# Probabilistic Forecasting of Electricity Demand Incorporating Mobility Data
## 3 Results
### Probabilistic Forecasting--Prediction Interval  
Predictions Interval (PI) under four confidence levels (CI) (95%, 90%, 80%, and 50%) are carried out by various kernel functions of the GPR and QR model in support of verifying the effectiveness of the proposed model. Both the PICP and MPIW of the PI for three scenarios are then calculated, and the performance comparisons are displayed in Table 6. The actual electricity demand is represented in Figure 5a-c by the black line and GPR prediction with a black dotted line. The 95%, 90%, 80%, and 50% values of PI are depicted in Figure 5. The results for RT, SVR, and BPNN are not shown since these are only point forecasting methods. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Scenario** & **Covariance Function** & \multicolumn{4}{c}{**PICP (\%)**} & \multicolumn{4}{c}{**MPIW (\%)**} \\  & & **95\%** & **90\%** & **80\%** & **50\%** & **95\%** & **90\%** & **80\%** & **50\%** \\ \hline \multirow{3}{*}{Scenario 1} & GPR (Exponential) & 98\% & 96\% & 96\% & 77\% & 30\% & 26\% & 20\% & 11\% \\  & GPR (Squared exp.) & 96\% & 96\% & 93\% & 64\% & 38\% & 32\% & 25\% & 13\% \\
2020 & GPR (Matern 3/20) & 97\% & 94\% & 93\% & 74\% & 35\% & 30\% & 23\% & 12\% \\  & QR & 95\% & 95\% & 92\% & 63\% & 39\% & 33\% & 26\% & 14\% \\ \hline \multirow{3}{*}{Scenario 2} & GPR (Exponential) & 97\% & 96\% & 87\% & 61\% & 33\% & 27\% & 21\% & 11\% \\  & GPR (Squared exp.) & 96\% & 96\% & 90\% & 64\% & 35\% & 29\% & 23\% & 12\% \\
2021 & GPR (Matern 3/20) & 96\% & 96\% & 93\% & 74\% & 34\% & 28\% & 22\% & 11\% \\  & QR & 95\% & 95\% & 89\% & 63\% & 36\% & 30\% & 24\% & 13\% \\ \hline \multirow{3}{*}{Scenario 3} & GPR (Exponential) & 96\% & 96\% & 96\% & 36\% & 37\% & 31\% & 24\% & 12\% \\  & GPR (Squared exp.) & 96\% & 93\% & 90\% & 58\% & 57\% & 37\% & 48\% & 19\% \\
2022 & GPR (Matern 3/20) & 96\% & 96\% & 90\% & 64\% & 56\% & 47\% & 37\% & 19\% \\  & QR & 95\% & 92\% & 89\% & 57\% & 58\% & 48\% & 38\% & 20\% \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Prediction Interval evaluation achieved by various kernel methods in three scenarios. Figure 5: Cont. It can be seen that the PI generated with the proposed method covers the actual value most of the time. For example, Figure 4(c) shows that the proposed method has a 95% confidence interval that can cover most of the actual electricity demands. Table 6 results show a good performance and indicates that the PICP and MPIW are exceeding the CI levels in most of the cases. For example, the PICP of scenario 1 (before the pandemic), scenario 2 and 3 (during the pandemic) are 98%, 97%, and 96%, respectively, exceeding the CI level of 95% with an MPIW of 30%, 33%, and 37%, respectively, using the exponential kernel function. In comparison, the 95% confidence level by the other two covariance functions (Squared exponential and Matern 3/2) does not cover many actual electricity demand points. However, the MPIW values by the Matern 3/2 covariance function of the GPR model is wider than that of the other two kernels in scenario 1 and 3. In addition, the PICP values of the QR model is marginally covering the preassigned PICP, and the MPIW values are also wider than that of the other GPR kernel models. The results show that the created PIs of the proposed model have a high probability of covering the target values and better captures uncertainty with sudden peak and descend values during the pandemic (scenario 2 and 3) that is hardly following any pattern. As shown in Figure 5, the actual values are mostly located in narrow PIs with 98%, 97%, and 96% PICP for scenarios 1, 2, and 3, respectively. As shown in Figure 4(b), the difference between the two peaks of some daily electricity demand curves is relatively small. For example, there is not much variance between days 15 and 16. Meanwhile, the peaks of some daily EDF curves have a large disparity; for example, day 17, 18, and 19 has a sudden peak and drop in electricity demand. The fluctuation range of the electricity demand varies daily, reflecting the uncertainty.