AN INVARIANT APPROACH TO STATISTICAL ANALYSIS OF SHAPES

By V. GISAO

1 INTRODUCTION 1 2 MORPHOMETRIC DATA 15 3 STATISTICAL MODELS FOR LANDMARK COORDINATE DATA 61 4 STATISTICAL METHODS FOR COMPARISON OF FORMS 122 5 THE STUDY OF GROWTH 213 6 CLASSIFICATION AND CLUSTERING APPLICATIONS 243 7 FURTHER APPLICATIONS OF EDMA 260 POSTLUDE 282 REFERENCES 284 CONTENTS CHAPTER 1 INTRODUCTION NATURAL SCIENTISTS HAVE LONG PERCEIVED AND CLASSIFIED ORGANISMS PRI MARILY ON THE BASIS OF THEIR APPEARANCE AND STRUCTURE OTHERWISE KNOWN AS THEIR FORM BEYOND CLASSIFICATION STUDENTS OF BIOLOGY STUDY FORM IN ORDER TO UNDERSTAND THOSE PROCESSES THAT UNDERLIE VARIATION IN FORM PROCESSES LIKE DISEASE GROWTH AND EVOLUTION THE TRUE FORM OF AN OBJECT DOES NOT CHANGE WHETHER AN ORGANISM IS MOVING ACROSS A SURFACE TRANSLATION IN MATHEMATICAL TERMS OR SPINNING ON AN AXIS ROTATION OUR PERCEPTION OF THE ORGANISM MAY CHANGE WHEN ORIENTATION CHANGES BUT PERCEPTION IS NOT OUR CONCERN HERE IT FOLLOWS THAT ANY QUANTITATIVE REPRESENTATION OF A FORM SHOULD NOT CHANGE IF THE COORDINATE SYSTEM USED TO REPRESENT THAT FORM CHANGES MOREOVER ANY COMPARISON OF FORMS SHOULD GIVE US EQUIVALENT RESULTS REGARDLESS OF WHETHER THE OPER ATIONS OF TRANSLATION AND ROTATION ARE USED IN THESE COMPARISONS THESE SIMPLE OBSERVATIONS UNDERSCORE THE IMPORTANCE OF THE PRINCIPLE OF INVARIANCE IN THE STUDY OF FORM THE INVARIANCE PRINCIPLE AND ITS IMPLICATIONS FOR THE STUDY OF BIO LOGICAL FORM HAVE PROMPTED US TO WRITE THIS BOOK THERE ARE NOW MANY MORPHOMETRIC METHODS AVAILABLE TO THE BIOLOGIST WHO WANTS TO STUDY FORM QUANTITATIVELY AND COMPUTER PROGRAMS FOR MOST OF THESE METHODS ARE EASILY OBTAINED HOWEVER UNTIL RECENTLY THE INVARIANCE PRINCIPLE AND ITS IMPLICATIONS FOR THE STUDY OF BIOLOGICAL FORM HAVE NOT BEEN CARE FULLY CONSIDERED IN THE FIELD OF MORPHOMETRICS WE HAVE WORKED TOGETH ER FOR OVER TEN YEARS DEVELOPING A UNIQUE APPROACH TO THE STUDY OF BIO NOR MUST WE FORGET THAT THE BIOLOGIST IS MUCH MORE EXACTING IN HIS REQUIREMENTS AS REGARDS FORM THAN THE PHYSICIST FOR THE LATTER IS USUAL LY CONTENT WITH EITHER AN IDEAL OR A GENERAL DESCRIPTION OF FORM WHILE THE STUDENT OF LIVING THINGS MUST BE SPECIFIC 1 LOGICAL FORM THIS APPROACH EUCLIDEAN DISTANCE MATRIX ANALYSIS OR EDMA PRONOUNCED ED MA HAS CONSIDERED INVARIANCE A CENTRAL PROP ERTY SINCE ITS INCEPTION LELE 1991 REAL BIOLOGICAL QUESTIONS AND THE ISSUES THAT SURROUND THEM PROVIDED THE MOTIVATION FOR