Directional Statistics

DIRECTIONAL STATISTICS......Page 1
Half-title......Page 2
Wiley Series in Probability and Statistics......Page 3
Title Page......Page 4
Copyright Page......Page 5
Dedication......Page 7
Contents......Page 12
Relationship with other books......Page 20
Survey of contents......Page 21
Acknowledgements......Page 22
1.2.1 Ungrouped Data......Page 23
1.2.2 Grouped Data......Page 24
1.3.1 Unimodal Distributions......Page 27
1.3.2 Multimodal Distributions......Page 28
1.4.1 Earth Sciences......Page 29
1.4.2 Meteorology......Page 30
1.4.3 Biology......Page 32
1.4.7 Medicine......Page 33
1.5 WRAPPING AND PROJECTING......Page 34
2.1.1 Preliminaries and Notation......Page 35
2.2.1 The Mean Direction......Page 37
2.3.1 The Mean Resultant Length and the Circular Variance......Page 39
2.3.2 Decomposition of Dispersion......Page 40
2.3.4 Other Measures of Dispersion......Page 41
2.4.1 Definitions......Page 42
2.4.2 Measures of Skewness and Kurtosis......Page 44
2.5 CORRECTIONS FOR GROUPING......Page 45
3.2 THE DISTRIBUTION FUNCTION......Page 47
3.3.1 Definition......Page 48
3.3.3 Independence and Convolution......Page 49
3.4.1 Trigonometric Moments......Page 50
3.4.2 Measures of Location and Dispersion......Page 51
3.4.4 Symmetrical Distributions......Page 53
3.5.1 Intduction......Page 54
3.5.2 Lattice Distributions......Page 56
3.5.3 Uniform Distribution......Page 57
3.5.4 Von Mises Distributions......Page 58
3.5.5 Cardioid Distributions......Page 67
3.5.6 Projected Normal Distributions......Page 68
3.5.7 Wrapped Distributions......Page 69
3.6.1 Wrapping the Circle onto Itself......Page 74
3.6.2 Mixtures......Page 75
3.7.1 Distributions on the Torus......Page 76
3.7.2 Distributions on the Cylinder......Page 77
4.2.1 Key Properties......Page 79
4.2.2 Polar Distributions and Characteristic Functions......Page 80
4.2.3 Further Properties of the Characteristic Function......Page 83
4.3.1 Central Limit Theorems......Page 84
4.3.2 Poincaré\'s Theorem......Page 85
4.4.1 The Distribution of θ̅ and R̅......Page 87
4.5 DISTRIBUTION OF C, S AND R FOR A VON MISES POPULATION......Page 90
4.5.2 Distributions of θ̅ and R̅......Page 91
4.5.3 Marginal Distributions of C and S......Page 92
4.6.1 The Distribution of R......Page 93
4.6.2 The Joint Distribution of (R, R)......Page 94
4.6.3 Distributions for the Homogeneous Case......Page 95
4.7 MOMENTS OF R̅......Page 97
4.8.1 Large-Sample Approximations......Page 98
4.8.2 High-Concentration Approximations......Page 100
4.8.3 Further Approximations to the Distribution of R̅......Page 102
5.2 UNBIASED ESTIMATORS AND A CRAMÉR–RAO BOUND......Page 105
5.3.1 Maximum Likelihood Estimation......Page 107
5.3.2 Estimation Using Marginal Likelihood of R̅......Page 110
5.4 WRAPPED CAUCHY DISTRIBUTIONS......Page 111
5.5 MIXTURES OF VON MISES DISTRIBUTIONS......Page 112
6.2 GRAPHICAL ASSESSMENT OF UNIFORMITY......Page 115
6.3.1 The Rayleigh Test......Page 116
6.3.2 Kuiper\'s Test......Page 121
6.3.3 Watson\'s U² Test......Page 125
6.3.4 Some Quick Tests......Page 127
6.3.5 Ajne\'s A n Test......Page 130
6.3.6 The Hermans–Rasson Test......Page 131
6.3.7 Beran\'s Class of Tests of Uniformity......Page 132
6.4.2 Tests of Goodness-of-Fit......Page 137
7.2.1 Tests for the Mean Direction......Page 141
7.2.2 Tests for the Concentration Parameter......Page 148
7.3.1 Tests for Equality of Mean Directions......Page 150
7.3.2 Tests of Equality of Concentration Parameters......Page 155
7.4.1 One-Way Classification......Page 156
7.4.2 Tests for the Homogeneity of Concentration Parameters......Page 161
7.4.3 The Heterogeneous Case......Page 163
7.5 TESTING VON MISESNESS......Page 164
8.2.1 Tests of Symmetry about a Given Axis......Page 167
8.3 TWO-SAMPLE TESTS......Page 168
8.3.1 The Uniform-Scores Test......Page 169
8.3.3 Watson\'s Two-Sample U² test......Page 172
8.3.4 The Runs Test......Page 174
8.3.5 Derivations of Two-Sample Tests from Tests of Uniformity......Page 176
8.4 MULTI-SAMPLE TESTS......Page 178
9.1 SPHERICAL DATA......Page 181
9.2.1 The Mean Direction and the Mean Resultant Length......Page 185
9.2.2 The Moment of Inertia......Page 187
9.3.2 Von Mises–Fisher Distributions......Page 189
