# Visual Assessment of Cluster Tendency with Variations of Distance Measures
## 1 Introduction
In a cluster heat map, dark areas located on the main diagonal enable us to determine the possible number of groups [16]. The VAT can be used for any numerical datasets. The VAT algorithm is similar to [PERSON]'s algorithm [13] for determining an MST of a weighted graph. The authors of the VAT algorithm noted the two following differences. Firstly, the VAT algorithm does not require their MST representation, only the order in which vertices are added as the graph grows is important. Secondly, in the VAT algorithm, the choice of the initial vertex depends on the maximum weight of the edge. In cases of a complex data structure, the efficiency of the VAT algorithm decreases quickly. However, VAT works well for datasets with dimension \(N\leq 500\). For datasets with higher dimensions, the time complexity of the algorithm (O(\(N^{2}\))) increases. In [17], the authors proposed an improved VAT (iVAT) algorithm. The iVAT images more clearly showed the number of groups, as well as their sizes. As the authors assure in [17]: \"Based on the iVAT image, the cluster structure in the data can be reliably estimated by visual inspection\". In addition, the authors proposed automated VAT (aVAT). The article [18] describes a method that handles asymmetric dissimilarity data (asymmetric iVAT (asiVAT)). In [19], the authors presented an approach for finding the number of groups by traversing the MST backwards by cutting \((k-1)\) the largest edge in the MST. VAT and iVAT have two restrictions:  
1. RDI may not be representative if the cluster structure in the data points is complex. 2. The quality of the RDI in VAT is significantly degraded due to the presence of outliers. The revised visual assessment of cluster tendency (reVAT) algorithm [20], using a quasi-ordering of the \(N\) points, reduces the time complexity to O(\(n\)). The reVAT does not require reordering the matrix of differences. The reVAT plots a set of \(c\) profile graphs of specific rows of an ordered difference matrix. An ensemble of a set of \(c\) profile graphs is used to visually determine the number of groups [20]. However, when the number of clusters is large and there are areas of points that do not have clear boundaries, a visual assessment of the number of clusters is difficult with the use of reVAT. Research [21] has presented a visual assessment of the clustering trend for large and relational datasets (bigVAT). This method uses the quasi-ordering from reVAT and builds an image similar to that built by VAT. The bigVAT solves the problems of the VAT and reVAT methods [20]. However, the image it builds may not be as visual as the image ordered by VAT. In [22], the authors presented the scalable VAT (sVAT) algorithm. To create an image, the sVAT algorithm generates a representative sample containing a structure of clusters similar to that of the original dataset. For this sample, sVAT generates a VAT-ordered image considering the following parameters: an overpriced grade of the actual number of clusters (\(k\)) and the size of representative sample (\(n\)) from the full set of \(N\) points. The sample is formed as follows: a set of \(k\) distinguished points is selected, then additional data are added next to each of the \(k\) objects. In addition, [23] described the method for visual assessment of the clustering tendency for rectangular matrices of differences (coVAT). The cluster count extraction (CCE) algorithm [24] creates a visual image using the VAT algorithm. This algorithm further processes the image to improve it for automatic verification, and extracts the number of clusters from the improved image. The article [25] demonstrates the dark block extraction (DBE) method. This method generates an RDI (VAT image) and then segments the desired areas into RDI and converts the filtered image to a distance transform image. The transformed image is superimposed on the RDI diagonal and the potential number of groups is determined. The CCE and DBE algorithms automatically determine the number of groups in unlabeled data points.