We introduce negative binomial matrix factorization (NBMF), a matrix factorization technique specially designed for analyzing over-dispersed count data. It can be viewed as an extension of Poisson matrix factorization (PF) perturbed by a multiplicative term which models exposure. This term brings a degree of freedom for controlling the dispersion, making NBMF more robust to outliers. We show that NBMF allows to skip traditional pre-processing stages, such as binarization, which lead to loss of information. Two estimation approaches are presented: maximum likelihood and variational Bayes inference. We test our model with a recommendation task and show its ability to predict user tastes with better precision than PF.
An automatic feed mechanism for conveying workpieces from a magazine to a position where they can be clamped between two rotatable spindles of a machine tool comprises a reciprocating workpiece carrier which moves radially relative to the spindle axis. In a first or loading position of the carrier, a workpiece retaining notch is aligned with the magazine to pick up a workpiece, and the carrier is movable forwardly toward a clamping position where the workpiece is placed into position between the temporarily axially separated spindles to be clamped therebetween. Means are also provided to assure that the workpiece and the machine tool are in a predetermined angular registry at the time of clamping. A number of methods have been proposed for dealing with extra-Poisson variation when doing regression analysis of count data. This paper studies negative-binomial regression models and examines efficiency and robustness properties of inference procedures based on them. The methods are compared with quasilikelihood methods. Plusieurs methodes ont ete proposees en vue de traiter le probleme de la variation extra-poissonnienne dans une analyse de regression pour donnees de denombrement. Cet article a pour objet l'etude de modeles de regression binomiale negative et se penche sur les proprietes d'efficacite et de robustesse des methodes inferentielles decoulant des modeles. Ces dernieres sont comparees aux methodes de quasi-vraisemblancce. In regression analysis of counts, a lack of simple and efficient algorithms for posterior computation has made Bayesian approaches appear unattractive and thus underdeveloped. We propose a lognormal and gamma mixed negative binomial (NB) regression model for counts, and present efficient closed-form Bayesian inference; unlike conventional Poisson models, the proposed approach has two free parameters to include two different kinds of random effects, and allows the incorporation of prior information, such as sparsity in the regression coefficients. By placing a gamma distribution prior on the NB dispersion parameter r, and connecting a lognormal distribution prior with the logit of the NB probability parameter p, efficient Gibbs sampling and variational Bayes inference are both developed. The closed-form updates are obtained by exploiting conditional conjugacy via both a compound Poisson representation and a Polya-Gamma distribution based data augmentation approach. The proposed Bayesian inference can be implemented routinely, while being easily generalizable to more complex settings involving multivariate dependence structures. The algorithms are illustrated using real examples. Preface 1. Introduction 2. The concept of risk 3. Overview of count response models 4. Methods of estimation and assessment 5. Assessment of count models 6. Poisson regression 7. Overdispersion 8. Negative binomial regression 9. Negative binomial regression: modeling 10. Alternative variance parameterizations 11. Problems with zero counts 12. Censored and truncated count models 13. Handling endogeneity and latent class models 14. Count panel models 15. Bayesian negative binomial models Appendix A. Constructing and interpreting interactions Appendix B. Data sets and Stata files References Index.