## Materials and Methods
### Multi-Objective Prediction Method Based on the Conditional Copula Function
#### 2.2.2 Interval Prediction Evaluation Index  
We adopt PICP and PIAW as the evaluation indices to validate the prediction accuracy of the proposed algorithm.  
(1) PICP  
Prediction interval coverage probability is the most important index to measure the quality of the predicted interval. It defines the probability that the actual observation value falls within the prediction interval which is enveloped by the upper and lower bounds. A larger PICP value indicates that more observation values are covered by the constructed prediction interval, and vice versa. Under ideal conditions, PICP = 100% and all observation values are within the prediction interval. The expected observation generally falls within the constructed prediction interval at a probability \(k\) above the preset confidence level, that is:  
\[k=\text{P}(V_{u}\in\big{[}\underline{V_{u}},\overline{V_{u}}\big{]}), \tag{9}\]  
where \(\text{P}()\) is the probability; \(\overline{V}_{u}\) and \(\underline{V}_{u}\) are the upper and lower bounds of the prediction interval. \(V_{u}\) is the corresponding observational value and \(\beta\) is the preset confidence level. According to [PERSON]'s law of large numbers, interval coverage can be expressed directly by the frequency of the predicted interval coverage observation value, which converges to \(k\) according to a certain probability.  
\[\text{PICP}=\frac{1}{U}\sum\limits_{u=1}^{U}A_{u}\times 100\%\ \ \rightarrow\ \ \text{P}(V_{u}\in\big{[}\underline{V_{u}},\overline{V_{u}}\big{]}), \tag{10}\]where \(U\) is the total amount of load to be predicted, \(u\) = 1,2... \(U\). \(A_{u}\) is a Boolean quantity that is defined as follows:  
\[A_{u}=\left\{\begin{array}{ll}1,&V_{u}\in\big{[}\frac{V_{u},\overline{V_{u}}}{ V_{u}}\big{]}\\ 0,&V_{u}\
otin\big{[}\frac{V_{u}}{V_{u}},\overline{V_{u}}\big{]}\end{array} \right\}. \tag{11}\]  
When the actual value of load to be predicted falls into the prediction interval, the value of \(A_{u}\) is 1. Otherwise, \(A_{u}\) is 0.  
To construct an effective prediction interval, the PICP should be as high as possible compared to the rated confidence level \(\beta\); if the PICP is much smaller than \(\beta\), the prediction interval is treated as an invalid interval and needs to be reconstructed.  
(2) PIAW  
PIAW is an important reference for evaluating the quality of the predicted interval. If the interval is sufficiently wide, it can easily satisfy the ideal coverage of PICP = 100%. However, such an interval is too conservative to give effective information about the uncertainty of the value to be predicted--this makes the interval prediction altogether impractical. It is necessary to measure the interval width in order to reasonably evaluate the prediction interval. The PIAW index is defined as follows:  
\[\text{PIAW}=\frac{1}{U}\sum_{u=1}^{U}\big{(}\overline{V_{u}}-\underline{V_{u }}\big{)}. \tag{12}\]  
The index measures the PIAW of each point to be predicted. The prediction interval width is positively correlated with the observed value; the interval width can be evaluated very accurately by PIAW. Under extreme conditions, if the upper and lower bounds of the prediction interval are the same for N points to be predicted, the width of the predicted interval is zero. The interval prediction is reduced to a point prediction and the evaluation index of the prediction interval loses its practical significance. Here, we strictly distinguish point prediction from interval prediction and ensure that the prediction interval satisfies PIAW \(\
eq\) 0.