Margin-free classification and new class detection using finite Dirichlet mixtures

We present a margin-free finite mixture model which allows us to simultaneously classify objects into known classes and to identify possible new object types using a set of continuous attributes. This application is motivated by the needs of identifying and possibly detecting new types of a particular kind of stars known as variable stars. We first suitably transform the physical attributes of the stars onto the simplex to achieve scale invariance while maintaining their dependence structure. This allows us to compare data collected by different sky surveys which can have different scales. The model hence combines a mixture of Dirichlet mixtures to represent the known classes with the semi-supervised classification strategy of Vatanen et al. (2012) for outlier detection. In line with previous work on semiparametric model-based clustering, the single Dirichlet distributions can be seen as providing the baseline pattern of the data. These are then combined to effectively model the complex distributions of the attributes for the different classes. The model is estimated using a hierarchical two-step procedure which combines a suitably adapted version of the Expectation-Maximization (EM) algorithm with Bayes' rule. We validate our model on a reliable sample of periodic variable stars available in the literature (Dubath et al., 2011) achieving an overall classification accuracy of 71.95%, a sensitivity of 86.11% and a specificity of 99.79% for new class detection.