Multivariate Bernoulli distribution

In this paper, we consider the multivariate Bernoulli distribution as a model to estimate the structure of graphs with binary nodes. This distribution is discussed in the framework of the exponential family, and its statistical properties regarding independence of the nodes are demonstrated. Importantly the model can estimate not only the main effects and pairwise interactions among the nodes but also is capable of modeling higher order interactions, allowing for the existence of complex clique effects. We compare the multivariate Bernoulli model with existing graphical inference models - the Ising model and the multivariate Gaussian model, where only the pairwise interactions are considered. On the other hand, the multivariate Bernoulli distribution has an interesting property in that independence and uncorrelatedness of the component random variables are equivalent. Both the marginal and conditional distributions of a subset of variables in the multivariate Bernoulli distribution still follow the multivariate Bernoulli distribution. Furthermore, the multivariate Bernoulli logistic model is developed under generalized linear model theory by utilizing the canonical link function in order to include covariate information on the nodes, edges and cliques. We also consider variable selection techniques such as LASSO in the logistic model to impose sparsity structure on the graph. Finally, we discuss extending the smoothing spline ANOVA approach to the multivariate Bernoulli logistic model to enable estimation of non-linear effects of the predictor variables.Comment: Published in at http://dx.doi.org/10.3150/12-BEJSP10 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Interactive high fidelity visualization of complex materials on the GPU

Documento submetido para revisão pelos pares. A publicar em Computers & Graphics. ISSN 0097-8493. 37:7 (nov. 2013) p. 809–819High fidelity interactive rendering is of major importance for footwear designers, since it allows experimenting with virtual prototypes of new products, rather than producing expensive physical mock-ups. This requires capturing the appearance of complex materials by resorting to image based approaches, such as the Bidirectional Texture Function (BTF), to allow subsequent interactive visualization, while still maintaining the capability to edit the materials' appearance. However, interactive global illumination rendering of compressed editable BTFs with ordinary computing resources remains to be demonstrated. In this paper we demonstrate interactive global illumination by using a GPU ray tracing engine and the Sparse Parametric Mixture Model representation of BTFs, which is particularly well suited for BTF editing. We propose a rendering pipeline and data layout which allow for interactive frame rates and provide a scalability analysis with respect to the scene's complexity. We also include soft shadows from area light sources and approximate global illumination with ambient occlusion by resorting to progressive refinement, which quickly converges to an high quality image while maintaining interactive frame rates by limiting the number of rays shot per frame. Acceptable performance is also demonstrated under dynamic settings, including camera movements, changing lighting conditions and dynamic geometry.Work partially funded by QREN project nbr. 13114 TOPICShoe and by National Funds through the FCT - Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) within projectPEst-OE/EEI/UI0752/2011

Better subset regression

To find efficient screening methods for high dimensional linear regression models, this paper studies the relationship between model fitting and screening performance. Under a sparsity assumption, we show that a subset that includes the true submodel always yields smaller residual sum of squares (i.e., has better model fitting) than all that do not in a general asymptotic setting. This indicates that, for screening important variables, we could follow a "better fitting, better screening" rule, i.e., pick a "better" subset that has better model fitting. To seek such a better subset, we consider the optimization problem associated with best subset regression. An EM algorithm, called orthogonalizing subset screening, and its accelerating version are proposed for searching for the best subset. Although the two algorithms cannot guarantee that a subset they yield is the best, their monotonicity property makes the subset have better model fitting than initial subsets generated by popular screening methods, and thus the subset can have better screening performance asymptotically. Simulation results show that our methods are very competitive in high dimensional variable screening even for finite sample sizes.Comment: 24 pages, 1 figur

The composite absolute penalties family for grouped and hierarchical variable selection

Extracting useful information from high-dimensional data is an important focus of today's statistical research and practice. Penalized loss function minimization has been shown to be effective for this task both theoretically and empirically. With the virtues of both regularization and sparsity, the $L_1$-penalized squared error minimization method Lasso has been popular in regression models and beyond. In this paper, we combine different norms including $L_1$ to form an intelligent penalty in order to add side information to the fitting of a regression or classification model to obtain reasonable estimates. Specifically, we introduce the Composite Absolute Penalties (CAP) family, which allows given grouping and hierarchical relationships between the predictors to be expressed. CAP penalties are built by defining groups and combining the properties of norm penalties at the across-group and within-group levels. Grouped selection occurs for nonoverlapping groups. Hierarchical variable selection is reached by defining groups with particular overlapping patterns. We propose using the BLASSO and cross-validation to compute CAP estimates in general. For a subfamily of CAP estimates involving only the $L_1$ and $L_{\infty}$ norms, we introduce the iCAP algorithm to trace the entire regularization path for the grouped selection problem. Within this subfamily, unbiased estimates of the degrees of freedom (df) are derived so that the regularization parameter is selected without cross-validation. CAP is shown to improve on the predictive performance of the LASSO in a series of simulated experiments, including cases with $p\gg n$ and possibly mis-specified groupings. When the complexity of a model is properly calculated, iCAP is seen to be parsimonious in the experiments.Comment: Published in at http://dx.doi.org/10.1214/07-AOS584 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org