From here to infinity - sparse finite versus Dirichlet process mixtures in model-based clustering

In model-based-clustering mixture models are used to group data points into clusters. A useful concept introduced for Gaussian mixtures by Malsiner Walli et al (2016) are sparse finite mixtures, where the prior distribution on the weight distribution of a mixture with $K$ components is chosen in such a way that a priori the number of clusters in the data is random and is allowed to be smaller than $K$ with high probability. The number of cluster is then inferred a posteriori from the data. The present paper makes the following contributions in the context of sparse finite mixture modelling. First, it is illustrated that the concept of sparse finite mixture is very generic and easily extended to cluster various types of non-Gaussian data, in particular discrete data and continuous multivariate data arising from non-Gaussian clusters. Second, sparse finite mixtures are compared to Dirichlet process mixtures with respect to their ability to identify the number of clusters. For both model classes, a random hyper prior is considered for the parameters determining the weight distribution. By suitable matching of these priors, it is shown that the choice of this hyper prior is far more influential on the cluster solution than whether a sparse finite mixture or a Dirichlet process mixture is taken into consideration.Comment: Accepted versio

Multivariate Bayesian semiparametric models for authentication of food and beverages

Food and beverage authentication is the process by which foods or beverages are verified as complying with its label description, for example, verifying if the denomination of origin of an olive oil bottle is correct or if the variety of a certain bottle of wine matches its label description. The common way to deal with an authentication process is to measure a number of attributes on samples of food and then use these as input for a classification problem. Our motivation stems from data consisting of measurements of nine chemical compounds denominated Anthocyanins, obtained from samples of Chilean red wines of grape varieties Cabernet Sauvignon, Merlot and Carm\'{e}n\`{e}re. We consider a model-based approach to authentication through a semiparametric multivariate hierarchical linear mixed model for the mean responses, and covariance matrices that are specific to the classification categories. Specifically, we propose a model of the ANOVA-DDP type, which takes advantage of the fact that the available covariates are discrete in nature. The results suggest that the model performs well compared to other parametric alternatives. This is also corroborated by application to simulated data.Comment: Published in at http://dx.doi.org/10.1214/11-AOAS492 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Generative Supervised Classification Using Dirichlet Process Priors.

Choosing the appropriate parameter prior distributions associated to a given Bayesian model is a challenging problem. Conjugate priors can be selected for simplicity motivations. However, conjugate priors can be too restrictive to accurately model the available prior information. This paper studies a new generative supervised classifier which assumes that the parameter prior distributions conditioned on each class are mixtures of Dirichlet processes. The motivations for using mixtures of Dirichlet processes is their known ability to model accurately a large class of probability distributions. A Monte Carlo method allowing one to sample according to the resulting class-conditional posterior distributions is then studied. The parameters appearing in the class-conditional densities can then be estimated using these generated samples (following Bayesian learning). The proposed supervised classifier is applied to the classification of altimetric waveforms backscattered from different surfaces (oceans, ices, forests, and deserts). This classification is a first step before developing tools allowing for the extraction of useful geophysical information from altimetric waveforms backscattered from nonoceanic surfaces

Identifying Mixtures of Mixtures Using Bayesian Estimation

The use of a finite mixture of normal distributions in model-based clustering allows to capture non-Gaussian data clusters. However, identifying the clusters from the normal components is challenging and in general either achieved by imposing constraints on the model or by using post-processing procedures. Within the Bayesian framework we propose a different approach based on sparse finite mixtures to achieve identifiability. We specify a hierarchical prior where the hyperparameters are carefully selected such that they are reflective of the cluster structure aimed at. In addition this prior allows to estimate the model using standard MCMC sampling methods. In combination with a post-processing approach which resolves the label switching issue and results in an identified model, our approach allows to simultaneously (1) determine the number of clusters, (2) flexibly approximate the cluster distributions in a semi-parametric way using finite mixtures of normals and (3) identify cluster-specific parameters and classify observations. The proposed approach is illustrated in two simulation studies and on benchmark data sets.Comment: 49 page

Sparse covariance estimation in heterogeneous samples

Standard Gaussian graphical models (GGMs) implicitly assume that the conditional independence among variables is common to all observations in the sample. However, in practice, observations are usually collected form heterogeneous populations where such assumption is not satisfied, leading in turn to nonlinear relationships among variables. To tackle these problems we explore mixtures of GGMs; in particular, we consider both infinite mixture models of GGMs and infinite hidden Markov models with GGM emission distributions. Such models allow us to divide a heterogeneous population into homogenous groups, with each cluster having its own conditional independence structure. The main advantage of considering infinite mixtures is that they allow us easily to estimate the number of number of subpopulations in the sample. As an illustration, we study the trends in exchange rate fluctuations in the pre-Euro era. This example demonstrates that the models are very flexible while providing extremely interesting interesting insights into real-life applications

Decorrelation of Neutral Vector Variables: Theory and Applications

In this paper, we propose novel strategies for neutral vector variable decorrelation. Two fundamental invertible transformations, namely serial nonlinear transformation and parallel nonlinear transformation, are proposed to carry out the decorrelation. For a neutral vector variable, which is not multivariate Gaussian distributed, the conventional principal component analysis (PCA) cannot yield mutually independent scalar variables. With the two proposed transformations, a highly negatively correlated neutral vector can be transformed to a set of mutually independent scalar variables with the same degrees of freedom. We also evaluate the decorrelation performances for the vectors generated from a single Dirichlet distribution and a mixture of Dirichlet distributions. The mutual independence is verified with the distance correlation measurement. The advantages of the proposed decorrelation strategies are intensively studied and demonstrated with synthesized data and practical application evaluations