# Visual Assessment of Cluster Tendency with Variations of Distance Measures
## 1 Introduction
The found minimal distances are summarized for all points and used for the Hopkins clustering index, which is calculated as follows:  
\[H_{ind}=\frac{\sum_{a=1}^{B}D_{a}}{\sum_{b=1}^{B}d_{b}+\sum_{a=1}^{B}D_{a}},\]  
where \(d_{b}=\sqrt{\left(X_{b}-X_{k}\right)^{2}}\) is the Euclidean distance between each of these real points \(b\) and their nearest real neighbor \(k\); \(D_{a}=\sqrt{\left(X_{a}-X_{k}\right)^{2}}\) is the Euclidean distance between these artificial points \(a\) and the nearest real point \(k\), \(H_{ind}\in[0,1]\). If \(H_{ind}=0.5\), then the data points are scattered randomly and it is impossible to extract the cluster structure. If \(H_{ind}=0\), then the data are homogeneous and there are no clusters, so we can say that all the data make up one big cluster. Approximation of \(H_{ind}\) to 1 means that a cluster structure can be identified in the data with a high degree of probability; if \(H_{ind}=1\), then this is the maximum degree of clustering. To obtain a stable Hopkins clustering index value (mean value), multiple calculations are carried out with new random samples. In [9; 12], the authors presented the modified Hopkins clustering index. This index is calculated as follows:  
\[H_{ind}^{*}=\lim_{B\rightarrow\infty}\frac{\left(\sum_{a=1}^{B}\frac{D_{a}}{B} \right)-\left(\sum_{n=1}^{N}d\frac{d_{n}}{N}\right)}{\left(\sum_{a=1}^{B} \frac{D_{a}}{B}\right)+\left(\sum_{n=1}^{N}d\frac{d_{n}}{N}\right)},\]  
where \(d_{n}=\sqrt{\left(X_{n}-X_{k}\right)^{2}}\) is the Euclidean distance between these points \(n\) and their real nearest point \(k\), i.e., \(d_{n}\) is computed for all the \(N\) points, \(H_{ind}^{*}\in[0,1]\). If \(H_{ind}^{*}=0\), then it is impossible to distinguish a cluster structure in the data points, if \(H_{ind}^{*}=1\), then this is the maximum degree of clustering [9]. In [9], the minimum spanning tree (MST) clustering index is proposed. The clustering algorithms associated with MST determine whether a point belongs to a cluster based on connectivity. The [PERSON]'s algorithm [13] and the Kruskal algorithm [14] are used to calculate the MST. The MST clustering index is defined as follows [9]:  
\[Index_{MST}=\sum_{d>d_{cell}}\frac{d}{d_{cell}}-1,\]where \(d_{crit}\) is the critical distance value and \(d\) is the tree distance. If \(Index_{MST}=0\), then we accept the null hypothesis \(H_{0}\), otherwise, the dataset is not homogeneous. The MST clustering index can be used to find groups, to measure the degree of heterogeneity of data points, and to detect the outliers. The authors [9] noted that the MST clustering index is sensitive to the empty space outside the multivariate limits of data points. In the case when there are areas of points in the dataset that do not have clear boundaries, the \(Index_{MST}\) and \(H_{ind}^{*}\) indexes are unable to determine the clustering tendency. However, all of these methods estimate only tendency to clustering, but not the number of clusters. For fuzzy clustering approach, there are several methods for estimating cluster validity. For example, [PERSON]'s separation index (XBI) [15], which expresses the inverse ratio between the total variation and N times the minimum separation of the clusters:  
\[XBI=\frac{\left(\frac{\sum_{k=1}^{K}\sum_{i=1}^{N}\mu_{ik}\|x_{i}-x_{k}\|^{2}} {N\cdot d_{min}^{2}}\right)}{N\cdot d_{min}^{2}},\]  
where \(d_{min}\) is the minimum distance of all distances between two clusters, \(K\) is the number of clusters, and \(\mu_{ik}\) is calculated as follows:  
\[\mu_{ik}=\left\{ \begin{array}{cc}1,&if\ \|x_{i}-u_{k}\|\leq\|x_{i}-u_{j}\|_{ \textbf{1}}\leq k,j\leq K,\ j\
eq k,\ 1\leq i\leq N.\\ &0,\ otherwise\end{array} \right.\]  
In [6], the authors presented a tool, named VAT (visual assessment of tendency), for visual assessment of the clustering tendency. At the first step, VAT algorithm reorders the matrix of differences. At the second step, it generates a reordered dissimilarity image (RDI)--a cluster heat map.