## 2 State of the Art
### Selected Classification and Regression Algorithms
There are neurons in the network with any number of inputs and outputs. In successive layers, operations on variables are performed until the result value is reached at the end of the graph. Figure 3 shows the structure of the neuron. The mathematical formula describing such a neuron is as follows [37]:  
\[P(c_{i}|x)\]1  
In the case of artificial neural networks, the values of the input weights for individual neurons as well as the values of the bias inputs are modified during the learning process, where a is the output value from the neuron, p is the neuron input value, f is the activation function, w is the neuron weight value, b is the bias input value (threshold), y is the output value from the neural network. * **Support vector machine**  
The support vector machine algorithm works on the principle of an abstract machine model. Classification is performed by using nonlinear decision boundaries. The decision-making space is divided by building boundaries that separate objects of different class affiliations. When determining partition hypersurfaces, this algorithm takes into consideration only the data that are located near the class boundary. Since there can be a very large number of functions separating the two classes, the algorithm for computing the hyperplane aims to obtain the largest margin possible. On this basis, the support vectors are determined on which the separating hypersurface is spread. An example showing the classification of data to appropriate classes is shown in \[P(c_{i}|x)\]2  
\[P(c_{i}|x)\]3 Structure of a simple biased neuron. The distance of a point from a plane is given by the formula:  
\[P(c_{i}|x)\]4  
The vector of the coefficients w = [w1, w2, \(\ldots\), wn]T is the normal vector of the plane (it is perpendicular to the plane), while ||w|| is the length of this vector. Then, it is established that the distance between each point xi and the discriminant plane is not less than a certain value of M called the margin. The final decision rule takes the following form:  
\[P(c_{i}|x)\]5  
where sgn (z) = 1, when z > 0 and then the point belongs to class (+1); sgn (z) = -1, when z < 0 and then the point belongs to class (-1); (x\({}_{i}\cdot\mathrm{x}\)) is the scalar product of two vectors. The machine learning methods can be successfully used to develop models used in the method described below. * **Linear regression** Linear regression is typically a first-choice algorithm when predicting continuous values. It assumes the existence of a relationship between model variables and predictors. The fitted regression line or curve, therefore, represents the estimated value of the expected output variable, i.e., the predictor. Thus, appropriately selected parameters constitute a solution to the prediction problem. These parameters are selected in such a way that the distances from the straight line to the observation are as short as possible. The least squares estimation method is usually used for this purpose. The smaller the distance between the real and theoretical values, the better the model, and the estimators of the model parameters minimize the sum of the distances between \(y_{i}\) and \(y_{i}\): \[P(c_{i}|x)\]6 (6)
* **Decision trees** This algorithm is often referred to as the classification tree (Figure 5). It is based on tree-shaped structures. Nodes contain conditional tests for attributes, and leaves contain categories or probabilities of categories. The decision tree is built in an iterative way from  
\[P(c_{i}|x)\]7 An example of classification according to the SVM method. the root to the leaves. Attributes are selected by the feature selection algorithm in such a sequence as to maximize the information gained from a given node. The model-building process ends when all the leaves are in the same class or when there are no classes to divide. In this case, the algorithm may be overfitted; therefore, the process of building such a tree can be limited to a specific depth.