Mixtures of Shifted Asymmetric Laplace Distributions

A mixture of shifted asymmetric Laplace distributions is introduced and used for clustering and classification. A variant of the EM algorithm is developed for parameter estimation by exploiting the relationship with the general inverse Gaussian distribution. This approach is mathematically elegant and relatively computationally straightforward. Our novel mixture modelling approach is demonstrated on both simulated and real data to illustrate clustering and classification applications. In these analyses, our mixture of shifted asymmetric Laplace distributions performs favourably when compared to the popular Gaussian approach. This work, which marks an important step in the non-Gaussian model-based clustering and classification direction, concludes with discussion as well as suggestions for future work

Unsupervised Learning via Mixtures of Skewed Distributions with Hypercube Contours

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 two unique features: they include a multivariate weight function, and the marginal distributions are also asymmetric Laplace. We use these mixtures of multiple scaled shifted asymmetric Laplace distributions for clustering applications, but they could equally well be used in the supervised or semi-supervised paradigms. The expectation-maximization algorithm is used for parameter estimation and the Bayesian information criterion is used for model selection. Simulated and real data sets are used to illustrate the approach and, in some cases, to visualize the skewed hypercube structure of the components

Parsimonious Shifted Asymmetric Laplace Mixtures

A family of parsimonious shifted asymmetric Laplace mixture models is introduced. We extend the mixture of factor analyzers model to the shifted asymmetric Laplace distribution. Imposing constraints on the constitute parts of the resulting decomposed component scale matrices leads to a family of parsimonious models. An explicit two-stage parameter estimation procedure is described, and the Bayesian information criterion and the integrated completed likelihood are compared for model selection. This novel family of models is applied to real data, where it is compared to its Gaussian analogue within clustering and classification paradigms

Model-Based Clustering, Classification, and Discriminant Analysis Using the Generalized Hyperbolic Distribution: MixGHD R package

The MixGHD package for R performs model-based clustering, classification, and discriminant analysis using the generalized hyperbolic distribution (GHD). This approach is suitable for data that can be considered a realization of a (multivariate) continuous random variable. The GHD has the advantage of being flexible due to skewness, concentration, and index parameters; as such, clustering methods that use this distribution are capable of estimating clusters characterized by different shapes. The package provides five different models all based on the GHD, an efficient routine for discriminant analysis, and a function to measure cluster agreement. This paper is split into three parts: the first is devoted to the formulation of each method, extending them for classification and discriminant analysis applications, the second focuses on the algorithms, and the third shows the use of the package on real datasets

Estimated discharge of microplastics via urban stormwater during individual rain events

Urban stormwater runoff is an important pathway for the introduction of microplastics and other anthropogenic pollutants into aquatic environments. Highly variable concentrations of microplastics have been reported globally in runoff, but knowledge of key factors within urban environments contributing to this variability remains limited. Furthermore, few studies to date have quantitatively assessed the release of microplastics to receiving waters via runoff. The objectives of this study were to assess the influence of different catchment characteristics on the type and amount of microplastics in runoff and to provide an estimate of the quantity of microplastics discharged during rain events. Stormwater samples were collected during both dry periods (baseflow) and rain events from 15 locations throughout the city of Calgary, Canada’s fourth largest city. These catchments ranged in size and contained different types of predominant land use. Microplastics were found in all samples, with total concentrations ranging from 0.7 to 200.4 pcs/L (mean = 31.9 pcs/L). Fibers were the most prevalent morphology identified (47.7 ± 33.0%), and the greatest percentage of microplastics were found in the 125–250 µm size range (26.6 ± 22.9%) followed by the 37–125 µm size range (24.0 ± 22.3%). Particles were predominantly black (33.5 ± 33.8%), transparent (22.6 ± 31.3%), or blue (16.0 ± 21.6%). Total concentrations, dominant morphologies, and size distributions of microplastics differed between rain events and baseflow, with smaller particles and higher concentrations being found during rain events. Concentrations did not differ significantly amongst catchments with different land use types, but concentrations were positively correlated with maximum runoff flow rate, catchment size, and the percentage of impervious surface area within a catchment. Combining microplastic concentrations with hydrograph data collected during rain events, we estimated that individual outfalls discharged between 1.9 million to 9.6 billion microplastics to receiving waters per rain event. These results provide further evidence that urban stormwater runoff is a significant pathway for the introduction of microplastics into aquatic environments and suggests that mitigation strategies for microplastic pollution should focus on larger urbanized catchments

Product selection for liking studies: The sensory informed design

AbstractLiking studies are designed to ascertain consumers likes and dislikes on a variety of products. However, it can be undesirable to construct liking studies where each panelist evaluates every target product. In such cases, an incomplete-block design, where each panelist evaluates only a subset of the target products, can be used. These incomplete blocks are often balanced, so that all pairs occur the same number of times. While desirable in many situations, balanced incomplete blocks have the disadvantage that, by their nature, they cannot favor placing dissimilar products next to one another. A novel incomplete-block design is introduced that utilizes the target product’s sensory profile to allocate products to each panelist so that they are, in general, as dissimilar as possible while also ensuring position balance. The resulting design is called a sensory informed design (SID). Herein, details on the formulation of SIDs are given. Data arising from these SIDs are analyzed using a simultaneous clustering and imputation approach, and the results are discussed