Topics in Circular Statistics

by Jammalamadaka, S. Rao; Sengupta, Ambar

• ISBN13: 9789810237783
• ISBN10: 9810237782
• Edition: Disk
• Format: Hardcover
• Copyright: 2001-06-01
• Publisher: WORLD SCIENTIFIC PUB CO INC

Summary

This research monograph on circular data analysis covers some recent advances in the field, besides providing a brief introduction to, and a review of, existing methods and models. The primary focus is on recent research into topics such as change-point problems, predictive distributions, circular correlation and regression, etc. An important feature of this work is the S-plus subroutines provided for analyzing actual data sets. Coupled with the discussion of new theoretical research, the book should benefit both the researcher and the practitioner.

Table of Contents

Introduction	1	(25)
Introduction	1	(3)
Applications and Background	4	(5)
Some Examples	4	(3)
The Need for Appropriate Analysis	7	(2)
Descriptive Statistics	9	(16)
Measure of Center	11	(4)
Circular Distance and Measure of Dispersion	15	(6)
Higher Moments	21	(4)
Circular Probability Distributions	25	(40)
Introduction	25	(8)
Some Methods of Obtaining Circular Distributions	29	(4)
Circular Distributions	33	(19)
Uniform Distribution	33	(1)
Cardioid Distribution	34	(1)
A Triangular Distribution	34	(1)
Circular Normal (CN) Distribution	35	(8)
Offset Normal Distribution	43	(1)
Wrapped Normal (WN) Distribution	44	(1)
Wrapped Cauchy (WC) Distribution	45	(1)
General Wrapped Stable (WS) Distributions	46	(2)
Variations of the CN Distribution	48	(2)
A Circular Beta Model	50	(2)
Asymmetric Circular Distributions	52	(1)
Bivariate Circular Distributions	52	(2)
A Bivariate von Mises Distribution	52	(1)
Wrapped Bivariate Normal Distribution	53	(1)
Circular-Linear Distribution	53	(1)
Generation of Circular Random Variables	54	(2)
Appendix: Curved Exponential Families and CND	56	(9)
Some Sampling Distributions	65	(20)
Introduction	65	(1)
Generalized Pearson's Random Walk Problem	66	(4)
The General Case	67	(3)
Sampling Distributions for CN Distribution	70	(3)
Distribution of (C, S)	71	(1)
Distribution of R	71	(2)
Distribution of V	73	(1)
Some Large Sample Results	73	(7)
Series Approximations for R and V	74	(1)
Central Limit Type Results	75	(1)
Large Sample Results for Statistics Based on Moments	76	(2)
First Significant Digit Phenomenon	78	(2)
Two or More Samples	80	(2)
Approximate Distributions for Large K	82	(3)
Estimation of Parameters	85	(22)
Introduction	85	(1)
CN Distribution	85	(5)
Estimating the Parameters of a CN Distribution	85	(3)
Optimal Properties of the MLEs	88	(2)
CN Mixtures	90	(1)
ML Estimation for the WC Distribution	91	(3)
Circular Beta Distribution	94	(1)
WS Family	95	(1)
Confidence Intervals	96	(1)
Appendix: Proofs	97	(10)
Tests for Mean Direction and Concentration	107	(24)
Introduction	107	(1)
Single Population	108	(17)
Tests for Mean Direction	108	(12)
Higher-Order Power Comparison	120	(3)
Tests for the Concentration Parameter	123	(2)
Two or More Populations	125	(6)
Comparing Mean Directions and Approximate ANOVA	125	(3)
Tests for Concentration Parameters	128	(3)
Tests for Uniformity	131	(20)
Introduction	131	(1)
Uniformity against WS Alternatives	132	(4)
LMP Test Against the WS Family	133	(1)
Monotonicity of Power Function	134	(1)
Consistency and Other Optimal Properties	135	(1)
Uniformity versus WSM Alternatives	136	(5)
Monotonicity of the Power Function	137	(2)
Consistency and Other Optimal Properties	139	(1)
Optimal Test with Unknown p	139	(2)
LMP Invariant Test for Unknown μ	141	(5)
Monotonicity of the Power Function	143	(2)
Consistency and Other Optimal Properties	145	(1)
Appendix: Proofs	146	(5)
Nonparametric Testing Procedures	151	(24)
Introduction	151	(1)
One-Sample Problem and Goodness-of-fit	151	(16)
Tests Based on Empirical Distribution Functions	152	(5)
χ2 and Other Tests	157	(4)
Tests Based on Sample Arc-Lengths	161	(6)
Two-Sample Problems	167	(4)
Two-Sample Tests based on Edf's	167	(2)
Wheeler and Watson Test	169	(1)
Tests based on Spacing-Frequencies	170	(1)
Multi-Sample Tests	171	(4)
Homogeneity Tests in Large Samples	172	(3)
Circular Correlation and Regression	175	(30)
Introduction	175	(1)
A Circular Correlation Measure, ρc	176	(7)
ρc for Some Parametric Models	180	(3)
Rank Correlation	183	(1)
Other Measures of Circular Correlation	184	(2)
Circular-Linear Correlation	186	(1)
Circular-Circular Regression	187	(8)
Estimation of Regression Coefficients	188	(3)
Determination of m	191	(4)
Circular-Linear Regression	195	(2)
Linear-Circular Regression	197	(2)
Testing Stochastic Independence of Circular Variables	199	(6)
Predictive Inference for Directional Data	205	(12)
Introduction	205	(2)
Prediction Under a von Mises Model	207	(2)
Generalized von Mises-type Distributions	209	(3)
l-Modal von Mises Distribution	209	(3)
Bayesian Predictive Density	212	(2)
Case of Large k	213	(1)
Robustness Against Symmetric Wrapped Stable Family	214	(1)
HPD Computation and An Example	215	(2)
Outliers and Related Problems	217	(24)
Introduction	217	(1)
Diagnostic Tools	218	(1)
Tests for Mixtures	219	(13)
Introduction	219	(2)
The LMPI Test	221	(2)
Asymptotic Distribution of T	223	(4)
Monotonicity and Consistency Properties	227	(3)
Non-regular Cases	230	(2)
Slippage Problem and Outliers	232	(7)
Introduction	232	(1)
A Decision Theoretic Approach	233	(3)
The LRT	236	(1)
Simulations and An Example	237	(2)
Concluding Remarks	239	(2)
Change-point Problems	241	(26)
Introduction	241	(2)
Tests for Change in Mean Direction	243	(15)
k Known Case	243	(2)
k Unknown Case	245	(1)
Simulated Critical Values	246	(9)
Power Comparisons	255	(3)
Robustness	258	(1)
Tests for Change in μ and/or k	258	(2)
An Example	260	(3)
Nonparametric Approaches	263	(1)
Other Approaches	264	(3)
Miscellaneous Topics	267	(20)
Introduction	267	(1)
An Entropy-based Test for the CND	267	(5)
Classification and Discriminant Analysis	272	(2)
Factorial Designs	274	(4)
Likelihood Ratio Tests	276	(2)
Bayesian Analysis	278	(1)
Sequential Methods	279	(1)
Shape Analysis	280	(1)
Stochastic Processes and Time Series	280	(2)
Density Estimation	282	(1)
Periodic Smoothing Splines	283	(4)
A Some Facts on Bessel Functions	287	(4)
B How to Use the CircStats Package	291	(4)
Installation Instructions for the CircStats Library	291	(2)
Windows Version	291	(1)
UNIX Version	292	(1)
CircStats Functions	293	(2)
Bibliography	295	(16)
Author Index	311	(4)
Subject Index	315	(6)
Notations and Abbreviations Index	321