Directional Statistics for Innovative Applications: A Bicentennial Tribute to Florence Nightingale (Forum for Interdisciplinary Mathematics)

Preface
Contents
Editors and Contributors
Quality of Life
Florence Nightingale, Quality of Life and Statistics
	1 Introduction
	2 Contributions of Florence Nightingale
	3 Florence Nightingale—The `Passionate' Statistician
	4 A Call on Quality of Life
	5 Burden of Disease
	6 Concluding Remarks
	References
Innovative Applications
Mixture Models for Spherical Data with Applications to Protein Bioinformatics
	1 Introduction
		1.1 Proteins and Hydrogen Bonding
	2 Fitting a Single Kent Distribution
		2.1 The Kent Distribution
		2.2 High Concentration Bivariate Normal Approximation
		2.3 Maximum Likelihood Estimators for the Kent Distribution
	3 Fitting Kent Mixtures Using the EM Algorithm
		3.1 The Mixture Distribution
		3.2 EM Algorithm
	4 Simulation Study
		4.1 Angular Separation of Components
		4.2 Model Selection
		4.3 Case 1: Four Equi-Sized Components
		4.4 Case 2: Five Unequally Sized Components
		4.5 Case 3: Case 2 with a Uniform Component Added
	5 Hydrogen Bond Data
		5.1 Exploratory Data Analysis
		5.2 Modelling the Helix-Helix Data
	6 Discussion
	References
Statistics of Orientation Relationships in Crystallography
	1 Crystallographic Orientations
	2 Orientations of Symmetrical Objects
		2.1 Symmetry Groups
		2.2 Symmetric Frames
		2.3 From Symmetric Frames to Symmetric Arrays
	3 Summary Statistics
	4 Distributions of Ambiguous Rotations
		4.1 A General Class of Distributions on SO(3)/K
		4.2 Some Simple Distributions of Cr-Frames and Dr-Frames
	5 Orientation Relationships and Variants
		5.1 Orientation Relationships
		5.2 Variants
	6 Estimation of Orientation Relationships
		6.1 Estimation of [ A]1,2
		6.2 Confidence Regions for [ A]1,2
	7 Assessing Spatial Homogeneity
	8 Common Parentage
	9 Reconstruction
	10 Analysis of Examples
	References
The Statistics of Circular Optimal Transport
	1 Introduction
	2 Circular Optimal Transport: Alternative Representation and Numerical Computation
	3 Limit Distributions
		3.1 Limit Laws for Empirical Circular Optimal Transport Distance
		3.2 Limit Laws for Bootstrapped Circular Optimal Transport Distance
	4 Simulations
	5 Testing for Goodness of Fit
		5.1 Testing for Uniformity
		5.2 Power Analysis Under von Mises Alternatives
		5.3 Power Analysis Under Stephens' Multimodal Alternatives
	6 Discussion and Outlook
	References
Modelling the Movement of Magnetic North
	1 Introduction
	2 Regression as a Rotation
	3 Prediction of Magnetic North
		3.1 Predicting More Steps Ahead
	References
Flexible Circular Modeling: A Case Study of Car Accidents
	1 Introduction. A Motivating Example
		1.1 Exploring the Data
		1.2 Assessing Some Distributional Characteristics
		1.3 Searching for a Circular Density Model
	2 The Classic Two-Parameter Distributions
		2.1 Data Modeling: Two-Parameter Distributions
	3 The Three-Parameter Distributions
		3.1 Symmetric Distributions
		3.2 Data Modeling: Three-Parameter Distributions (I)
		3.3 Asymmetric Distributions
		3.4 Data Modeling: Three-Parameter Distributions (II)
	4 The Four-Parameter Distributions
		4.1 Data Modeling: Four-Parameter Distributions
	5 The Very Flexible Models
		5.1 Data Modeling: Very Flexible Distributions
	6 Conclusions
	References
Data Visualization, Simulation and Transformations
Simulation and Visualization of 3D-Spherical Distributions
	1 Introduction
	2 GFB Family of Distributions and Their Interrelationships
		2.1 Model GFB6
		2.2 Kent Model GFB5,K
		2.3 Bingham Model GFB5,B
		2.4 Model GFB4,β
		2.5 Model GFB4
		2.6 Dimroth-Watson Model GFB3,DW
	3 A Spherical Histogram
	4 Simulation of Random Variates on the Sphere
		4.1 Acceptance-Rejection Method
		4.2 Simulation of FB Families
		4.3 Simulating Spherical Distributions Resulting from Spherical Harmonics
