Geometry Driven Statistics

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A timely collection of advanced, original material in the area of statistical methodology motivated by geometric problems, dedicated to the influential work of Kanti V. Mardia

This volume celebrates Kanti V. Mardia's long and influential career in statistics. A common theme unifying much of Mardia’s work is the importance of geometry in statistics, and to highlight the areas emphasized in his research this book brings together 16 contributions from high-profile researchers in the field.

Geometry Driven Statistics covers a wide range of application areas including directional data, shape analysis, spatial data, climate science, fingerprints, image analysis, computer vision and bioinformatics. The book will appeal to statisticians and others with an interest in data motivated by geometric considerations.

Summarizing the state of the art, examining some new developments and presenting a vision for the future, Geometry Driven Statistics will enable the reader to broaden knowledge of important research areas in statistics and gain a new appreciation of the work and influence of Kanti V. Mardia. Preface xiii

List of Contributors xv

Part I Kanti Mardia 1

1 A Conversation with Kanti Mardia 3
Nitis Mukhopadhyay

1.1 Family background 4

1.2 School days 6

1.3 College life 7

1.4 Ismail Yusuf College —University of Bombay 8

1.5 University of Bombay 10

1.6 A taste of the real world 12

1.7 Changes in the air 13

1.8 University of Rajasthan 14

1.9 Commonwealth scholarship to England 15

1.10 University of Newcastle 16

1.11 University of Hull 18

1.12 Book writing at the University of Hull 20

1.13 Directional data analysis 21

1.14 Chair Professorship of Applied Statistics, University of Leeds 25

1.15 Leeds annual workshops and conferences 28

1.16 High profile research areas 31

1.16.1 Multivariate analysis 32

1.16.2 Directional data 33

1.16.3 Shape analysis 34

1.16.4 Spatial statistics 36

1.16.5 Applied research 37

1.17 Center of Medical Imaging Research (CoMIR) 40

1.18 Visiting other places 41

1.19 Collaborators, colleagues and personalities 44

1.20 Logic, statistics and Jain religion 48

1.21 Many hobbies 50

1.22 Immediate family 51

1.23 Retirement 2000 53

Acknowledgments 55

References 55

2 A Conversation with Kanti Mardia: Part II 59
Nitis Mukhopadhyay

2.1 Introduction 59

2.2 Leeds, Oxford, and other affiliations 60

2.3 Book writing: revising and new ones 61

2.4 Research: bioinformatics and protein structure 63

2.5 Research: not necessarily linked directly with bioinformatics 66

2.6 Organizing centers and conferences 68

2.7 Memorable conference trips 71

2.8 A select group of special colleagues 73

2.9 High honors 74

2.10 Statistical science: thoughts and predictions 76

2.11 Immediate family 78

2.12 Jain thinking 80

2.13 What the future may hold 81

Acknowledgment 84

References 84

3 Selected publications 86
K V Mardia

Part II Directional Data Analysis 95

4 Some advances in constrained inference for ordered circular parameters in oscillatory systems 97
Cristina Rueda, Miguel A. Fernández, Sandra Barragán and Shyamal D. Peddada

