18.929 \\
**MAPE** & 18.591 & - & 18.591 & 18.591 & 18.591 & 18.591 & 18.591 & 18.591 & 18.591 & 18.591 & 18.591 \\
**MaxAPE** & 107.196 & - & 107.196 & 107.196 & 107.196 & 107.196 & 107.196 & 107.196 & 107.196 & 107.196 \\
**MAE** & 14.703 & - & 14.703 & 14.703 & 14.703 & 14.703 & 14.703 & 14.703 & 14.703 & 14.703 \\
**MaxAE** & 45.534 & - & 45.534 & 45.534 & 45.534 & 45.534 & 45.534 & 45.534 & 45.534 & 45.534 \\
**Normalized BIC** & 6.039 & - & 6.039 & 6.039 & 6.039 & 6.039 & 6.039 & 6.039 & 6.039 & 6.039 & 6.039 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Numerical characteristics of ARIMA (2,0,1) model: specific PV system yield. Figure 6: The values and 95% level confidence interval of ACF and PACF errors of the ARIMA (2,0,1) model. Specific PV system yield: (**a**) residual ACF; (**b**) residual PACF. Table 6 (analogously to Table 5) shows the main numerical characteristics of the ARIMA (3,0,4) model for the specific PV system yield time series. The same estimates of the model are given. The coefficient of determination \(R^{2}\) is 85.3%, which is higher than the same metric of the ARIMA (2,0,1) model, and the value of the Normalized BIC is 5.93, which is lower than its counterpart. These results suggest that the second model achieves a better fit and thus we expect better forecasts. Figure 8 (analogously to Figure 6) presents the values and the confidence interval of ACF and PACF errors under the ARIMA (3,0,4) model for the specific PV system yield time series. The number of the ACF errors and the PACF errors outside the confidence interval is less than those for the ARIMA (2,0,1) model and they are less than \(p\) and \(q\), respectively. This indicates a good model, and trying to make it better wo \begin{table}
\begin{tabular}{l c c c c c c c c c c} \hline \hline  & & \multicolumn{10}{c}{**Model Fit**} \\ \hline \multirow{2}{*}{**Fit Statistic**} & \multirow{2}{*}{**Mean**} & \multirow{2}{*}{**SE**} & \multirow{2}{*}{**Minimum**} & \multirow{2}{*}{**Maximum**} & \multicolumn{10}{c}{**Percentile**} \\ \cline{5-10}  & & & & & **5** & **10** & **25** & **50** & **75** & **90** & **95** \\ \hline
**Stationary** & & & & & & & & & & & \\
**R-Squared** & 0.853 &. & 0.853 & 0.853 & 0.853 & 0.853 & 0.853 & 0.853 & 0.853 & 0.853 & 0.853 \\
**RMSE** & 16.568 &. & 16.568 & 16.568 & 16.568 & 16.568 & 16.568 & 16.568 & 16.568 & 16.568 & 16.568 \\
**MAPE** & 15.957 &. & 15.957 & 15.957 & 15.957 & 15.957 & 15.957 & 15.957 & 15.957 & 15.957 & 15.957 \\
**MaxAPE** & 97.457 &. & 97.457 & 97.457 & 97.457 & 97.457 & 97.457 & 97.457 & 97.457 & 97.457 & 97.457 \\
**MAE** & 12.657 &. & 12.657 & 12.657 & 12.657 & 12.657 & 12.657 & 12.657 & 12.657 & 12.657 & 12.657 \\
**MaxAE** & 44.682 &. & 44.682 & 44.682 & 44.682 & 44.682 & 44.682 & 44.682 & 44.682 & 44.682 & 44.682 \\
**Normalized BIC** & 5.930 &. & 5.930 & 5.930 & 5.930 & 5.930 & 5.930 & 5.930 & 5.930 & 5.930 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Numerical characteristics of ARIMA (3,0,4) model: specific PV system yield. Figure 7: An analysis of real data, approximated values, and predicted values, along with their confidence intervals for the ARIMA (2,0,1) model: specific PV system yield. Figure 9 (analogous to Figure 7) shows the real data for the specific PV system yield time series in red, the approximated values by the ARIMA (2,0,1) model in blue, the predicted values in bold blue, and their confidence intervals with the dashed line in purple.