Angular Statistics

Description

Directional data arise in the form of circular / semicircular / axial, symmetric / asymmetric, uni / bimodal data, in practical situations of varied fields. For the purpose of modeling such kind of data sets, the data scientists found that existing models as inadequate. As there is paucity of angular models, and to fill the gap, this book is designed at constructing new angular models with the existing techniques and to develop new tools of constructing angular models with an application to control charts in angular models. This book is planned to cover the following topics in nine chapters un function, positive definite sequences, discretization and through differential approach Extemporaneous Semicircular / arc and asymmetric l - axial models Choice of angular models as an inferential aspect and construction of control charts for angular data as an application are presented. This graduate level book will be useful for data scientists, researchers and research students of Statistics and allied fields. Cover Half Title Title Page Copyright Page Dedication Table of Contents Foreword Preface Acknowledgments Authors 1: Introduction to Angular Data and Descriptive Statistics 1.1 Introduction to Circular Data 1.2 Applications and Nature of Angular Data Sets 1.2.1 Long-Axis Orientations of Feldspar Laths (Semicircular) Data 1.2.2 Face-Cleat in a Coal Seam (Semicircular) Data 1.2.3 Movements of Turtles (Circular) Data 1.2.4 Thirteen Homing Pigeon (Circular) Data 1.2.5 Movements of a Hundred Ants (Circular) Data 1.2.6 Noisy Scrub Birds Data 1.2.7 Cross-Bed Azimuths of Palaeocurrents (Circular) Data 1.2.8 Plant Phenology 1.2.9 Study of Muscular Activity 1.2.10 Spatial and Temporal Performance Analysis 1.2.11 Study of Neurology 1.2.12 Political Science 1.2.13 Psychophysical Research 1.2.14 Geography 1.2.15 Archaeology 1.2.16 Remote Sensing 1.2.17 Spatial Analysis 1.2.18 Plant Biology 1.2.19 Kernel Regression for Directional Data 1.3 Limitations of Linear Statistics and Need for Angular Statistics 1.4 Descriptive Statistics for Directional Data 1.4.1 Mean Direction 1.4.2 Circular Distance and Measure of Dispersion 1.4.3 Distance between Any Two Points A, B on the Circle 1.4.4 Circular Distance between Any Point to Several Points 1.4.5 Circular Distribution 1.4.6 Trigonometric Moments 1.4.7 Population Characteristics of Circular Models 1.4.8 The Mean Direction and the Resultant Length 1.4.9 Circular Variance and Standard Deviation 1.4.10 Central Trigonometric Moments 1.4.11 Skewness and Kurtosis 1.5 Existing Circular Models References 2: Wrapped Circular Models 2.1 Introduction 2.2 Methodology of Wrapping for Continuous Linear Models 2.2.1 Proposition 2.3 Wrapped Exponentiated Inverted Weibull Distribution (WEIWD) 2.3.1 Probability Density Function (pdf) of Wrapped Exponentiated Inverted Weibull Distribution (WEIWD) 2.3.2 Cumulative Distribution Function (cdf) of WEIWD 2.3.3 Characteristic Function of WEIWD 2.3.4 Population Characteristics of WEIWD 2.4 Wrapped New Weibull–Pareto Distribution (WNWPD) 2.4.1 Probability Density Function (pdf) of Wrapped New Weibull–Pareto Distribution (WNWPD) 2.4.2 Cumulative Distribution Function (cdf) of WNWPD 2.4.3 Characteristic Function of WNWPD 2.4.4 Population Characteristics of WNWPD 2.5 Wrapped Lognormal Distribution (WLND) 2.5.1 Probability Density Function (pdf) of Wrapped Lognormal Distribution (WLND) 2.5.2 Cumulative Distribution Function (cdf) of WNWPD 2.5.3 Characteristic Function and Trigonometric Moments of WLND 2.6 Other Continuous Wrapped Circular Models 