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.
Author(s): Patrick Suppes, David H. Krantz, R. Duncan Luce, Amos Tversky
Series: Foundations of Measurement, Vol. II
Publisher: Academic Press
Year: 1989
Language: English
Pages: C+vi+493+B
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
