Dattatreya Rao A.V., Girija S.V.S. Angular Statistics

Boca Raton: CRC Press, 2020. — 255 p.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.Half Title
Title Page
Copyright Page
DedicationForewordAuthors
: Introduction to Angular Data and Descriptive Statistics
Introduction to Circular Data
Applications and Nature of Angular Data Sets
Long-Axis Orientations of Feldspar Laths (Semicircular) Data
Face-Cleat in a Coal Seam (Semicircular) Data
Movements of Turtles (Circular) Data
Thirteen Homing Pigeon (Circular) Data
Movements of a Hundred Ants (Circular) Data
Noisy Scrub Birds Data
Cross-Bed Azimuths of Palaeocurrents (Circular) Data
Plant Phenology
Study of Muscular Activity
Spatial and Temporal Performance Analysis
Study of Neurology
Political Science
Psychophysical Research
Geography
Archaeology
Remote Sensing
Spatial Analysis
Plant Biology
Kernel Regression for Directional Data
Limitations of Linear Statistics and Need for Angular Statistics
Descriptive Statistics for Directional Data
Mean Direction
Circular Distance and Measure of Dispersion
Distance between Any Two Points A, B on the Circle
Circular Distance between Any Point to Several Points
Circular Distribution
Trigonometric Moments
Population Characteristics of Circular Models
The Mean Direction and the Resultant Length
Circular Variance and Standard Deviation
Central Trigonometric Moments
Skewness and Kurtosis
Existing Circular Models: Wrapped Circular ModelsMethodology of Wrapping for Continuous Linear Models
Proposition
Wrapped Exponentiated Inverted Weibull Distribution (WEIWD)
Probability Density Function (PDF) of Wrapped Exponentiated Inverted Weibull Distribution (WEIWD)
Cumulative Distribution Function (cdf) of WEIWD
Characteristic Function of WEIWD
Population Characteristics of WEIWD
Wrapped New Weibull–Pareto Distribution (WNWPD)
Probability Density Function (PDF) of Wrapped New Weibull–Pareto Distribution (WNWPD)
Cumulative Distribution Function (cdf) of WNWPD
Characteristic Function of WNWPD
Population Characteristics of WNWPD
Wrapped Lognormal Distribution (WLND)
Probability Density Function (PDF) of Wrapped Lognormal Distribution (WLND)
Cumulative Distribution Function (cdf) of WNWPD
Characteristic Function and Trigonometric Moments of WLND
Other Continuous Wrapped Circular Models
Wrapped Logistic Distribution (WLGD)
Wrapped Weibull Distribution (WWBD)
Wrapped Extreme-Value Distribution (WEVD)
Wrapped Binormal Distribution (WBNRD)
Wrapped Half Logistic Distribution (WHLD)
Methodology of Wrapping for Discrete Linear Models
Probability Mass Function
Cumulative Distribution Function
Characteristic Function
Central Trigonometric Moments
Wrapped Binomial Distribution (WBD)
Probability Mass Function of Wrapped Binomial Distribution
Cumulative Distribution Function of Wrapped Binomial Distribution
Characteristic Function and Trigonometric Moments
Population Characteristics of Wrapped Binomial Distribution
Other Discrete Wrapped Circular Models
Wrapped Poisson Distribution (WPD)
Wrapped Logarithmic Distribution (WLGRD): Stereographic Circular ModelsMethodology of Inverse Stereographic Projection
The Characteristic Function of a Stereographic Circular Model
Stereographic Logistic Distribution (SLGD)
Probability Density Function (PDF) of Stereographic Logistic Distribution
Cumulative Distribution Function (cdf) of Stereographic Logistic Distribution
Characteristic Function and Trigonometric Moments
Population Characteristics of SLGD
Stereographic Lognormal Distribution (SLND)
Probability Density Function (PDF) of Stereographic Lognormal Distribution
Cumulative Distribution Function (cdf) of Stereographic Lognormal Distribution
Characteristic Function
Population Characteristics of SLND
Stereographic Double-Weibull Distribution (SDWD)
Probability Density Function (PDF) of Stereographic Double-Weibull Distribution
Cumulative Distribution Function (cdf) of Stereographic Double-Weibull Distribution
Characteristic Function and Trigonometric Moments of SDWD
Trigonometric Moments of the Stereographic Double-Weibull Model
Population Characteristics
Other Stereographic Circular Models
Stereographic Extreme-Value Distribution (SEVD)
Stereographic Reflected Gamma Distribution (SRGD): Offset Circular ModelsMethodology of Offsetting
Offset Cauchy Model
