## 2 Developed Methodology
### Computed Error Metrics
#### 2.3.2 PIF Error Metrics  
From previous published studies [30; 31; 33; 37; 39], it is known that the PIF's error metrics fall into two categories: reliability or sharpness indexes. Reliability error metrics, such as prediction interval coverage percentage (PICP), examines how many of the actual values fall into the predicted intervals, whereas the sharpness error metrics such as skill score (SS) analyze the predicted intervals' width. In this study, we calculated four error metrics, a couple of each category. Before describing the computed error metrics, it is necessary to define the concept of prediction interval bound (PIB), which is composed of the upper (UB) and lower bounds (LB). Therefore, the PIB for a chosen error rate \(\alpha\) and sample \(i\) is mathematically defined as \(\text{PIB}_{i}^{\alpha}=(\text{LB}_{i}^{\alpha};\text{UB}_{i}^{\alpha})\) pair. From Equation (14), \(\text{LB}_{i}^{\alpha}\), and \(\text{UB}_{i}^{\alpha}\) are expressed as below,  
\[\text{LB}_{i}^{\alpha}=\ \hat{y}_{i}-\ t_{F-L}^{\alpha/2}s\sqrt{1+\big{(}D\big{(} \hat{\beta}\big{)}^{T}\big{[}\mathcal{J}\big{(}\hat{\beta}\big{)}^{T}\mathcal{ J}\big{(}\hat{\beta}\big{)}\big{]}^{-1}D\big{(}\hat{\beta}\big{)}}, \tag{17}\]  
\[\text{UB}_{i}^{\alpha}=\ \hat{y}_{i}+\ t_{F-L}^{\alpha/2}s\sqrt{1+\big{(}D\big{(} \hat{\beta}\big{)}^{T}\big{[}\mathcal{J}\big{(}\hat{\beta}\big{)}^{T}\mathcal{ J}\big{(}\hat{\beta}\big{)}\big{]}^{-1}D\big{(}\hat{\beta}\big{)}}. \tag{18}\]  
The most commonly computed and best-known PIF error metric is the PICP index. For a given set of samples with their respective PIBs, the PICP index analyzes and calculatesas a percentage the number of samples that fall into their respective PIBs. This index is mathematically expressed as,  
\[\text{PICP}_{\alpha}=\frac{1}{N}\ \sum_{i=1}^{N}\gamma_{i}^{\alpha}\ \ \gamma_{i}^{ \alpha}=\left\{ \begin{array}{cc}1,&y_{i}\ \in\ \text{PIB}_{i}^{\alpha}\\ 0,&y_{i}\ \
otin\ \text{PIB}_{i}^{\alpha}\end{array} \right. \tag{19}\]  
In addition, as is supported by literature [55; 56], a proposed model is considered acceptable when the obtained \(\text{PICP}_{\alpha}\) is equal to or greater than the confidence level (CL) chosen by the user \(\text{CL}_{\alpha}=1-\alpha\). Therefore, the higher the \(\text{PICP}_{\alpha}\), the better the model's reliability. Moreover, if this condition is not fulfilled, the proposed model must be reanalyzed until this condition is met. Once the required condition is met, other error metrics can be computed, such as the coverage error (CE), whose value is computed through  
\[\text{CE}_{\alpha}=\text{PICP}_{\alpha}-\text{CL}_{\alpha} \tag{20}\]  
Based on the fact that \(\text{PICP}_{\alpha}\geq\text{CL}_{\alpha}\), \(\text{CE}_{\alpha}\) must always be at least equal to or greater than zero. The higher the \(\text{CE}_{\alpha}\), the better model's reliability. Moreover, as noted in the literature, it is also necessary to examine the sharpness of the predicted intervals [33; 56], i.e., the interval width. In order to understand the computed sharpness error metrics, it is necessary to define the mathematical concept prediction interval width (PIW), for a random sample \(i\) and chosen \(\alpha\) error rate:  
\[\text{PIW}_{i}^{\alpha}=\ \text{UB}_{i}^{\alpha}-\ \text{LB}_{i}^{\alpha} \tag{21}\]  
With regard to the interval sharpness error metrics, a skill score (SS) was computed to quantify predicted interval width for a set of \(N\) samples. SS is mathematically expressed as,  
\[\text{SS}_{\alpha}=\frac{1}{N}\sum_{i=1}^{N}\lvert\gamma_{i}^{\alpha}-(1- \alpha)\lvert\text{max}(\lvert\text{LB}_{i}^{\alpha}-y_{i}\rvert,\lvert y_{t }-\text{UB}_{i}^{\alpha}\rvert) \tag{22}\]  
\(\text{SS}_{\alpha}\) must always be positive, and the closer this value is to zero, the narrower the computed intervals are. Often \(\text{SS}_{\alpha}\) is normalized by parameter \(P\) in order to facilitate comparison with other studies. \(P\)'s order of magnitude is related to the forecasted variable.