## 4 Methodology
This is not peaceable, and it is necessary to transform all of the values of different variables to one scale. To accomplish the transformation, a normalization technique known as \"min-max normalization\" is used. Min-max normalization performs a linear transformation on the original data mapped into a new range using Equation (7) [26; 27]. \[v^{\prime}=\left((v-min_{A})/\left(max_{A}-min_{A}\right)\right)\times\left( new\_{max_{A}}-new\_min_{A}\right)+new\_min_{A} \tag{7}\]  
where \(v^{\prime}\) is the transformed value of \(v\), \(min_{A}\) is the minima among all of the data points, \(max_{A}\) is the maxima among all of the data points, \(new\_max_{A}\) is the ceiling value of the new range and \(new\_min_{A}\) is the floor value of the new range. If the pixel at coordinates (0, 0) of a bitmap is the origin, then \(new\_min_{A}=0\). Then, Equation (7) becomes:  
Figure 2:0  
The width of the bitmap is used for representing the independent variable, while the height of the bitmap is used for representing multiple dependent variables. Thus, the width of the bitmap is proportional to the range of independent variables, and the height is proportional to the maximum range among the ranges of all dependent variables. If the height and the width of the bitmap is \(h\) and \(w\), respectively, then:  
Figure 2:1  
Figure 2:2  
where \(R_{maxD}\) is the maximum range among the ranges of all dependent variables, \(R_{I}\) is the range of the independent variable, \(F_{maxD}\) is the scaling factor of the dependent variable that has the maximum range among all ranges of all dependent variables and \(F_{I}\) is the scaling factor of the independent variable. In the concept of the GKU, it always uses the integer part of a number for mapping on the GKU. Because of that, it is a must to scale up data that have a small range and decimal values to minimize the influence due to truncation. Selecting appropriate values for \(new\_max_{A}\) for Equation (8) will establish the new set of transformed data. The method of selecting a suitable value for \(new\_max_{A}\) is mentioned in the next paragraph. Furthermore, min-max normalization transforms the data series into a positive series, which is another requirement of the GKU. These transformation data are saved in the \"GKU specific data area\" as a color [25]. With multiple variables, there are two possible ways of scaling. The first method is to scale all of the parameters using a single scaling factor (\(F\)). This will change all of the values of the variables while keeping the original ratio between variable values. The name \"absolute scaling\" will refer to such a scaling technique, from now onwards. In \"absolute scaling\", \(new\_max_{A}\) for all dependent variables is not the same, but \(F_{iD}\) is the same for all dependent variables, where \(F_{iD}\) is the scaling factor of \(i\)-th dependent variable. The scaling factor of the variable with the height range can be considered as the common scaling factor for all of the dependent variables. Define \(R_{iD}\) as the range of the \(i\)-th dependent variable, and \(new\_max_{Ai}\) is the new maximum of the \(i\)-th dependent variable. Then,  
Figure 2:3  
From Equation (9):  
Figure 2:4  
Figure 2:5  
The second method is to scale all of the variables in a manner such that the maximum and minimum of all of the variables coincide with each other. The name \"relative scaling\" will refer to such scaling techniques, from now onwards. In \"relative scaling\", \(new\_max_{A}\) for all dependent variables is the same, but \(F_{iD}\) is not the same for all dependent variables. \"\(h\)\", which is the previously decided height of the image, can be considered as the common value for \(new\_max_{Ai}\). Thus, for \"relative scaling\":  
Figure 2:6  
Depending on the scaling method, a compatible \(new\_max_{A}\) can be calculated using either Equation (13) or Equation (14). To have a visually meaningful GKU, it is necessary to have a higher number of overlapping clusters.