9.3.3 Other Distributions......Page 195
9.4.1 Introduction......Page 202
9.4.3 Bingham Distributions......Page 203
9.4.4 Angular Central Gaussian Distributions......Page 204
9.5.1 The Uniform Distribution......Page 205
9.5.2 Von Mises–Fisher Distributions......Page 206
9.6.1 Moments of the Uniform Distribution......Page 208
9.6.2 Large-Sample Asymptotics......Page 209
9.6.3 High-Concentration Asymptotics......Page 211
9.6.4 High-Dimensional Asymptotics......Page 214
10.2 EXPLORATORY DATA ANALYSIS......Page 215
10.3.1 Von Mises–Fisher Distribution......Page 219
10.3.2 Watson Distribution......Page 224
10.3.3 Bingham Distributions......Page 225
10.3.4 Kent Distributions......Page 227
10.4.1 Tests of Uniformity......Page 228
10.4.2 Tests for the Mean Direction......Page 232
10.4.3 Mean Direction in a Given Subspace......Page 238
10.4.4 A Test for the Concentration Parameter......Page 240
10.5.1 Tests for Equality of Mean Directions......Page 241
10.5.2 Tests for Equality of Concentration Parameters......Page 242
10.6.1 One-Way Classification......Page 244
10.6.2 Tests for the Homogeneity of Concentrations......Page 248
10.6.4 Tests for Mean Directions in a Subspace......Page 250
10.7.1 Tests of Uniformity......Page 254
10.7.2 Testing Rotational Symmetry......Page 256
10.7.3 One-Sample Tests on Watson Distributions......Page 258
10.7.4 Multi-Sample Tests on Watson Distributions......Page 260
10.8 A GENERAL FRAMEWORK FOR TESTING UNIFORMITY......Page 262
11.2.1 Linear–Circular Correlation......Page 267
11.2.2 Circular–Circular Correlation......Page 270
11.2.3 Spherical–Spherical Correlation......Page 276
11.3.2 Circular Response......Page 279
11.3.3 Spherical Response......Page 280
11.4 BIVARIATE MODELS......Page 284
11.5.2 Time Series Models......Page 285
12.2.1 Tests Based on Mean Resultant Length......Page 289
12.2.3 Tests Based on Exponential Distributions......Page 290
12.3.1 Tests Based on the Tangent-Normal Decomposition......Page 291
12.3.2 Score Tests against Specified Larger Models......Page 294
12.4.1 Estimation of Mean Direction......Page 296
12.5 BOOTSTRAP METHODS......Page 298
12.6.1 Kernel Density Estimation......Page 299
12.6.2 Density Estimators Based on the Embedding Approach......Page 300
12.7 BAYESIAN METHODS......Page 301
12.8.1 Scalar Predictor and Spherical Response......Page 302
12.8.4 Circular Predictor and Scalar Response......Page 303
12.9 OTHER METHODS......Page 304
13.1 INTRODUCTION......Page 305
13.2.1 Stiefel Manifolds......Page 307
13.2.2 Uniformity......Page 309
13.2.3 Matrix Fisher Distributions......Page 311
13.2.4 Other Distributions on Stiefel Manifolds......Page 314
13.3.1 Grassmann Manifolds......Page 315
13.3.2 Uniformity......Page 316
13.3.4 Other Distributions on Grassmann Manifolds......Page 317
13.4.1 Hyperboloids......Page 319
13.4.2 General Manifolds......Page 321
14.1 INTRODUCTION......Page 325
14.2 MEAN SHAPE AND VARIATION......Page 330
14.2.1 Directional Averages......Page 335
14.3 SHAPE COORDINATE SYSTEMS AND UNIFORM DISTRIBUTIONS......Page 336
14.3.1 The General Case......Page 337
14.3.2 The Triangle Case......Page 339
14.3.3 The Shape Sphere......Page 340
14.4 A TEST OF UNIFORMITY......Page 343
14.5 SHAPE DISTRIBUTIONS......Page 346
14.5.2 Distributions of Triangle Shapes obtained by Isometry......Page 347
14.6.1 The Distributions......Page 348
14.6.2 The Normalising Constant......Page 350
14.6.3 High Concentrations......Page 353
14.6.4 Inference......Page 354
14.6.5 Relationship with the Fisher Distributions......Page 355
14.7.1 The Density......Page 356
14.7.2 The Normalising Constant......Page 357
14.7.4 Large Concentration......Page 358
14.7.5 Maximum Likelihood Estimation......Page 359
14.7.6 Hypothesis Testing......Page 360
14.7.7 Likelihood Ratio Tests......Page 362
14.8 THE COMPLEX ANGULAR GAUSSIAN MODEL......Page 365
14.9 A ROTATIONALLY SYMMETRIC SHAPE FAMILY......Page 366
14.10 SHAPES IN HIGHER DIMENSIONS......Page 367
Appendix 1. Special Functions......Page 371
Appendix 2. Tables and Charts for the Circular Case......Page 375
Appendix 3. Tables for the Spherical Case......Page 403
Appendix 4. Notation......Page 413
References and Author Index......Page 417
Index......Page 437