		4.4 Simulation of a U-Distribution
	References
Transformations to Improve the Approximation by a von Mises Distribution
	1 Introduction
	2 Transformation Procedure with Examples
	3 Computational Limits to the Approach
	4 Large Sample Behavior of the ML Estimators
		4.1 Main Results
		4.2 First- and Second-Order Partial Derivatives of Log-Likelihood
	5 Concluding Remark
	References
Distribution Theory and Parametric Inference
Generalized Skew-Symmetric Circular and Toroidal Distributions
	1 Introduction
	2 Generalized Circular Skew-Symmetric Model
	3 Generalized k-Sine-Skewed Model
		3.1 Trigonometric Moments and Range of Skewness for k=1
		3.2 Maximum Likelihood Estimation
		3.3 Monte Carlo Simulation Study
		3.4 Real Data Analysis
	4 Generalized Bivariate Skew-Symmetric Distributions on the Torus
		4.1 Real Data Analysis
	5 Concluding Remarks
	References
Information Theoretic Results  for Stationary Time Series  and the Gaussian-Generalized  von Mises Time Series
	1 Introduction
	2 The GvM and the Gaussian-GvM Time Series
		2.1 General Considerations
		2.2 Spectral Kullback–Leibler Information and Entropy
		2.3 Temporal Entropy
	3 Some Computational Aspects
		3.1 Integral Functions of the GvM2 Time Series
		3.2 Estimation of the GvM Spectral Distribution
		3.3 GvM Spectral Distribution Function
	4 Concluding Remarks
	References
A Circular Distribution Constructed from the Product of Cardioid-Type Densities with (Hyper-) Toroidal Extension
	1 Introduction
	2 Properties of CTP2
		2.1 Trigonometric Moments of CTP2
		2.2 Some Figures of CTP2 Densities
	3 A Distribution on the (Hyper-) Torus
		3.1 A Hypertoroidal Distribution
		3.2 A New Toroidal Distribution
		3.3 Some Properties of TCTP2
	4 Discretization
	5 Illustration
		5.1 Thunder Data
		5.2 Circular Genome Data
	6 Discussion and Conclusion
	References
On Some Flexible Models for Circular, Toroidal, and Cylindrical Data
	1 Introduction
	2 Skewed Circular Distributions
		2.1 Simple Construction of Skewed Distributions
		2.2 Trigonometric Moments
		2.3 Random Number Generation for the Skewed Circular Distribution
	3 Toroidal Distribution
		3.1 Simple Construction of Toroidal Distribution
		3.2 Circular Correlation
		3.3 Random Number Generation for the Toroidal Distribution
		3.4 Fisher Information
		3.5 Joint Cylindrical Distribution
	4 Identifiability of Sine-Correlated Toroidal Model
	5 Summary
	References
Bivariate Cardioid Distributions
	1 Introduction
	2 Mixture Formulations for Bivariate Distributions
	3 Bivariate Cardioid Distributions
		3.1 Univariate Cardioid Distribution as a Mixed Distribution
		3.2 Bivariate Cardioid Distributions
		3.3 Fourier Series Expansion and Moments
	4 Test for Bivariate Isotropy
	5 Parameter Estimation
	6 Test of Independence
	7 Concluding Remarks
	8 Appendix
	References
Statistical Inference Using the Three-Parameter Generalized von Mises Distribution and Outlier Detection Method for Asymmetrically Distributed Circular Data
	1 Introduction
	2 Asymptotic Simultaneous Inference of µ, κ and λ in GvM3 Distribution
		2.1 Review of GvM3
		2.2 Asymptotic Simultaneous Inference Using Confidence Ellipsoid
	3 Detection of Outliers Using the Angular Simplicial Depth
		3.1 Simulation Study
		3.2 Discussion of the Simulation Results
	4 Real Data Example
		4.1 Outlier Detection
		4.2 Confidence Ellipsoid
	5 Conclusion
	References
Regression Analysis
15 Modeling Wind Direction Using von Mises Regression on Wind Speed
	15.1 Introduction
	15.2 Model
		15.2.1 Handling covariates
	15.3 Model Fitting
	15.4 Mixtures of von Mises Regressions
	15.5 Conclusion and Discussion
	References
Spatial Autoregressive Models for Circular Data
	1 Introduction
	2 Marine Currents
	3 von Mises Autoregressive Models
	4 Markov Chain Monte Carlo Maximum Likelihood
		4.1 Simulating from a Multivariate von Mises Distribution
		4.2 Maximum Pseudo-likelihood Estimation
