# Visual Assessment of Cluster Tendency with Variations of Distance Measures
## 1 Introduction  
The identification of the cluster structure in data is useful for discovering the dependencies between the sample objects. Dividing the sample into groups of similar objects simplifies further data processing and decision making, and enables us to apply a separate analysis method to each cluster. A decrease in the number of groups can help us to select the most common patterns in the data. Finding the cluster structure is essential for analyzing self-organized networking structures, such as social networks. In such problems, different measures of distance may be applied. For evaluating the similarity between objects, we can use the internal and external similarity measures [1]. Measures of internal similarity are based on the proximity of objects within a group and their distance from objects in other groups. Internal similarity measures rely on information obtained only from a dataset. External similarity measures, on the other hand, are based on comparing clustering results with reference results, usually created with the assistance of experts. In this paper, we focus on internal similarity measures. The modern literature provides a wide range of evaluation criteria, such as the Calinski-Harabasz index [2], Dunn index, Davies-Bouldin index [3], silhouette index [4, 5], and others. In the work of [PERSON] and [PERSON] [6], the authors raised an important question: \"Do clusters exist?\" Clustering algorithms divide data points into groups according to some criterion so that objects from one group are more similar to each other than to objects from other groups. Clustering algorithms are usually applied in an unsupervised way when vectors of object parameter values or a matrix of differences between objects are used as initial data. In unsupervised learning, clustering methods form groups of objects, even if the analyzed dataset is a completely random structure. This is one of the most importantproblems of clustering. Therefore, the first validation task that is recommended to be performed before clustering is to assess the general propensity of the data to cluster (clustering tendency). Before evaluating the performance of clustering, we must make sure that the dataset we are working with tends to cluster and does not contain uniformly distributed points (objects). If the data do not tend to cluster, then the groups identified by any modern clustering algorithm may be meaningless. The term \"clustering tendency\" is introduced in [7]. Using it, we can estimate the propensity of the data points to clustering. Some methods for assessing clustering propensity are discussed in [7; 8]. They can be roughly divided into following categories: statistical and visual. In [9], the authors noted that \"A clustering index should be an indicator of the degree of non-uniformity of the distribution of the objects. It should be used both as an automatic warning or, better, as a quantitative measure of the quality of the dataset, original or extracted\". The Hopkins clustering index [10; 11] has these properties [9]. It is based on the null hypothesis \(H_{0}\): \"The points in a dataset are uniformly distributed in multidimensional space\". In a known dataset, X\({}_{1}\),..., X\({}_{N}\), in a \(M\)-dimensional space, \(B\) real points are randomly selected and artificial points are added using a distribution with the standard deviation of known data points. For each real and artificial point, the Euclidean distance to the nearest point is determined. The distance between a real point to its nearest point \(d_{b}\) is compared with the distance between an artificial point to nearest real point \(D_{a}\). If there is a clustering tendency, then \(d_{b}<D_{a}\)[12].