Angular Statistics

A V Dattatreya Rao, S V S Girija

• Publisher: Chapman and Hall/CRC
• Year: 2019
• Print ISBN: 9780367030001
• eBook ISBN: 9781000739930
• Edition: 1
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

Directional data arise in the form of circular / semicircular / axial, symmetric / asymmetric, uni / bimodal data, in practical situations of varied fields. For the purpose of modeling such kind of data sets, the data scientists found that existing models as inadequate. As there is paucity of angular models, and to fill the gap, this book is designed at constructing new angular models with the existing techniques and to develop new tools of constructing angular models with an application to control charts in angular models.

This book is planned to cover the following topics in nine chapters

• Wrapped, stereographic and offset circular models
• Construction of angular models using Rising Sun function, positive definite sequences, discretization and through differential approach
• Extemporaneous Semicircular / arc and asymmetric l – axial models
• Choice of angular models as an inferential aspect and construction of control charts for angular data as an application are presented.

This graduate level book will be useful for data scientists, researchers and research students of Statistics and allied fields.

1

Introduction to Angular Data and Descriptive Statistics

1.1 Introduction to Circular Data

The observations in several diverse scientific fields are “directions.” Such directions arise in statistical problems where the data is in the form of angular measurements giving orientations or angles in the plane (two dimensional) or in space (three or more dimensional). A set of such observations on directions is referred to as “DIRECTIONAL DATA”; in particular, directional data of two dimensions is called “CIRCULAR DATA.”

If the directional data consists of only two-dimensional observations, then it is called circular data. Since it deals with two-dimensional angular data, the words “angular data” and “circular data” are used interchangeably. Since the direction has no magnitude, these can be conveniently represented as points on the circumference of a unit circle centered at the origin or as unit vectors connecting the origin to these points.

Directional statistics is an important branch of statistics for analyzing and modeling angular data. Angular data have many novel and distinctive characteristics and challenges in terms of its treatment and statistical analysis. These special features make angular data analysis distinct from the standard linear statistical analysis.

Circular data arise in many scientific fields, including Biology, Geology, Geography, Meteorology, and Physics. Many examples of circular data coming from widely differing scientific fields are detailed in the book Directional Statistics by Mardia and Jupp (2000).

The frequently quoted examples for the angular data are the measurements obtained from the clock and the compass. Typical examples of angular data from a compass are the measurements of geographical directions, wind directions, orientations of birds and animals, orientation of geological phenomena, etc. By taking a calendar year as a one-year clock, the data, such as number of tourists visiting a country in a year or the number of cases of a specified disease diagnosed at a hospital, can be considered as angular data.

1.2 Applications and Nature of Angular Data Sets

Directional data arise in various natural, physical, and social sciences in different kinds of phenomena, such as circular/semicircular/axial, symmetric/asymmetric, and unimodal/bimodal data. The following are some of the applications of angular statistics and the methods of circular statistics in analyzing the angular data.

1.2.1 Long-Axis Orientations of Feldspar Laths (Semicircular) Data

Measurement of long-axis orientations of Feldspar laths: the data of 133 directional measurements of feldspar laths in basalt reported by Smith (1988) and presented by Fisher (1993) on page 240 (in degrees) are as follows (Figure 1.1):

176 162 49 174 174 49 54 63 59 61 66 104 97 58 121 5 178 3 168 0 18 39 140 63 55 170 169 37 152 73 53 176 72 170 113 56 87 161 164 21 50 6 59 140 54 64 56 38 61 143 51 144 148 44 60 98 86 145 38 168 39 134 68 57 129 68 132 82 54 119 131 50 93 160 127 124 65 108 52 61 86...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Table of Contents
• Foreword
• Preface
• Acknowledgments
• Authors
• 1. Introduction to Angular Data and Descriptive Statistics
• 2. Wrapped Circular Models
• 3. Stereographic Circular Models
• 4. Offset Circular Models
• 5. Angular Models with New Techniques
• 6. Extemporaneous Semicircular/Axial Models
• 7. Asymmetric l-Axial Models
• 8. Choice of Angular Models
• 9. Control Charts for Angular Data
• Appendix
• Index