# Probabilistic Forecasting of Electricity Demand Incorporating Mobility Data
## 1 Introduction
GPR has been used for electricity demand prediction in several studies, and it can generate better estimates than benchmark methods for time series data, such as NN and SVR [16; 28]. Moreover, when the Gaussian process (GP) is used to solve the regression issue, the accuracy of the forecast is sensitive to the choice of the covariance function. The kernel function solves the high-dimensional prediction distribution covariance matrix without extensive computing [29]. For the prediction of future electricity demand, a covariance kernel is built that incorporates daily/weekly trends and meteorological variables [27]. The effectiveness of three distinct kernels has been assessed in [16]. Decision-makers can pick kernel functions by comparing their performance across categories in time series data, which improves probability prediction accuracy [30]. The deep Gaussian process (DGP) was used for EDF in [31]. The authors of [31] used sparse GP (SGP) and double stochastic variational inference DGP with mobility data to improve computational efficiency. However, SGP captures only higher-level latent space uncertainty [32]. DGP learns deep features from variable-sized data. Hence, they scale poorly with data size [33]. In [34], Gaussian process quantile regression (GPQR) was used to quantify the uncertainty in electricity demand forecasting since GPR is inherently capable of handling the complex interactions seen in time series. In [35], applied weighted GPR was used to forecast the uncertainty of solar power, while [36] used GPR to predict the residential electricity demand rather than the system-level EDF. The researchers in [25] integrated both forecast combination and residual simulation approaches to model the forecast uncertainty. In [37], a hybrid model was proposed, which consists of kernel density estimation and Quantile regression (QR). QR has certain drawbacks; however, parameter estimation is more difficult than in Gaussian or generalized regression, which is a significant limitation [38]. In addition to that, the derived quantile curves may cross each other, resulting in an invalid response distribution [38]. This issue is mostly driven by the fact that these strategies estimate models individually for each quantile. Interval prediction approaches provide lower and upper limits of the future forecast based on a confidence level yet cannot represent the full probability distribution of electricity demand [37]. By creating a probability density function (PDF) of forecasting results, probability density forecasting helps quantify uncertainty. Density estimation methods should be applied to enable GPR to obtain the PDF of the predicted electricity demand. This study takes advantage of the Gaussian kernel density estimation (KDE) approach to construct a probability density forecasting method due to its excellent generalization capabilities that greatly affect the distribution of response variables [39]. Moreover, the literature reports that numerous articles incorrectly assumed the GPR approach was deterministic and failed to fully use its benefits [40; 41; 42; 43; 44; 45]. Using deterministic error metrics with the probabilistic EDF methods is one reason why they have not been developed enough. When using deterministic error metrics, the probabilistic EDF methods may have been underestimated because they did not work as well as their deterministic counterparts [7]. This paper aims to develop a comprehensive short-term probabilistic electricity demand forecasting method based on GPR-KDE that can deal with anomalous events. The contributions of this paper are summarized as follows:  
* Considering the EDF uncertainty during unprecedented events, GPR is effectively applied to deal with the regression problem with limited historical data to perform successful inductive reasoning. Incorporating mobility data as an important feature that better represents the underlying shifts in practice theory [1; 46] further improves its performance.