Foundations of Measurement Volume II: Geometrical, Threshold, and Probabilistic Representations

by David H. Krantz, R. Duncan Luce, Amos Tversky, Patrick Suppes

Overview

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, and Volume III examines representation as expressed in axiomatization and invariance.

Product Details

ISBN-13:	9780486453156
Publisher:	Dover Publications
Publication date:	12/15/2006
Series:	Dover Books on Mathematics , #2
Pages:	512
Product dimensions:	5.50(w) x 8.50(h) x (d)

About the Author

David H. Krantz is affiliated with Columbia University; R. Duncan Luce with the University of California, Irvine, and Patrick Suppes with Stanford University. Amos Tversky is deceased.

Table of Contents

Preface     xiii
Acknowledgments     xv
Overview
Geometry Unit     1
Geometrical Representations (Chapter 12)     3
Axiomatic Synthetic Geometry (Chapter 13)     4
Proximity Measurement (Chapter 14)     7
Color and Force Measurement (Chapter 15)     9
Threshold and Error Unit     9
Representations with Thresholds (Chapter 16)     10
Representations of Choice Probabilities (Chapter 17)     11
Geometrical Representations
Introduction     13
Vector Representations     14
Vector Spaces     16
Analytic Affine Geometry     21
Analytic Projective Geometry     24
Analytic Euclidean Geometry     31
Meaningfulness in Analytic Geometry     35
Minicowski Geometry     42
General Projective Metrics     46
Metric Representations     51
General Metrics with Geodesies     52
Elementary Spaces and the Helmholtz-Lie Problem     57
Riemannian Metrics     59
Other Metrics     71
Exercises     77
Axiomatic Geometry and Applications
Introduction     80
Order on the Line     83
Betweenness: Affine Order     83
Separation: Projective Order     85
Proofs     89
Projective Planes     93
Projective Spaces     102
Affine and Absolute Spaces     104
Ordered Geometric Spaces     105
Affine Spaces     107
Absolute Spaces     109
Euclidean Spaces     110
Hyperbolic Spaces     111
Elliptic Spaces     114
Double Elliptic Spaces     115
Single Elliptic Spaces     117
Classical Space-Time     118
Space-Time of Special Relativity     121
Other Axiomatic Approaches     127
Perceptual Spaces     131
Historical Survey through the Nineteenth Century     131
General Considerations Concerning Perceptual Spaces     134
Experimental Work before Luneburg's Theory     138
Luneburg Theory of Binocular Vision     139
Experiments Relevant to Luneburg's Theory     145
Other Studies     150
Exercises     153
Proximity Measurement
Introduction     159
Metrics with Additive Segments     163
Collinearity      163
Constructive Methods     165
Representation and Uniqueness Theorems     168
Proofs     169
Theorem 2     169
Reduction to Extensive Measurement     170
Theorem 3     171
Theorem 4     172
Multidimensional Representations     175
Decomposability     178
Intradimensional Subtractivity     179
Interdimensional Additivity     181
The Additive-Difference Model     184
Additive-Difference Metrics     185
Proofs     188
Theorem 5     188
Theorem 6     189
Theorem 7     191
Theorem 9     192
Preliminary Lemma     194
Theorem 10     197
Experimental Tests of Multidimensional Representations     200
Relative Curvature     201
Translation Invariance     202
The Triangle Inequality     205
Feature Representations     207
The Contrast Model     209
Empirical Applications     215
Comparing Alternative Representations     218
Proofs     222
Theorem 11     222
Exercises      224
Color and Force Measurement
Introduction     226
Grassmann Structures     229
Formulation of the Axioms     229
Representation and Uniqueness Theorems     234
Discussion of Proofs of Theorems 3 and 4     235
Proofs     237
Theorem 3     237
Theorem 4     239
Color Interpretations     240
Metameric Color Matching     240
Tristimulus Colorimetry     243
Four Ways to Misunderstand Color Measurement     250
Asymmetric Color Matching     252
The Dimensional Structure of Color and Force     255
Color Codes and Metamer Codes     256
Photopigments     260
Force Measurement and Dynamical Theory     263
Color Theory in a Measurement Framework     268
The Konig and Hurvich-Jameson Color Theories     271
Representations of 2-Chromatic Reduction Structures     271
The Konig Theory and Alternatives     277
Codes Based on Color Attributes     279
The Cancellation Procedure     280
Representation and Uniqueness Theorems     283
Tests and Extensions of Quantitative Opponent-Colors Theory      286
Proofs     291
Theorem 6     291
Theorem 9     294
Theorem 10     295
Exercises     296
Representations with Thresholds
Introduction     299
Three Approaches to Nontransitive Data     300
Idea of Thresholds     301
Overview     302
Ordinal Theory     303
Upper, Lower, and Two-Sided Thresholds     304
Induced Quasiorders: Interval Orders and Semiorders     306
Compatible Relations     311
Biorders: A Generalization of Interval Orders     313
Tight Representations     314
Constant-Threshold Representations     318
Interval and Indifference Graphs     322
Proofs     326
Theorem 2     326
Lemma 1     326
Theorem 6     327
Theorem 9     327
Theorem 10     328
Theorem 11     329
Theorems 14 and 15     330
Ordinal Theory for Families of Orders     331
Finite Families of Interval Orders and Semiorders     332
Order Relations Induced by Binary Probabilities     336
Dimension of Partial Orders      339
Proofs     341
Theorem 16     341
Theorem 17     342
Theorem 18     343
Theorem 19     344
Semiordered Additive Structures     344
Possible Approaches to Semiordered Probability Structures     345
Probability with Approximate Standard Families     349
Axiomatization of Semiordered Probability Structures     352
Weber's Law and Semiorders     353
Proof of Theorem 24     354
Random-Variable Representations     359
Weak Representations of Additive Conjoint and Extensive Structures     364
Variability as Measured by Moments     366
Qualitative Primitives for Moments     368
Axiom System for Qualitative Moments     371
Representation Theorem and Proof     374
Exercises     379
Representation of Choice Probabilities
Introduction     383
Empirical Interpretations     384
Probabilistic Representations     385
Ordinal Representations for Pair Comparisons     388
Stochastic Transitivity     388
Difference Structures     390
Local Difference Structures     391
Additive Difference Structures      393
Intransitive Preferences     397
Proofs     401
Theorem 2     401
Theorem 3     405
Theorem 4     407
Constant Repmsentations for Multiple Choice     410
Simple Scalabihty     410
The Strict-Utility Model     413
Proofs     419
Theorem 5     419
Theorem 7     420
Random Variable Representations     421
The Random-Utility Model     421
The Independent Double-Exponential Model     424
Error Tradeoff     427
Proofs     432
Theorem 9     432
Theorem 12     433
Theorem 13     435
Markovian Elimination Processes     436
The General Model     436
Elimination by Aspects     437
Preference Trees     444
Proofs     450
Theorem 15     450
Theorem 16     452
Theorem 17     455
Exercises     457
References     459
Author Index     481
Subject Index     487