SenGupta A., Arnold B.C. (eds.) Directional Statistics for Innovative Applications: A Bicentennial Tribute to Florence Nightingale

New York: Springer, 2022. — 487 p.In commemoration of the bicentennial of the birth of the “lady who gave the rose diagram to us”, this special contributed book pays a statistical tribute to Florence Nightingale. This book presents recent phenomenal developments, both in rigorous theory as well as in emerging methods, for applications in directional statistics, in 25 chapters with contributions from 65 renowned researchers from 25 countries. With the advent of modern techniques in statistical paradigms and statistical machine learning, directional statistics has become an indispensable tool. Ranging from data on circles to that on the spheres, tori and cylinders, this book includes solutions to problems on exploratory data analysis, probability distributions on manifolds, maximum entropy, directional regression analysis, spatio-directional time series, optimal inference, simulation, statistical machine learning with big data, and more, with their innovative applications to emerging real-life problems in astro-statistics, bioinformatics, crystallography, optimal transport, statistical process control, and so on.Editors and Contributors
Quality of Life
Florence Nightingale, Quality of Life and StatisticsContributions of Florence Nightingale
Florence Nightingale—The `Passionate' Statistician
A Call on Quality of Life
Burden of Disease
Concluding RemarksInnovative Applications
Mixture Models for Spherical Data with Applications to Protein BioinformaticsProteins and Hydrogen Bonding
Fitting a Single Kent Distribution
The Kent Distribution
High Concentration Bivariate Normal Approximation
Maximum Likelihood Estimators for the Kent Distribution
Fitting Kent Mixtures Using the EM Algorithm
The Mixture Distribution
EM Algorithm
Simulation Study
Angular Separation of Components
Model Selection
Case : Four Equi-Sized Components
Case : Five Unequally Sized Components
Case : Case with a Uniform Component Added
Hydrogen Bond Data
Exploratory Data Analysis
Modelling the Helix-Helix Data
DiscussionStatistics of Orientation Relationships in Crystallography
Crystallographic Orientations
Orientations of Symmetrical Objects
Symmetry Groups
Symmetric Frames
From Symmetric Frames to Symmetric Arrays
Summary Statistics
Distributions of Ambiguous Rotations
A General Class of Distributions on SO()/K
Some Simple Distributions of Cr-Frames and Dr-Frames
Orientation Relationships and Variants
Orientation Relationships
Variants
Estimation of Orientation Relationships
Estimation of [ A],
Confidence Regions for [ A],
Assessing Spatial Homogeneity
Common Parentage
Reconstruction
Analysis of ExamplesThe Statistics of Circular Optimal TransportCircular Optimal Transport: Alternative Representation and Numerical Computation
Limit Distributions
Limit Laws for Empirical Circular Optimal Transport Distance
Limit Laws for Bootstrapped Circular Optimal Transport Distance
Simulations
Testing for Goodness of Fit
Testing for Uniformity
Power Analysis Under von Mises Alternatives
Power Analysis Under Stephens' Multimodal Alternatives
Discussion and OutlookModelling the Movement of Magnetic NorthRegression as a Rotation
Prediction of Magnetic North
Predicting More Steps AheadFlexible Circular Modeling: A Case Study of Car Accidents
Introduction A Motivating Example
Exploring the Data
Assessing Some Distributional Characteristics
Searching for a Circular Density Model
The Classic Two-Parameter Distributions
Data Modeling: Two-Parameter Distributions
The Three-Parameter Distributions
Symmetric Distributions
Data Modeling: Three-Parameter Distributions (I)
Asymmetric Distributions
Data Modeling: Three-Parameter Distributions (II)
The Four-Parameter Distributions
Data Modeling: Four-Parameter Distributions
The Very Flexible Models
Data Modeling: Very Flexible Distributions
ConclusionsData Visualization, Simulation and Transformations
Simulation and Visualization of D-Spherical DistributionsGFB Family of Distributions and Their Interrelationships
Model GFB
Kent Model GFB,K
Bingham Model GFB,B
Model GFB,β
Model GFB
Dimroth-Watson Model GFB,DW
A Spherical Histogram
Simulation of Random Variates on the Sphere
Acceptance-Rejection Method
Simulation of FB Families
Simulating Spherical Distributions Resulting from Spherical Harmonics
Simulation of a U-DistributionTransformations to Improve the Approximation by a von Mises DistributionTransformation Procedure with Examples
Computational Limits to the Approach
Large Sample Behavior of the ML Estimators
Main Results
First- and Second-Order Partial Derivatives of Log-Likelihood
Concluding RemarkDistribution Theory and Parametric Inference
Generalized Skew-Symmetric Circular and Toroidal DistributionsGeneralized Circular Skew-Symmetric Model
Generalized k-Sine-Skewed Model
Trigonometric Moments and Range of Skewness for k=
Maximum Likelihood Estimation
Monte Carlo Simulation Study
Real Data Analysis
Generalized Bivariate Skew-Symmetric Distributions on the Torus
Real Data Analysis
Concluding RemarksInformation Theoretic Results for Stationary Time Series and the Gaussian-Generalized von Mises Time SeriesThe GvM and the Gaussian-GvM Time Series
General Considerations
Spectral Kullback–Leibler Information and Entropy
Temporal Entropy
Some Computational Aspects
