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

Some contributions to the analysis of skew data on the line and circle

In the first part of this thesis we consider the skew-normal class of distributions on the line and its limiting general half-normal distribution. Inferential procedures based on the methods of moments and maximum likelihood are developed and their performance assessed using simulation. Data on the strength of glass fibre and the body fat of elite athletes are used to illustrate some of the inferential issues raised. The second part of the thesis is devoted to a consideration of the analysis of skew circular data. First, we derive the large-sample distribution of certain key circular statistics and show how this result provides a basis for inference for the corresponding population measures. Next, tests for circular reflective symmetry about an unknown central direction are investigated. A large-sample test and computer intensive variants of it are developed, and their operating characteristics explored both theoretically and empirically. Subsequently, we consider tests for circular reflective symmetry about a known or specified median axis. Two new procedures are developed for testing for symmetry about a known median axis against skew alternatives, and their operating characteristics compared in a simulation experiment with those of the circular analogues of three linear tests. On the basis of the results obtained from the latter, a simple testing strategy is identified. The performance of the tests against rotation alternatives is also investigated. Throughout, the use of the various tests of symmetry is illustrated using a wide range of circular data sets. Finally, we propose the wrapped skew-normal distribution on the circle as a potential model for circular data. The distribution’s fundamental properties are presented and inference based on the methods of moments and maximum likelihood is explored. Tests for limiting cases of the class are proposed, and a potential use of the distribution is illustrated in the mixture based modelling of data on bird migration

Sinh-arcsinh distributions

We introduce the sinh-arcsinh transformation and hence, by applying it to a generating distribution with no parameters other than location and scale, usually the normal, a new family of sinh-arcsinh distributions. This four-parameter family has symmetric and skewed members and allows for tailweights that are both heavier and lighter than those of the generating distribution. The central place of the normal distribution in this family affords likelihood ratio tests of normality that are superior to the state-of-the-art in normality testing because of the range of alternatives against which they are very powerful. Likelihood ratio tests of symmetry are also available and are very successful. Three-parameter symmetric and asymmetric subfamilies of the full family are also of interest. Heavy-tailed symmetric sinh-arcsinh distributions behave like Johnson SU distributions, while their light-tailed counterparts behave like sinh-normal distributions, the sinh-arcsinh family allowing a seamless transition between the two, via the normal, controlled by a single parameter. The sinh-arcsinh family is very tractable and many properties are explored. Likelihood inference is pursued, including an attractive reparameterization. Illustrative examples are given. A multivariate version is considered. Options and extensions are discussed

２変量角度データのための確率分布

Open House, ISM in Tachikawa, 2012.6.15統計数理研究所オープンハウス（立川）、H24.6.15ポスター発

Tractable circula densities from Fourier series

This article proposes an approach, based on infinite Fourier series, to constructing tractable densities for the bivariate circular analogues of copulas recently coined ‘circulas’. As examples of the general approach, we consider circula densities generated by various patterns of nonzero Fourier coefficients. The shape and sparsity of such arrangements are found to play a key role in determining the properties of the resultant models. The special cases of the circula densities we consider all have simple closed-form expressions involving no computationally demanding normalizing constants and display wide-ranging distributional shapes. A highly successful model identification tool and methods for parameter estimation and goodness-of-fit testing are provided for the circula densities themselves and the bivariate circular densities obtained from them using a marginal specification construction. The modelling capabilities of such bivariate circular densities are compared with those of five existing models in a numerical experiment, and their application illustrated in an analysis of wind directions

Inverse Batschelet distributions for circular data

We provide four-parameter families of distributions on the circle which are unimodal and display the widest ranges of both skewness and peakedness yet available. Our approach is to transform the scale of a generating distribution, such as the von Mises, using various nontrivial extensions of an approach first used in Batschelet’s (1981, Circular Statistics in Biology) book. The key is to employ inverses of Batschelet-type transformations in certain ways; these exhibit considerable advantages over direct Batschelet transformations. The skewness transformation is especially appealing as it has no effect on the normalizing constant. As well as a variety of interesting theoretical properties, when likelihood inference is explored these distributions display orthogonality between elements of a pairing of parameters into (location, skewness) and (concentration, peakedness). Further, the location parameter can sometimes be made approximately orthogonal to all the other parameters. Profile likelihoods come to the fore in practice. Two illustrative applications, one concerning the locomotion of a Drosophila fly larva, the other analyzing a large set of sudden infant death syndrome data, are investigated

Problems of inference for Azzalini's skewnormal distribution

This paper considers various unresolved inference problems for the skewnormal distribution. We give reasons as to why the direct parameterization should not be used as a general basis for estimation, and consider method of moments and maximum likelihood estimation for the distribution's centred parameterization. Large sample theory results are given for the method of moments estimators, and numerical approaches for obtaining maximum likelihood estimates are discussed. Simulation is used to assess the performance of the two types of estimation. We also present procedures for testing for departures from the limiting folded normal distribution. Data on the percentage body fat of elite athletes are used to illustrate some of the issues raised.