Recent advances in directional statistics

Mainstream statistical methodology is generally applicable to data observed in Euclidean space. There are, however, numerous contexts of considerable scientific interest in which the natural supports for the data under consideration are Riemannian manifolds like the unit circle, torus, sphere and their extensions. Typically, such data can be represented using one or more directions, and directional statistics is the branch of statistics that deals with their analysis. In this paper we provide a review of the many recent developments in the field since the publication of Mardia and Jupp (1999), still the most comprehensive text on directional statistics. Many of those developments have been stimulated by interesting applications in fields as diverse as astronomy, medicine, genetics, neurology, aeronautics, acoustics, image analysis, text mining, environmetrics, and machine learning. We begin by considering developments for the exploratory analysis of directional data before progressing to distributional models, general approaches to inference, hypothesis testing, regression, nonparametric curve estimation, methods for dimension reduction, classification and clustering, and the modelling of time series, spatial and spatio-temporal data. An overview of currently available software for analysing directional data is also provided, and potential future developments discussed.Comment: 61 page

High-dimensional Structured Additive Regression Models: Bayesian Regularisation, Smoothing and Predictive Performance

Data structures in modern applications frequently combine the necessity of flexible regression techniques such as nonlinear and spatial effects with high-dimensional covariate vectors. While estimation of the former is typically achieved by supplementing the likelihood with a suitable smoothness penalty, the latter are usually assigned shrinkage penalties that enforce sparse models. In this paper, we consider a Bayesian unifying perspective, where conditionally Gaussian priors can be assigned to all types of regression effects. Suitable hyperprior assumptions on the variances of the Gaussian distributions then induce the desired smoothness or sparseness properties. As a major advantage, general Markov chain Monte Carlo simulation algorithms can be developed that allow for the joint estimation of smooth and spatial effects and regularised coefficient vectors. Two applications demonstrate the usefulness of the proposed procedure: A geoadditive regression model for data from the Munich rental guide and an additive probit model for the prediction of consumer credit defaults. In both cases, high-dimensional vectors of categorical covariates will be included in the regression models. The predictive ability of the resulting high-dimensional structure additive regression models compared to expert models will be of particular relevance and will be evaluated on cross-validation test data