Preface xvii

1 Fundamental Concepts 1
1.1 Preliminary Notions 1
1.2 Characteristic Functions and Trigonometric Moments 2
1.3 Derivations of Probability Density Functions 4
1.4 Statistical Inference and Applications 6

2 Preliminary Notions for Circular Distributions 9
2.1 Circular Distributions 9
2.2 Probability Density Function and Probability Mass Function 9
2.3 Density Plots 11
2.4 Characteristic Functions and Trigonometric Moments 12
2.5 Summary Statistics 13
2.6 Symmetry, Skewness, and Kurtosis 16
2.7 Tail Behavior and Points of Inflection 17
2.8 Flexible Family of Distributions 18

3 General Methods of Construction for Circular Distributions 21
3.1 Ad Hoc 21
3.2 Characterizations 22
3.3 Wrapping 24
3.4 Projection 26
3.5 Conditioning 27
3.6 Linear Fractional Transformation 28
3.7 Brownian Motion on the Circle 31
3.8 Characteristic Function-Based Method 31
3.9 Mixtures 32
3.10 Scale Mixtures/Elliptically Contoured Distributions 33
3.11 Transformations from Bivariate Distributions 34
3.12 Construction of Skew or Asymmetric Circular Distributions 35
3.13 Arithmetic and Geometric Methods 36
3.14 Shortcomings and Future Work 37

4 Symmetric Distributions on the Circle 43
4.1 Preliminary Notions 43
4.2 Uniform Distribution 44
4.3 Ad Hoc 49
4.4 Wrapping Method 66
4.5 Characteristic Function-Based Method 75
4.6 (Möbius) Transformation-Based Method 78
4.7 Circular Beta Distributions 84
4.8 Circular Transformed Beta Distributions 85
4.9 A Circular t-Distribution and Its Extension 86
4.10 Projection Method 88
4.11 Mixtures 93
4.12 Statistical Inference 95
4.13 Applications 96

5 Asymmetric and Multimodal Distributions on the Circle 105
5.1 Preliminary Notions 105
5.2 Ad Hoc 106
5.3 Exponential Family-Based Generalizations 107
5.4 Power Batschelet Distributions 114
5.5 Generalized Power Cardioid Family 118
5.6 Wrapping-Based Distributions and Families 120
5.7 Characteristic Function-Based Method 122
5.8 Wrapped Exponential and Laplace Distributions 127
5.9 Wrapped Gamma Distribution 133
5.10 Other Wrapped Distributions 136
5.11 Sine-Skewed Distribution 136
5.12 A Family of Cosine-Perturbed Distributions 140
5.13 Transforming Circular Distributions via Möbius Transformation 141
5.14 Projected Lognormal Distribution 144
5.15 Other Distributions 145
5.16 Statistical Inference 146
5.17 Applications 146

6 Miscellaneous Distributions on the Circle 151
6.1 Preliminary Notions 151
6.2 Discrete Distributions 153
6.3 Compound Distributions 160
6.4 Miscellaneous Distributions 164
6.5 Statistical Inference 169
6.6 Applications 169

7 Axial Distributions 173
7.1 Preliminary Notions 173
7.2 General Methods of Construction for Axial Distributions 174
7.3 Skew Distributions 177
7.4 Bivariate Axial Distributions 184
7.5 Multivariate Axial Distributions 189
7.6 Statistical Inference 190
7.7 Applications 191

8 Distributions on the Sphere, Hemi-Sphere and Hyper-Sphere 195
8.1 Preliminary Notions for Spherical, Hemi-Spherical and Hyper-Spherical Distributions 195
8.2 General Methods of Construction 199
8.3 Distributions on the Sphere 204
8.4 Distributions on the Hyper-Sphere 207
8.5 Distributions on the Complex Sphere 227
8.6 Statistical Inference 229
8.7 Applications 231

9 Distributions on the (Ring) Torus and Hyper-Torus 237
9.1 Preliminary Notions 237
9.2 General Methods of Construction 244
9.3 Uniform Distribution 250
9.4 Families of Bivariate von Mises Distributions 251
9.5 Bivariate Gamma-Mixed von Mises Distributions 252
9.6 Wrapped Bivariate Normal Distribution 254
9.7 Wrapped Bivariate Heavy Tail Distributions 256
9.8 Bivariate Cardioid Distributions 257
9.9 Bivariate Asymmetric Generalized von Mises Distribution 260
9.10 A Model Generated from Brownian Motion 262
9.11 Distributions on the Hyper-Torus 264
9.12 Statistical Inference 267
9.13 Applications 268

10 Distributions on the (Vertical) Cylinder and Hyper-Cylinder 275
10.1 Preliminary Notions 275
10.2 Uniform Distribution 277
10.3 General Methods for Obtaining Distributions on the Cylinder 277
10.4 Distributions on a Discrete Cylinder 291
10.5 Distributions on the Hyper-Cylinder 293
10.6 Statistical Inference 295
10.7 Applications 296

11 Distributions on the Disc and Hyper-Disc 303
11.1 Preliminary Notions 303
11.2 Uniform Distribution 305
11.3 General Methods of Constructing Distributions on the Disc 306
11.4 Some More Distributions on the Disc 312
11.5 Distributions on the Hyper-Disc 314
11.6 Statistical Inference 321
11.7 Applications 322

Appendix A Some Mathematical Concepts for Directional Data 325
A.1 Fourier Series 325
A.2 A Chebishev Inequality 327
A.3 Measures of Similarity or Dissimilarity 327
A.4 Central Limit Theorem 328

Appendix B Mathematical Formulae 331

References 336
Index 337