## 2 Data and Methods
### Data
#### 2.1.2 Daily Maximum and Minimum Temperature  
The daily maximum and minimum temperatures at 2 m (\(T_{\text{max}}\) and \(T_{\text{min}}\)) were measured at the same station from which the radiosonde is launched (i.e., Ljubljana-Bezigrad station). The data were available for the same period as the radiosonde data and were without any missing data. The daily minimum typically occurs early in the morning, sometime after sunrise, while the daily maximum temperature is generally observed in mid-afternoon; however, the minimums and maximums can occasionally also occur at other times, e.g., if the frontal system passes the station during the day.  
We also calculated the daily climatological values for the \(T_{\text{max}}\) and \(T_{\text{min}}\). The climatological values were computed as average values in a 7-day window centered on a specific day in the period 2000-2020. For example, the value corresponding to June 5 th is the average value of \(T_{\text{max}}\) or \(T_{\text{min}}\) in days between June 2 nd and June 8 th in the period 2000-2020.  
Figure 2 shows the correlations of the vertical profiles of temperature, dew-point temperature and relative humidity with measurements of daily maximum and minimum temperature performed on the same day. As could be expected, the correlations are high for radiosonde temperatures near the ground, with correlations starting to noticeably decrease above the altitude of 1.2 km, although they are larger than 0.6 in most of the troposphere, while they rapidly decrease in the stratosphere. The correlations near the ground are also large for dew-point temperature but start to decrease much more rapidly with altitude. For relative humidity, the correlations are generally low. For \(T_{\text{max}}\), the correlations with \(RH\) are weakly negative in most of the troposphere, which could be linked to the possible existence of clouds that weaken downward shortwave radiation near the ground during the day, which reduces the temperature. For \(T_{\text{min}}\) the correlations with \(RH\) are weakly positive, which could be linked to clouds and high humidity causing an increase of downward longwave radiation near the ground during the night, which decreases the night-time radiation cooling and thus causes an increase in temperature.  
Figure 1: The number of radiosonde measurements in each month that passed the quality control and were used for neural network training, validation and testing; shown with blue bars. Grey bars indicate number of days in each month.  
Note that \(T\), \(T_{d}\) and \(RH\) are related via the following equation (derived from Clausius-Clapeyron relation):  
\[\log RH=\frac{L}{R_{v}}\bigg{(}\frac{1}{T}-\frac{1}{T_{d}}\bigg{)} \tag{1}\]  
and thus they are not independent (uncorrelated) variables. In Equation (1), \(L=2.5\) MJ kg\({}^{-1}\) denotes latent heat of condensation, while \(R_{v}=461.5\) kg\({}^{-1}\) K\({}^{-1}\) is the specific gas constant for water vapor.  
Previous-day temperature extremes, \(T_{\max}(t-1)\) and \(T_{\min}(t-1)\), show very high correlation (correlation coefficient 0.95) with present-day extremes, \(T_{\max}(t)\) and \(T_{\min}(t)\), while the correlation with the same-day climatological extremes is approximately 0.9.