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In model-based clustering, the population is assumed to be a combination of sub-populations. Typically, each sub-population is modeled by a mixture model component, distributed according to a known probability distribution. Each component is considered a cluster. Two primary approaches have been used in the literature when clusters are skewed: (1) transforming the data within each cluster and applying a mixture of symmetric distributions to the transformed data, and (2) directly modeling each cluster using a skewed distribution. Among skewed distributions, the generalized hyperbolic distribution is notably flexible and includes many other known distributions as special or limiting cases. This paper achieves two goals. First, it extends the flexibility of transformation-based methods as outlined in approach (1) by employing a flexible symmetric generalized hyperbolic distribution to model each transformed cluster. This innovation results in the introduction of two new models, each derived from distinct within-cluster data transformations. Second, the paper benchmarks the approaches listed in (1) and (2) for handling skewness using both simulated and real data. The findings highlight the necessity of both approaches in varying contexts. 

Abstract Data clustering has a long history and refers to a vast range of models and methods that exploit the ever-more-performing numerical optimization algorithms and are designed to find homogeneous groups of observations in data. In this framework, the probability distance clustering (PDC) family methods offer a numerically effective alternative to model-based clustering methods and a more flexible opportunity in the framework of geometric data clustering. GivennJ-dimensional data vectors arranged in a data matrix and the numberKof clusters, PDC maximizes the joint density function that is defined as the sum of the products between the distance and the probability, both of which are measured for each data vector from each center. This article shows the capabilities of the PDC family, illustrating the package .