## 2 Materials and Methods
### Machine Learning Techniques  
In the framework of the present work, the applicability of artificial neural networks to the task of reconstructing the relationships between meteorological parameters of the atmosphere and optical and geometric characteristics of HLCs was investigated. Neural networks are able to take into account the complex interaction of meteorological parameters with each other, as well as to reconstruct practically any dependence of the HLC characteristics on these parameters. In the process of prototyping the software product (SP), the greatest difficulty was determining the network architecture, the loss function, and the method of supplying the input parameters (features). Each of these problems affected the most important problem of neural networks--the overtraining problem, which occurs when the neural network stops summarizing data and starts to tune to them.  
When choosing a neural network architecture and one of the possible machine learning models, we were guided by a practical heuristic: the number of parameters used to tune the model should be smaller than the set of data used for training. When this condition is met, the neural network begins to generalize the data rather than memorize it. This also allows us to limit the set of potential architectures that can be used for training. For this purpose, a preliminary analysis was made of lidar measurements of HLC BSPM for the period 2016-2023 (2009-2015 differed in the way HLCs were measured, so the data require long and more complex processing and will be available soon). The dataset was summarized into a database (DB), which contains information on 1177 measurement series (the duration of one measurement series was about 17 min). In 312 cases of these, HLCs were detected. In addition, in another 172 series, HLCs were also registered, but the lidar signal from them was insufficient for statistical processing. These data were used to verify the prediction by the neural network of the altitudes of HLC formation depending on meteorological conditions. The results of lidar measurements were aligned with the ERA5 reanalysis data [22] at each point of the vertical profile from 0 to 15 km with a step of 150 m. Vertical profiles of the following meteorological parameters were generated for each session, taking into account the ERA5 data: relative humidity, absolute humidity, wind speed, wind direction, and temperature [18]. After creating the database and bringing the dataset to a format that allows for the training of algorithms, we created a neural network to solve the following atmospheric optics problems:  
1. Determine the probability of observation of HLCs depending on meteorological parameters (classification task);
2. Make a preliminary estimation of the observation altitude and boundaries of HLCs depending on meteorological parameters (regression task);
3. Estimate BSPM values using meteorological parameters (regression task).  
We encountered a problem; despite the substantial number of lidar experiments, the number of measurement series in the 2016-2023 period, during which HLCs were recorded and which are suitable for training the neural network, was only 312. The problem of statistical smallness of the experimental dataset was solved by software, which is acceptable at the step of software prototype development. The next step requires more data to refine the result and to add more information about the environment in the area of the experiment.  
The augmentation of the experimental dataset with data is ongoing. In addition, the information about the environment in the experimental area is being investigated, the characteristics of which will be added as input parameters for training the neural network.