An Invariant Approach to the Statistical Analysis of Shapes

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 StatisticalAnalysis 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.

Contents

INTRODUCTION

• A Brief History of Morphometrics
• Foundations for the Study of Biological Forms
• Description of the data Sets

MORPHOMETRIC DATA

• Types of Morphometric Data
• Landmark Homology and Correspondence
• Collection of Landmark Coordinates
• Reliability of Landmark Coordinate Data
• Summary

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
• Summary
• 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
• Computational Algorithms

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
• Summary
• 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
• Computational Algorithms

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
• Summary

CLASSIFICATION, CLUSTERING AND MISCELLANEOUS TOPICS

• Classification Problem
• Methods of Classification
• Dissimilarity measures for Landmark Coordinate Data
• Classification Example Analysis
• Cluster Analysis
• Clustering Example Analysis

FURTHER APPLICATIONS OF EDMA

• The Study of Asymmetry
• Comparisons of Molecular Structures
• Detection of Phylogenetic Signal