4.1 Introduction 97

4.2 Oscillatory data and the problems of interest 99

4.3 Estimation of angular parameters under order constraint 101

4.4 Inferences under circular restrictions in von Mises models 103

4.5 The estimation of a common circular order from multiple experiments 105

4.6 Application: analysis of cell cycle gene expression data 107

4.7 Concluding remarks and future research 111

Acknowledgment 111

References 112

5 Parametric circular–circular regression and diagnostic analysis 115
Orathai Polsen and Charles C. Taylor

5.1 Introduction 115

5.2 Review of models 116

5.3 Parameter estimation and inference 118

5.4 Diagnostic analysis 119

5.4.1 Goodness-of-fit test for the von Mises distribution 120

5.4.2 Influential observations 121

5.5 Examples 123

5.6 Discussion 126

References 127

6 On two-sample tests for circular data based on spacing-frequencies 129
Riccardo Gatto and S. Rao Jammalamadaka

6.1 Introduction 129

6.2 Spacing-frequencies tests for circular data 130

6.2.1 Invariance, maximality and symmetries 131

6.2.2 An invariant class of spacing-frequencies tests 134

6.2.3 Multispacing-frequencies tests 136

6.2.4 Conditional representation and computation of the null distribution 137

6.3 Rao’s spacing-frequencies test for circular data 138

6.3.1 Rao’s test statistic and a geometric interpretation 139

6.3.2 Exact distribution 139

6.3.3 Saddlepoint approximation 140

6.4 Monte Carlo power comparisons 141

Acknowledgments 144

References 144

7 Barycentres and hurricane trajectories 146
Wilfrid S. Kendall

7.1 Introduction 146

7.2 Barycentres 147

7.3 Hurricanes 149

7.4 Using k-means and non-parametric statistics 151

7.5 Results 155

7.6 Conclusion 158

Acknowledgment 159

References 159

Part III Shape Analysis 161

8 Beyond Procrustes: a proposal to save morphometrics for biology 163
Fred L. Bookstein

8.1 Introduction 163

8.2 Analytic preliminaries 165

8.3 The core maneuver 168

8.4 Two examples 173

8.5 Some final thoughts 178

8.6 Summary 180

Acknowledgments 180

References 180

9 Nonparametric data analysis methods in medical imaging 182
Daniel E. Osborne, Vic Patrangenaru, Mingfei Qiu and Hilary W. Thompson

9.1 Introduction 182

9.2 Shape analysis of the optic nerve head 183

9.3 Extraction of 3D data from CT scans 187

9.3.1 CT data acquisition 187

9.3.2 Object extraction 189

9.4 Means on manifolds 190

9.4.1 Consistency of the Fre´chet sample mean 190

9.4.2 Nonparametric bootstrap 192

9.5 3D size-and-reflection shape manifold 193

9.5.1 Description of SRΣk 3,0 193

9.5.2 Schoenberg embeddings of SRΣk 3,0 193

9.5.3 Schoenberg extrinsic mean on SRΣk 3,0 194

9.6 3D size-and-reflection shape analysis of the human skull 194

9.6.1 Confidence regions for 3D mean size-and-reflection shape landmark configurations 194

9.7 DTI data analysis 196

9.8 MRI data analysis of corpus callosum image 200

Acknowledgments 203

References 203

10 Some families of distributions on higher shape spaces 206
Yasuko Chikuse and Peter E. Jupp

10.1 Introduction 206

10.1.1 Distributions on shape spaces 207

10.2 Shape distributions of angular central Gaussian type 209

10.2.1 Determinantal shape ACG distributions 209

10.2.2 Modified determinantal shape ACG distributions 211

10.2.3 Tracial shape ACG distributions 212

10.3 Distributions without reflective symmetry 213

10.3.1 Volume Fisher–Bingham distributions 213

10.3.2 Cardioid-type distributions 215

10.4 A test of reflective symmetry 215

10.5 Appendix: derivation of normalising constants 216

References 216

11 Elastic registration and shape analysis of functional objects 218
Zhengwu Zhang, Qian Xie, and Anuj Srivastava

11.1 Introduction 218

11.1.1 From discrete to continuous and elastic 219

11.1.2 General elastic framework 220

11.2 Registration in FDA: phase-amplitude separation 221

11.3 Elastic shape analysis of curves 223

11.3.1 Mean shape and modes of variations 225

11.3.2 Statistical shape models 226

11.4 Elastic shape analysis of surfaces 228

11.5 Metric-based image registration 231

11.6 Summary and future work 235

References 235

Part IV Spatial, Image and Multivariate Analysis 239

12 Evaluation of diagnostics for hierarchical spatial statistical models 241
Noel Cressie and Sandy Burden

12.1 Introduction 241

12.1.1 Hierarchical spatial statistical models 242

12.1.2 Diagnostics 242

12.1.3 Evaluation 243

12.2 Example: Sudden Infant Death Syndrome (SIDS) data for North Carolina 244

12.3 Diagnostics as instruments of discovery 247

12.3.1 Nonhierarchical spatial model 250

12.3.2 Hierarchical spatial model 251

12.4 Evaluation of diagnostics 252

12.4.1 DSC curves for nonhierarchical spatial models 253

12.4.2 DSC curves for hierarchical spatial models 254

12.5 Discussion and conclusions 254

Acknowledgments 254

References 255

13 Bayesian forecasting using spatiotemporal models with applications to ozone concentration levels in the Eastern United States 260
Sujit Kumar Sahu, Khandoker Shuvo Bakar and Norhashidah Awang

