Directional Statistics

Transcription

1 Directional Statistics Kanti V. Mardia University of Leeds, UK Peter E. Jupp University of St Andrews, UK I JOHN WILEY & SONS, LTD Chichester New York Weinheim Brisbane Singapore Toronto

2 Contents Preface xix 1 Circular Data INTRODUCTION DIAGRAMMATICAL REPRESENTATION Ungrouped Data Grouped Data Axial Data FORMS OF FREQUENCY DISTRIBUTIONS Unimodal Distributions Multimodal Distributions FURTHER EXAMPLES OF DIRECTIONAL DATA Earth Sciences Meteorology Biology Physics Psychology Image Analysis Medicine Astronomy Wrapping and Projecting.S 12 2 Summary Statistics INTRODUCTION Preliminaries and Notation MEASURES OF LOCATION The Mean Direction The Median Direction MEASURES OF CONCENTRATION AND DISPERSION The Mean Resultant Length and the Circular Variance Decomposition of Dispersion The Circular Standard Deviation 19

3 Xll Other Measures of Dispersion TRIGONOMETRIC MOMENTS Definitions Measures of Skewness and Kurtosis CORRECTIONS FOR GROUPING 23 3 Basic Concepts and Models INTRODUCTION THE DISTRIBUTION FUNCTION THE CHARACTERISTIC FUNCTION Definition Fourier Series Independence and Convolution MOMENTS AND MEASURES OF LOCATION AND DIS PERSION Trigonometrie Moments Measures of Location and Dispersion A Chebyshev Inequality Symmetrical Distributions CIRCULAR MODELS Introduction Lattice Distributions Uniform Distribution Von Mises Distributions Cardioid Distributions Projected Normal Distributions Wrapped Distributions MULTIPLY-WRAPPED DISTRIBUTIONS Wrapping the Circle onto Itself Mixtures DISTRIBUTIONS ON THE TORUS AND THE CYLINDER Distributions on the Torus Distributions on the Cylinder 55 4 Fundamental Theorems and Distribution Theory INTRODUCTION PROPERTIES OF CHARACTERISTIC FUNCTIONS Key Properties Polar Distributions and Characteristic Functions Further Properties of the Characteristic Function LIMIT THEOREMS Central Limit Theorems Poincare's Theorem 63

4 4.4 THE DISTRIBUTION OF 6 AND R FROM THE UNIFORM DISTRIBUTION The Distribution of 6 and R The Distribution of C and S DISTRIBUTION OF C, S AND R FOR A VON MISES POPULATION The Joint Distribution of C and S Distributions of 6 and R Marginal Distributions of C and S DISTRIBUTIONS RELATED TO THE MULTI-SAMPLE PROBLEM FOR VON MISES POPULATIONS The Distribution of R The Joint Distribution of (R, R) Distributions for the Homogeneous Case MOMENTS OF R LIMITING DISTRIBUTIONS OF CIRCULAR STATISTICS Large-Sample Approximations High-Concentration Approximations Further Approximations to the Distribution of R Point Estimation INTRODUCTION UNBIASED ESTIMATORS AND A CRAMER-RAO BOUND VON MISES DISTRIBUTIONS Maximum Likelihood Estimation Estimation Using Marginal Likelihood of R WRAPPED CAUCHY DISTRIBUTIONS MIXTURES OF VON MISES DISTRIBUTIONS 90 6 Tests of Uniformity and Tests of Goodness-of-Fit INTRODUCTION GRAPHICAL ASSESSMENT OF UNIFORMITY TESTS OF UNIFORMITY The Rayleigh Test Kuiper's Test Watson's U 2 Test Some Quick Tests../ Ajne's A n Test../ The Hermans-Rasson Test Beran's Class of Tests of Uniformity Relative Performances of Various Tests of Uniformity TESTS OF GOODNESS-OF-FIT The Probability Integral Transformation Tests of Goodness-of-Fit 115 xin

