Unsupervised learning via mixtures of skewed distributions with hypercube contours

• A multivariate generalization of the shifted asymmetric Laplace distribution is formulated.
• This distribution has convex upper level sets, making it excellent for cluster analysis.
• Finite mixtures of this generalization are developed for unsupervised learning.
• Parameter estimation is carried out via an EM algorithm.
• These mixtures give excellent results compared to the current state-of-the-art.

Mixture models whose components have skewed hypercube contours are developed via a generalization of the multivariate shifted asymmetric Laplace density. Specifically, we develop mixtures of multiple scaled shifted asymmetric Laplace distributions. The component densities have a unique combination of features: they include a multivariate weight function and the marginal distributions are asymmetric Laplace. We use these mixtures of multiple scaled shifted asymmetric Laplace distributions for clustering applications, but they could be used in the supervised or semi-supervised paradigms. Parameter estimates are obtained via an expectation-maximization algorithm and the Bayesian information criterion is used for model selection. Simulated and real data sets are utilized to illustrate the approach and, in some cases, to visualize the skewed hypercube structure of the components.