## 2 Materials and Methods
### Calculation of Benchmark Values and Reasonable Intervals  
According to the data quality analysis, the original datasets contain a large quantile of outliers, which are far from the rational range, causing it much difficult to obtain an accurate result. Therefore, the task of this study is to utilize robust learning strategy and achieve a stable regression result away from outliers, as shown in Figure 4.  
Commonly, there is a robust learning solution that one can set thresholds manually and delete outliers from the dataset according to those thresholds, where the rest of the dataset can be applied to train a machine learning model. However, it still remains a problem how to decide precise thresholds. Besides, computed bounds of the reasonable interval from the learning model may be close to the manual thresholds, which breaks the distribution of the original dataset and makes it meaningless to train a probabilistic learning model. In this case, the calculation method based on RNN is proposed, as shown in Figure 5, which consists of the following steps:  
1. Build a RNN. In order to fully expand its robustness, DAE, multi-path architecture, L2 regularization, dropout layer and Huber loss function are applied. It is noted that a RNN possesses ten output nodes, where each node is connected to one layer with a dissimilar dropout rate (from 0.05 to 0.50).
2. Calculate an average value according to the ten different outputs, which is the final benchmark value of line loss rate, as: \[\overline{y}_{i}=\frac{1}{10}{\sum_{n=1}^{10}y_{i}^{n}},\] (7) where \(\overline{y}_{i}\) is the \(i\)th benchmark value, and \(y_{i}^{n}\) is the \(n\)th output of the \(i\)th line loss rate.
3. Operate error analysis to acquire a reasonable interval. Not only the absolute error between the benchmark value and the actual line loss rates is computed, the variance of different outputs is calculated as well. According to the interval result, data points that are not involved between the bounds of the interval are considered as outliers. The operation can be described under the following equations: \[e_{1}=\sqrt{\frac{1}{n_{s}}{\sum_{i=1}^{n_{s}}\left(\overline{y}_{i}-y_{i}^{n} \right)^{2}}},\] (8) \[e_{2}=\sqrt{\frac{1}{10-1}{\sum_{i=1}^{10}}{\left(\overline{y}_{i}-y_{i}^{n} \right)^{2}}},\] (9) \[\left\{\begin{array}{l}l_{i}=\overline{y}_{i}-e_{1}-e_{2}\\ u_{i}=\overline{y}_{i}+e_{1}+e_{2}\end{array}\right.,\] (10) where \(e_{1}\) and \(e_{2}\) are the results of error analysis. \(e_{1}\) is a constant and \(e_{2}\) changes according to the number \(i\). \(n_{s}\) is the number of training samples; \(y_{i}^{n}\) is the \(i\)th actual line loss rate; \(l_{i}\) and \(u_{i}\) are the lower and upper bounds of the reasonable interval, respectively. Furthermore, as outliers exist in the actual line loss rate values, which may influence the result of \(e_{1}\), a two-tailed test is utilized to  
Figure 4: Conventional learning methods may be easily affected by outliers. (**a**) Common condition; (**b**) affected by outliers.  
eliminate the possibly abnormal \(y_{i}^{\star}\) values that are smaller than 0.7 percentile and bigger than 99.3 percentile, as shown in Figure 6: