Topics in Circular Statistics-Vol 5.

336 Pages·2001·10.541 MB·English

by  Jammalamadaka, S. Rao, Sengupta, Ashis

S e r i es on
Multivariate
Analysis
• Vol. 5 •
OPICS IN
/ •
CIRCULAR
STATISTICS
S Rao Jammalamadaka
A SeaGupta
World Scientific
TOPICS IN
CIRCULAR
STATISTICS
SERIES ON MULTIVARIATE ANALYSIS
Editor: M M Rao
Published
Vol. 1: Martingales and Stochastic Analysis
J. Yeh
Vol. 2: Multidimensional Second Order Stochastic Processes
Y. Kakihara
Vol. 3: Mathematical Methods in Sample Surveys
H. G. Tucker
Vol. 4: Abstract Methods in Information Theory
Y. Kakihara
Vol. 5: Topics in Circular Statistics
S. R. Jammalamadaka and A. SenGupta
Forthcoming
Convolution Structures and Stochastic Processes
R. Lasser
IffEUfflSkjui
Vol. 5
T
OPICS IN
CIRCULAR
STATISTICS
S Rao Jammalamadaka
Department oj Statistics and Applied Probability
University oj Calijornia, Santa Barbara
USA
A SenGupta
Applied Statistics Unit
Indian Statistical Institute
India
l I Sj World Scientific
wn SSiinnqgaappoorree* •N Neeww J Jeersrseeyy L • London • Hong Kong
Published by
World Scientific Publishing Co. Pte. Ltd.
P O Box 128, Farrer Road, Singapore 912805
USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publication Data
Jammalamadaka, S. Rao.
Topics in circular statistics / S. Rao Jammalamadaka, A. SenGupta.
p. cm. — (Series on multivariate analysis ; vol. 5)
Includes bibliographical references and indexes.
ISBN 9810237782
1. Circular data. 2. Mathematical statistics. I. SenGupta, Ambar, 1963- II. Title. III.
Series.
QA276 ,J36 2001
519.5-dc21 2001024071
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 2001 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof may not be reproduced in any form or by any means,
electronic or mechanical, including photocopying, recording or any information storage and retrieval
system now known or to be invented, without written permission from the Publisher.
Printed in Singapore by Uto-Print
Preface
This research monograph on circular data analysis covers recent theoretical
advances in the field after providing an introduction to this novel area. No
attempt has been made to cover spherical models or their analysis although
there are many essential common features with circular data analysis. The
first seven chapters provide an up-to-date, if not exhaustive, coverage of
circular modeling and analysis. The focus then shifts to more recent research
in this field in such areas as change-point problems, predictive distributions,
circular correlation and regression, classification and discriminant analysis,
etc., highlighting mainly the authors' own research interests and that of their
co-workers. The statistical methods in this area, as in many others, evolved
as a series of practical and sometimes ad-hoc tools. Whenever possible, we
attempt to analyze these methods in the framework of optimal statistical
inference, as for example, when we examine the circular normal distribution
as a member of the curved exponential family. Besides learning about the
novelty of circular data analysis and the essential techniques, we hope a
careful reader can pick up possible topics for future research on topics such
as optimal properties of procedures, modeling, and robustness. As stated
before, it is not our intent to provide comprehensive coverage of all the tools
and techniques in the field but to give an introduction to the area and describe
some current research.
Along with such theoretical discussions, we also provide a package of
computational subroutines in SPlus, called CircStats, which allows users
to analyze their own data sets. This important feature should make this
monograph very useful to practitioners of circular data analysis.
Chapter 1 starts with an introduction to the field while Chapter 2 provides
a discussion of circular models and methods for generating them. Sampling
distributions are covered in Chapter 3 and parametric estimation and test
ing in Chapters 4, 5, and 6. Chapter 7 covers nonparametric methods and
VI PREFACE
Chapter 8 discusses circular correlation and regression. Recent research on
predictive densities, outliers, and change point problems are covered in the