2.6.1 Wrapped Logistic Distribution (WLGD) 2.6.2 Wrapped Weibull Distribution (WWBD) 2.6.3 Wrapped Extreme-Value Distribution (WEVD) 2.6.4 Wrapped Binormal Distribution (WBNRD) 2.6.5 Wrapped Half Logistic Distribution (WHLD) 2.7 Methodology of Wrapping for Discrete Linear Models 2.7.1 Probability Mass Function 2.7.2 Cumulative Distribution Function 2.7.3 Characteristic Function 2.7.4 Central Trigonometric Moments 2.8 Wrapped Binomial Distribution (WBD) 2.8.1 Probability Mass Function of Wrapped Binomial Distribution 2.8.2 Cumulative Distribution Function of Wrapped Binomial Distribution 2.8.3 Characteristic Function and Trigonometric Moments 2.8.4 Population Characteristics of Wrapped Binomial Distribution 2.9 Other Discrete Wrapped Circular Models 2.9.1 Wrapped Poisson Distribution (WPD) 2.9.2 Wrapped Logarithmic Distribution (WLGRD) References 3: Stereographic Circular Models 3.1 Introduction 3.2 Methodology of Inverse Stereographic Projection 3.3 The Characteristic Function of a Stereographic Circular Model 3.4 Stereographic Logistic Distribution (SLGD) 3.4.1 Probability Density Function (pdf) of Stereographic Logistic Distribution 3.4.2 Cumulative Distribution Function (cdf) of Stereographic Logistic Distribution 3.4.3 Characteristic Function and Trigonometric Moments 3.4.4 Population Characteristics of SLGD 3.5 Stereographic Lognormal Distribution (SLND) 3.5.1 Probability Density Function (pdf) of Stereographic Lognormal Distribution 3.5.2 Cumulative Distribution Function (cdf) of Stereographic Lognormal Distribution 3.5.3 Characteristic Function 3.5.4 Population Characteristics of SLND 3.6 Stereographic Double-Weibull Distribution (SDWD) 3.6.1 Probability Density Function (pdf) of Stereographic Double-Weibull Distribution 3.6.2 Cumulative Distribution Function (cdf) of Stereographic Double-Weibull Distribution 3.6.3 Characteristic Function and Trigonometric Moments of SDWD 3.6.3.1 Trigonometric Moments of the Stereographic Double-Weibull Model 3.6.4 Population Characteristics 3.7 Other Stereographic Circular Models 3.7.1 Stereographic Extreme-Value Distribution (SEVD) 3.7.2 Stereographic Reflected Gamma Distribution (SRGD) References 4: Offset Circular Models 4.1 Introduction 4.2 Methodology of Offsetting 4.3 Offset Cauchy Model 4.3.1 Probability Density Function (pdf) of Offset Cauchy Distribution 4.3.2 Cumulative Distributive Function (cdf) of Offset Cauchy Distribution 4.3.3 Characteristic Function and Trigonometric Moments 4.4 Offset Pearson-Type II Model 4.4.1 Probability Density Function (pdf) of Offset Pearson-Type II Model 4.4.2 Cumulative Distribution Function (cdf) of Offset Pearson-Type II Model 4.4.3 Characteristic Function and Trigonometric Moments 4.5 Offset t-Distribution 4.5.1 Probability Density Function (pdf) and Cumulative Distribution Function (cdf) of Offset t-Distribution References 5: Angular Models with New Techniques 5.1 Introduction to the Rising Sun Circular Models 5.2 Methodology of Constructing the Rising Sun Circular Models 5.3 Rising Sun von Mises Model 5.3.1 Probability Density Function (pdf) of the Rising Sun von Mises Distribution (RSVMD) 5.3.2 The Characteristic Function and the Population Characteristics of the Rising Sun von Mises Model 5.4 Rising Sun Wrapped Cauchy Distribution (RSWCD) 5.4.1 Probability Density Function (pdf) of the Rising Sun Wrapped Cauchy Distribution 5.4.2 The Characteristic Function and the Population Characteristics of the Rising Sun Wrapped Cauchy Model 5.5 Other Rising Sun Circular Models 5.5.1 The Rising Sun Wrapped Lognormal Distribution (RSWLGND) 