13.1 Introduction 260

13.2 Test data set 262

13.3 Forecasting methods 264

13.3.1 Preliminaries 264

13.3.2 Forecasting using GP models 265

13.3.3 Forecasting using AR models 267

13.3.4 Forecasting using the GPP models 268

13.4 Forecast calibration methods 269

13.5 Results from a smaller data set 272

13.6 Analysis of the full Eastern US data set 276

13.7 Conclusion 278

References 279

14 Visualisation 282
John C. Gower

14.1 Introduction 282

14.2 The problem 284

14.3 A possible solution: self-explanatory visualisations 286

References 287

15 Fingerprint image analysis: role of orientation patch and ridge structure dictionaries 288
Anil K. Jain and Kai Cao

15.1 Introduction 288

15.2 Dictionary construction 292

15.2.1 Orientation patch dictionary construction 292

15.2.2 Ridge structure dictionary construction 293

15.3 Orientation field estimation using orientation patch dictionary 296

15.3.1 Initial orientation field estimation 296

15.3.2 Dictionary lookup 297

15.3.3 Context-based orientation field correction 297

15.3.4 Experiments 298

15.4 Latent segmentation and enhancement using ridge structure dictionary 301

15.4.1 Latent image decomposition 302

15.4.2 Coarse estimates of ridge quality, orientation, and frequency 303

15.4.3 Fine estimates of ridge quality, orientation, and frequency 305

15.4.4 Segmentation and enhancement 305

15.4.5 Experimental results 305

15.5 Conclusions and future work 307

References 307

Part V Bioinformatics 311

16 Do protein structures evolve around ‘anchor’ residues? 313
Colleen Nooney, Arief Gusnanto, Walter R. Gilks and Stuart Barber

16.1 Introduction 313

16.1.1 Overview 313

16.1.2 Protein sequences and structures 314

16.2 Exploratory data analysis 315

16.2.1 Trypsin protein family 315

16.2.2 Multiple structure alignment 316

16.2.3 Aligned distance matrix analysis 317

16.2.4 Median distance matrix analysis 319

16.2.5 Divergence distance matrix analysis 320

16.3 Are the anchor residues artefacts? 325

16.3.1 Aligning another protein family 325

16.3.2 Aligning an artificial sample of trypsin structures 325

16.3.3 Aligning Cα atoms of the real trypsin sample 329

16.3.4 Aligning the real trypsin sample with anchor residues removed 330

16.4 Effect of gap-closing method on structure shape 331

16.4.1 Zig-zag 331

16.4.2 Idealised helix 331

16.5 Alternative to multiple structure alignment 332

16.6 Discussion 334

References 335

17 Individualised divergences 337
Clive E. Bowman

17.1 The past: genealogy of divergences and the man of Anek¯antav¯ada 337

17.2 The present: divergences and profile shape 338

17.2.1 Notation 338

17.2.2 Known parameters 339

17.2.3 The likelihood formulation 342

17.2.4 Dealing with multivariate data – the overall algorithm 343

17.2.5 Brief new example 345

17.2.6 Justification for the consideration of individualised divergences 347

17.3 The future: challenging data 348

17.3.1 Contrasts of more than two groups 348

17.3.2 Other data distributions 351

17.3.3 Other methods 352

References 353

18 Proteins, physics and probability kinematics: a Bayesian formulation of the protein folding problem 356
Thomas Hamelryck, Wouter Boomsma, Jesper Ferkinghoff-Borg, Jesper

Foldager, Jes Frellsen, John Haslett and Douglas Theobald

18.1 Introduction 356

18.2 Overview of the article 359

18.3 Probabilistic formulation 360

18.4 Local and non-local structure 360

18.5 The local model 362

18.6 The non-local model 363

18.7 The formulation of the joint model 364

18.7.1 Outline of the problem and its solution 364

18.7.2 Model combination explanation 365

18.7.3 Conditional independence explanation 366

18.7.4 Marginalization explanation 366

18.7.5 Jacobian explanation 367

18.7.6 Equivalence of the independence assumptions 367

18.7.7 Probability kinematics explanation 368

18.7.8 Bayesian explanation 369

18.8 Kullback–Leibler optimality 370

18.9 Link with statistical potentials 371

18.10 Conclusions and outlook 372

Acknowledgments 373

References 373

19 MAD-Bayes matching and alignment for labelled and unlabelled configurations 377
Peter J. Green

19.1 Introduction 377

19.2 Modelling protein matching and alignment 378

19.3 Gap priors and related models 379

19.4 MAD-Bayes 381

19.5 MAD-Bayes for unlabelled matching and alignment 382

19.6 Omniparametric optimisation of the objective function 384

19.7 MAD-Bayes in the sequence-labelled case 384

19.8 Other kinds of labelling 385

19.9 Simultaneous alignment of multiple configurations 385

19.10 Beyond MAD-Bayes to posterior approximation? 386

19.11 Practical uses of MAD-Bayes approximations 387

Acknowledgments 388

References 388

Index 391

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