An Invariant Approach to Statistical Analysis of Shapes
Chapman and Hall/CRC – 2001 – 328 pages
Natural scientists perceive and classify organisms primarily on the basis of their appearance and structure- their form , defined as that characteristic remaining invariant after translation, rotation, and possibly reflection of the object. The quantitative study of form and form change comprises the field of morphometrics. For morphometrics to succeed, it needs techniques that not only satisfy mathematical and statistical rigor but also attend to the scientific issues.
An Invariant Approach to the Statistical Analysis of Shapes results from a long and fruitful collaboration between a mathematical statistician and a biologist. Together they have developed a methodology that addresses the importance of scientific relevance, biological variability, and invariance of the statistical and scientific inferences with respect to the arbitrary choice of the coordinate system. They present the history and foundations of morphometrics, discuss the various kinds of data used in the analysis of form, and provide justification for choosing landmark coordinates as a preferred data type. They describe the statistical models used to represent intra-population variability of landmark data and show that arbitrary translation, rotation, and reflection of the objects introduce infinitely many nuisance parameters. The most fundamental part of morphometrics-comparison of forms-receives in-depth treatment, as does the study of growth and growth patterns, classification, clustering, and asymmetry.
Morphometrics has only recently begun to consider the invariance principle and its implications for the study of biological form. With the advantage of dual perspectives, An Invariant Approach to the Statistical Analysis of Shapes stands as a unique and important work that brings a decade's worth of innovative methods, observations, and insights to an audience of both statisticians and biologists.
"The appearance of this book by Subhash Lele and Joan Richtsmeier is to be welcomed. In recent years there has been much discussion of the relative advantages of morphometric methodology developed by Fred Bookstein and his colleagues versus the EDMA approach advocated by Lele and Richtsmeier. Now readers can decide for themselves."
-Short Book Reviews, Vol. 21, No. 2, August 2001
"The invariance principle, a beautiful mathematical concept, is used, alongside statistical techniques, to analyze various biological shapes and forms… Landmark coordinate data technique is used throughout, with topics covered ranging from the study of growth and form to Euclidean distance matrix analysis and applications. In addition to end-of-chapter summaries, useful algorithms, and end-of-text bibliography, various applications are provided of a wide range of problems that transcend disciplinary boundaries. Highly recommended. Graduates through professionals."
-CHOICE, January 2002
"This book is a result of a successful, interdisciplinary collaboration between a statistician and a biologist. Most chapters are broken into two clearly identified parts-the first part is strongly rooted in biological applications and the second part contains the accompanying formal mathematical analyses. Despite the advanced level of this monograph, the writing is clear and well organized. The book is a highly recommended resource for scholars who are interested in mathematical and statistical analyses of shape information."
-Journal of Mathematical Psychology, Vol. 46 (2002)
"This book describes statistical methods that are applicable to analyse morphometric data. …The closing part offers new ideas to extend Euclidian distance matrix analysis procedures to complex biological problems. The book is an important practical guide for the analysis of morphometric data."
- Zentralblatt fur Mathematik, August 2002
"…this is a useful and complementary addition to the recent series of books on statistical shape analysis."
-I. L. Dryden, Biometrics, Vol 58, June 2002
"This book is an unusual book in that it is a collaborative work by a statistician and an anthropologist. … useful for applied statisticians who are interested in analyzing the shapes of biological organisms."
- Technometrics, August 2004, Vol. 46, No. 3
A Brief History of Morphometrics
Foundations for the Study of Biological Forms
Description of the data Sets
Types of Morphometric Data
Landmark Homology and Correspondence
Collection of Landmark Coordinates
Reliability of Landmark Coordinate Data
STATISTICAL MODELS FOR LANDMARK COORDINATE DATA
Statistical Models in General
Models for Intra-Group Variability
Effect of Nuisance Parameters
Invariance and Elimination of Nuisance Parameters
A Definition of Form
Coordinate System Free Representation of Form
Estimability of the Mean Form and Variance
Analysis of Example Data Sets
Perspective: Some Comments of EDMA versus other Morphometric Methods
Part 2: Statistical Theory for the Analysis of Single Population
The Perturbation Model
Invariance and the elimination of Nuisance Parameters
Estimation of Parameters in the Single Sample Case
STATSTICAL METHODS FOR COMPARISON OF FORMS
Limiting Factors in Morphometrics
Comparing Two Forms: General Set-Up
Superimposition-Based Approaches and Invariance
Transformational Grids for Deformation-Based Approaches and Invariance
The Relationship between Mathematical and Scientific Invariance
An Invariant Approach: Euclidean Distance Matrix Analysis (EDMA)
Statistical Hypothesis Testing for Shape Difference
Methods for Exploring the Form Difference Matrix
Example Data Analyses
Part 2: Statistical Theory for the Comparison of Two Forms
Deformation Approach to Form Difference and Lack of Invariance
Superimposition Methods for Comparison of Forms and Lack of Invariance
Matrix Transformations, Side Conditions, Likelihood, and Identifiability Issues
Form Comparisons Based on Distances
Statistical Properties of the Estimators of Mean Form, Mean Form Difference, and Mean Shape Difference Matrices
THE STUDY OF GROWTH
Longitudinal versus Cross-Sectional Data
Assigning Age and Forming Age-Related Groups
EDMA Applied to the Study of Growth
Growth Difference Matrix Analysis: Comparing Patterns of Growth using Growth Matrices
Example Data Analyses
Producing Hypothetical Morphologies from Forms and Growth Patterns
CLASSIFICATION, CLUSTERING AND MISCELLANEOUS TOPICS
Methods of Classification
Dissimilarity measures for Landmark Coordinate Data
Classification Example Analysis
Clustering Example Analysis
FURTHER APPLICATIONS OF EDMA
The Study of Asymmetry
Comparisons of Molecular Structures
Detection of Phylogenetic Signal