5 XIV 7 Tests on von Mises Distributions INTRODUCTION SINGLE-SAMPLE TESTS Tests for the Mean Direction Tests for the Concentration Parameter TWO-SAMPLE TESTS.../ Tests for Equality of Mean Directions Tests of Equality of Concentration Parameters MULTI-SAMPLE TESTS One-Way Classification Tests for the Homogeneity of Concentration Parameters The Heterogeneous Case TESTING VON MISESNESS Non-parametric Methods INTRODUCTION TESTS OF SYMMETRY Tests of Symmetry about a Given Axis Tests of 2-fold Symmetry TWO-SAMPLE TESTS The Uniform-Scores Test Kuiper's Two-Sample Test Watson's Two-Sample U 2 test The Runs Test Derivations of Two-Sample Tests from Tests of Uniformity Tests Obtained from Two-Sample Rank Tests on the Linel MULTI-SAMPLE TESTS Distributions on Spheres SPHERICAL DATA DESCRIPTIVE MEASURES The Mean Direction and the Mean Resultant Length The Moment of Inertia MODELS FOR SPHERICAL DATA The Uniform Distribution Von Mises-Fisher Distributions Other Distributions MODELS FOR AXIAL DATA Introduction Watson Distributions Bingham Distributions Angular Central Gaussian Distributions Other Axial Distributions DISTRIBUTION THEORY 183

6 xv The Uniform Distribution Von Mises-Fisher Distributions MOMENTS AND LIMITING DISTRIBUTIONS Moments of the Uniform Distribution Large-Sample Asymptotics High-Concentration Asymptotics High-Dimensional Asymptotics Inference on Spheres INTRODUCTION EXPLORATORY DATA ANALYSIS POINT ESTIMATION Von Mises-Fisher Distributions Watson Distributions Bingham Distributions Kent Distributions Angular Central Gaussian Distributions SINGLE-SAMPLE TESTS Tests of Uniformity Tests for the Mean Direction Mean Direction in a Given Subspace A Test for the Concentration Parameter TWO-SAMPLE TESTS Tests for Equality of Mean Directions Tests for Equality of Concentration Parameters MULTI-SAMPLE TESTS One-Way Classification Tests for the Homogeneity of Concentrations The Heterogeneous Case Tests for Mean Directions in a Subspace TESTS ON AXIAL DISTRIBUTIONS Tests of Uniformity Testing Rotational Symmetry One-Sample Tests on Watson Distributions Multi-sample Tests on Watson Distributions A GENERAL FRAMEWORK FOR TESTING UNIFORMITY Correlation and Regression INTRODUCTION MEASURES OF CORRELATION Linear-Circular Correlation Circular-Circular Correlation Spherical-Spherical Correlation REGRESSION MODELS 257

7 xvi Linear Response Circular Response Spherical Response BIVARIATE MODELS DIRECTIONAL TIME SERIES Assessing Serial Dependence Time Series Models Modern Methodology INTRODUCTION OUTLIERS Tests Based on Mean Resultant Length Likelihood Ratio Tests Tests Based on Exponential Distributions GOODNESS-OF-FIT Tests Based on the Tangent-Normal Decomposition Score Tests against Specified Larger Models Tests Based on Density Estimates ROBUST METHODS Estimation of Mean Direction Estimation of Concentration BOOTSTRAP METHODS DENSITY ESTIMATION Kernel Density Estimation Density Estimators Based on the Embedding Approach BAYESIAN METHODS CURVE FITTING AND SMOOTHING Scalar Predictor and Spherical Response Scalar Predictor and Rotational Response Planar Predictor and Circular Response Circular Predictor and Scalar Response OTHER METHODS General Sample Spaces INTRODUCTION FRAMES AND ROTATIONS Stiefel Manifolds Uniformity Matrix Fisher Distributions Other Distributions on Stiefel Manifolds SUBSPACES Grassmann Manifolds Uniformity Matrix Bingham Distributions 295

8 xvii Other Distributions on Grassmann Manifolds OTHER SAMPLE SPACES Hyperboloids General Manifolds Shape Analysis INTRODUCTION MEAN SHAPE AND VARIATION Directional Averages Form Average SHAPE COORDINATE SYSTEMS AND UNIFORM DISTRI BUTIONS The General Case The Triangle Case The Shape Sphere A TEST OF UNIFORMITY SHAPE DISTRIBUTIONS Offset Shape Distributions Distributions of Triangle Shapes obtained by Isometry COMPLEX BINGHAM DISTRIBUTIONS The Distributions The Normalising Constant High Concentrations Inference Relationship with the Fisher Distributions COMPLEX WATSON DISTRIBUTIONS The Density The Normalising Constant Relationship with the Complex Bingham Distributions Large Concentration Maximum Likelihood Estimation Hypothesis Testing Likelihood Ratio Tests THE COMPLEX ANGULAR GAUSSIAN MODEL A ROTATIONALLY SYMMETRIC SHAPE FAMILY SHAPES IN HIGHER DIMENSIONS 345 Appendix 1 Special Functions 349 Appendix 2 Tables and Charts for the Circular Case 353 Appendix 3 Tables for the Spherical Case 381 Appendix 4 List of Notation 391

9 xviii References and Author Index 395 Index 415