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

Fast DD-classification of functional data

A fast nonparametric procedure for classifying functional data is introduced. It consists of a two-step transformation of the original data plus a classifier operating on a low-dimensional hypercube. The functional data are first mapped into a finite-dimensional location-slope space and then transformed by a multivariate depth function into the $DD$-plot, which is a subset of the unit hypercube. This transformation yields a new notion of depth for functional data. Three alternative depth functions are employed for this, as well as two rules for the final classification on $[0,1]^q$. The resulting classifier has to be cross-validated over a small range of parameters only, which is restricted by a Vapnik-Cervonenkis bound. The entire methodology does not involve smoothing techniques, is completely nonparametric and allows to achieve Bayes optimality under standard distributional settings. It is robust, efficiently computable, and has been implemented in an R environment. Applicability of the new approach is demonstrated by simulations as well as a benchmark study

Building nonparametric nn-body force fields using Gaussian process regression

Constructing a classical potential suited to simulate a given atomic system is a remarkably difficult task. This chapter presents a framework under which this problem can be tackled, based on the Bayesian construction of nonparametric force fields of a given order using Gaussian process (GP) priors. The formalism of GP regression is first reviewed, particularly in relation to its application in learning local atomic energies and forces. For accurate regression it is fundamental to incorporate prior knowledge into the GP kernel function. To this end, this chapter details how properties of smoothness, invariance and interaction order of a force field can be encoded into corresponding kernel properties. A range of kernels is then proposed, possessing all the required properties and an adjustable parameter $n$ governing the interaction order modelled. The order $n$ best suited to describe a given system can be found automatically within the Bayesian framework by maximisation of the marginal likelihood. The procedure is first tested on a toy model of known interaction and later applied to two real materials described at the DFT level of accuracy. The models automatically selected for the two materials were found to be in agreement with physical intuition. More in general, it was found that lower order (simpler) models should be chosen when the data are not sufficient to resolve more complex interactions. Low $n$ GPs can be further sped up by orders of magnitude by constructing the corresponding tabulated force field, here named "MFF".Comment: 31 pages, 11 figures, book chapte