Geometry Driven Statistics

Ian L. Dryden, University of Nottingham, UKJohn T. Kent, University of Leeds, UK

• Preface xiiiList of Contributors xvPart I Kanti Mardia 11 A Conversation with Kanti Mardia 3Nitis Mukhopadhyay1.1 Family background 41.2 School days 61.3 College life 71.4 Ismail Yusuf College — University of Bombay 81.5 University of Bombay 101.6 A taste of the real world 121.7 Changes in the air 131.8 University of Rajasthan 141.9 Commonwealth scholarship to England 151.10 University of Newcastle 161.11 University of Hull 181.12 Book writing at the University of Hull 201.13 Directional data analysis 211.14 Chair Professorship of Applied Statistics, University of Leeds 251.15 Leeds annual workshops and conferences 281.16 High profile research areas 311.16.1 Multivariate analysis 321.16.2 Directional data 331.16.3 Shape analysis 341.16.4 Spatial statistics 361.16.5 Applied research 371.17 Center of Medical Imaging Research (CoMIR) 401.18 Visiting other places 411.19 Collaborators, colleagues and personalities 441.20 Logic, statistics and Jain religion 481.21 Many hobbies 501.22 Immediate family 511.23 Retirement 2000 53Acknowledgments 55References 552 a Conversation with Kanti Mardia: Part II 59Nitis Mukhopadhyay2.1 Introduction 592.2 Leeds, Oxford, and other affiliations 602.3 Book writing: revising and new ones 612.4 Research: bioinformatics and protein structure 632.5 Research: not necessarily linked directly with bioinformatics 662.6 Organizing centers and conferences 682.7 Memorable conference trips 712.8 A select group of special colleagues 732.9 High honors 742.10 Statistical science: thoughts and predictions 762.11 Immediate family 782.12 Jain thinking 802.13 What the future may hold 81Acknowledgment 84References 843 Selected publications 86K V MardiaPart II Directional Data Analysis 954 Some advances in constrained inference for ordered circular parameters in oscillatory systems 97Cristina Rueda, Miguel A. Fernández, Sandra Barragán and Shyamal D. Peddada4.1 Introduction 974.2 Oscillatory data and the problems of interest 994.3 Estimation of angular parameters under order constraint 1014.4 Inferences under circular restrictions in von Mises models 1034.5 The estimation of a common circular order from multiple experiments 1054.6 Application: analysis of cell cycle gene expression data 1074.7 Concluding remarks and future research 111Acknowledgment 111References 1125 Parametric circular–circular regression and diagnostic analysis 115Orathai Polsen and Charles C. Taylor5.1 Introduction 1155.2 Review of models 1165.3 Parameter estimation and inference 1185.4 Diagnostic analysis 1195.4.1 Goodness-of-fit test for the von Mises distribution 1205.4.2 Influential observations 1215.5 Examples 1235.6 Discussion 126References 1276 On two-sample tests for circular data based on spacing-frequencies 129Riccardo Gatto and S. Rao Jammalamadaka6.1 Introduction 1296.2 Spacing-frequencies tests for circular data 1306.2.1 Invariance, maximality and symmetries 1316.2.2 An invariant class of spacing-frequencies tests 1346.2.3 Multispacing-frequencies tests 1366.2.4 Conditional representation and computation of the null distribution 1376.3 Rao’s spacing-frequencies test for circular data 1386.3.1 Rao’s test statistic and a geometric interpretation 1396.3.2 Exact distribution 1396.3.3 Saddlepoint approximation 1406.4 Monte Carlo power comparisons 141Acknowledgments 144References 1447 Barycentres and hurricane trajectories 146Wilfrid S. Kendall7.1 Introduction 1467.2 Barycentres 1477.3 Hurricanes 1497.4 Using k-means and non-parametric statistics 1517.5 Results 1557.6 Conclusion 158Acknowledgment 159References 159Part III Shape Analysis 1618 Beyond Procrustes: a proposal to save morphometrics