# Probabilistic Forecasting of Electricity Demand Incorporating Mobility Data
## 3 Results
### Probabilistic Density Prediction  
From the analysis of the previous sections, Predictive intervals provide a lower and upper bound between which the actual future value will fall within a certain probability (confidence interval). Finally, probabilistic forecasts can be provided as predictive probability density functions, which provide a full estimation of the probability distribution around a point forecast. The lower and upper bounds of the PI in Section 2.1 are used to estimate the KDE as a range of possible values and create a histogram with the values falling within this range. Using KDE, the underlying density function can be estimated after the histogram is obtained [8]. To estimate the underlying density, KDE includes integrating the histogram with the Gaussian kernel function. This study uses the CRPS to quantify the calibration and sharpness for density forecast evaluation. Table 7 compares the CRPS values with different kernel functions of GPR, and QR probabilistic forecasting techniques are also tested for comparison. Figure 6 provides the complete electricity demand probability density curves. To make the comparisons, the actual electricity demand values are also presented with the point forecasting value of different baseline methods of three scenarios. To provide the proposed method's potential to incorporate uncertainty, the prediction results from Figure 5a-c are illustrated in Figure 6 (density curve). From the prediction interval (Figure 5), the highest actual electricity demand values from three scenarios are day 7 in 2020, day 18 in 2021, and day 13 in 2022. Based on Figure 6, most of the actual values distribute in the middle of probability density curves. This indicates these values are anticipated in the forecasting distributions with high probability. Moreover, Figure 6 shows that the baseline point forecasting approaches do not yield reliable results and cannot capture the uncertainty caused by unprecedented events, such as the COVID-19 pandemic. It is also evident that the observation lines in scenario 1 are closer to the center of the PDF while others (scenarios 2 and 3) are slightly distant from the center, which indicates that the probabilistic forecasting is reliable. Concurrently, the other possible explanation is that many variables influence consumption patterns and become critical to EDF with precision. Hence, the suggested technique adopts a conservative method to reach conservative decisions under varying impacts. Figure 7 shows the histograms and the PDFs based on the Gaussian kernel density estimation (KDE) (red line) functions of residuals between the actual and the predicted electricity demand under three scenarios. It can be seen that the estimated KDE functions well matched the histograms of the EDF residuals in three scenarios, approximately obeying normal distribution. As shown in Figure 7, the forecast error values (residuals) for all testing sets are normally distributed around the zero value. The forecast error histogram plots showed a high number of cases centered on zero error, indicating that the suggested model achieved excellent performance with less bias. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Scenario** & **Covariance Function** & **CRPS** \\ \hline \multirow{4}{*}{Scenario 1} & GPR (Exponential) & 21.35 \\  & GPR (Squared expo.) & 43.23 \\
2020 & GPR (Matern 3/2) & 27.48 \\  & QR & 44.74 \\ \hline \multirow{4}{*}{Scenario 2} & GPR (Exponential) & 36.80 \\  & GPR (Squared expo.) & 38.76 \\
2021 & GPR (Matern 3/2) & 30.26 \\  & QR & 39.85 \\ \hline \multirow{4}{*}{Scenario 3} & GPR (Exponential) & 22.60 \\  & GPR (Squared expo.) & 69.55 \\
2022 & GPR (Matern 3/2) & 53.21 \\  & QR & 70.78 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: This Probabilistic density prediction results from three scenarios. Figure 6: Probabilistic density curves based on GPR-KDE and point forecasting results of different methods for three scenarios.