## 2 Materials and Methods
### Statistical Indexes  
The reliability of the MEC algorithm for the LST prediction has been assessed comparing the observed LST data with both forecasted and simulated values. Different skill scores commonly known in scientific literature [31; 32; 33] are here used for the performance evaluation, in particular, the Nash-Sutcliffe efficiency, Root Mean Square Error, and the Pearson and determination coefficients.  
The Nash-Sutcliffe Efficiency (NSE) explains the capability of the forecast to predict the observed time series, and its formula is expressed in Equation (5):  
\[\text{NSE}=1-\frac{\sum_{1}^{n}\left(x_{\text{i,oss}}-\hat{x_{\text{i}}}\right) ^{2}}{\sum_{1}^{n}\left(x_{\text{i,oss}}-m_{\text{oss}}\right)^{2}} \tag{5}\]  
where:  
n is the observations number, \(\hat{x_{\text{i}}}\) is i-th simulated or forecasted value, and \(m_{\text{oss}}\) is equal to the average value of observations.  
If NSE = 0, then the model forecast is no more accurate than the mean of the observations; if NSE < 0 the mean observed value is a more accurate predictor than the model. The Root Mean Square Error measures the \"average\" error, weighted according to the square of the error, while the Pearson Correlation Coefficient addresses the question how well did the forecast values correspond to the observed values. This is sensitive to outliers and it does not take forecast bias into account.  
Finally, the determination coefficient summarizes in a single value how much the analyzed quantity differs from the regression model: the more the index approaches the unit, the better is the performance of the model used. In addition to the LST prediction, the estimation of the ice alert code probability is valued through a contingency table (Figure 2), in order to evaluate the algorithm with familiar skill scores such as the accuracy, bias, false alarm ratio (FAR), miss alarm ratio (MAR), and hit rate (HR).  
The accuracy, which is simple and intuitive, can be misleading, since it is heavily influenced by the most common category, usually \"no event\" in the case of infrequent weather patterns. To resolve this issue, only days with air temperatures below an arbitrary threshold of 4 \({}^{\circ}\)C were selected in the whole dataset for the performance evaluation of ice warning probability: hence, the number of \"no event\" has been largely reduced.  
As shown in Table 4, the algorithm issues four different ice alert codes between 0 (no ice) and 3 (high risk). To simplify the comparison, and to create a \(2\times 2\) contingency table, only one level of ice warning is considered, independently from its value.  
Finally, to better highlight the differences between different sky conditions, statistical indexes are calculated dividing the dataset in function of the ratio between the observed incoming solar radiation and the radiation we would measure at clear sky conditions: if the ratio remains higher than 0.75, most  
Figure 2: Contingency tables and statistical indices used in the evaluation for the ice alert code probability.  
of the radiative transfer is able to occur, and we call it \"clear day\" and if it is lower than 0.25, most of the radiative transfer is not able to occur, and we call it \"cloudy day\".  
Intermediate conditions i.e., partly cloudy days, are here omitted because an accurate forecast of the incoming and outgoing solar radiation is a hard task [34]. This decision has been taken also considering that the majority of ice formation cases in northern Italy happens during clear sky nights, and so the choice to avoid partly cloudy days does not undermine the applicability of the algorithm. Further improvements in the predictability of cloud formation need to be done in order to implement the algorithm in every sky condition.