Model-based approach for household clustering with mixed scale variables

The Ministry of Social Development in Mexico is in charge of creating and assigning social programmes targeting specific needs in the population for the improvement of the quality of life. To better target the social programmes, the Ministry is aimed to find clusters of households with the same needs based on demographic characteristics as well as poverty conditions of the household. Available data consists of continuous, ordinal, and nominal variables, all of which come from a non-i.i.d complex design survey sample. We propose a Bayesian nonparametric mixture model that jointly models a set of latent variables, as in an underlying variable response approach, associated to the observed mixed scale data and accommodates for the different sampling probabilities. The performance of the model is assessed via simulated data. A full analysis of socio-economic conditions in households in the Mexican State of Mexico is presented

Flexible sampling of discrete data correlations without the marginal distributions

Learning the joint dependence of discrete variables is a fundamental problem in machine learning, with many applications including prediction, clustering and dimensionality reduction. More recently, the framework of copula modeling has gained popularity due to its modular parametrization of joint distributions. Among other properties, copulas provide a recipe for combining flexible models for univariate marginal distributions with parametric families suitable for potentially high dimensional dependence structures. More radically, the extended rank likelihood approach of Hoff (2007) bypasses learning marginal models completely when such information is ancillary to the learning task at hand as in, e.g., standard dimensionality reduction problems or copula parameter estimation. The main idea is to represent data by their observable rank statistics, ignoring any other information from the marginals. Inference is typically done in a Bayesian framework with Gaussian copulas, and it is complicated by the fact this implies sampling within a space where the number of constraints increases quadratically with the number of data points. The result is slow mixing when using off-the-shelf Gibbs sampling. We present an efficient algorithm based on recent advances on constrained Hamiltonian Markov chain Monte Carlo that is simple to implement and does not require paying for a quadratic cost in sample size.Comment: An overhauled version of the experimental section moved to the main paper. Old experimental section moved to supplementary materia

Spatial probit models for multivariate ordinal data: computational efficiency and parameter identifiability

2013 Summer.Includes bibliographical references.The Colorado Natural Heritage Program (CNHP) at Colorado State University evaluates Colorado's rare and at-risk species and habitats and promotes conservation of biological resources. One of the goals of the program is to determine the condition of wetlands across the state of Colorado. The data collected are measurements, or metrics, representing landscape condition, biotic condition, hydrologic condition, and physiochemical condition in river basins statewide. The metrics differ in variable type, including binary, ordinal, count, and continuous response data. It is common practice to uniformly discretize the metrics into ordinal values and combine them using a weighted-average to obtain a univariate measure of wetland condition. The weights assigned to each metric are based on best professional judgement. The motivation of this work was to improve on the user-defined weights by developing a statistical model to estimate the weights using observed data. The challenges of creating a model that fulfills this requirement are many. First, the observed data are multivariate and consist of different variable types which we wish to preserve. Second, the multivariate response data are not independent across river basin because wetlands at close proximity are correlated. Third, we want the model to provide a univariate measure of wetland condition that can be compared across the state. Lastly, it is of interest to the ecologists to predict the univariate measure of wetland condition at unobserved locations requiring covariate information to be incorporated into the model. We propose a multivariate multilevel latent variable model to address these challenges. Latent continuous response variables are used to model the different types of response variables. An additional latent variable, or common factor, is used as a univariate measure of wetland condition. The mean of the common factor contains observable covariate data in order to predict at unobserved locations. The variance of the common factor is defined by a spatial covariance function to account for the dependence between wetlands. The majority of the metrics reported by the CNHP are ordinal. Therefore, our primary focus is modeling multivariate ordinal response data where binary data is a special case. Probit linear models and probit linear mixed models are examples of models for ordinal response data. Probit models are attractive in that they can be defined in terms of latent variables. Computational efficiency is a major issue when fitting multivariate latent variable models in a Bayesian framework using Markov chain Monte Carlo (MCMC). There is also a high computation cost for running MCMC when fitting geostatistical spatial models. Data augmentation and parameter expansion are both modeling techniques that can lead to optimal iterative sampling algorithms for MCMC. Data augmentation allows for simpler and more feasible simulation from a posterior distribution. Parameter expansion is a method for accelerating convergence of iterative sample algorithms and can enhance data augmentation algorithms. We propose data augmentation and parameter-expanded data augmentation algorithms for fitting MCMC to spatial probit models for binary and ordinal response data. Parameter identifiability is another challenge when fitting multivariate latent variable models due to the multivariate response data, number of parameters, unobserved latent variables, and spatial random effects. We investigate parameter identifiability for the common factor model for multivariate ordinal response data. We extend the common factor model to include covariates and spatial correlation so we can predict wetland condition at unobserved locations. The partial sill and range parameter of a spatial covariance function are difficult to estimate because they are near-nonidentifiable. We propose a new parameterization for the covariance function of the spatial probit model that leads to better mixing and faster convergence of the MCMC. Whereas our spatial probit model for ordinal response data follows the common factor model approach, there are other forms of the spatial probit model. We give a comprehensive comparison of two types of spatial probit models, which we refer to as the first-stage and second-stage spatial probit model. We discuss the implications of fitting each model and compare them in terms of their impact on parameter estimation and prediction at unobserved locations. We propose a new approximation for predicting ordinal response data that is both accurate and efficient. We apply the multivariate multilevel latent variable model to data collected in the North Platte and Rio Grande River Basins to evaluate wetland condition. We obtain statistically derived weights for each of the response metrics with confidence limits. Lastly, we predict the univariate measure of wetland condition at unobserved locations