5.5.2 The Rising Sun Wrapped Exponential Distribution 5.6 Circular Models Using Positive Definite Sequences 5.7 Methodology of Construction of Circular Models through Positive Definite Sequences 5.8 Discretization of Continuous Circular Models 5.9 Discrete Wrapped Exponential Distribution 5.9.1 Probability Mass Function 5.9.2 Cumulative Distribution Function 5.9.3 Characteristic Function 5.9.4 Population Characteristics of Discrete Wrapped Exponential Distribution 5.10 Construction of the Circular Model through Differential Equation References 6: Extemporaneous Semicircular/Axial Models 6.1 Introduction 6.2 Stereographic Semicircular Weibull Distribution (SSCWBD) 6.2.1 Probability Density Function and Cumulative Distribution Function 6.2.2 Characteristic Function and Trigonometric Moments 6.3 Stereographic Semicircular Half Logistic Distribution (SSCHLD) 6.3.1 Probability Density Function and Cumulative Distribution Function 6.3.2 Characteristic Function and Trigonometric Moments 6.4 Stereographic Semicircular Exponentiated Inverted Weibull Distribution 6.4.1 Probability Density Function and Cumulative Distributive Function 6.4.2 Characteristic Function and Trigonometric Moments 6.4.3 Population Characteristics of Stereographic Semicircular Exponentiated Inverted Weibull Distribution 6.5 Arc Offset Beta Model 6.5.1 Probability Density Function and Cumulative Distribution Function of Offset Beta Model 6.6 Other Extemporaneous Semicircular/Arc Models 6.6.1 Stereographic Semicircular Exponential Distribution (SSCEXPD) 6.6.2 Stereographic Semicircular Gamma Distribution (SSCGD) Model 6.6.3 Stereographic Semicircular New Weibull–Pareto Distribution (SSCNWPD) 6.6.4 Arc Offset Exponential (AOEXP) Type Model References 7: Asymmetric l-Axial Models 7.1 Introduction 7.2 Stereographic l-Axial Distributions 7.2.1 Stereographic l-Axial Generalized Gamma Model 7.2.2 Stereographic l-Axial Weibull Distribution 7.2.3 Stereographic l-Axial Exponential Distribution 7.3 Marshall–Olkin Circular Models 7.3.1 Marshall–Olkin Transformation for Circular Data 7.3.2 Marshall–Olkin Stereographic Circular Logistic Distribution 7.3.3 Wrapped Marshall–Olkin Logistic Distribution 7.4 Marshall–Olkin Stereographic l-Axial Logistic Distribution 7.5 Wrapped l-Axial Marshall–Olkin Logistic Distribution 7.6 Other Skewed l-Axial Models 7.6.1 Offset l-Axial Beta Model 7.6.2 Sine Skewed l-Axial von Mises Model References 8: Choice of Angular Models 8.1 Introduction 8.2 Live Data Sets 8.2.1 Live Data Set 1: Movements of Turtles (Circular) 8.2.2 Live Data Set 2: Long-Axis Orientation of Feldspar Laths (Semicircular) 8.2.3 Uniform Probability Plot 8.3 Estimation of Parameters 8.4 Goodness of Fit 8.4.1 Kuiper’s Test 8.4.2 Watson’s U2 Test 8.4.2.1 Application of the Tests References 9: Control Charts for Angular Data 9.1 Introduction 9.2 Methodology Adopted for Construction of Control Charts 9.2.1 Finding CR, ACR, and CCR Angles for Different Simulation Sizes 9.2.2 Finding Estimates and Variances of CR, ACR, and CCR Angles for a Given Simulation Size 9.2.3 Finding CR, ACR, and CCR Angles for Different Sample Sizes 9.2.4 Finding Estimates and Variances of CR, ACR, and CCR Angles for a Given Sample Size 9.2.5 Finding Theoretical Values of CR, ACR, and CCR Angles 9.3 Construction of Control Charts for Circular Distributions 9.3.1 Control Charts for Wrapped Exponentiated Inverted Weibull Distribution 9.4 Construction of Control Charts for Semicircular Distributions 9.4.1 Control Charts for Stereographic Semicircular New Weibull–Pareto Distribution References Appendix Index