## 6 Results and Discussion
This leads to difficulties in identifying clusters visually, especially in low density areas. When examining Layer 1 of Figure 8, this phenomenon is more visible. In Layer 1, neither clusters between different variables nor density clusters of individual variables are possible to identify. Furthermore, in Layer 0 of Figure 8, except high density areas, all of the other areas are in black or nearly black color, and this prevents visual identification of low density areas in relation to different variables. This phenomenon can be easily identified when considering the points P\({}_{A}\) and P\({}_{B}\), as shown in Figure 8. It is not possible to distinguish between P\({}_{A}\) and P\({}_{B}\) visually due to the colors of the DPSs. However, Figure 6, these two points can be easily identified. Nevertheless, in reality, all black areas contain different colors (e.g., (0, 0, 40), (0, 40, 0) and (40, 0, 0)), which is visible only to an algorithm. Thus, selection of this type of color is not a problem for identifying clusters in low density areas with a computer. As a  
recommendation, it is stated that the different colors close to white are the best selection for the MLMV-GKU for visually capturing clusters between different variables, as well as density clusters of individual variables. Plot (a) of Figure 9 shows the compression of the conventional scattered plot for one independent and three dependent variables and plot (b) of Figure 9 the MLMV-GKU of two layers for the same variables. The plots represent 21,951 data points, where the effective bitmap width and height are 400 and 625 pixels, respectively. Plots represent data in relation to three dependent variables (TVFA, AA and PA) against the concentration of volatile solids (VS). All of the data are normalized using the max-min normalization method and used 625, 550 and 225 pixels as \(new\_max_{A}\) for TVFA, AA and PA, respectively. In the MLMV-GKU, the initial color of each variable is (255, 255, 10), (255, 10, 255) and (10, 255, 255). A circle of a radius of 10 pixels is used as the DPS. The MLMV-GKU shows density clusters of each variable by means of automatically-generated contour lines, which can be easily identified by the naked eye. This is a special feature of the GKU, which is not possible with a scatter plot. As mentioned in Section 1, star coordinates directly visualize dimensions as groups in a form of high density clusters in a two-dimensional plot. It is one of the better visualization techniques. However, star coordinates suffer from cluster overlapping of clusters of variables [17; 20]. This is the major drawback of the star coordinates. If there are no overlapping regions, star coordinates can be used to visualize high density data clusters. However, the data we used have overlapping regions and cannot be visualized using star coordinates. Figure 8: The MLMV-GKU representation of the same data shown in Figure 6 with a different color scheme. **(a)** Layer 0 of the MLMV-GKU; **(b)** Layer 1 of the MLMV-GKU. As in Figure 6, the effective bitmap width and height are 400 and 400 pixels for representing the data of three dependent variables (concentrations of total volatile fatty acids (TVFA), acetic acid (AA) and propionic acid (PA)) against the concentration of volatile solids where each dependent variable consists of 2885 data points. All of the data are normalized using the max-min normalization method and used 400, 250 and 250 pixels as \(new\_max_{A}\) for TVFA, AA and PA, respectively. The initial color of each variable is (0, 0, 20), (0, 20, 0) and (20, 0, 0). A circle of a radius of 10 pixels is used as the DPS. The MLMV-GKU shows visual density clusters only in high overlapping areas. When there is low overlapping, it is not possible to distinguish points in relation to variables, such as \(P_{A}\), \(P_{B}\) and Layer 0. The nature of the MLMV-GKU allows the usage of a higher number of layers for representing variables. Unfortunately, this will lead to problems in understanding the whole picture.