Probability Distributions for Directional Data on Smooth Manifolds

Ashis SenGupta, PhD, is Distinguished Professor in the Department of Statistics at Middle East Technical University, Ankara, Turkey; Adjunct Professor in the Department of Population Health Sciences at Augusta University, Georgia, USA; CSIR Emeritus Scientist, Government of India, and Advisor/Consultant and former Professor (Higher Allowance Grade) and Head, ASU, at Indian Statistical Institute, Kolkata, India. Kunio Shimizu, PhD, is Project Professor in the Center for Training Professors in Statistics at The Institute of Statistical Mathematics, Tokyo, Japan, and Professor Emeritus at Keio University, Yokohama, Japan.

• Preface xvii1 Fundamental Concepts 11.1 Preliminary Notions 11.2 Characteristic Functions and Trigonometric Moments 21.3 Derivations of Probability Density Functions 41.4 Statistical Inference and Applications 62 Preliminary Notions for Circular Distributions 92.1 Circular Distributions 92.2 Probability Density Function and Probability Mass Function 92.3 Density Plots 112.4 Characteristic Functions and Trigonometric Moments 122.5 Summary Statistics 132.6 Symmetry, Skewness, and Kurtosis 162.7 Tail Behavior and Points of Inflection 172.8 Flexible Family of Distributions 183 General Methods of Construction for Circular Distributions 213.1 Ad Hoc 213.2 Characterizations 223.3 Wrapping 243.4 Projection 263.5 Conditioning 273.6 Linear Fractional Transformation 283.7 Brownian Motion on the Circle 313.8 Characteristic Function-Based Method 313.9 Mixtures 323.10 Scale Mixtures/Elliptically Contoured Distributions 333.11 Transformations from Bivariate Distributions 343.12 Construction of Skew or Asymmetric Circular Distributions 353.13 Arithmetic and Geometric Methods 363.14 Shortcomings and Future Work 374 Symmetric Distributions on the Circle 434.1 Preliminary Notions 434.2 Uniform Distribution 444.3 Ad Hoc 494.4 Wrapping Method 664.5 Characteristic Function-Based Method 754.6 (Möbius) Transformation-Based Method 784.7 Circular Beta Distributions 844.8 Circular Transformed Beta Distributions 854.9 A Circular t-Distribution and Its Extension 864.10 Projection Method 884.11 Mixtures 934.12 Statistical Inference 954.13 Applications 965 Asymmetric and Multimodal Distributions on the Circle 1055.1 Preliminary Notions 1055.2 Ad Hoc 1065.3 Exponential Family-Based Generalizations 1075.4 Power Batschelet Distributions 1145.5 Generalized Power Cardioid Family 1185.6 Wrapping-Based Distributions and Families 1205.7 Characteristic Function-Based Method 1225.8 Wrapped Exponential and Laplace Distributions 1275.9 Wrapped Gamma Distribution 1335.10 Other Wrapped Distributions 1365.11 Sine-Skewed Distribution 1365.12 A Family of Cosine-Perturbed Distributions 1405.13 Transforming Circular Distributions via Möbius Transformation 1415.14 Projected Lognormal Distribution 1445.15 Other Distributions 1455.16 Statistical Inference 1465.17 Applications 1466 Miscellaneous Distributions on the Circle 1516.1 Preliminary Notions 1516.2 Discrete Distributions 1536.3 Compound Distributions 1606.4 Miscellaneous Distributions 1646.5 Statistical Inference 1696.6 Applications 1697 Axial Distributions 1737.1 Preliminary Notions 1737.2 General Methods of Construction for Axial Distributions 1747.3 Skew Distributions 1777.4 Bivariate Axial Distributions 1847.5 Multivariate Axial Distributions 1897.6 Statistical Inference 1907.7 Applications 1918 Distributions on the Sphere, Hemi-Sphere and Hyper-Sphere 1958.1 Preliminary Notions for Spherical, Hemi-Spherical and Hyper-Spherical Distributions 1958.2 General Methods of Construction 1998.3 Distributions on the Sphere 2048.4 Distributions on the Hyper-Sphere 2078.5 Distributions on the Complex Sphere 2278.6 Statistical Inference 2298.7 Applications 2319 Distributions on the (Ring) Torus and Hyper-Torus 2379.1 Preliminary Notions 2379.2 General Methods of Construction 2449.3 Uniform Distribution 2509.4 Families of Bivariate von Mises Distributions 2519.5 Bivariate Gamma-Mixed von Mises Distributions 2529.6 Wrapped Bivariate Normal Distribution 2549.7 Wrapped Bivariate Heavy Tail Distributions 2569.8 Bivariate Cardioid Distributions 2579.9 Bivariate Asymmetric Generalized von Mises Distribution 2609.10 A Model Generated from Brownian Motion 2629.11 Distributions on the Hyper-Torus 2649.12 Statistical Inference 2679.13 Applications 26810 Distributions on the (Vertical) Cylinder and Hyper-Cylinder 27510.1 Preliminary Notions 27510.2 Uniform Distribution 27710.3 General Methods for Obtaining Distributions on the Cylinder 27710.4 Distributions on a Discrete Cylinder 29110.5 Distributions on the Hyper-Cylinder 29310.6 Statistical Inference 29510.7 Applications 29611 Distributions on the Disc and Hyper-Disc 30311.1 Preliminary Notions 30311.2 Uniform Distribution 30511.3 General Methods of Constructing Distributions on the Disc 30611.4 Some More Distributions on the Disc 31211.5 Distributions on the Hyper-Disc 31411.6 Statistical Inference 32111.7 Applications 322Appendix A Some Mathematical Concepts for Directional Data 325A.1 Fourier Series 325A.2 A Chebishev Inequality 327A.3 Measures of Similarity or Dissimilarity 327A.4 Central Limit Theorem 328Appendix B Mathematical Formulae 331References 336Index 337