# Predictive Modeling of Photovoltaic Energy Yield Using an ARIMA Approach
## 3 Results
### Specific PV System Yield  
Tables 3 and 4 present the values of the autocorrelations (ACFs), the partial autocorrelations (PACFs), and their standard deviations of the specific PV system yield series. The statistics show that all quantities are statistically significant. The values of autocorrelations and partial autocorrelations are shown in Figure 5.  
The values of autocorrelations and partial autocorrelations as well as their confidence intervals are useful for constructing a better ARIMA model. In ARIMA \((p,d,q)\) models, the model parameters \(p,\ d\), and \(q\) are chosen by the researcher. These values can account for the number of lags for which errors are outside the confidence interval of the ACF and PACF errors. In the program product IBM SPSS Statistics, there is an option for the system to propose an ARIMA model. Thus, the first model proposed is the one proposed by the system itself: in this case, it is ARIMA (2,0,1). Table 5 shows the main numerical characteristics of the ARIMA (2,0,1) model, obtaining 122 data samples overall from the specific PV system yield series. Various estimates of the model are given (coefficient of determination, RMSE (Root Mean Square Error), etc.). The coefficient of determination \(R^{2}\) equals 80.2%, which is a good value. Another one of \begin{table}
\begin{tabular}{c c c} \hline \hline
**Lag** & **Partial Autocorrelation** & **Std. Error** \\ \hline
1 & 0.789 & 0.091 \\
2 & \(-\)0.493 & 0.091 \\
3 & \(-\)0.441 & 0.091 \\
4 & \(-\)0.299 & 0.091 \\
5 & \(-\)0.149 & 0.091 \\
6 & \(-\)0.174 & 0.091 \\
7 & \(-\)0.151 & 0.091 \\
8 & 0.080 & 0.091 \\
9 & 0.169 & 0.091 \\
10 & 0.257 & 0.091 \\
11 & 0.239 & 0.091 \\
12 & 0.067 & 0.091 \\
13 & \(-\)0.013 & 0.091 \\
14 & 0.000 & 0.091 \\
15 & \(-\)0.076 & 0.091 \\
16 & 0.202 & 0.091 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Partial autocorrelations. Series: specific PV system yield. Figure 5: The values of the autocorrelations, the partial autocorrelations, and their confidence intervals. Specific PV system 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. the most important indicators is the information criterion of Bayes-Normalized BIC. Its value is 6.039. Through it, different proposed models are compared. Figure 6 presents the values and the confidence interval of ACF and PACF errors under the ARIMA (2,0,1) model for the specific PV system yield time series. Based on this figure, there are three ACF errors and four PACF errors outside the confidence interval for 16 lags. Therefore, a model with a parameter value of at least \(p=3\) and \(q=4\) could be considered for improvement. 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. We can see that the forecast respects the seasonality, but the magnitude of the range must be higher than suggested. Nevertheless, we opt for a better model.