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Reinhard Suck

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RMIT University

Vrije Universiteit Brussel

Norwegian University of Science and Technology

Armando Marques-Guedes

UNL - New University of Lisbon

Ucla

Macquarie University

Dublin Institute of Technology

Roshan Chitrakar

Nepal College of Information Technology

Hugo Klappenbach

CONICET

University of Rochester

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Papers by Reinhard Suck

Ordered Relational Structures

Abstract The concepts of a relation, a mapping, and an operation are introduced. An ordered relat... more Abstract The concepts of a relation, a mapping, and an operation are introduced. An ordered relational structure consists of a set and a few relations defined on it. The formal definition is presented and illustrated by numerous examples from social sciences and from mathematics. Homomorphisms and isomorphisms between structures are defined. These are mappings of one structure into another which, in a way, keep its structural properties, while possibly changing the meaning of its primitives. Automorphisms are mappings of the structure onto itself. They reveal most of its important properties. The group of automorphisms of a structure is described leading to the concept of homogeneity and uniqueness of a structure. The Alper/Narens theorem characterizes structures on the reals with specified homogeneity and uniqueness properties of their automorphism group. This famous theorem is discussed along with a recent generalization. The order topology and some results in connection with it are presented. Finally, the problems which arise when a structure is embedded into a Dedekind complete ordered relational structure, are briefly mentioned. This article is closely related to the article on representational measurement theory. Furthermore, the section on the completion of structures draws on Section Order Topology of the article Partial Orders .

The Normal Distribution Derived from Qualitative Conditions

Journal of Mathematical Psychology, Apr 1, 2001

The normal distribution is characterized in a measurement theoretic framework. The qualitative co... more The normal distribution is characterized in a measurement theoretic framework. The qualitative conditions guarantee that representations can be regarded as random variables. Additional axioms, also qualitative in the measurement sense, yield the normal. One characterization draws on a limit theorem. The main result derives the normal distribution from conjoint measurement axioms. This approach consists of formulating properties of a linear model as a component structure with error as one component. The normal distribution of errors is shown to be a consequence of the measurement theoretic assumptions. The possible impact of these results on statistical models is discussed.

Distance correlation: Discovering meta-analytic relationships between variables when other correlation coefficients fail

Assessing nonlinear relationships  Comparison of four different coefficients • Distance correlat... more Assessing nonlinear relationships  Comparison of four different coefficients • Distance correlation, Pearson's ρ, Kendall's τ, and Spearman's ρ Linear relationship Inverted-U relationship Distance correlation in meta-analysis Lukasz Stasielowicz & Reinhard Suck Research Synthesis, Dubrovnik Methods in meta-analysis (29.05.2019

Probabilistic Biclassification and Random Variable Representations

Journal of Mathematical Psychology, Mar 1, 1997

Random utility theory has been generalized by various authors to a large variety of relations. A ... more Random utility theory has been generalized by various authors to a large variety of relations. A fairly general description of this theory is given and its combinatorial, geometric, and probabilistic ingredients are outlined. Subsequently, the technique is applied to the class of biclassification relations. Furthermore, a modification is presented which is applicable in cases in which the standard theory breaks down. The principles of applying the modified version to the problem of interrater reliability are outlined. The paper closes with considering statistical questions both in the standard and in the modified version.

Factorization and Additive Decomposition of A Weak Order

Advances in psychology, 1984

Abstract Possibilities of deriving a Cartesian nroduct structure from a weak order are investigat... more Abstract Possibilities of deriving a Cartesian nroduct structure from a weak order are investigated. As an intermediate step a “comoensatory structure” is introduced and a renresentation theorem for this special type proved. Conditions are formulated that guarantee the desired factorization and yield an additive reoresentation. The naner closes with some remarks on possible applications.

Algebraic Representation of Additive Structures with an Infinite Number of Components

Journal of Mathematical Psychology, Dec 1, 1993

Representation theorems for additive conjoint structures with an in!inite number of components we... more Representation theorems for additive conjoint structures with an in!inite number of components were !irst developed by T. C. Koopmans (1960, Econometrica 28, 287-309) for modelling dynamic decision behavior. However, he proved his theorems within a topological framework. Here, the theorems are adapted to and proved within a more general algebraic framework. Moreover, while Koopmans provided a representation only for bounded component sequences, here the representation is extended to unbounded sequences satisfying polynomial growth. Additive conJomt structures with an infinte number of components were intro duced by Koopmans (1960) to model dynamic decision behavior. He derived representations for preference orderings of consumption programs (see also Koopmans, Diamond, and Williamson, 1964; Diamond, 1965; Koopmans, 1972). Recently, Hubner (l989a) showed that such structures are not only useful in modelling other dynamic behavior as well, but that they are also valuable in combining axiomatic measurement theory and linear system theory. By successfully modelling loudness adaption he demonstrated that his ideas are even suitable for system identification (Hubner, 1989b). A disadvantage of Koopmans' results is that they are developed within the topological framework of Debreu (1960). It is not only that this formalism is uncommon for psychologically motivated measurement theories, but, as has been mentioned by Krantz, Luce, Suppes, and Tversky (1971) and recently emphasized by Wakker (1988), the algebraic approach of Krantz e/ al. (1971) is more general than the topological; i.e., the algebraic approach is applicable to more cases.