Integral Functions of the GvM Time Series
Estimation of the GvM Spectral Distribution
GvM Spectral Distribution Function
Concluding RemarksA Circular Distribution Constructed from the Product of Cardioid-Type Densities with (Hyper-) Toroidal ExtensionProperties of CTP
Trigonometric Moments of CTP
Some Figures of CTP Densities
A Distribution on the (Hyper-) Torus
A Hypertoroidal Distribution
A New Toroidal Distribution
Some Properties of TCTP
Discretization
Illustration
Thunder Data
Circular Genome Data
Discussion and ConclusionOn Some Flexible Models for Circular, Toroidal, and Cylindrical DataSkewed Circular Distributions
Simple Construction of Skewed Distributions
Trigonometric Moments
Random Number Generation for the Skewed Circular Distribution
Toroidal Distribution
Simple Construction of Toroidal Distribution
Circular Correlation
Random Number Generation for the Toroidal Distribution
Fisher Information
Joint Cylindrical Distribution
Identifiability of Sine-Correlated Toroidal ModelBivariate Cardioid DistributionsMixture Formulations for Bivariate Distributions
Bivariate Cardioid Distributions
Univariate Cardioid Distribution as a Mixed Distribution
Bivariate Cardioid Distributions
Fourier Series Expansion and Moments
Test for Bivariate Isotropy
Parameter Estimation
Test of Independence
Concluding Remarks
AppendixStatistical Inference Using the Three-Parameter Generalized von Mises Distribution and Outlier Detection Method for Asymmetrically Distributed Circular DataAsymptotic Simultaneous Inference of µ, κ and λ in GvM Distribution
Review of GvM
Asymptotic Simultaneous Inference Using Confidence Ellipsoid
Detection of Outliers Using the Angular Simplicial Depth
Simulation Study
Discussion of the Simulation Results
Real Data Example
Outlier Detection
Confidence EllipsoidRegression Analysis
Modeling Wind Direction Using von Mises Regression on Wind SpeedModel
Handling covariates
Model Fitting
Mixtures of von Mises Regressions
Conclusion and DiscussionSpatial Autoregressive Models for Circular DataMarine Currents
von Mises Autoregressive Models
Markov Chain Monte Carlo Maximum Likelihood
Simulating from a Multivariate von Mises Distribution
Maximum Pseudo-likelihood Estimation
Markov Chain Monte Carlo Approximation of the Log-Likelihood
Marine Currents
DiscussionComplex Multiplication Model for Circular RegressionComplex Multiplication Regression Framework
Linear Complex Multiplication Regression Model
General T-Linear Relationship
Minimum Norm Estimation
Tests for Dependence Parameters
Simulation Study
Real Data Analysis
Concluding RemarksRegression Models for Directional VariablesCircular-Circular Regression
Trigonometric Polynomial Model
Rotation Model
Decentred Predictor Model
Möbius Transformation Model
Inverse-Circular Regression Model
Multiple Circular-Circular Regression Models
Spherical-Spherical Regression
Stereographic Projection Model
Rigid-Rotation Model
Projective Linear Transformation Model
Geometric Model
Nonparametric Regression Models
Diffeomorphism Model
Kernel-Based Regression on HyperspheresNon-parametric Inference
On Nonparametric Density Estimation for Circular Data: An OverviewMotivation for the Circular Kernel Density Estimator
Alternative Circular Density Estimators
Approximation Methods by Orthogonal Functions
Density Estimator Derived from Smooth Estimator of the Distribution Function
A Connection Between the Circular Kernel Density Estimator and the Orthogonal Series on Circle
Selection of the Concentration Parameter
Examples and Remarks
Illustration of Circular Kernel Density Estimator
RemarksOn Weighted Sign Tests for Rotational Symmetry on HyperspheresRotational Symmetry
Testing Rotational Symmetry
Simulations
Conclusions and Research Perspectives
AppendixTime Series and Change-Point Analysis
Long-Range Dependence in Directional Data
Stationary Circular Time Series with Long-Range Dependence
Estimation of the Mean Direction Under Long-Range Dependence
Extension to Nonparametric Regression with Deterministic Explanatory Variables
Extension to Nonparametric Regression with Random Explanatory VariablesModelling Circular Time Series with ApplicationsReview of Some Circular Time Series Models
Model Fitting of Circular Time Series
Transformation Methods
Score-Driven Models
Model Selection and Identification
Applications
Wind Direction for the Period of Wet Season
Wind Direction for the Period of Dry SeasonStatistical Process Control on the Circle: A Review and Some New ResultsThe Change Point Problem
Nonparametric Methods
Parametric Methods
Segmentation Procedures
Sequential Procedures
Parametric Sequential Procedures
Nonparametric Sequential ProceduresOnline SupplementStatistical Machine Learning
Angular-Angular and Linear-Angular Regression Using ANNCircular Regression
Linear-Angular Regression
Angular-Angular Regression
Artificial Neural Network
Estimating ANN Model Parameters
ANN-Based Linear-Angular Regression
ANN-Based Angular-Angular Regression
Approximate Prediction Interval
Performance of Linear-Angular and Angular-Angular ANN
Datasets Used
Linear-Angular Regression
Angular-Angular RegressionWind Speed and Wind Direction Prediction: An Implementation of a Deep Learning Algorithm Enriched by SWT and Circular PCAStationary Wavelet Transform
Kernel Principal Component Analysis
Circular Principal Component Analysis
Deep Learning for Prediction
Data Preparation
Tests and Results