## 2 Materials and Methods
### Line Passing through Two Centroids in the Data Particle Geometrical Divide Approaches  
In the publication [1], it was demonstrated that it is possible to create new data particles based on already existing data particles through their geometrical divide. Such a strategy provides better matching of the decision boundary to the characteristics of data set than the one applied by the popular method of data particle creation based on class by a compound of \(1\div 1\) cardinality (1 CT1P). A line passing through two points is an elementary tool applied in the geometrical divide of data particles belonging to two-dimensional feature space [1]. In space of dimension \(R^{2}\), the line passing through two points \(P_{i}=[x_{i},y_{i}]\) and \(P_{j}=[x_{j},y_{j}]\) is expressed by Equation (1). \[(y-y_{i})(x_{j}-x_{i})-(y_{j}-y_{i})(x-x_{i})=0. \tag{1}\]  
Taking into consideration that the Geometrical Divide method [1] in each iteration executes the divide of all data particles, this can be a problem in the classification of imbalanced data sets, in which a minority class can limit the possibility to perform the sufficiently deep divide of data particles. To solve the problem mentioned in this article, the new method was developed, which is a modification of the Geometrical Divide approach. In the proposed algorithm, the selection process of data particles to be divided depends on their size. In the geometric division methods, in order to divide the data particles with respect to the line, in the first step, two points have to be ascertained: the centre of mass and the geometric centre [1]. Determining the geometric centre \(c=[c_{1},c_{2}]\) consists in finding the range of a given feature, and then in reducing its maximum value by a half of the range value. The procedure of determining the centre of mass \(\mu=[\mu_{1},\mu_{2}]\) is based on establishing the mean value for each component of the feature vector. The vector ascertained in that way is the centre of mass for the analysed data particle. As a result of the actions described in the previous paragraphs, two points are created for an individual data particle: the geometric centre and the centre of mass. An assumption for the geometrical divide methods, which was not clearly defined in a previous research concerning this field [1], should be introduced at this point. According to it, the vectors expressing the mentioned centres have to be different. It is necessary to find the line which will divide the data particle in the next step of the method. Otherwise the process of data particle geometrical divide will not be possible. In accordance to the above-mentioned the Lemma 1 is defined as:  
**Lemma 1. **_Geometrical divide of data particle in a two-dimensional feature space is possible if the vectors expressing the centre of mass and the geometric centre are different._  
In the next step of the algorithm, a line dividing the analysed data particle will pass through those two points. By inserting the created points into the equation of the straight line expressed by Equation (1), the equation of the straight line expressed by the Equation (2) was obtained. \[(y-\mu_{2})(c_{1}-\mu_{1})-(c_{2}-\mu_{2})(x-\mu_{1})=0. \tag{2}\]  
The above-described stages of the data particle geometrical divide algorithms for an exemplary data set were visualized in Figure 1.  
The last stage in the data particle geometrical divide algorithms is to verify whether each of the atomic data particles belonging to the analysed sample lie under, on, or above the drawn line. This step is realised by inserting the vector components of an analysed atomic data particle to the equation of line passing through two centroids (Equation (2)). The result of the described stage, namely the information which concerns the belonging of each atomic data particle to one of two newly created data particles, is shown in Figure 2. The red and blue dots represent the atomic data particles and are creating two groups relating to the two new data particles.