THE DEVELOP MENT OF THIS APPROACH WE HAVE HAD THE ADVANTAGE OF BOTH STATISTICAL AND BIOLOGICAL PERSPECTIVES FROM THE BEGINNING OF OUR COLLABORATIVE WORK AND BELIEVE THAT THIS HAS RESULTED IN A METHOD THAT IS NOT JUST STATISTICALLY CORRECT BUT ALSO VALUABLE TO BIOLOGISTS WE HAVE STRUGGLED WITH COMMUNICATING COGENT CONCERNS FROM TWO DIFFERENT DISCIPLINES THROUGHOUT OUR COLLABORATION AND HAVE REPEATEDLY STUMBLED ONTO NEW INSIGHTS THAT WOULD NOT HAVE COME WITHOUT SCIENTIFIC QUESTIONS MOTI VATING THE DEVELOPMENT OF THE METHODS SOME OF THESE REALIZATIONS WERE SUBTLE AND COMPLEX OTHERS WERE SIMPLER SINCE OUR COLLABORATIVE PUBLICATIONS HAVE APPEARED IN A NUMBER OF DIFFERENT JOURNALS OVER MANY YEARS WE FELT IT TIMELY TO BRING OUR METHODS OBSERVATIONS AND INSIGHTS TOGETHER INTO A PRESENTATION SUITED TO BOTH THE BIOLOGICAL AND STATISTICAL COMMUNITY BIOLOGISTS MEASURE FORMS TO OBTAIN QUANTITATIVE INFORMATION ABOUT SPECIMENS THE TERM MEASUREMENT CONNOTES QUANTITATIVE OBSERVATION ASCERTAINMENT OF DIMENSIONS AND COMPARISON WITH STANDARDS REFERRING TO AN ORGANISM AS A LARGE DOG IS NOT PARTICULARLY INFORMA TIVE UNLESS WE HAVE AN IDEA OF WHAT CONSTITUTES LARGENESS OR HAVE A COMMON FRAME OF REFERENCE FOR DOGS A RECORD OF WEIGHT AND STANDING HEIGHT OF THE LARGE DOG ADDS PRECISION TO THE STATEMENT AND ENABLES EXPLICIT COMPARISON AMONG DOGS MEASUREMENTS CAN BE USED TO VALI DATE RELATIONSHIPS THAT ARE QUALITATIVELY EVIDENT BUT CAN ALSO BE USED TO DISCOVER ASPECTS OF BIOLOGICAL PHENOMENA THAT MAY NOT BE DISCERN ABLE FROM CASUAL OBSERVATION WHETHER METRIC DIFFERENCES ARE OBVIOUS OR OBSCURE QUANTITATIVE DATA CAN ULTIMATELY BE USED TO REVEAL INFOR MATION PERTAINING TO GROWTH PATTERNS EVOLUTIONARY PROCESSES BIOME CHANICAL DESIGN OR PHYLOGENETIC RELATIONSHIPS THESE PHENOMENA ARE NOT ALWAYS DIRECTLY OBSERVABLE BUT MAY BE REFLECTED IN MORPHOLOGY MOST SCIENTIFIC STUDIES OF BIOLOGICAL MORPHOLOGY FOCUS ON ONE OF TWO PRIMARY COMPONENTS DESIGN AND DIVERSITY LAUDER 1982 DESIGN REFERS TO THE RELATIONSHIP BETWEEN FORM AND FUNCTION A RELATIONSHIP THAT SERVES AS THE BASIS FOR SEVERAL DISCIPLINES FUNCTIONAL ANATOMY BIOMECHANICS BIOCHEMISTRY CRYSTALLOGRAPHY AND BIOENGINEERING ARE EXAMPLES OF FIELDS BUILT AROUND IDEAS PERTINENT TO BIOLOGICAL DESIGN THE OTHER COMPONENT DIVERSITY CONCERNS VARIATION OBSERVED AMONG ORGAN ISMS VARIATION IS OBSERVABLE BETWEEN AND WITHIN GROUPS OF BIOLOGICAL ORGANISMS THAT CURRENTLY POPULATE THIS PLANET