Applied Directional Statistics Modern Methods and Case Studies

Cover
......Page 1
Half title
......Page 2
Series editors......Page 3
Title
......Page 4
Copyright
......Page 5
Contents......Page 6
Contributors......Page 12
Editors......Page 14
Introduction......Page 16
Kanti V. Mardia, Jesper Illemann Foldager, and Jes Frellsen 1.1 Introduction......Page 18
1.2 Protein Structure......Page 19
1.3 Protein Geometry......Page 21
1.4 Structure Determination and Prediction......Page 23
1.4.1 Markov Chain Monte Carlo Simulations of Proteins .......Page 25
1.5 Generative Models for the Polypeptide Backbone......Page 27
1.5.1.2 Histograms and Fourier Series......Page 28
1.5.2 A Dynamical Bayesian Network Model: TorusDBN . .......Page 29
1.6 Generative Models for Amino Acids Side Chains......Page 32
1.7 Discussion......Page 34
Richard Arnold and Peter Jupp 2.1 Ambiguous Rotations......Page 42
2.1.1 Symmetry Groups......Page 48
2.1.2 Symmetric Frames......Page 49
2.2 From Symmetric Frames to Symmetric Arrays......Page 50
2.3 Summary Statistics......Page 52
2.4 Testing Uniformity......Page 53
2.5.1 A General Class of Distributions on SO(3)=K......Page 55
2.5.2 Concentrated Distributions......Page 56
2.6.1 One-Sample Tests......Page 57
2.7 Further Developments......Page 58
2.8.3 Analysis of Example 3......Page 59
Chapter
3 Correlated Cylindrical Data......Page 62
Francesco Lagona 3.1 Correlated Cylindrical Data......Page 64
3.2.2 Modeling a Cylindrical Time Series......Page 66
3.2.3 Modelling a Cylindrical Spatial Series......Page 68
3.3 Identication of Sea Regimes......Page 69
3.4 Segmentation of Current Fields......Page 71
3.5 Outline......Page 72
Chapter
4 Toroidal Di usions and Protein Structure Evolution......Page 78
4.1.1 Protein Structure......Page 79
4.1.2 Protein Evolution......Page 81
4.1.3 Toward a Generative Model of Protein Evolution......Page 83
4.2 Toroidal Diusions......Page 84
4.2.1 Toroidal Ornstein{Uhlenbeck Analogues......Page 87
4.2.2 Estimation for Toroidal Diusions......Page 89
4.2.3 Empirical Performance......Page 92
4.3.1 Hidden Markov Model Structure......Page 94
4.3.2 Site-Classes: Constant Evolution and Jump Events . .......Page 97
4.3.3 Model Training......Page 98
4.3.4 Benchmarks......Page 100
4.4 Case Study: Detection of a Novel Evolutionary Motif......Page 104
4.5 Conclusions......Page 107
Thanh Mai Pham Ngoc 5.1 Introduction......Page 112
5.2 Some Preliminaries about Harmonic Analysis on SO(3) and S2......Page 113
5.3.1 Null and Alternative Hypotheses......Page 115
5.3.2 Noise Assumptions......Page 116
5.4 Test Constructions......Page 117
5.5.1 The Testing Procedures......Page 119
5.5.2 Alternatives......Page 120
5.5.3 Simulations......Page 121
5.5.4 Real Data: Paleomagnetism......Page 124
5.5.5 Real Data: UHECR......Page 125
Louis-Paul Rivest and Karim Oualkacha 6.1 Introduction......Page 128
6.2.2 A Geometric Construction of the One Axis Model . .......Page 130
6.3 Modeling Data from SE(3)......Page 132
and B3......Page 133
6.4.1 The Rotation Only Estimator of the Rotation Axis A......Page 20
6.4.2 The Translation Only Estimator of the Parameters . .......Page 134
6.4.3 The Rotation-Translation Estimator of the Parameters......Page 135
6.5.1 Simulations......Page 136
6.5.2 Data Analysis......Page 138
6.6 Discussion......Page 140
Chapter
7 Spatial and Spatio-Temporal Circular Processes with Application to Wave Directions......Page 146
Giovanna Jona-Lasinio, Alan E. Gelfand, and Gianluca Mastrantonio 7.1 Introduction......Page 147
7.2 The Wrapped Spatial and Spatio-Temporal Process......Page 148
7.2.1 Wrapped Spatial Gaussian Process......Page 149
7.2.3 Kriging and Forecasting......Page 150
7.2.4 Wave Data for the Examples......Page 151
7.2.5 Wrapped Skewed Gaussian Process......Page 153
7.2.5.1 Space-Time Analysis Using the Wrapped Skewed Gaussian Process......Page 157
7.3.1 Univariate Projected Normal Distribution......Page 158
7.3.2 Projected Gaussian Spatial Processes......Page 159
7.3.3 Model Fitting and Inference......Page 162
