Foundations of measurement. vol. 2, Geometrical, threshold, and probabilistic representations

All of the sciences―physical, biological, and social―have a need for quantitative measurement. This influential series, Foundations of Measurement, established the formal foundations for measurement, justifying the assignment of numbers to objects in terms of their structural correspondence. Volume I introduces the distinct mathematical results that serve to formulate numerical representations of qualitative structures. Volume II extends the subject in the direction of geometrical, threshold, and probabilistic representations, Volume III examines representation as expressed in axiomatization and invariance. Reprint of the Academic Press, New York and London, 1989 edition. Cover Foundations of Measurement, Volume II Frontispiece by Ruth Weisberg FOUNDATIONS OF MEASUREMENT, Volume II: Geometrical, Threshold, and Probabilistic Representations COPYRIGHT © 1989 BY ACADEMIC PRESS, ISBN 0124254020 QA465. F68 530.1' 6 LCCN 72-154365 Table of Contents Preface Acknowledgments Chapter 11 Overview 11.1 GEOMETRY UNIT 11.1.1 Geometrical Representations (Chapter 12) 11.1.2 Axiomatic Synthetic Geometry (Chapter 13) 11.1.3 Proximity Measurement (Chapter 14) 11.1.4 Color and Force Measurement (Chapter 15) 11.2 THRESHOLD AND ERROR UNIT 11.2.1 Representations with Thresholds (Chapter 16) 11.2.2 Representations of Choice Probabilities (Chapter 17) Chapter 12 Geometrical Representations 12.1 INTRODUCTION 12.2 VECTOR REPRESENTATIONS 12.2.1 Vector Spaces 12.2.2 Analytic Affine Geometry 12.2.3 Analytic Projective Geometry 12.2.4 Analytic Euclidean Geometry 12.2.5 Meaningfulness in Analytic Geometry 12.2.6 Minkowski Geometry 12.2.7 General Projective Metrics 12.3 METRIC REPRESENTATIONS 12.3.1 General Metrics with Geodesics 12.3.2 Elementary Spaces and the Helmholtz-Lie Problem 12.3.3 Riemannian Metrics 12.3.4 Other Metrics EXERCISES Chapter 13 Axiomatic Geometry and Applications 13.1 INTRODUCTION 13.2 ORDER ON THE LINE 13.2.1 Betweenness: Affine Order 13.2.2 Separation: Projective Order 13.3 PROOFS 13.4 PROJECTIVE PLANES 13.5 PROJECTIVE SPACES 13.6 AFFINE AND ABSOLUTE SPACES 13.6.1 Ordered Geometric Spaces 13.6.2 Affine Space 13.6.3 Absolute Spaces 13.6.4 Euclidean Spaces 13.6.5 Hyperbolic Spaces 13.7 ELLIPTIC SPACES 13.7.1 Double Elliptic Spaces 13.7.2 Single Elliptic Spaces 13.8 CLASSICAL SPACE-TIME 13.9 SPACE-TIME OF SPECIAL RELATIVITY 13.9.1 Other Axiomatic Approaches 13.10 PERCEPTUAL SPACES 13.10.1 Historical Survey through the Nineteenth Century 13.10.2 General Considerations Concerning Perceptual Spaces 13.10.3 Experimental Work before Luneburg's Theory 13.10.4 Luneburg Theory of Binocular Vision 13.10.5 Experiments Relevant to Luneburg's Theory 13.10.6 Other Studies EXERCISES Chapter 14 Proximity Measurement 14.1 INTRODUCTION 14.2 METRICS WITH ADDITIVE SEGMENTS 14.2.1 Collinearity 14.2.2 Constructive Methods 14.2.3 Representation and Uniqueness Theorems 14.3 PROOFS 14.3.1 Theorem 2 (p. 167) 14.3.2 Reduction to Extensive Measurement 14.3.3 Theorem 3 (p. 168) 14.3.4 Theorem 4 (p. 169) 14.4 MULTIDIMENSIONAL REPRESENTATIONS 14.4.1 Decomposability 14.4.2 Intradimensional Subtractivity 14.4.3 Interdimensional Additivity 14.4.4 The Additive-Difference Model 14.4.5 Additive-Difference Metrics 14.5 PROOFS 14.5.1 Theorem 5 (p. 179) 14.5.2 Theorem 6 (p. 181) 14.5.3 Theorem 7 (p. 183) 14.5.4 Theorem 9 (p. 186) 14.5.5 Preliminary Lemma 14.5.6 Theorem 10 (p. 187). 14.6 EXPERIMENTAL TESTS OF MULTIDIMENSIONAL REPRESENTATIONS 14.6.1 Relative Curvature 14.6.2 Translation Invariance 14.6.3 The Triangle Inequality 14.7 FEATURE REPRESENTATIONS 14.7.1 The Contrast Model 14.7.2 Empirical Applications 14.7.3 Comparing Alternative Representations 14.8 PROOFS 14.8.1 Theorem 11 EXERCISES Chapter 1 5 Color and Force Measurement 15.1 INTRODUCTION 15.2 GRASSMANN STRUCTURES 15.2.1 Formulation of the Axioms 15.2.2 Representation and Uniqueness Theorems 15.2.3 Discussion of Proofs of Theorems 3 and 4 15.3 PROOFS 15.3.1 Theorem 3 (p. 234) 15.3.2 Theorem 4 (p. 235) 15.4 COLOR INTERPRETATIONS 15.4.1 Metameric Color Matching 15.4.2 Tristimulus Colorimetry 15.4.3 Four Ways to Misunderstand