## 3 Experimental Setup  
Next, we test the indices in Table 1 with different distance measures, i.e., with different prototype-based clustering algorithms. The index values are calculated with the distance corresponding to the clustering method used, i.e., city-block with the K-medians, squared Euclidean with the K-means, and Euclidean with the K-spatialmedians. All datasets were scaled to the range of \([-1,1]\). All the tests were run on MATLAB (R2014a), where a reference implementation on both the validation indices and the general clustering Algorithm 1 with the initialization given in Algorithm 2 were prepared. The impact of the necessary amount of repetitions of Algorithms 1 and 2 was tested with multiple datasets (\(S\)-sets, _Dim_-sets, _A1_, _Unbalance_), comparing the clustering error and the cluster assignments. With 100 repetitions, the minimum clustering error and the corresponding cluster assignments were stabilized and an appropriate clustering result was found. This result with the minimum clustering error was used for computing the CVI value. To test the indices, we used the basic benchmark datasets described in detail in [48], with the two other synthetic datasets [[http://users.jyu.fi/~jookriha/CVI/Data/](http://users.jyu.fi/~jookriha/CVI/Data/)]([http://users.jyu.fi/~jookriha/CVI/Data/](http://users.jyu.fi/~jookriha/CVI/Data/)) as given in [30] (see Figure 1). These benchmark sets are synthetic datasets, suggested for use when testing any algorithm dealing with clustering spherical or Gaussian data. Here, we restrict ourselves to the benchmarks with at most 20 clusters, because the interpretation and knowledge discovery from the clustering results with a large number of prototypes might become tedious [6; 49]. Therefore, the number of clusters was also tested with \(K=2-25\). In addition, we use six real datasets that include Steel Plates, Ionosphere, and Satimage (Train) from the UCI Machine Learning Repository, [[https://archive.ics.uci.edu/ml/datasets.html](https://archive.ics.uci.edu/ml/datasets.html)]([https://archive.ics.uci.edu/ml/datasets.html](https://archive.ics.uci.edu/ml/datasets.html)) Iris and Arrhythmia from MATLAB's sample datasets, [[https://se.mathworks.com/help/stats/_bq9](https://se.mathworks.com/help/stats/_bq9) uxn4.html]([https://se.mathworks.com/help/stats/_bq9](https://se.mathworks.com/help/stats/_bq9) uxn4.html), and the USPS dataset. [[https://www.otexts.org/1577](https://www.otexts.org/1577)]([https://www.otexts.org/1577](https://www.otexts.org/1577)) A summary of all these datasets can be seen in Table 2. For these real datasets, even if there are class labels provided, we do not compare the clustering results with the class information because of the completely unsupervised scenario with the internal validation indices. Moreover, the classes need not correspond to the clusters determined by the prototype-based clustering algorithm, since the probability density functions of the classes are not necessarily spherically symmetric. For these datasets, we therefore only study and compare the stability of the suggestions on the numbers of clusters for different indices. Finally, our datasets include the following: _A1_, with 20 spherical clusters and some overlap; the \(S\)-sets, including four datasets with 15 Gaussian clusters, and cluster overlap increasing gradually from _S1_ to _S4_; _Dim_-sets, including six datasets with 16 well-separated clusters in high-dimensional space and dimensions varying from 32 to 1024; a subset of Birch2, including 19 datasets