# Predictive Modeling of Photovoltaic Energy Yield Using an ARIMA Approach
## 3 Results
### Specific PV System Yield
Now, the forecast looks very plausible--both the seasonality and the range are coherent with the historical values; thus, we stop at this model. We will comment on the forecasts and their significance in the next section. Now, we continue with the total yield. Figure 8: The values and confidence interval of ACF and PACF errors of the ARIMA (3,0,4) model. Specific PV system yield: (**a**) residual ACF; (**b**) residual PACF. Figure 9: An analysis of real data, approximated values, predicted values, and their confidence intervals of the ARIMA (3,0,4) model: specific PV system yield. As discussed earlier, IBM SPSS Statistics suggests a model. In our case, now, it is again ARIMA (2,0,1). Table 9 shows the main numerical characteristics of the ARIMA (3,0,4) model for the total yield time series, obtained over all 122 data samples. Now, we can see that the value of \(R^{2}\) is again 80.2% and the value of the Normalized BIC is a bit higher, but this does not necessarily mean that the fit is worse. We will later compare it to a model, selected by the authors. Figure 11 presents the values and the confidence interval of ACF and PACF errors under the ARIMA (2,0,1) model for the total yield time series. Based on this figure, there are three ACF errors and five PACF errors outside the confidence interval for 16 lags. As discussed earlier, a model with a parameter value of at least \(p=3\) and \(q=5\) can be considered for improvement. Figure 10: The values of the autocorrelations, the partial autocorrelations, and their confidence intervals, for total yield: (**a**) autocorrelations for the last 16 lags and their confidence intervals for ACF; (**b**) partial autocorrelations for the last 16 lags and their confidence intervals for partial ACF. \begin{table}
\begin{tabular}{l r r r r r r r r r r} \hline \hline  & & \multicolumn{10}{c}{**Model Fit**} \\ \hline
**Fit Statistic** & **Mean** & **SE** & **Minimum** & **Maximum** & \multicolumn{6}{c}{**Percentile**} \\ \cline{5-12}
**Stationary** & 0.802 & - & 0.802 & 0.802 & 0.802 & 0.802 & 0.802 & 0.802 & 0.802 & 0.802 \\
**R-Squared** & 0.802 & - & 0.802 & 0.802 & 0.802 & 0.802 & 0.802 & 0.802 & 0.802 & 0.802 \\
**RMSE** & 106.009 & - & 106.009 & 106.009 & 106.009 & 106.009 & 106.009 & 106.009 & 106.009 & 106.009 & 106.009 \\
**MAPE** & 18.591 & - & 18.591 & 18.591 & 18.591 & 18.591 & 18.591 & 18.591 & 18.591 & 18.591 & 18.591 & 18.591 \\
**MaxAPE** & 107.209 & - & 107.209 & 107.209 & 107.209 & 107.209 & 107.209 & 107.209 & 107.209 & 107.209 \\
**MAE** & 82.340 & - & 82.340 & 82.340 & 82.340 & 82.340 & 82.340 & 82.340 & 82.340 & 82.340 \\
**MaxAE** & 255.018 & - & 255.018 & 255.018 & 255.018 & 255.018 & 255.018 & 255.018 & 255.018 & 255.018 \\
**Normalized BIC** & 9.485 & - & 9.485 & 9.485 & 9.485 & 9.485 & 9.485 & 9.485 & 9.485 & 9.485 & 9.485 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Numerical characteristics of ARIMA (2,0,1) model: total yield. Figure 12 shows the real data for the total 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. Again, we can observe that the forecast respects the seasonality, but the magnitude of the range is much lower than one of the past peaks. So, we continue to look for a better model. Table 10 (analogously to Table 9) shows the main numerical characteristics of the ARIMA (3,0,5) model for the total yield time series, which is selected by the authors, based on the aforementioned considerations. The same estimates of the model are given, so the  
Figure 11: The values and confidence intervals of ACF and PACF errors of the ARIMA (2,0,1) model, for total yield: (**a**) residual ACF; (**b**) residual PACF. Figure 12: An analysis of real data, approximated values, predicted values, and their confidence intervals of the ARIMA (2,0,1) model: total yield. value of \(R^{2}\) is 85.6% (>80.2%) and the value of the Normalized BIC is 9.405 (<9.485).