BUT ALSO ACROSS TIME IN AN INVARIANT APPROACH TO STATISTICAL ANALYSIS OF SHAPES 2 THE STUDY OF EVOLUTIONARY TRENDS OR DEVELOPMENTAL TRAJECTORIES PROPER STUDY OF DESIGN AND DIVERSITY REQUIRES OBJECTIVE CRITERIA UPON WHICH TO BASE DECISIONS REGARDING SIMILARITIES OR DIFFERENCES AMONG ORGANISMS ONE CLASS OF OBJECTIVE CRITERIA IS MEASUREMENT 1 1 A BRIEF HISTORY OF MORPHOMETRICS THE QUANTITATIVE STUDY OF FORM AND FORM CHANGE COMPRISES THE FIELD OF MORPHOMETRICS MORPHOMETRICS EXPERIENCED TREMENDOUS ACTIVITY DUR ING THE 1980S AND 1990S E G BOOKSTEIN 1978 1982 1990 1991 ROHLF AND BOOKSTEIN 1990 REYMENT 1991 MARCUS BELLO ET AL 1993 ROHLF AND MARCUS 1993 MARCUS CORTI ET AL 1996 SMALL 1996 DRYDEN AND MARDIA 1998 BUT ITS FOUNDATIONS CAN BE TRACED TO THE EARLY PART OF THE TWENTIETH CENTURY SIR D ARCY THOMPSON IS REGARDED BY SOME AS THE FATHER OF MORPHOMETRICS EVEN THOUGH A TERM FOR THIS FIELD OF STUDY HAD NOT BEEN COINED AT THAT TIME ACCORDING TO THOMPSON THOMPSON 1992 THE PURPOSE OF HIS VOLUME ON GROWTH AND FORM WAS TO CORRELATE WITH MATHEMATICAL STATEMENT AND PHYSICAL LAW CERTAIN OF THE SIMPLER OUTWARD PHENOMENA OF ORGANIC GROWTH AND STRUCTURE OR FORM WHILE ALL THE WHILE REGARDING THE FABRIC OF THE ORGANISM EX HYPOTH ESI AS A MATERIAL AND MECHANICAL CONFIGURATION THOMPSON S WORK IS STILL REGARDED AS ADVANCED FOR HIS TIME AND HIS TRANSFORMATIONAL GRIDS ARE MOST MODERN MORPHOMETRICIANS ENVY THE INFORMATION COMMUNICATED BY THOMPSON S TRANSFORMATIONAL GRIDS REMAINS THE GOAL OF MOST MORPHOMETRIC STUDIES E G BOOKSTEIN 1978 CHEVERUD LEWIS ET AL 1983 BOOKSTEIN 1988 1989 O HIGGINS 1999 RICHTSMEIER AND LELE 1993 TWO EARLY STUDIES BY FRANZ BOAS 1905 AND HIS STUDENT ELEANOR PHELPS 1932 DEMONSTRATE THAT THE BIRTH OF MORPHOMETRICS ALSO LIES IN PART IN THE WORK OF THESE EARLY ANTHROPOLOGISTS COLE 1996 THE REASON BOA S CONTRIBUTION IS SO SIGNIFICANT IS THAT HE EXPLICITLY RECOGNIZED THE ARBITRARY NATURE OF REGISTRATION SYSTEMS USED IN CRANIOMETRY E G THE FRANKFURT HORIZONTAL PLANE AS WELL AS THE IMPLICIT ASSUMPTION THAT THE SELECTED REGISTRATION POINTS ARE BIOLOGICALLY STABLE BOAS SUGGEST ED THAT THE MOST FAVORABLE SUPERIMPOSITION OF ANY TWO FORMS WOULD BE OBTAINED WHEN ALL POINTS ARE CONSIDERED AS HAVING EQUAL WEIGHT IN THE COMPARISON AS POINTED OUT BY COLE 1996 BOAS 1905 SOLUTION RECOG NIZED THE ARBITRARY NATURE OF REGISTRATION SYSTEMS AND ANTICIPATED AN INVARIANT APPROACH TO STATISTICAL ANALYSIS OF SHAPES 3