An Invariant Approach to Statistical Analysis of Shapes (Hardback) book cover

An Invariant Approach to Statistical Analysis of Shapes

By Subhash R. Lele, Joan T. Richtsmeier

© 2001 – Chapman and Hall/CRC

328 pages

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pub: 2001-01-19
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About the Book

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

Table of Contents


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 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

Computational Algorithms


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

Computational Algorithms


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 Problem

Methods of Classification

Dissimilarity measures for Landmark Coordinate Data

Classification Example Analysis

Cluster Analysis

Clustering Example Analysis


The Study of Asymmetry

Comparisons of Molecular Structures

Detection of Phylogenetic Signal

About the Series

Chapman & Hall/CRC Interdisciplinary Statistics

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Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Probability & Statistics / General
SCIENCE / Life Sciences / Biology / Molecular Biology