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.

Table of Contents:
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 Frećhet sample mean 190 9.4.2 Nonparametric bootstrap 192 9.5 3D size-and-reflection shape manifold 193 9.5.1 Description of SRΣ k 3,0 193 9.5.2 Schoenberg embeddings of SRΣ k 3,0 193 9.5.3 Schoenberg extrinsic mean on SRΣ k 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 C α 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 Anekāntavāda 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