## 2 Materials and Methods
### Mathematical Description of the Method  
The algorithm of the method (Figure 2) requires providing it with a set of Cartesian coordinates describing the position of the analyzed points \(P_{i}\) (\(x_{i};y_{i}\)) from the analyzed set of points (E1). The provided set is used to determine the linear trend function \(y(x)\) (6) first, with the inclination angle of the trendline \(\alpha\) (7) (E2) being determined subsequently based on it. The inclination angle of the trendline is the target ellipse inclination angle, which is the angle between the ellipse's longer semiaxis and the horizontal axis of the reference frame. \[y(x)=\left(\frac{\sum_{i=1}^{n}(x_{i}-\overline{x})(y_{i}-\overline{y})}{ \sum_{i=1}^{n}(x_{i}-\overline{x})^{2}}\right)x+\left(\overline{y}-\left(\frac {\sum_{i=1}^{n}(x_{i}-\overline{x})(y_{i}-\overline{y})}{\sum_{i=1}^{n}(x_{i} -\overline{x})^{2}}\right)\overline{x}\right) \tag{6}\]  
Figure 1: Schematic illustration of the method replacing an arbitrary set of points (**a**) with an ellipse defining the area of presence for points on the analyzed plane (**b**). \[\alpha=\tan^{-1}\left(\frac{\sum_{i=1}^{n}(x_{i}-\overline{x})(y_{i}-\overline{y})}{ \sum_{i=1}^{n}(x_{i}-\overline{x})^{2}}\right) \tag{7}\]  
where \(y(x)\)--the value of the linear function of the trend, \(n\)--the number of points in the analyzed set, \(x_{i}\)--the coordinate of the position on the horizontal axis of any point from the set, \(\overline{x}\)--the mean coordinate of the position on the horizontal axis from the coordinates of the analyzed set of points, \(y_{i}\)--the coordinate of the position on the vertical axis of any point from the set, \(\overline{y}\)--the mean coordinate of the position on the vertical axis for the coordinates of the analyzed set of points, \(x\)--any coordinate on the horizontal axis, and \(a\)--the angle of inclination of the trendline with respect to the horizontal axis. In the next step (E3), the position of the ellipse center on the horizontal \(x_{COG}\) (8) and vertical \(y_{COG}\) (9) axes is calculated. These coordinates correspond to the position of the center of the plotted COG ellipse. \[x_{COG}=\frac{\sum_{i=1}^{n}x_{i}}{n} \tag{8}\]  
\[y_{COG}=\frac{\sum_{i=1}^{n}y_{i}}{n} \tag{9}\]  
where \(x_{COG}\)--mean position on the horizontal axis of the analyzed set of points, \(y_{COG}\)--mean position on the vertical axis of the analyzed set of points, \(x_{i}\)--position coordinate on the horizontal axis of any point from the set, \(y_{i}\)--position coordinate on the vertical axis of any point from the set, and \(n\)--the number of points in the analyzed set. The next step of the algorithm (E4) requires rotating the analyzed points by the negative value of the angle of inclination of the ellipse. This step is only a mathematical transformation necessary to calculate the dimensions of the ellipse and it does not affect the shape or position of the area approximating the set of points in the next steps. The rotation  
Figure 2: Schematic of the algorithm allowing to describe a set of points with an ellipse. should result in a set of points with their trendline parallel to the horizontal axis. The rotation of the analyzed points is performed using the rotation matrix \(R(-\alpha)\) (10), which is multiplied by all the analyzed points \(P_{i}\), resulting in points \(P_{i}^{\prime}\) (11) with coordinates rotated by angle \(\alpha\). \[R(-\alpha)=\left[ \begin{array}{cc}\cos(-\alpha)&-\sin(-\alpha)\\ \sin(-\alpha)&\cos(-\alpha)\end{array} \right] \tag{10}\]  
\[\left[ \begin{array}{cc}\cos(-\alpha)&-\sin(-\alpha)\\ \sin(-\alpha)&\cos(-\alpha)\end{array} \right]\left[ \begin{array}{c}x_{i}\\ y_{i}\end{array} \right]=\left[ \begin{array}{c}x_{i}\cos(-\alpha)-y_{i}\sin( -\alpha)\\ x_{i}\sin(-\alpha)+y_{i}\cos(-\alpha)\end{array}