## 4 Methodology
If the total color value in relation to a certain variable corresponding to the bit block \([p,q]\) is \(C_{[p,q]}^{(i,j)}\), then:  
\[C_{[p,q]}^{(i,j)}=\sum_{l=0}^{L-1}c_{[p,q]|l}^{(i,j)|l}\times PV^{l} \tag{3}\]  
Substituting from Equation (1):  
\[C_{[p,q]}^{(i,j)}=\sum_{l=0}^{L-1}c_{[p,q]|l}^{(i,j)|l}\times{(2^{(q-p+1)}-1)} ^{l} \tag{4}\]  
We reserved the last number of the bit block for representing the initial color of the bitmap. Therefore, we used \((2^{(q-p+1)}-1)^{l}\) instead of using \((2^{(q-p+1)})^{l}\) as the place-value of a certain layer. For example, in the case of equal bit block allocation, if the BPV are eight, we used \(255^{l}\) as the place-value of the layer \(l\) instead of \(256^{l}\). This will reserve the last value \"255\" for the initial color of the bit block. Whenever the whole pixel is considered, \"white\" is the initial color of the pixel and the bitmap. In the concept of the GKU, one data point is represented by a shape that consists of more than one pixel (e.g., a circle) [25] as a data point shape (DPS). While adding a data point, the existing color value of all of the pixels that are overlapped by the new data point need to be updated by adding the color of the newly-added data point. In a multi-variable environment, separate data points are used for each variable. It is possible to use different DPSs for different variables. However, in this paper, the usage of the same DPS (e.g., circle) for all of the variables is discussed. In every new data addition, the color increment of the intersected area needs to be updated with a previously decided value and named the \"color increment\" (CI) [25]. When there is more than one variable, it is possible to use the same or different CI values for variables. In all of the DPSs, the bit block that is assigned for a certain variable is set to its initial color value while keeping all of the bits that do not belong to the considered  
Figure 4: Bit numbering method of a pixel in a layer. bit block at value 1. For example, consider the situation of representing data points of four variables (V1, V2, V3 and V4) on the MVML-GKU by means of circles (radius = \(r>0\)), which has a 32-bit RGB pixel format. Then, in the equal bit allocation, each variable is assigned eight bits as alpha, red, green and blue, respectively. Four separate circles are used to represent V1, V2, V3 and V4 and are colored using different \"shape colors\" (SC) as (CI, 255, 255, 255), (255, CI, 255, 255), (255, 255, CI, 255) and (255, 255, 255, CI), respectively. Furthermore, using (CI, 0, 0, 0), (0, CI, 0), (0, 0, CI, 0) and (0, 0, 0, CI) is another possible way of implementing the four different variables. If bit block [\(p\), \(q\)] represents the CI of variable \(P\), then \(P\) can be represented as \(P_{[p,q]}\). Then, \(P_{[p,q]}^{(\vec{i},\vec{j})}\) represents the pixel color of pixel \(\left(\vec{i},\vec{j}\right)\) of the DPS, where \(\left(\vec{i},\vec{j}\right)\) is the pixel of the DPS, which coincides with the pixel \((i,j)\) of the MVML-GKU. When updating the MVML-GKU, first, the \(C_{[p,q]}^{(i,j)}\) is calculated using Equation (4), and then, \(P_{[p,q]}^{(\vec{i},\vec{j})}\) is added. Therefore:  
\[C_{[p,q]}^{(i,j)}=C_{[p,q]}^{(i,j)}+P_{[p,q]}^{(\vec{i},\vec{j})} \tag{5}\]  
Finally, the value \(C_{[p,q]}^{(i,j)}\) is used to update the layers of the MVML-GKU using Equation (6). \[c_{[p,q]}^{(i,j)|t}=\text{QUOTIENT}\left(C_{[p,q]}^{(i,j)}-\sum_{m=l+1}^{L-1}c _{[p,q]|m^{\prime}}^{(i,j)|m}\left(2^{(q-p+1)}-1\right)^{l}\right) \tag{6}\]  
where \(\sum_{m=l+1}^{L-1}c_{[p,q]|m}^{(i,j)|m}=0\) when \(m>L\)  
When using Equation (6), it updates the MVML-GKU starting from the last layer and ends with the first layer (layer 0). However, depending on the requirements, it is also possible to update the MVML-GKU starting from Layer 0 with different approaches. Variables with different scales or different value ranges require a larger bitmap for representing the actual value ranges of the variable values.