## 2 Problem Outlines
### The Objective Function
#### 2.3.9 Load Forecast Uncertainty
The value of \(v\) is set as 5/2 or 3/2 in most previously conducted research studies, which subsequently and, respectively, are named Mat5 and Mat3. The probabilistic prediction model also has another covariance function named the period covariance, which is useful to model periodic phenomena and can be described as follows:  
\[COV_{Mat}\big{(}l,l^{\prime}\big{)}=\theta_{f}^{2}\exp\left(-\frac{2}{\theta_ {length}^{2}}\sin^{2}\bigg{(}\pi\frac{l-l^{\prime}}{p}\bigg{)}\right) \tag{28}\]  
Typically, the load forecasts are highly influenced by a variety of features that can be defined by Equation (29). According to this equation, load at time \(t\) depends on the similar hours (\(t\in\{0,24\}\)) throughout the historical records, the day of the year (\(d\in\{1,\ldots,365\}\)), the value of the load at similar intervals, the value of weather variables such as temperature, and the price at similar intervals. \[\hat{y}=f(t,d,v_{t},v_{w},\textit{price}) \tag{29}\]  
GPQR method seeks for the relationships and correlation between inputs and output based on a probabilistic framework. Quantile regression (QR) delineates a type of regression analysis that detects and exploits the relationships between quantiles of the conditional distribution of a response variable and input variables. The least absolute deviationsregression integrates the median of the conditional distribution, which is called the norm regression case of quantile regression shown by \(L_{1}\). Unlike least-squares regression and norm regression, the quantile regression encompasses minimization of the summation of asymmetrically weighted absolute residuals. Hence, QR can find more sophisticated relationships between inputs and outputs with better precision. QR has more flexibility and compatibility to deal with large datasets, such as market analysis or econometrics. With regard to the accuracy of predictions, the loss function can be described as follows:  
\[L_{\tau}(\varepsilon_{i})=\left\{ \begin{array}{l l l}\tau\varepsilon_{i}&if& \varepsilon_{i}\geq 0\\ (\tau-1)\varepsilon_{i}&if&\varepsilon_{i}\geq 0\end{array} \right. \tag{30}\]  
The required quantile is defined by \(\tau\in\)[0, 1] and \(\varepsilon_{i}=y_{i}-\hat{y}_{i}\) so that \(y_{i}\) is the actual model and \(\hat{y}_{i}\) denotes the predicted quantile model. So far, different linear programming methods are employed to achieve the desired quantiles through direct loss function minimization, which leads to the maximization of a likelihood. To solve this drawback, the Gaussian process is incorporated into the QR model. The density function of loss based on this model can be represented as follows:  
\[L(t|\mu,\sigma,\tau)=\frac{\tau(1-\tau)}{\sigma}\exp\left[-\frac{t-\mu}{ \sigma}(t-I(t\leq\mu))\right] \tag{31}\]  
where \(\tau\in\)[0, 1] is responsible for shaping and controlling the skewness of the distribution curve, \(\mu\) stands for the mean value, and \(\sigma\) denotes the standard deviation, which should always be positive. The binary variable \(I\) takes the value of 1 when the condition is true; otherwise, it takes 0. \[U_{\tau}(y,q)=Z\exp\left[-\sum_{i=1}^{N}L_{\tau}(y_{i},q_{i})\right] \tag{32}\]  
In Equation (32), \(q\) denotes the predicted value of the \(\tau\) quantile, \(Z\) is the normalization constant. Afterward, a Gaussian process is placed on the QR function:  
\[p(q)=GP(q|0,K) \tag{33}\]  
The GPQR model training can be conducted by integral maximization. The expectation propagation algorithm can be employed to locally approximate this integral. \[\text{argmax}\int\limits_{q}U_{\tau}(y,q)p(q)\ d(q) \tag{34}\]  
Suppose a dataset of historical load records as \(l=\{x_{1},\,x_{2},\,\ldots,\,x_{N}\}\), which are independently distributed samples. The estimated shape of this distribution function can be obtained through a Gaussian kernel density estimator model.