for biology 163Fred L. Bookstein8.1 Introduction 1638.2 Analytic preliminaries 1658.3 The core maneuver 1688.4 Two examples 1738.5 Some final thoughts 1788.6 Summary 180Acknowledgments 180References 1809 Nonparametric data analysis methods in medical imaging 182Daniel E. Osborne, Vic Patrangenaru, Mingfei Qiu and Hilary W. Thompson9.1 Introduction 1829.2 Shape analysis of the optic nerve head 1839.3 Extraction of 3D data from CT scans 1879.3.1 CT data acquisition 1879.3.2 Object extraction 1899.4 Means on manifolds 1909.4.1 Consistency of the Frećhet sample mean 1909.4.2 Nonparametric bootstrap 1929.5 3D size-and-reflection shape manifold 1939.5.1 Description of SRΣ k 3,0 1939.5.2 Schoenberg embeddings of SRΣ k 3,0 1939.5.3 Schoenberg extrinsic mean on SRΣ k 3,0 1949.6 3D size-and-reflection shape analysis of the human skull 1949.6.1 Confidence regions for 3D mean size-and-reflection shape landmark configurations 1949.7 DTI data analysis 1969.8 MRI data analysis of corpus callosum image 200Acknowledgments 203References 20310 Some families of distributions on higher shape spaces 206Yasuko Chikuse and Peter E. Jupp10.1 Introduction 20610.1.1 Distributions on shape spaces 20710.2 Shape distributions of angular central Gaussian type 20910.2.1 Determinantal shape ACG distributions 20910.2.2 Modified determinantal shape ACG distributions 21110.2.3 Tracial shape ACG distributions 21210.3 Distributions without reflective symmetry 21310.3.1 Volume Fisher–Bingham distributions 21310.3.2 Cardioid-type distributions 21510.4 A test of reflective symmetry 21510.5 Appendix: derivation of normalising constants 216References 21611 Elastic registration and shape analysis of functional objects 218Zhengwu Zhang, Qian Xie, and Anuj Srivastava11.1 Introduction 21811.1.1 From discrete to continuous and elastic 21911.1.2 General elastic framework 22011.2 Registration in FDA: phase-amplitude separation 22111.3 Elastic shape analysis of curves 22311.3.1 Mean shape and modes of variations 22511.3.2 Statistical shape models 22611.4 Elastic shape analysis of surfaces 22811.5 Metric-based image registration 23111.6 Summary and future work 235References 235Part IV Spatial, Image and Multivariate Analysis 23912 Evaluation of diagnostics for hierarchical spatial statistical models 241Noel Cressie and Sandy Burden12.1 Introduction 24112.1.1 Hierarchical spatial statistical models 24212.1.2 Diagnostics 24212.1.3 Evaluation 24312.2 Example: Sudden Infant Death Syndrome (SIDS) data for North Carolina 24412.3 Diagnostics as instruments of discovery 24712.3.1 Nonhierarchical spatial model 25012.3.2 Hierarchical spatial model 25112.4 Evaluation of diagnostics 25212.4.1 DSC curves for nonhierarchical spatial models 25312.4.2 DSC curves for hierarchical spatial models 25412.5 Discussion and conclusions 254Acknowledgments 254References 25513 Bayesian forecasting using spatiotemporal models with applications to ozone concentration levels in the Eastern United States 260Sujit Kumar Sahu, Khandoker Shuvo Bakar and Norhashidah Awang13.1 Introduction 26013.2 Test data set 26213.3 Forecasting methods 26413.3.1 Preliminaries 26413.3.2 Forecasting using GP models 26513.3.3 Forecasting using AR models 26713.3.4 Forecasting using the GPP models 26813.4 Forecast calibration methods 26913.5 Results from a smaller data set 27213.6 Analysis of the full Eastern US data set 27613.7 Conclusion 278References 27914 Visualisation 282John C. Gower14.1 Introduction 28214.2 The problem 28414.3 A possible solution: self-explanatory visualisations 286References 28715 Fingerprint image analysis: role of orientation patch and ridge structure dictionaries 288Anil K. Jain and Kai Cao15.1 Introduction 28815.2 Dictionary construction 29215.2.1 