Independent random utility representations

Mathematical Social Sciences, Jul 1, 2002

ABSTRACT The question of the existence of independent random variables which represent a given se... more ABSTRACT The question of the existence of independent random variables which represent a given set of binary choice data is investigated. It is related to the linear ordering polytope. The subset of it which is independently representable, denoted by ILOn, is characterized for three elements. For the general case, necessary conditions are given and their geometric meaning is discussed. A procedure, called mixture technique, is developed which allows one to construct a new point in ILOn and its independent representation from known points in ILOn. Finally, a few results on parametric representations are derived.

Stress Caused by Waiting: A Theoretical Evaluation of a Mathematical Model

Journal of Mathematical Psychology, Sep 1, 1997

According to cognitive stress theories, stress caused by waiting is influenced by two components,... more According to cognitive stress theories, stress caused by waiting is influenced by two components, the (psychological) cost of waiting, that is, a function C of time, and a probability distribution over waiting times. Osuna (1985a) suggested a model by which stress as a function of time could be calculated from these constituents. The aim of this paper is (1) to generalize the model, (2) to investigate its mathematical properties, (3) to derive predictions for experimental tests of the model, and (4) to give a precise meaning to the variability and duration hypothesis discussed in the experimental literature. In particular, several theorems are derived which specify situations in which the model enables the user to compare conditions for their stress inducing potential. Such conditions are the duration of the waiting time and the predictability of the length of a waiting period. The core of the mathematical derivations is an identity for expected stress which simplifies the calculations of the Osuna model considerably.

Conjointness as a derived property

Journal of Mathematical Psychology, Mar 1, 1990

Abstract In this paper we investigate the question of which conditions imposed on a weak order ( ... more Abstract In this paper we investigate the question of which conditions imposed on a weak order ( A , ≾) allow for a representation of ( A , ≾) as a Cartesian product structure ( X 1 , X 2 , ≾′) with independent components X 1 , X 2 and a weak order ≾′ on X 1 × X 2 . Two possibilities are analyzed: first, two weak orders on A are supposed to exist in addition to ≾ and a compatibility condition between all three is shown to yield the decomposition; second, one additional weak order, together with a relation which reflects the compensatory trade-off components typically exhibit in an additive representation, is demonstrated to give essentially the same result. Furthermore, the paper deals with the automorphism groups of these structures and relates them to the important concept of a factorizable automorphism of a conjoint structure. Next, a generalization to more than two components is presented and it is shown that the compatibility between the weak orders gives rise to a partial order or, if additional assumptions are met, even to a lattice. The final section of the paper discusses an application of one of our theorems to the theory of psychological tests.

Parsimonious set representations of orders, a generalization of the interval order concept, and knowledge spaces

Discrete Applied Mathematics, Apr 1, 2003

Partial orders can be represented by subsets of a given set ordered by inclusion. Special kinds o... more Partial orders can be represented by subsets of a given set ordered by inclusion. Special kinds of such set representations are investigated because they facilitate the exploration of properties of knowledge spaces. A new type of order relation is deÿned which is closely related to interval orders. These orders and the interval orders are investigated with respect to their parsimonious set representations. The theorems are applied to knowledge space theory.

Partial Orders

Elsevier eBooks, 2015

Erratum to: “Media: A mathematical structure, its development, forerunners and applications” [J. Math. Psychol. 52 (2008) 202–205]

Journal of Mathematical Psychology, Oct 1, 2008

The publisher regrets that the following corrections requested by the author at the proof stage w... more The publisher regrets that the following corrections requested by the author at the proof stage were not incorporated. On page 202, the book author’s name should read ‘‘David Eppstein’’ instead of ‘‘David Epstein’’. On page 203, in line 6 of the third full paragraph in the right column, the extraneous ‘‘the’’ before ‘‘these sets’’ should be deleted. On page 203, in lines 2 and 3 from the bottom of the page, ‘‘twodimensional’’ should read ‘‘two dimension’’. On page 204, in line 5 from the top, ‘‘their close relation to graphs’’ should read ‘‘the close relation between media and graphs’’. On page 204, in line 13 of the second full paragraph in the right column, ‘‘space is a union of a closed collection of subsets’’ should read ‘‘space is a union closed collection of subsets’’. On page 204, in the fourth paragraph in the right column, the

Media: A mathematical structure, its development, forerunners and applications

Journal of Mathematical Psychology, May 1, 2008

An ordinal evaluation of categorical judgement data by random utilities and a corresponding correlation analysis

Journal of Mathematical Psychology, Dec 1, 2005

ABSTRACT Categorical judgement data are analyzed along the lines of random utility theory. A clas... more ABSTRACT Categorical judgement data are analyzed along the lines of random utility theory. A class of orders is introduced (categorical weak orders); their characteristic vectors are regarded as points in a Euclidean space; their convex hull forms a polytope whose facets are fully characterized. This polytope is shown to correspond to an order polytope. Furthermore, its relation to the biorder polytope is pointed out. The convex representations of a given point of the polytope are discussed. The impact of these results on the methods of analyzing data arising from a categorical judgement procedure is outlined. In particular, some consequences are drawn with respect to the usual evaluation of correlations of such data.

