## 4 Methodology  
A GKU with a 24-bit bitmap, the RGB pixel format, is used for representing data points. Furthermore, it is possible to use a 32-bit bitmap with the alpha-RGB (ARGB) pixel format for representing the GKU. In both situations, a maximum of 24 bits or 32 bits can be allocated for representing a single variable against another variable. Instead of allocating the whole bit range of a pixel to a single variable, it is possible to divide the bits among several variables. Figure 2 shows an example of the equal allocation of bits of a 32-bit bitmap with the ARGB pixel format for four variables. Furthermore, unequal allocation is possible depending on the requirements. Here, the equal allocation of bits for the variables was mainly considered. Figure 2 elaborates on the allocation of alpha, red, green and blue portions for representing four variables, where eight bits per variable (BPV) are used. However, when the numbers of variables are high, BPV is low. When the BPV is low, it leads to fewer numbers of overlapping data points. If BPV = 8 bits, it supports the representation of the maximum of 256 overlaps and will not allow the representation of a higher number of data points. This will discourage the usage of multiple variables in the GKU. As a solution, a multi-layer bitmap GKU is proposed, which consists of several equal-sized bitmaps (Figure 3). In the GKU, when one block is full (e.g., blue), it is possible to use the adjacent block (green) to represent overlapping [25]. In contrast, in a multi-layer GKU, when an allocated block is full, it uses the relevant pixel block of another bitmap of the same size (Figure 3), which is named a layer. If the number of layers is \(L\) and the number of bits allocated for a variable is \(k\), this technique provides \(k\times L\) bits for one variable. In the GKU, the whole bit range of a pixel is allocated for one variable, which can be considered as a horizontal array. In contrast, in the MVML-GKU concept, a single variable is a vertical array with \(k\times L\) bits (Figure 3). Depending on the requirement, it is possible to decide the number of layers in the MVML-GKU. Figure 3: Four variables (V1, V2, V3 and V4) are represented using multiple bitmaps of a 32-bit RGB pixel format for forming a multi-variable multi-layer graphical knowledge unit (MVML-GKU). Each pixel is divided into four equal portions, where each portion consists of eight bits. When the 32-bit RGB pixel format is divided into four equal parts, each variable represents the alpha, red, green and blue sections of the pixel. This provides a vertical array of \(k\times L\) bits for one variable. Figure 2: A single pixel of a 32-bit RGB format is split into four equal portions for representing four variables (V1, V2, V3 and V4). A, R, G and B represents the alpha, red, green and blue portions of the pixel, respectively. Furthermore, each layer is numbered starting from 0, and the \"place-value\" is assigned for each layer according to the layer number. This assigns the same \"place-value\" for each bit in the relevant layer. The base of the \"place-value\" is in relation to the number of assigned bits for one variable, as shown in Equation (1). \[PV^{l}={(2^{(q-p+1)}-1)}^{l} \tag{1}\]  
where \(PV^{l}\) is the positional value of a bit in the layer \(l\), \(p\) is the index of the starting bit of the bit block and \(q\) is the index of the ending bit of the bit block; thus, \(p\) = 0, 1, 2,... and \(q\) = 0, 1, 2,... If the \(c_{[p,q]|l|}^{(i,j)|l}\) represents the color value of the variable, it corresponds to bit block \([p,q]\) of the pixel \((i,j)\) of the layer \(l\) (Figure 4). Therefore:  
\[c_{[p,q]|l}^{(i,j)|l}=\sum_{k=q}^{p}b_{k}^{(i,j)|l}\times 2^{q-k} \tag{2}\]  
where \(b_{k}^{(i,j)|l}\) is the bit value (0 or 1) of the bit \(k\) of the bit block of a variable of pixel \((i,j)\) of the layer \(l\).