Foundations of measurement, vol.1: Additive and polynomial representations

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