# Visual Assessment of Cluster Tendency with Variations of Distance Measures
## 2 Materials and Methods
The VAT and iVAT images with Mahalanobis distance did not show actual clusters. The iVAT image with distance correlation and cosine similarity showed better performance (Appendix A, Figure A4). For Microchips 1526 IE10_002, four-batch mixed lot, the computational experiment showed that the VAT and iVAT algorithms with Mahalanobis distance, Chebyshev distance, and iVAT with squared Euclidean distance did not cope with the task. The VAT and iVAT images with Mahalanobis distance and the iVAT with squared Euclidean distance showed no clusters (Appendix A, Figure A5). For Microchips 1526 IE10_002, three-batch mixed lot, the computational experiment showed that the resulting iVAT images more clearly showed the number of clusters than the resulting VAT images (Appendix A, Figure A6). The cosine similarity and squared Euclidean distance had an advantage over the other distances (Appendix A, Figure A6). The VAT and iVAT algorithms with Mahalanobis distance and Chebyshev distance did not cope with the task. The VAT and iVAT images with Mahalanobis showed no clusters (Appendix A, Figure A6). For Microchips 1526 IE10_002, two-batch mixed lot, the computational experiment showed



Mahalanobis distance and Chebyshev distance did not cope with the task. The VAT and iVAT images with Mahalanobis showed no clusters (Appendix A, Figure A7). ### Siberian Forest Compartments Datasets

For Chunsky forestry, the computational experiment showed that the resulting iVAT images more clearly showed the number of clusters than the resulting VAT images. However, the Euclidean distance and squared Euclidean distance had an advantage over the other distances (Appendix A, Figure A8). For Irbey forestry, the computational experiment showed that the resulting iVAT images more clearly showed the number of clusters than the resulting VAT images. The Canberra distance, distance correlation, and cosine similarity had an advantage over the other distances (Appendix A, Figure A9). For Lower Yenisei forestry, computational experiment showed that the resulting iVAT images more clearly showed the number of clusters than the resulting VAT images (Appendix A, Figure A10). ## 4 Discussion

Before applying any clustering method, it is important to evaluate cluster validity, i.e., to assess whether the datasets contain meaningful clusters. If clusters exist, then the assessment of the cluster tendency is a good tool to get prior knowledge of the number of clusters in problems such as \(k\)-means, where calculating the \(k\) value is a challenge. Despite the fact that there are many similarity measures that have been considered in the literature, it is quite difficult to choose the right one for a specific dataset. Due to the multidimensionality of the data, the cluster structure is not visible explicitly, so the choice of the most suitable distance measure makes it possible to clearly identify this structure. In this research, we modified the VAT and iVAT algorithms in such a way to be able to calculate many possible distance measures besides the Euclidean. The computational experiments showed that using different similarity measures in VAT and iVAT algorithms allows the expert to more confidently estimate the clustering tendency of the dataset. The computational experiments showed that cluster heat maps produced by the iVAT algorithm in all cases had higher contrast than produced by the VAT algorithm and, hence, were more clear for interpretation. Our studies have shown that distance measure significantly affects the visual tendency of clustering. In most cases, the best results were obtained with squared Euclidean distance unlike Euclidean distance, which is usually used by default in the VAT family of algorithms. Conceptualization, G.S. and E.T. ; methodology, G.S. ; software, N.R. ; validation, G.S., N.R. and E.T. ; formal analysis, N.R. ; investigation, N.R. ; resources, L.K. ; data curation, L.K. ; writing--original draft preparation, G.S. and E.T. ; writing--review and editing, E.T. and L.K. ; visualization, N.R. ; supervision, G.S.