Orientation patch dictionary construction 29215.2.2 Ridge structure dictionary construction 29315.3 Orientation field estimation using orientation patch dictionary 29615.3.1 Initial orientation field estimation 29615.3.2 Dictionary lookup 29715.3.3 Context-based orientation field correction 29715.3.4 Experiments 29815.4 Latent segmentation and enhancement using ridge structure dictionary 30115.4.1 Latent image decomposition 30215.4.2 Coarse estimates of ridge quality, orientation, and frequency 30315.4.3 Fine estimates of ridge quality, orientation, and frequency 30515.4.4 Segmentation and enhancement 30515.4.5 Experimental results 30515.5 Conclusions and future work 307References 307Part V Bioinformatics 31116 Do protein structures evolve around ‘anchor’ residues? 313Colleen Nooney, Arief Gusnanto, Walter R. Gilks and Stuart Barber16.1 Introduction 31316.1.1 Overview 31316.1.2 Protein sequences and structures 31416.2 Exploratory data analysis 31516.2.1 Trypsin protein family 31516.2.2 Multiple structure alignment 31616.2.3 Aligned distance matrix analysis 31716.2.4 Median distance matrix analysis 31916.2.5 Divergence distance matrix analysis 32016.3 Are the anchor residues artefacts? 32516.3.1 Aligning another protein family 32516.3.2 Aligning an artificial sample of trypsin structures 32516.3.3 Aligning C α atoms of the real trypsin sample 32916.3.4 Aligning the real trypsin sample with anchor residues removed 33016.4 Effect of gap-closing method on structure shape 33116.4.1 Zig-zag 33116.4.2 Idealised helix 33116.5 Alternative to multiple structure alignment 33216.6 Discussion 334References 33517 Individualised divergences 337Clive E. Bowman17.1 The past: genealogy of divergences and the man of Anekāntavāda 33717.2 The present: divergences and profile shape 33817.2.1 Notation 33817.2.2 Known parameters 33917.2.3 The likelihood formulation 34217.2.4 Dealing with multivariate data – the overall algorithm 34317.2.5 Brief new example 34517.2.6 Justification for the consideration of individualised divergences 34717.3 The future: challenging data 34817.3.1 Contrasts of more than two groups 34817.3.2 Other data distributions 35117.3.3 Other methods 352References 35318 Proteins, physics and probability kinematics: a Bayesian formulation of the protein folding problem 356Thomas Hamelryck, Wouter Boomsma, Jesper Ferkinghoff-Borg, Jesper Foldager, Jes Frellsen, John Haslett and Douglas Theobald18.1 Introduction 35618.2 Overview of the article 35918.3 Probabilistic formulation 36018.4 Local and non-local structure 36018.5 The local model 36218.6 The non-local model 36318.7 The formulation of the joint model 36418.7.1 Outline of the problem and its solution 36418.7.2 Model combination explanation 36518.7.3 Conditional independence explanation 36618.7.4 Marginalization explanation 36618.7.5 Jacobian explanation 36718.7.6 Equivalence of the independence assumptions 36718.7.7 Probability kinematics explanation 36818.7.8 Bayesian explanation 36918.8 Kullback–Leibler optimality 37018.9 Link with statistical potentials 37118.10 Conclusions and outlook 372Acknowledgments 373References 37319 MAD-Bayes matching and alignment for labelled and unlabelled configurations 377Peter J. Green19.1 Introduction 37719.2 Modelling protein matching and alignment 37819.3 Gap priors and related models 37919.4 MAD-Bayes 38119.5 MAD-Bayes for unlabelled matching and alignment 38219.6 Omniparametric optimisation of the objective function 38419.7 MAD-Bayes in the sequence-labelled case 38419.8 Other kinds of labelling 38519.9 Simultaneous alignment of multiple configurations 38519.10 Beyond MAD-Bayes to posterior approximation? 38619.11 Practical uses of MAD-Bayes approximations 387Acknowledgments 388References 388Index 391