Geometry Driven Statistics
Description
More details
Other editions
Additional editions
Persons
Content
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 Frechet sample mean 190
9.4.2 Nonparametric bootstrap 192
9.5 3D size-and-reflection shape manifold 193
9.5.1 Description of SRS k 3,0 193
9.5.2 Schoenberg embeddings of SRS k 3,0 193
9.5.3 Schoenberg extrinsic mean on SRS 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 a 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 Anekantavada 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
Chapter 1
A Conversation with Kanti Mardia
Nitis Mukhopadhyay
Department of Statistics, University of Connecticut, Storrs, CT, USA
This paper originally appeared in Statistical Science 2002, Vol. 17, No. 1, 113-148.
Kantilal Vardichand Mardia was born on April 3, 1935, in Sirohi, Rajasthan, India. He earned his B.Sc. degree in mathematics from Ismail Yusuf College-University of Bombay, in 1955, M.Sc. degrees in statistics and in pure mathematics from University of Bombay in 1957 and University of Poona in 1961, respectively, and Ph.D. degrees in statistics from the University of Rajasthan and the University of Newcastle, respectively, in 1965 and 1967. For significant contributions in statistics, he was awarded a D.Sc. degree from the University of Newcastle in 1973. He started his career as an Assistant Lecturer in the Institute of Science, Bombay and went to Newcastle as a Commonwealth Scholar. After receiving the Ph.D. degree from Newcastle, he joined the University of Hull as a lecturer in statistics in 1967, later becoming a reader in statistics in 1971. He was appointed a Chair Professor in Applied Statistics at the University of Leeds in 1973 and was the Head of the Department of Statistics during 1976-1993, and again from 1997 to the present. Professor Mardia has made pioneering contributions in many areas of statistics including multivariate analysis, directional data analysis, shape analysis, and spatial statistics. He has been credited for path-breaking contributions in geostatistics, imaging, machine vision, tracking, and spatio-temporal modeling, to name a few. He was instrumental in the founding of the Center of Medical Imaging Research in Leeds and he holds the position of a joint director of this internationally eminent center. He has pushed hard in creating exchange programs between Leeds and other scholarly centers such as the University of Granada, Spain, and the Indian Statistical Institute, Calcutta. He has written several scholarly books and edited conference proceedings and other special volumes. But perhaps he is best known for his books: Multivariate Analysis (coauthored with John Kent and John Bibby, 1979, Academic Press), Statistics of Directional Data (second edition with Peter Jupp, 1999, Wiley) and Statistical Shape Analysis (coauthored with Ian Dryden, 1998, Wiley). The conferences and workshops he has been organizing in Leeds for a number of years have had significant impacts on statistics and its interface with IT (information technology). He is dynamic and his sense of humor is unmistakable. He is a world traveler. Among other places, he has visited Princeton University, the University of Michigan, Harvard University, the University of Granada, Penn State and the University of Connecticut. He has given keynote addresses and invited lectures in international conferences on numerous occasions. He has been on the editorial board of statistical, as well as image related, journals including the IEEE Transactions on Pattern Analysis and Machine Intelligence, Journal of Environmental and Ecological Statistics, Journal of Statistical Planning and Inference and Journal of Applied Statistics. He has been elected a Fellow of the American Statistical Association, a Fellow of the Institute of Mathematical Statistics, and a Fellow of the American Dermatoglyphic Association. He is also an elected member of the International Statistical Institute and a Senior Member of IEEE. Professor Mardia retired on September 30, 2000 to take a full-time post as Senior Research Professor at Leeds-a new position especially created for him.
In April, 1999, Professor Kanti V. Mardia was invited to the University of Connecticut as a short-term guest professor for four weeks. This conversation began on Monday, April 19, 1999 in Nitis Mukhopadhyay's office in the Department of Statistics, University of Connecticut, Storrs.
1.1 Family background
Mukhopadhyay: Kanti, shall we start at the origin, so to speak? Where were you born?
