## 2 Materials and Methods
### The Design of the Combined Metrics Based on Neural Networks
It should be noted that a simple linear model is used here, and the selection of results will be approximate. Therefore, it is recommended to validate and carry out single replacements as needed;
* Type of the neural network. * As the results in [72] have shown, the use of more complex modifications with non-linear dependencies between layers as compared to feed-forward networks leads to longer training but may lead to a noteworthy increase in the accuracy of the designed metric. At the same time, the results obtained for the best types--cascade and Elman networks--gave comparable efficiency. Therefore, in the paper, the feed-forward and cascade-forward neural networks are analyzed. The Lasso algorithm has an important feature that should be taken into account during the calculations. It performs a sequential optimization of the weights, so its result depends on the starting point of reference. In the context of neural networks, this means that the order of the metrics in the input data is essential. Therefore, the training procedure was carried out in two stages. At the first stage, 100 sequences have been randomly generated from the considered metrics. For the initial configuration, from 5 input metrics, neural networks have been calculated assuming the use of both types of neural networks and a different number of layers and neurons (20 training rounds with a randomly subdivided test set). As shown in previously obtained results [11], the efficiency is enhanced with an increase in the number of metrics, and for the maximum size (all metrics) the influence of Lasso is minimized. Therefore, the minimum set of input data is the most informative. It should be noted that the presence of only 3 input data vectors (elementary metrics) does not allow to ensure the optimal choice of metrics (SROCC < 0.7), and more accurate results have been obtained when searching for 5 metrics (SROCC > 0.8). Depending on the configuration and the type of the neural network (feed-forward or cascade-forward), the difference in the SROCC value can reach 0.04 for a smaller number of inputs. In this case, the advantages over the elementary metrics can be provided by both of them, however, more often by the cascade-forward networks, used also in the resulting network. At the second stage, for each type of neural network, several configurations are considered to select the optimal structure, similarly to [11]: from 1 to 5 hidden layers, with an equal or evenly decreasing number of neurons in each layer. Similarly, it has been previously determined that the appropriate selection of suitable NN activation functions eliminates the need for preliminary data linearization. The use of a higher number of neurons does not provide significantly better results. They are very similar for each number of layers and mostly dependent on the used datasets, therefore the decision to use 5 layers has been made. As activation function in hidden layers, we used tansig that allows normalizing the infinitive range of values to the limited range [\(-1\);\(1\)] and the linear one for the last layer. After [PERSON]'s calculations, the combinations of metrics shown in Table 2 are considered (NNZ means the number of non-zero values). Another advantage of the application of Lasso is that the use of even the smallest thresholds eliminates the weakest metrics. Since there are some other databases for which it is possible to check the performance of the optimized combined metrics, such an analysis has been performed for the combination of 5 and 10 elementary metrics. To verify the universality of the proposed approach, it has been checked for widely known LIVE database [28] (for all types of distortions), for several subsets of distortions that can be formed for TID2013, as well as for several subsets for the KADID-10k database [73] that contain distortions typical for the RS applications. For comparison, the elementary metrics used as inputs of the combined NN metric have been taken.