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

Entropy-based goodness-of-fit tests for multivariate distributions

Entropy is one of the most basic and significant descriptors of a probability distribution. It is still a commonly used measure of uncertainty and randomness in information theory and mathematical statistics. We study statistical inference for Shannon and Rényi’s entropy-related functionals of multivariate Gaussian and Student-t distributions. This thesis investigates three themes. First, we provide a non-parametric test of goodness-of-fit for a class of multivariate generalized Gaussian distributions based on maximum entropy principle via using the k-th nearest neighbour (NN) distance estimator of the Shannon entropy. Its asymptotic unbiasedness and consistency are demonstrated. Second, we show a class of estimators of the Rényi entropy based on an independent identical distribution sample drawn from an unknown distribution f on R m. We focus on the maximum Rényi entropy principle for multivariate Student-t and Pearson type II distributions. We also consider the entropy-based test for multivariate Student-t distribution using the k-th NN distances estimator of entropy and employ the properties of entropy estimators derived from NN distances. Third, we introduce a new classes of unimodal rotational invariant directional distributions, which generalize von Mises-Fisher distribution. We propose three types of distributions in which one of them represents the axial data. We provide all of the formula together with a short computational study of parameter estimators for each new type via the method of moments and method of maximum likelihood. We also offer the goodness-of-fit test to detect that the sample entries follow one of the introduced generalized von Mises-Fisher distribution based on the maximum entropy principle using the k-th NN distances estimator of Shannon entropy and to prove its L2 -consistence

Harmonic analysis and distribution-free inference for spherical distributions

Fourier analysis, and representation of circular distributions in terms of their Fourier coefficients, is quite commonly discussed and used for model-free inference such as testing uniformity and symmetry, in dealing with 2-dimensional directions. However, a similar discussion for spherical distributions, which are used to model 3-dimensional directional data, is not readily available in the literature in terms of their harmonics. This paper, in what we believe is the first such attempt, looks at probability distributions on a unit sphere through the perspective of spherical harmonics, analogous to the Fourier analysis for distributions on a unit circle. Representation of any continuous spherical density in terms of spherical harmonics is given, and such series expansions provided for some commonly used spherical distributions, as well as for two new spherical distributions that are introduced. Through the prism of harmonic analysis, one can look at the mean direction, dispersion, and various forms of symmetry for these models in a nonparametric setting. Aspects of distribution-free inference such as estimation and large-sample tests for various symmetries, are provided, each type of symmetry being characterized through its harmonics. The paper concludes with a real-data example analyzing the longitudinal sunspot activity

Harmonic analysis and distribution-free inference for spherical distributions

Fourier analysis and representation of circular distributions in terms of their Fourier coefficients, is quite commonly discussed and used for model-free inference such as testing uniformity and symmetry etc. in dealing with 2-dimensional directions. However a similar discussion for spherical distributions, which are used to model 3-dimensional directional data, has not been fully developed in the literature in terms of their harmonics. This paper, in what we believe is the first such attempt, looks at the probability distributions on a unit sphere, through the perspective of spherical harmonics, analogous to the Fourier analysis for distributions on a unit circle. Harmonic representations of many currently used spherical models are presented and discussed. A very general family of spherical distributions is then introduced, special cases of which yield many known spherical models. Through the prism of harmonic analysis, one can look at the mean direction, dispersion, and various forms of symmetry for these models in a generic setting. Aspects of distribution free inference such as estimation and large-sample tests for these symmetries, are provided. The paper concludes with a real-data example analyzing the longitudinal sunspot activity.Comment: 26 pages, 2 figure