The important role of finite mixture models in the statistical analysis of data is underscored by the ever-increasing rate at which articles on mixture applications appear in the statistical and general scientific literature. The aim of this article is to provide an up-to-date account of the theory and methodological developments underlying the applications of finite mixture models. Because of their flexibility, mixture models are being increasingly exploited as a convenient, semiparametric way in which to model unknown distributional shapes. This is in addition to their obvious applications where there is group-structure in the data or where the aim is to explore the data for such structure, as in a cluster analysis. It has now been three decades since the publication of the monograph by McLachlan & Basford (1988) with an emphasis on the potential usefulness of mixture models for inference and clustering. Since then, mixture models have attracted the interest of many researchers and have found many new and interesting fields of application. Thus, the literature on mixture models has expanded enormously, and as a consequence, the bibliography here can only provide selected coverage.

Mixture distributions, in particular normal mixtures, are applied to data with two main purposes in mind. One is to provide an appealing semiparametric framework in which to model unknown distributional shapes, as an alternative to, say, the kernel density method. The other is to use the mixture model to provide a probabilistic clustering of the data into g clusters corresponding to the g components in the mixture model. In both situations, there is the question of how many components to include in the normal mixture model. We review various methods that have been proposed to answer this question. WIREs Data Mining Knowl Discov 2014, 4:341–355. doi: 10.1002/widm.1135 This article is categorized under: Technologies > Machine Learning

Gaussian processes, such as Brownian motion and the Ornstein-Uhlenbeck process, have been popular models for the evolution of quantitative traits and are widely used in phylogenetic comparative methods. However, they have drawbacks that limit their utility. Here we describe new, non-Gaussian stochastic differential equation (diffusion) models of quantitative trait evolution. We present general methods for deriving new diffusion models and develop new software for fitting non-Gaussian evolutionary models to trait data. The theory of stochastic processes provides a mathematical framework for understanding the properties of current and future phylogenetic comparative methods. Attention to the mathematical details of models of trait evolution and diversification may help avoid some pitfalls when using stochastic processes to model macroevolution.

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Studies have shown that algorithms based on single-channel airflow records are effective in screening for sleep-disordered breathing diseases (SDB). In this study, we investigate the diagnostic effectiveness of a classifier trained on a set of features derived from single-channel airflow measurements. The features considered are based on recurrence quantification analysis (RQA) of the measurement time series and are optionally augmented with single measurements of neck circumference and body mass index. The airflow measurement utilized is the nasal pressure (NP). The study used an overnight recording from each of 77 patients undergoing PSG testing. Mixture discriminant analysis was used to obtain a classifier, which predicts whether or not a measurement segment contains an SDB event. Patients were diagnosed as having SDB disease if the recording contained measurement segments predicted to include an SDB event at a rate exceeding a threshold value. A patient can be diagnosed as having SDB disease if the rate of SDB events per hour of sleep, the respiratory disturbance index (RDI), is > or = 15 or sometimes > or = 5. Here we trained and evaluated the classifier under each assumption, obtaining areas under receiver operating curves using fivefold cross-validation of 0.96 and 0.93, respectively. We used a two-layer structure to select the optimal operating point and assess the resulting classifier to avoid unbiased estimates. The resulting estimates for diagnostic sensitivity/specificity were 71.5%/89.5% for disease classification when RDI > or = 15 and 63.3%/100% for RDI > or = 5. These results were found assuming that the costs of misclassifying healthy and diseased subjects are equal, but we provide a framework to vary these costs. The results suggest that a classifier based on RQA features derived from NP measurements could be used in an automated SDB screening device.