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

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

Nonparametric inference with directional and linear data

The term directional data refers to data whose support is a circumference, a sphere or, generally, an hypersphere of arbitrary dimension. This kind of data appears naturally in several applied disciplines: proteomics, environmental sciences, biology, astronomy, image analysis or text mining. The aim of this thesis is to provide new methodological tools for nonparametric inference with directional and linear data (i.e., usual Euclidean data). Nonparametric methods are obtained for both estimation and testing, for the density and the regression curves, in situations where directional random variables are present, that is, directional, directional-linear and directional-directional random variables. The main contributions of the thesis are collected in six papers briefly described in what follows. In García-Portugués et al. (2013a) different ways of estimating circular-linear and circularcircular densities via copulas are explored for an environmental application. A new directionallinear kernel density estimator is introduced in García-Portugués et al. (2013b) together with its basic properties. Three new bandwidth selectors for the kernel density estimator with directional data are given in García-Portugués (2013) and compared with the available ones. The directional-linear estimator is used in García-Portugués et al. (2014a) for constructing an independence test for directional and linear variables that is applied to study the dependence between wildfire orientation and size. In García-Portugués et al. (2014b) a central limit theorem for the integrated squared error of the directional-linear estimator is presented. This result is used to derive the asymptotic distribution of the independence test and of a goodness-of-fit test for parametric directional-linear and directional-directional densities. Finally, a local linear estimator with directional predictor and linear response is given in García-Portugués et al. (2014) jointly with a goodness-of-fit test for parametric regression functions

An introduction to statistical methods for circular data

Angles, directions, events, occurrences along time... all of them can be viewed as data on a circle (circular data). The particular nature of this type of data requires specific and adapted inferential and modelling procedures. Although there are quite a few references on this topic, and despite circular data are quite common in many applied sciences, they are frequently overlooked. This brief introduction aims to give the reader just some basic ideas on circular data analysis (with some mentions to the general case of spherical or directional data), providing some relevant references and tools for their application in practiceS

Statistical characterisation of wind fields over complex terrain with applications in bushfire modelling

The propagation of bushfire across the landscape is dependent on a variety of environmental factors, but the wind, in particular, has a major effect on both the speed and direction of fire propagation. As such, bushfire spread models, which underpin successful bushfire management, require accurate knowledge of the pattern of winds across the landscape. This can be problematic over complex terrain where winds exhibit considerable spatial variability due to wind-terrain interactions, and where detailed measurements of wind characteristics are comparatively rare. This thesis contributes two new wind datasets to address the previous lack of data available to develop and validate wind models over complex terrain. It also details analyses that focus on the statistical characterisation of wind as joint wind direction distributions, which represent the directional wind response to changing topography and surface roughness. A novel method for toroidal surface fitting is introduced and implemented to estimate the true continuous response from discrete observed data. This new method, which relies on a conceptually simple adaptation of planar techniques, is compared to the limited range of available toroidal surface estimation techniques and is shown to perform as well as, if not better than, these more sophisticated methods. Monte Carlo simulations are employed to highlight the sensitivity of statistical comparison tests to alternative distribution structures, and to validate bivariate and circular extensions of the Kolmogorov-Smirnov test. These tests are applied to directional wind response pairs, showing that vegetation regrowth has a significant but varying impact across complex terrain. Finally, this thesis demonstrates how statistical approaches can be used to complement current physics-based wind modelling methods. The resulting probabilistic representations provide more accurate predictions of wind direction variability, and are better suited to emerging ensemble-based bushfire prediction frameworks. As such, they provide a superior characterisation of uncertainty across the fire modelling process; ultimately enabling fire managers to make more informed decisions

Measures of goodness of fit obtained by almost-canonical transformations on Riemannian manifolds

The standard method of transforming a continuous distribution on the line to the uniform distribution on $[0,1]$ is the probability integral transform. Analogous transforms exist on compact Riemannian manifolds, \scr X, in that for each distribution with continuous positive density on \scr X, there is a continuous mapping of \scr X to itself that transforms the distribution into the uniform distribution. In general, this mapping is far from unique. This paper introduces the construction of an almost-canonical version of such a probability integral transform. The construction is extended to shape spaces, Cartan–Hadamard manifolds, and simplices. The probability integral transform is used to derive tests of goodness of fit from tests of uniformity. Illustrative examples of these tests of goodness of fit are given involving (i) Fisher distributions on $S^2$, (ii) isotropic Mardia–Dryden distributions on the shape space $Σ^5_2$. Their behaviour is investigated by simulation