Probability Density Function (PDF) of Offset Cauchy Distribution
Cumulative Distributive Function (cdf) of Offset Cauchy Distribution
Characteristic Function and Trigonometric Moments
Offset Pearson-Type II Model
Probability Density Function (PDF) of Offset Pearson-Type II Model
Cumulative Distribution Function (cdf) of Offset Pearson-Type II Model
Characteristic Function and Trigonometric Moments
Offset t-Distribution
Probability Density Function (PDF) and Cumulative Distribution Function (cdf) of Offset t-Distribution: Angular Models with New Techniques
Introduction to the Rising Sun Circular Models
Methodology of Constructing the Rising Sun Circular Models
Rising Sun von Mises Model
Probability Density Function (PDF) of the Rising Sun von Mises Distribution (RSVMD)
The Characteristic Function and the Population Characteristics of the Rising Sun von Mises Model
Rising Sun Wrapped Cauchy Distribution (RSWCD)
Probability Density Function (PDF) of the Rising Sun Wrapped Cauchy Distribution
The Characteristic Function and the Population Characteristics of the Rising Sun Wrapped Cauchy Model
Other Rising Sun Circular Models
The Rising Sun Wrapped Lognormal Distribution (RSWLGND)
The Rising Sun Wrapped Exponential Distribution
Circular Models Using Positive Definite Sequences
Methodology of Construction of Circular Models through Positive Definite Sequences
Discretization of Continuous Circular Models
Discrete Wrapped Exponential Distribution
Probability Mass Function
Cumulative Distribution Function
Characteristic Function
Population Characteristics of Discrete Wrapped Exponential Distribution
Construction of the Circular Model through Differential Equation: Extemporaneous Semicircular/Axial ModelsStereographic Semicircular Weibull Distribution (SSCWBD)
Probability Density Function and Cumulative Distribution Function
Characteristic Function and Trigonometric Moments
Stereographic Semicircular Half Logistic Distribution (SSCHLD)
Probability Density Function and Cumulative Distribution Function
Characteristic Function and Trigonometric Moments
Stereographic Semicircular Exponentiated Inverted Weibull Distribution
Probability Density Function and Cumulative Distributive Function
Characteristic Function and Trigonometric Moments
Population Characteristics of Stereographic Semicircular Exponentiated Inverted Weibull Distribution
Arc Offset Beta Model
Probability Density Function and Cumulative Distribution Function of Offset Beta Model
Other Extemporaneous Semicircular/Arc Models
Stereographic Semicircular Exponential Distribution (SSCEXPD)
Stereographic Semicircular Gamma Distribution (SSCGD) Model
Stereographic Semicircular New Weibull–Pareto Distribution (SSCNWPD)
Arc Offset Exponential (AOEXP) Type Model: Asymmetric l-Axial ModelsStereographic l-Axial Distributions
Stereographic l-Axial Generalized Gamma Model
Stereographic l-Axial Weibull Distribution
Stereographic l-Axial Exponential Distribution
Marshall–Olkin Circular Models
Marshall–Olkin Transformation for Circular Data
Marshall–Olkin Stereographic Circular Logistic Distribution
Wrapped Marshall–Olkin Logistic Distribution
Marshall–Olkin Stereographic l-Axial Logistic Distribution
Wrapped l-Axial Marshall–Olkin Logistic Distribution
Other Skewed l-Axial Models
Offset l-Axial Beta Model
Sine Skewed l-Axial von Mises Model: Choice of Angular ModelsLive Data Sets
Live Data Set : Movements of Turtles (Circular)
Live Data Set : Long-Axis Orientation of Feldspar Laths (Semicircular)
Uniform Probability Plot
Estimation of Parameters
Goodness of Fit
Kuiper’s Test
Watson’s U Test
Application of the Tests: Control Charts for Angular DataMethodology Adopted for Construction of Control Charts
Finding CR, ACR, and CCR Angles for Different Simulation Sizes
Finding Estimates and Variances of CR, ACR, and CCR Angles for a Given Simulation Size
Finding CR, ACR, and CCR Angles for Different Sample Sizes
Finding Estimates and Variances of CR, ACR, and CCR Angles for a Given Sample Size
Finding Theoretical Values of CR, ACR, and CCR Angles
Construction of Control Charts for Circular Distributions
Control Charts for Wrapped Exponentiated Inverted Weibull Distribution
Construction of Control Charts for Semicircular Distributions
Control Charts for Stereographic Semicircular New Weibull–Pareto DistributionAppendix