next three chapters, while the final chapter reviews some important miscel
laneous topics.
We have had considerable help in completing this work and we would
like to thank Dr. Kaushik Ghosh, A. Laha, C. Pal, Debashis Paul, Huaxin
You, and Anna Valeva for helping with IMgX and Dr. Saralees Nadarajah
for reading the manuscript. The package of computer routines, CircStats
based on SPlus, is the work of Dr. Ulric Lund whom we would like to thank
for letting us incorporate it as part of this monograph.
Contents
1 Introduction 1
1.1 Introduction 1
1.2 Applications and Background 4
1.2.1 Some Examples 4
1.2.2 The Need for Appropriate Analysis 7
1.3 Descriptive Statistics 9
1.3.1 Measure of Center 11
1.3.2 Circular Distance and Measure of Dispersion 15
1.3.3 Higher Moments 21
2 Circular Probability Distributions 25
2.1 Introduction 25
2.1.1 Some Methods of Obtaining Circular Distributions . . 29
2.2 Circular Distributions 33
2.2.1 Uniform Distribution 33
2.2.2 Cardioid Distribution 34
2.2.3 A Triangular Distribution 34
2.2.4 Circular Normal (CN) Distribution 35
2.2.5 Offset Normal Distribution 43
2.2.6 Wrapped Normal (WN) Distribution 44
2.2.7 Wrapped Cauchy (WC) Distribution 45
2.2.8 General Wrapped Stable (WS) Distributions 46
2.2.9 Variations of the CN Distribution 48
2.2.10 A Circular Beta Model 50
2.2.11 Asymmetric Circular Distributions 52
2.3 Bivariate Circular Distributions 52
2.3.1 A Bivariate von Mises Distribution 52
2.3.2 Wrapped Bivariate Normal Distribution 53
viii CONTENTS
2.3.3 Circular-Linear Distribution 53
2.4 Generation of Circular Random Variables 54
2.5 Appendix: Curved Exponential Families and CND 56
3 Some Sampling Distributions 65
3.1 Introduction 65
3.2 Generalized Pearson's Random Walk Problem 66
3.2.1 The General Case 67
3.3 Sampling Distributions for CN Distribution 70
3.3.1 Distribution of (C, S) 71
3.3.2 Distribution of R 71
3.3.3 Distribution of V 73
3.4 Some Large Sample Results 73
3.4.1 Series Approximations for R and V 74
3.4.2 Central Limit Type Results 75
3.4.3 Large Sample Results for Statistics Based on Moments 76
3.4.4 First Significant Digit Phenomenon 78
3.5 Two or More Samples 80
3.6 Approximate Distributions for Large K 82
4 Estimation of Parameters 85
4.1 Introduction 85
4.2 CN Distribution 85
4.2.1 Estimating the Parameters of a CN Distribution . . .. 85
4.2.2 Optimal Properties of the MLEs 88
4.3 CN Mixtures 90
4.4 ML Estimation for the WC Distribution 91
4.5 Circular Beta Distribution 94
4.6 WS Family 95
4.7 Confidence Intervals 96
4.8 Appendix: Proofs 97
5 Tests for Mean Direction and Concentration 107
5.1 Introduction 107
5.2 Single Population 108
5.2.1 Tests for Mean Direction 108
5.2.2 Higher-Order Power Comparison 120
5.2.3 Tests for the Concentration Parameter 123
CONTENTS ix
5.3 Two or More Populations 125
5.3.1 Comparing Mean Directions and Approximate ANOVA 125
5.3.2 Tests for Concentration Parameters 128
6 Tests for Uniformity 131
6.1 Introduction 131
6.2 Uniformity against WS Alternatives 132
6.2.1 LMP Test Against the WS Family 133
6.2.2 Monotonicity of Power Function 134
6.2.3 Consistency and Other Optimal Properties 135
6.3 Uniformity versus WSM Alternatives 136
6.3.1 Monotonicity of the Power Function 137
6.3.2 Consistency and Other Optimal Properties 139
6.3.3 Optimal Test with Unknown p 139
6.4 LMP Invariant Test for Unknown \i 141
6.4.1 Monotonicity of the Power Function 143
6.4.2 Consistency and Other Optimal Properties 145
6.5 Appendix: Proofs 146
7 Nonparametric Testing Procedures 151
7.1 Introduction 151
7.2 One-Sample Problem and Goodness-of-fit 151
7.2.1 Tests Based on Empirical Distribution Functions . . . 152
7.2.2 x2 and Other Tests 157
7.2.3 Tests Based on Sample Arc-Lengths 161
7.3 Two-Sample Problems 167
7.3.1 Two-Sample Tests based on Edf's 167
7.3.2 Wheeler and Watson Test 169
7.3.3 Tests based on Spacing-Frequencies 170
7.4 Multi-Sample Tests 171
7.4.1 Homogeneity Tests in Large Samples 172
8 Circular Correlation and Regression 175
8.1 Introduction 175
8.2 A Circular Correlation Measure, p 176
c
8.2.1 p for Some Parametric Models 180
c
8.3 Rank Correlation 183
8.4 Other Measures of Circular Correlation 184