are median and spatial median, whose underlying spherically symmetric distributions are uniform and Laplace distributions, respectively. If the type of data is discrete, for instance, an integer variable with uniform quantization error [16], then the Gaussian assumption is not valid. Median, given by the middle value of the ordered univariate sample (unique only for odd numbers of points [17]), can, like the mean, be estimated from the marginal distribution being inherently univariate. The spatial median, on the other hand, is truly a multivariate, orthogonally equivariant location estimate [18]. These location estimates and their intrinsic properties are illustrated and more thoroughly discussed in [17; 19]. The median and spatial median have many attractive statistical properties, especially since their so-called breakdown point is 0.5, i.e., they can handle up to 50% of contaminated and erroneous data.  
In a typical unsupervised scenario, one does not possess any prior knowledge of the number of clusters \(K\). Finding the best possible representation of data with \(K\) groups is difficult because the number of all possible groupings is the sum of Stirling numbers of the second kind [19]. Defining validation measures for clustering results has been, therefore, a challenging problem that different approaches have tried to overcome [20; 21; 22; 23; 24; 25]. The quality of a clustering result can be measured with a Clustering Validation Index (CVI). The aim of a CVI is to estimate the most appropriate \(K\) based on the compactness and separation of the clusters. Validation indices can be divided into three categories [26]: internal, external, and relative. An external validation index uses prior knowledge, an internal index is based on information from the data only, and in a relative CVI, multiple clustering results are compared. A comprehensive review of clustering validation techniques up to 2001 was provided in [27]. There exists also alternative approaches for determining the number of clusters, e.g., by measuring the stability of the clustering method [28] or using multiobjective evolutionary approaches [11; 29].  
In this paper, we continue the previous work reported in [30] by focusing on a comparison of the seven best internal CVIs, as identified in [30] and augmented by [22]. The earlier comparisons, typically reported when suggesting novel CVIs, only include K-means as the partitional clustering algorithm [22; 30; 31; 32; 33; 34; 35; 36]. Here, this treatment is generalized by using multiple statistical estimates as a cluster prototype and to define the clustering error, under the currently most common initialization strategy as proposed in [9] (which is also generalized). Note that prototype-based clustering can also be conducted with an incremental fashion [37; 38; 39]. However, here were restrict ourselves on the batch versions of the algorithms, which can be guaranteed to converge in a finite number of iterations (see Section 2.1). The definitions of the considered validation indices are also extended and empirically compared with K-means, K-medians, and K-spatialmedians (using spatial median as a prototype estimate) clustering results for a large pool of benchmark datasets. According to our knowledge, there exists no previous work that compares CVIs with multiple different distance metrics. Our aim is to sample the main characteristics of the indices considered and to identify what indices most reliably refer to ground truth values of the benchmark datasets. Note that by their construction, all CVIs considered here can also be used to suggest the number of clusters in hierarchical clustering.  
The structure of the article is as follows. After this introductory section, we describe generalized prototype-based clustering, discuss its convergence, and also present the generalized versions of cluster initialization and indices in Section 2. Our experimental setup is described in Section 3, and the results are given and discussed in Section 4. Finally, conclusions are drawn in Section 5.