		4.3 Markov Chain Monte Carlo Approximation of the Log-Likelihood
	5 Marine Currents
	6 Discussion
	References
Complex Multiplication Model  for Circular Regression
	1 Introduction
	2 Complex Multiplication Regression Framework
	3 Linear Complex Multiplication Regression Model
		3.1 General T-Linear Relationship
		3.2 Minimum Norm Estimation
	4 Tests for Dependence Parameters
	5 Simulation Study
	6 Real Data Analysis
	7 Concluding Remarks
	References
Regression Models for Directional Variables
	1 Introduction
	2 Circular-Circular Regression
		2.1 Trigonometric Polynomial Model
		2.2 Rotation Model
		2.3 Decentred Predictor Model
		2.4 Möbius Transformation Model
		2.5 Inverse-Circular Regression Model
		2.6 Multiple Circular-Circular Regression Models
	3 Spherical-Spherical Regression
		3.1 Stereographic Projection Model
		3.2 Rigid-Rotation Model
		3.3 Projective Linear Transformation Model
		3.4 Geometric Model
	4 Nonparametric Regression Models
		4.1 Diffeomorphism Model
		4.2 Kernel-Based Regression on Hyperspheres
	5 Conclusion
	References
Non-parametric Inference
On Nonparametric Density Estimation for Circular Data: An Overview
	1 Introduction
	2 Motivation for the Circular Kernel Density Estimator
	3 Alternative Circular Density Estimators
		3.1 Approximation Methods by Orthogonal Functions
		3.2 Density Estimator Derived from Smooth Estimator of the Distribution Function
	4 A Connection Between the Circular Kernel Density Estimator and the Orthogonal Series on Circle
	5 Selection of the Concentration Parameter
	6 Examples and Remarks
		6.1 Illustration of Circular Kernel Density Estimator
		6.2 Remarks
	References
On Weighted Sign Tests for Rotational Symmetry on Hyperspheres
	1 Introduction
	2 Rotational Symmetry
	3 Testing Rotational Symmetry
	4 Simulations
	5 Conclusions and Research Perspectives
	6 Appendix
	References
Time Series and Change-Point Analysis
Long-Range Dependence in Directional Data
	1 Stationary Circular Time Series with Long-Range Dependence
	2 Estimation of the Mean Direction Under Long-Range Dependence
	3 Extension to Nonparametric Regression with Deterministic Explanatory Variables
	4 Extension to Nonparametric Regression with Random Explanatory Variables
	References
Modelling Circular Time Series with Applications
	1 Introduction
	2 Review of Some Circular Time Series Models
	3 Model Fitting of Circular Time Series
		3.1 Transformation Methods
		3.2 Score-Driven Models
	4 Model Selection and Identification
	5 Applications
		5.1 Wind Direction for the Period of Wet Season
		5.2 Wind Direction for the Period of Dry Season
	6 Conclusion
	References
Statistical Process Control on the Circle: A Review and Some New Results
	1 Introduction
	2 The Change Point Problem
		2.1 Nonparametric Methods
		2.2 Parametric Methods
	3 Segmentation Procedures
	4 Sequential Procedures
		4.1 Parametric Sequential Procedures
		4.2 Nonparametric Sequential Procedures
	5 Summary
	6 Online Supplement
	References
Statistical Machine Learning
Angular-Angular and Linear-Angular Regression Using ANN
	1 Introduction
	2 Circular Regression
		2.1 Linear-Angular Regression
		2.2 Angular-Angular Regression
	3 Artificial Neural Network
		3.1 Estimating ANN Model Parameters
		3.2 ANN-Based Linear-Angular Regression
		3.3 ANN-Based Angular-Angular Regression
		3.4 Approximate Prediction Interval
	4 Performance of Linear-Angular and Angular-Angular ANN
		4.1 Datasets Used
		4.2 Linear-Angular Regression
		4.3 Angular-Angular Regression
	5 Conclusion
	References
Wind Speed and Wind Direction Prediction: An Implementation of a Deep Learning Algorithm Enriched by SWT and Circular PCA
	1 Introduction
	2 Stationary Wavelet Transform
	3 Kernel Principal Component Analysis
	4 Circular Principal Component Analysis
	5 Deep Learning for Prediction
	6 Data Preparation
	7 Tests and Results
	8 Conclusion
	References