Modern Directional Statistics: 1st Edition (Hardback) book cover

Modern Directional Statistics

1st Edition

By Christophe Ley, Thomas Verdebout

Chapman and Hall/CRC

176 pages

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Hardback: 9781498706643
pub: 2017-09-15
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Description

Modern Directional Statistics collects important advances in methodology and theory for directional statistics over the last two decades. It provides a detailed overview and analysis of recent results that can help both researchers and practitioners. Knowledge of multivariate statistics eases the reading but is not mandatory.

The field of directional statistics has received a lot of attention over the past two decades, due to new demands from domains such as life sciences or machine learning, to the availability of massive data sets requiring adapted statistical techniques, and to technological advances. This book covers important progresses in distribution theory,high-dimensional statistics, kernel density estimation, efficient inference on directional supports, and computational and graphical methods.

Christophe Ley is professor of mathematical statistics at Ghent University. His research interests include semi-parametrically efficient inference, flexible modeling, directional statistics and the study of asymptotic approximations via Stein’s Method. His achievements include the Marie-Jeanne Laurent-Duhamel prize of the Société Française de Statistique and an elected membership at the International Statistical Institute. He is associate editor for the journals Computational Statistics & Data Analysis and Econometrics and Statistics.

Thomas Verdebout is professor of mathematical statistics at Université libre de Bruxelles (ULB). His main research interests are semi-parametric statistics, high- dimensional statistics, directional statistics and rank-based procedures. He has won an annual prize of the Belgian Academy of Sciences and is an elected member of the International Statistical Institute. He is associate editor for the journals Statistics and Probability Letters and Journal of Multivariate Analysis.

Reviews

"The book is definitely handy for researchers and graduate students in statistics as well as for scientists and practical users in bioscience, ecological and environmental sciences, social sciences and other applied areas where directional data analysis is needed and even high-dimensional data analytics is encountered." ~Shuangzhe Liu, Stat Papers

Table of Contents

Introduction

Overview

A brief introduction to directional statistics

A brief outline of the theoretical advances presented in this book

Directional datasets

Paleomagnetism

Political sciences

Text mining

Wildfire orientation

Life sciences and bioinformatics

Basics and notations

Plan of the book

Advances in flexible parametric distribution theory

Introduction

Flexible parametric modeling: an active research area on RP

Organization of the remainder of the chapter

Flexible circular distributions

Four ways to construct circular densities

The classics: von Mises, cardioid and wrapped Cauchy distributions

Beyond the classics modern flexible circular modeling

Flexible modeling of symmetric data: the Jones-Pewsey distribution

Sine-skewing: a simple tool to skew any symmetric distribution

Skewness combined with unimodality: the scale-transforming approach

A general device for building symmetric bipolar distributions

A brief description of three other flexible models

Flexible spherical distributions

Classical spherical distributions

Rotationally symmetric distributions

A general method to skew-rotationally symmetric distributions

Flexible toroidal and cylindrical distributions

Some history, motivations and goals

The bivariate von Mises distribution and its variants

Mardia-Sutton type cylindrical distributions

Johnson-Wehrly type cylindrical distributions

The copula approach

Further reading

Advances in kernel density estimation on directional supports

Introduction

Kernel density estimation on the real line

Organization of the remainder of the chapter

Definitions and main properties

Spherical kernel density estimation

Cylindrical kernel density estimation

A delicate yet crucial issue: bandwidth choice

Spherical AMISB and bandwidth selection

Rule of thumb based on the FvML distribution

A gain in generality: AMISE via mixtures of FvML densities

Three further proposals

Bandwidth selection in the cylindrical setting

Inferential procedures

Non-parametric goodness-of-fit test for directional data

Non-parametric independence test for cylindrical data

An overview of non-parametric regression

Further reading

Computational and graphical methods

Ordering data on the sphere: quantiles and depth functions

Ordering on R and RP, and organization of the remainder of the section

Classical depth functions on the sphere

Projected quantiles and related asymptotic results

The angular Mahalanobis depth

Statistical procedures based on projected quantities and the angular Mahalanobis depth

Statistical inference under order restrictions on the circle

Isotomic regression estimation and organization of the remainder of the section

Order restrictions on the circle

Circular isotonic regression

Exploring data features with the CircSiZer

The SiZer, scale space theory and organization of the remainder of the section

The CircSiZer

Kernel choice based on causality: the special role of the wrapped normal

Computationally fast estimation for high-dimensional FvML distributions

Maximum likelihood expressions for the parameters of FvML distributions and organization of the section

Approximations for the concentration parameter from Mardia & Jupp (200) and their limitations in high dimensions

New (high-dimensional) approximations for the concentration parameter

Further reading

Local asymptotic normality for directional data

Introduction

The LAN property on R and its deep impact on asymptotic statistics

Organization of the remainder of the chapter

Local asymptotic normality and optimal testing

Contiguity

Local asymptotic normality

Optimal testing in LAN experiments

LAN, semiparametric efficiency and invariance

LAN for directional data

The Le Cam methodology for curved experiments and associated efficient tests

LAN property for rotationally symmetric distributions

Application 1: Optimal inference based on signed-ranks

Application 2: ANOVA on spheres

Application 3: Asymptotic power of tests of concentration

Further reading

Recent results for tests of uniformity and symmetry

Introduction

Organization of the remainder of the chapter

Recent advances concerning the Rayleigh test of uniformity

Sobolev tests of uniformity

Uniformity tests based on random projections

Testing for uniformity with noisy data

Tests of reflective symmetry on the circle

Tests of rotational symmetry on hyperspheres

Testing for spherical location in the vicinity of the uniform distribution

Further reading

High-dimensional directional statistics

Introduction

High-dimensional techniques in RP

Organization of the remainder of the chapter

Distributions on high-dimensional spheres

Testing uniformity in the high-dimensional case

Location tests in the high-dimensional case

Concentration tests in the high-dimensional case

Principal nested spheres

Further reading

About the Authors

Christophe Ley is professor of mathematical statistics at Ghent University. His research interests include semi-parametrically efficient inference, flexible modeling, directional statistics and the study of asymptotic approximations via Stein’s Method. His achievements include the Marie-Jeanne Laurent-Duhamel prize of the Société Française de Statistique and an elected membership at the International Statistical Institute. He is associate editor for the journals Computational Statistics & Data Analysis and Econometrics and Statistics.

Thomas Verdebout is professor of mathematical statistics at Université libre de Bruxelles (ULB). His main research interests are semi-parametric statistics, high- dimensional statistics, directional statistics and rank-based procedures. He has won an annual prize of the Belgian Academy of Sciences and is an elected member of the International Statistical Institute. He is associate editor for the journals Statistics and Probability Letters and Journal of Multivariate Analysis.

About the Series

Chapman & Hall/CRC Interdisciplinary Statistics

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Subject Categories

BISAC Subject Codes/Headings:
COM037000
COMPUTERS / Machine Theory
MAT029000
MATHEMATICS / Probability & Statistics / General
SCI008000
SCIENCE / Life Sciences / Biology / General