Color Measurement 15.4.4 Asymmetric Color Matching 15.5 THE DIMENSIONAL STRUCTURE OF COLOR AND FORCE 15.5.1 Color Codes and Metamer Codes 15.5.2 Photopigments 15.5.3 Force Measurement and Dynamical Theory 15.5.4 Color Theory in a Measurement Framework 15.6 THE KONIG AND HURVICH-JAMESON COLOR THEORIES 15.6.1 Representations of 2-Chromatic Reduction Structures 15.6.2 The Konig Theory and Alternatives 15.6.3 Codes Based on Color Attributes 15.6.4 The Cancellation Procedure 15.6.5 Representation and Uniqueness Theorems 15.6.6 Tests and Extensions of Quantitative Opponent-Colors Theory 15.7 PROOFS 15.7.1 Theorem 6 (p. 266) 15.7.2 Theorem 9 (p. 283) 15.7.3 Theorem 10 (p. 283) EXERCISES Chapter 16 Representations with Thresholds 16.1 INTRODUCTION 16.1.1 Three Approaches to Nontransitive Data 16.1.2 Idea of Thresholds 16.1.3 Overview 16.2 ORDINAL THEORY 16.2.1 Upper, Lower, and Two-Sided Thresholds 16.2.2 Induced Quasiorders: Interval Orders and Semiorders 16.2.3 Compatible Relations 16.2.4 Biorders: A Generalization of Interval Orders 16.2.5 Tight Representations 16.2.6 Constant-Threshold Representations 16.2.7 Interval and Indifference Graphs 16.3 PROOFS 16.3.1 Theorem 2 (p. 310) 16.3.2 Lemma 1 (p. 315) 16.3.3 Theorem 6 (p. 327) 16.3.4 Theorem 9 (p. 318) 16.3.5 Theorem 10 (p. 319) 16.3.6 Theorem 11 (p. 320) 16.3.7 Theorems 14 and 15 (p. 325) 16.4 ORDINAL THEORY FOR FAMILIES OF ORDERS 16.4.1 Finite Families of Interval Orders and Semiorders 16.4.2 Order Relations Induced by Binary Probabilities 16.4.3 Dimension of Partial Orders 16.5 PROOFS 16.5.1 Theorem 16 (p. 333) 16.5.2 Theorem 17 (p. 337) 16.5.3 Theorem 18 (p. 338) 16.5.4 Theorem 19 (p. 338) 16.6 SEMIORDERED ADDITIVE STRUCTURES 16.6.1 Possible Approaches to Semiordered Probability Structures 16.6.2 Probability with Approximate Standard Families 16.6.3 Axiomatization of Semiordered Probability Structures 16.6.4 Weber's Law and Semiorders 16.7 PROOF OF THEOREM 24 (p. 351) 16.8 RANDOM-VARIABLE REPRESENTATIONS 16.8.1 Weak Representations of Additive Conjoint and Extensive Structures 16.8.2 Variability as Measured by Moments 16.8.3 Qualitative Primitives for Moments 16.8.4 Axiom System for Qualitative Moments 16.8.5 Representation Theorem and Proof EXERCISES Chapter 17 Representation of Choice Probabilities 17.1 INTRODUCTION 17.1.1 Empirical Interpretations 17.1.2 Probabilistic Representations 17.2 ORDINAL REPRESENTATIONS FOR PAIR COMPARISONS 17.2.1 Stochastic Transitivity 17.2.2 Difference Structures 17.2.3 Local Difference Structures 17.2.4 Additive Difference Structures 17.2.5 Intransitive Preferences 17.3 PROOFS 17.3.1 Theorem 2 (p. 392) 17.3.2 Theorem 3 (p. 395) 17.3.3 Theorem 4 (p. 397) 17.4 CONSTANT REPRESENTATIONS FOR MULTIPLE CHOICE 17.4.1 Simple Scalability 17.4.2 The Strict-Utility Model 17.5 PROOFS 17.5.1 Theorem 5 (p. 412). 17.5.2 Theorem 7 (p. 417) 17.6 RANDOM VARIABLE REPRESENTATIONS 17.6.1 The Random-Utility Model 17.6.2 The Independent Double-Exponential Model 17.6.3 Error Tradeoff 17.7 PROOFS 17.7.1 Theorem 9 (p. 422) 17.7.2 Theorem 12 (p. 424) 17.7.3 Theorem 13 (p. 430) 17.8 MARKOVIAN ELIMINATION PROCESSES 17.8.1 The General Model 17.8.2 Elimination by Aspects 17.8.3 Preference Trees 17.9 PROOFS 17.9.1 Theorem 15 (p. 439) 17.9.2 Theorem 16 (p. 440) 17.9.3 Theorem 17 (p. 449) EXERCISES References Author Index Subject Index Back Cover Foundations of Measurement offers the most coherently organized treatment of the topics and issues central to measurement. Much of the research involved has been scattered over several decades and a multitude of journals--available in many instances only to specialties. With the publication of Volumes two and three of this important work, Foundations of Measurement is the most comprehensive presentation in the area of measurement. "A classic series in the field of quantitative measurement, Volume I introduces the distinct mathematical results that serve to formulate numerical representations of qualitative structures. Volume II extends the subject in the direction of geometrical, threshold, and probabilistic representations, and Volume III examines representation as expressed in axiomatization and invariance. 1971 edition." -- Publisher