## 6 Results and Discussion  
The results included in this section represent different forms of the MLMV-GKU representations, which were obtained using different techniques. Furthermore, this section deliberates the methods and interpretation of the MLMV-GKUs. Figure 5 shows a conventional scatter plot, which presents the 2885 occurrences of three dependent variables (TVFA, AA and PA) against the independent variable (VS). Effective value ranges of TVFA, AA and PA are [0, 8], [0, 5] and [0, 5], respectively. The plot itself does not convey strong evidence on data clustering or data density. On the other hand, due to overlapping, it is difficult to identify some of the data points that may lie under other data points. Plots in Figure 6 show the plotting of the same dataset shown in Figure 7, on the MLMV-GKU of two layers using \"relative scaling\". The MLMV-GKU was implemented using a circle with a ten-pixel radius DPS and CI = 20. Three different RGB colors, (255, 255, 20), (255, 20, 255) and (20, 255, 255), were used for representing TVFA, AA and PA, respectively. The effective width and height of the bitmap is 400 and 400 pixels, respectively. Since \"relative scaling\" is used for scaling the values of variables, according to the Equation (14), \(new\_max_{A}\) for each dependent variable is 400 pixels. Based on this, all existing ranges of all of the variables ([0, 8], [0, 5] and [0, 5]) were transformed. In Layer 0 of Figure 6, the formation of clusters that belong to different variables can be identified with the naked eye. This is one advantage of using the concept of the GKU, because it creates visibly dense clusters when there is an adequate amount of overlapping. Figure 5: A conventional scatter plot 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. Each dependent variable consists of 2885 data points. The ranges of TVFA, AA and PA are [0, 8], [0, 5], and [0, 5], respectively. Due to over-plotting, the large number of data points and the multi-variable environment, this plot itself does not provide direct useful information about data clusters. This is the major drawback of using a scatter plot for representing multi-variable big data. Figure 6: The MLMV-GKU of two layers. (**a**) Layer 0 of the MLMV-GKU; (**b**) Layer 1 of the MLMV-GKU. 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 (relative scaling). The initial color of each variable is (255, 255, 20), (255, 20, 255) and (20, 255, 255). A circle of a radius of 10 pixels is used as the data point shape (DPS). The MLMV-GKU shows density clusters of both high and low density areas of each variable by means of automatically-generated contour lines, which can be easily identified by the naked eye. Figure 7: The MLMV-GKU of two layers. (**a**) Layer 0 of the MLMV-GKU; (**b**) Layer 1 of the MLMV-GKU. 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 using 400 pixels as _new_max\({}_{A}\)_ for all dependent variables (absolute scaling). The initial color of each variable is (255, 255, 20), (255, 20, 255) and (20, 255, 255). A circle of a radius of 10 pixels is used as the data point shape (DPS).