# Visual Assessment of Cluster Tendency with Variations of Distance Measures
## 1 Introduction
Paper [26] presents the CLODD (cluster in ordered difference data) method. In this algorithm, to identify potential clusters, an objective function is determined that combines contrast and edginess measures and is optimized using particle swarm optimization (PSO) [27]. With the help of the objective function, the block structure is recognized in the reordered dissimilarity matrix. The CLODD algorithm is used both to search for clusters in unlabeled data and to represent the validity index of these clusters. The effectiveness of the considered methods based on the SDS largely depends on the quality of the SDS images. SDS is effective only for defining compact clearly marked groups. However, there are datasets that have a very complex structure of clusters. In [28], the authors presented a novel silhouette-based clustering propensity score (SACT) algorithm for determining the potential number of clusters and their centroids in applications for hyperspectral image analysis. The SACT algorithm was inspired by the VAT algorithm. The SACT algorithm builds a weighted matrix and a weighted graph using the Euclidean distance measure, then, a minimum spanning tree is constructed using [PERSON]'s algorithm. The original minimum spanning tree is then hashed into branches corresponding to data clusters. Finally, the number of potential clusters is determined using the silhouette index. Clustering datasets with different cluster density is a difficult problem. The VAT and iVAT algorithms use the difference between intra-cluster and inter-cluster distances to discover the data structure. These algorithms do not work well for data points consisting of groups with different levels of density. In [29], the authors introduced the locally scaled VAT (LS-VAT) algorithm and locally scaled iVAT (LS-iVAT) algorithm. In the LS-VAT algorithm, before generating the MST, the distance matrix is converted into an adjacency matrix using local scaling [30]. The algorithm builds a similarity graph for a dataset; for this, instead of connecting the data point that has the smallest distance from the current tree, it connects the data point that has the highest adjacency to the existing tree. In case a set of points with different inter-cluster densities is used, the LS-iVAT algorithm must be used to provide higher quality iVAT images. LS-VAT and LS-iVAT outperform algorithms [6; 16; 31; 32] in terms of clustering quality. In [33], the authors presented a semi-supervised constraint-based approach for the iVAT (coniVAT) algorithm. To improve VAT/iVAT for complex data points, coniVAT uses partial background knowledge in the form of constraints. ConiVAT uses the input constraints to learn a basic similarity measure, builds a minimum transitive matrix of differences, and then applies VAT to it. For big datasets, algorithms ClusiVAT [34] and S-MVCS-VAT [35] were developed. In [36], the spectral technique was applied to ClusiVAT and S-MVCS-VAT algorithms. An algorithm based on VAT for streaming data processing was presented in [37]. [PERSON] and [PERSON] [15] presented a detailed and systematic review of a variety of VAT and iVAT algorithms and models. This article describes 25 algorithms. Thus, VAT/iVAT algorithms are methods for visually extracting some information about the structure of clusters from the initial dataset before applying any clustering algorithm. They do not change the initial dataset, but rearrange the objects in such a way as to emphasize the possible structure of the cluster. Most of the considered algorithms use the Euclidean distance measure. However, it is known that, depending on the shape of the clusters, different distance measures are better suited for different datasets. Despite the fact that there are many distance measures (similarity measures) in the literature [16; 18; 38], it is quite difficult to choose the right one for a specific dataset. The choice of the distance measure that is most appropriate in each specific case allows obtaining clearer RDIs and, accordingly, obtaining a more valid result of pre-clustering.