Mardia: I was born in Sirohi on April 3, 1935. Sirohi, was the capital of the Sirohi State about ten thousand square miles in area, in Rajasthan, before India's independence. Subsequently, the Sirohi State became the Sirohi district. Sirohi is situated about four hundred miles east of Bombay. One of the greatest wonders near my place of birth has been the hill station, Mount Abu. It has one of the finest Jain temples, Delwara, with gorgeous Indian architecture from the eleventh century. The exquisite details are all meticulously hand-curved on marble, without parallels anywhere else in India. Those shapes and formations on the ceiling and columns with intricate details influenced me even when I was small child. Much later in my life, some of those incredible shapes made deeper and more tangible impacts on my research career.
Mukhopadhyay: Please tell me about your parents.
Mardia: I come from a business family. My father's and mother's names are, respectively, Vardichand and Sanghari. My father inherited the business of moneylending from my grandfather and he had a pawnbroker's shop in Bombay. My grandfather started with practically nothing but through his business acumen acquired a large fortune.
But, my father had to live through some tragedies. He lost his father, two brothers and their families in the span of one year in an epidemic. Due to the spread of some severe unknown disease in that particular area, many in his family perished. My father, about sixteen, was practically the lone survivor in his family.
Figure 1.1 Kanti Mardia on his uncle's lap, Bombay, 1940.
Figure 1.2 Ceiling from Jain Temple at Mount Abu (Rajasthan), Sirohi District. Original in white marble with tendrils circling in a fractal form, 1031 AD.
Mukhopadhyay: How did this episode affect your father and the family?
Mardia: It had a devastating effect. My father started taking life very philosophically and decided to take everything easy. His whole perspective of life changed. He passed on the family businesses to my uncles. One uncle was a compulsive gambler who piled up huge debts. Eventually, many of the family businesses and other assets (e.g., several buildings and movie theaters in and around Bombay) were lost as loan payments on those debts. By the time I turned ten, our family had already slipped down from a very rich status and joined the upper middle class.
Mukhopadhyay: What about your mother's side of the family?
Mardia: My maternal grandfather was a lawyer and writer. He was an original thinker. He wrote a number of novels. Any writing skills I may have, I probably inherited from him.
Mukhopadhyay: How about your brothers and sisters?
Mardia: I have four brothers and one sister. It is a large family. I am the one in the middle, a kind of the "median," a robust estimator. (Laughs.)
I became the first college graduate in the family. My brother Mangesh Kumarji looked after the family-run businesses. He earned real money to support the family while I had to study for my degrees! (Laughs.)
Mukhopadhyay: Was any of your siblings mathematically oriented?
Mardia: My younger brother Babu followed my footsteps and got a masters degree in pure mathematics. He is an Associate professor of Mathematics in Rajasthan University, Udaipur, India. During my childhood and school days, we lived in Sirohi as well as in Bombay, a major city center for all the businesses. We had to shuttle between these two places.
1.2 School days
Mukhopadhyay: Where and how did your schooling begin?
Mardia: In kindergarten, we learned numbers and even simple fractions. For example, at the age of four or five, we learned the concept of what is one-half of ten or one-quarter of eight! We had to memorize such multiplication tables and the teachers were very strict. We also had to learn to speak and write in Hindi, but this had to be mastered with the Rajasthani script and dialect, even though those styles were practically dead by then. It did feel like I was mastering a foreign language. This was on top of learning English.
Mukhopadhyay: Did you happen to have some inspiring teachers?
Mardia: In my time, there was only one high school in Sirohi, which I had to attend. Neither the teachers nor the curriculum had any flexibility and I did not like most of the subjects very much, except for mathematics. In the lower grades, we had a mathematics teacher who hailed from Ajmer, another part of Rajasthan, and he had an interesting habit. He used to assign challenging mathematical puzzles to the class and gave small prizes to whoever could solve the puzzles first. I was pretty good in solving such mathematical puzzles and won many prizes along the way. This math teacher had a big influence on me. I also enjoyed plane Euclidean geometry very much. I went through these constructions and proofs of theorems based on axioms. However, I have to confess that I preferred algebraic derivations and proofs with equations to the geometry-based arguments. (Laughs.)
Mukhopadhyay: (Laughs) Kanti, in quiet moments, sometimes you probably think what an irony of life that was!
Mardia: (Laughs) Nitis, you are right. Later in life, "geometry" became my mantra. What an irony indeed! I was not very interested nor considered particularly bright by others in nonmathematical subjects. I...
