## 3 Results and Discussion
### Artificial Neural Networks
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**No.**} & **Architecture** & **Quality of** & **Quality of** & **Error for** & **Error for** & **Error for** \\  & **of RBF ANN** & **Training** & **Testing** & **Validation** & **Training Set** & **Testing Set** & **Validation Set** \\ \hline
1 & RBF-8-8-1 & 0.894587 & 0.974533 & 0.627975 & 0.000743 & 0.002112 & 0.000280 \\ \hline
2 & RBF-8-9-1 & 0.934342 & 0.974127 & 0.323185 & 0.000472 & 0.000778 & 0.000374 \\ \hline
3 & RBF-8-8-1 & 0.874248 & 0.987826 & 0.782566 & 0.000878 & 0.003250 & 0.000074 \\ \hline
4 & RBF-8-8-1 & 0.925234 & 0.982205 & 0.579015 & 0.000535 & 0.001283 & 0.000071 \\ \hline
5 & RBF-8-7-1 & 0.591974 & 0.710101 & 0.346976 & 0.002416 & 0.006794 & 0.000429 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Statistics of the selected network for input data set ANN_2. better results than a model with a potentially more appropriate architecture because a more favourable initial set of weights will be randomly selected for it. In general, however, networks with a similar architecture are characterised by similar efficiency in solving a specific task. As mentioned earlier, a group of five structures of different RBF networks was obtained with the help of the AND. Their effectiveness was then compared. On this basis, the architecture potentially most beneficial for the operation of the network was selected. In the case of neural networks, it is practically impossible to say with certainty that the developed model is 'the best' possible solution to the problem under consideration. When evaluating and comparing the efficiency of the neural networks of the models presented in this paper, the smallest training and validation errors were selected as indicators of the ANN's quality. In issues referring to the use of neural networks, the error measure referred to as the mean square error (MSE) is most often used. It is a measure based on the sum of the squares of the differences between the obtained and the actual value of the network output for all individual cases included in a given set of data [40]:  
\[\text{MSE}=\sum_{i=1}^{n}\frac{\left(y_{i}-\hat{y}_{i}\right)^{2}}{n-\text{p}} \tag{2}\]  
where \(y_{i}\) represents the actual COF values, \(\hat{y}_{i}\) represents the predicted COF values, \(n\) represents the number of data, and p represents the number of parameters within the model. Errors during summation are squared to avoid the effect of compensating positive errors by negative errors (after squaring, every error is positive), and moreover, squaring makes the measure used sensitive to large errors. The resulting SSE is divided by the number of elements in the considered set of cases in order to obtain the MSE (per single case). Often, the square root of the MSE is taken to obtain error values in the same units in which the network output value is measured, which facilitates their interpretation. Network training consists of minimising the MSE for the training set. The MSE is very intuitive and is mainly used in regression tasks when the trained network returns output numerical values that are a solution to the analysed problem. However, this error is less useful in classification tasks, where the network should indicate the correct category at the output. Table 8 summarises 'the best' networks in terms of error values, one for each of the data sets ANN_1-ANN_4. In this way, it was determined which sheet surface roughness parameter, Sa, Sq, Ssk, or Sku, carries important information apart from the other parameters, determining the value of the COF. From this, the best network was determined as the RBF-8-9-1 network (Table 8) for the ANN_2 input parameters (Table 2): nominal pressure, oil pressure, oil viscosity, and the root mean square height Sq surface roughness parameter. The next part of the article will concern this network. Figures 11-13 show the dependence of the COF and the input variables for the experimental data and the response surfaces of the RBF-8-9-1 network.