7.3.4 Kriging with the Projected Gaussian Processes......Page 163
7.3.4.1 An Example Using the Projected Gaussian Process......Page 164
7.3.6 A Separable Space-Time Wave Direction Data Example......Page 166
7.4 Space-Time Comparison of the WN and PN Models......Page 168
7.5 Joint Modeling of Wave Height and Direction......Page 170
7.6 Concluding Remarks......Page 174
Chapter
8 Cylindrical Distributions and Their Applications to Biological Data......Page 180
Toshihiro Abe and Ichiro Ken Shimatani 8.1 Introduction......Page 181
8.3.1 Probability Distributions on [0;1)......Page 182
8.3.2 Circular Distributions......Page 184
8.4.1 The Johnson{Wehrly Distribution......Page 185
8.4.2 The Weibull{von Mises Distribution......Page 186
8.4.3 Gamma{von Mises Distribution......Page 187
8.4.4 Generalized Gamma{von Mises Distribution......Page 188
8.4.5 Sine-Skewed Weibull{von-Mises Distribution......Page 189
8.5 Application 1: Quantication of the Speed/Turning Angle Patterns of a Flying Bird......Page 190
8.6 Application of Cylindrical Distributions 2: How Trees Are Expanding Crowns......Page 191
8.6.1 Crown Asymmetry in Boreal Forests......Page 193
8.6.2 Crown Asymmetry Model......Page 194
8.6.3 Results of the Cylindrical Models......Page 197
8.7 Concluding Remarks......Page 201
Jose Ameijeiras{Alonso, Rosa M. Crujeiras, and Alberto Rodrguez Casal 9.1 Introduction to Wildre modeling......Page 204
9.2 Fires' Seasonality......Page 205
9.2.1 Landscape Scale......Page 206
9.2.2 Global Scale......Page 212
9.3 Fires' Orientation......Page 214
9.3.1 Main Spread on the Orientation of Fires......Page 216
9.3.2 Orientation{Size Joint Distribution......Page 217
9.3.3 Orientation{Size Regression Modeling......Page 222
9.4 Open Problems......Page 223
Chapter
10 Bayesian Analysis of Circular Data in Social and Behavioral Sciences......Page 228
Irene Klugkist, Jolien Cremers, and Kees Mulder 10.1 Introduction......Page 229
10.2 Introducing Two Approaches Conceptually......Page 230
10.2.1 Intrinsic......Page 231
10.2.2 Embedding......Page 232
10.3 Bayesian Modeling......Page 233
10.3.1 Intrinsic......Page 234
10.4.1 The Data......Page 235
10.4.2 Bayesian Inference......Page 236
10.4.3 Inequality Constrained Hypotheses......Page 242
10.5 Basic Human Values in the European Social Survey......Page 244
10.5.1 The Data......Page 245
10.5.2 The Model......Page 246
10.5.3 Variable Selection......Page 247
10.5.4 Bayesian Inference......Page 248
10.5.5 Comparison of Approaches......Page 251
10.6 Discussion......Page 253
Chapter 11 Nonparametric Classification
for Circular Data......Page 258
Marco Di Marzio, Stefania Fensore, Agnese Panzera, and Charles C. Taylor 11.1 Density Estimation on the Circle......Page 260
11.2 Classication via Density Estimation......Page 262
11.3.1 Binary Regression via Density Estimation......Page 263
11.3.2 Local Polynomial Binary Regression......Page 265
11.4 Numerical Examples......Page 267
11.5 Classication of Earth's Surface......Page 268
11.6 Conclusion......Page 270
Suvrit Sra 12.1 Introduction......Page 276
12.2.1 Uniform Distribution......Page 277
12.2.3 Watson Distribution......Page 278
12.2.4 Other Distributions......Page 279
12.4 Modeling Directional Data: Maximum-Likelihood Estimation......Page 280
12.4.1 Maximum-Likelihood Estimation for vMF......Page 281
12.4.2 Maximum-Likelihood Estimation for Watson......Page 282
12.5 Mixture Models......Page 283
12.5.1 EM Algorithm......Page 284
12.5.2 Limiting Versions......Page 285
12.5.3 Application: Clustering Using movMF......Page 286
12.5.4 Application: Clustering Using moW......Page 288
12.6 Conclusion......Page 289
Arthur Pewsey 13.1 Introduction......Page 294
13.2 The Circular Package......Page 295
13.3 Packages that Use the Circular Package......Page 297
13.5 The Directional Package......Page 298
13.6 Other Packages for Directional Statistics......Page 300
13.7 Unsupported Directional Statistics Methodologies......Page 301
13.8 Conclusions......Page 302
Index......Page 308