A Note on the Eigensystem of the Covariance Matrix of Dichotomous Guttman Items

Frontiers in Psychology, Dec 1, 2015

We consider the covariance matrix for dichotomous Guttman items under a set of uniformity conditi... more We consider the covariance matrix for dichotomous Guttman items under a set of uniformity conditions, and obtain closed-form expressions for the eigenvalues and eigenvectors of the matrix. In particular, we describe the eigenvalues and eigenvectors of the matrix in terms of trigonometric functions of the number of items. Our results parallel those of Zwick (1987) for the correlation matrix under the same uniformity conditions. We provide an explanation for certain properties of principal components under Guttman scalability which have been first reported by Guttman (1950).

Ordered Relational Structures

Elsevier eBooks, 2001

Abstract The concepts of a relation, a mapping, and an operation are introduced. An ordered relat... more Abstract The concepts of a relation, a mapping, and an operation are introduced. An ordered relational structure consists of a set and a few relations defined on it. The formal definition is presented and illustrated by numerous examples from social sciences and from mathematics. Homomorphisms and isomorphisms between structures are defined. These are mappings of one structure into another which, in a way, keep its structural properties, while possibly changing the meaning of its primitives. Automorphisms are mappings of the structure onto itself. They reveal most of its important properties. The group of automorphisms of a structure is described leading to the concept of homogeneity and uniqueness of a structure. The Alper/Narens theorem characterizes structures on the reals with specified homogeneity and uniqueness properties of their automorphism group. This famous theorem is discussed along with a recent generalization. The order topology and some results in connection with it are presented. Finally, the problems which arise when a structure is embedded into a Dedekind complete ordered relational structure, are briefly mentioned. This article is closely related to the article on representational measurement theory. Furthermore, the section on the completion of structures draws on Section Order Topology of the article Partial Orders .

A Qualitative Characterization of the Exponential Distribution

Journal of Mathematical Psychology, Dec 1, 1998

An axiomization of a random variable representation is developed. The existence of the probabilit... more An axiomization of a random variable representation is developed. The existence of the probability measure with respect to which the representations are measurable is derived from qualitative conditions in the sense of measurement theory. The structure is based on a combination of a difference with an extensive measurement structure. Furthermore, a special condition, qualitative as the others, is shown to yield the result that the real valued representations of the structure are random variables with an exponential distribution. The key property is a condition relating the difference order and the concatenation in such a way that shifting two intervals by concatenating their endpoints with the same element does not affect their order relation.

Progress in mathematical psychology

Elsevier eBooks, 1987

Measurement Theory. Cognitive Processes. Performance and Timing. Perception and Psychophysics. Ps... more

The Empirical Claim of Probability Statements, Idealized Bernoulli Experiments and their Approximate Version

BRILL eBooks, 1995

Ordering orders

Mathematical Social Sciences, Sep 1, 1998

This paper is concerned with structures of the form (A, O) where A is a nonempty set and O a set ... more This paper is concerned with structures of the form (A, O) where A is a nonempty set and O a set of order relations on A. In particular we investigate their representability as a product structure with a weak order defined on a Cartesian product satisfying independence in the Conjoint Measurement sense. It is shown by a number of examples

Ordered Relational Structures

Abstract The concepts of a relation, a mapping, and an operation are introduced. An ordered relat... more Abstract The concepts of a relation, a mapping, and an operation are introduced. An ordered relational structure consists of a set and a few relations defined on it. The formal definition is presented and illustrated by numerous examples from social sciences and from mathematics. Homomorphisms and isomorphisms between structures are defined. These are mappings of one structure into another which, in a way, keep its structural properties, while possibly changing the meaning of its primitives. Automorphisms are mappings of the structure onto itself. They reveal most of its important properties. The group of automorphisms of a structure is described leading to the concept of homogeneity and uniqueness of a structure. The Alper/Narens theorem characterizes structures on the reals with specified homogeneity and uniqueness properties of their automorphism group. This famous theorem is discussed along with a recent generalization. The order topology and some results in connection with it are presented. Finally, the problems which arise when a structure is embedded into a Dedekind complete ordered relational structure, are briefly mentioned. This article is closely related to the article on representational measurement theory. Furthermore, the section on the completion of structures draws on Section Order Topology of the article Partial Orders .

The Normal Distribution Derived from Qualitative Conditions

Journal of Mathematical Psychology, Apr 1, 2001

The normal distribution is characterized in a measurement theoretic framework. The qualitative co... more The normal distribution is characterized in a measurement theoretic framework. The qualitative conditions guarantee that representations can be regarded as random variables. Additional axioms, also qualitative in the measurement sense, yield the normal. One characterization draws on a limit theorem. The main result derives the normal distribution from conjoint measurement axioms. This approach consists of formulating properties of a linear model as a component structure with error as one component. The normal distribution of errors is shown to be a consequence of the measurement theoretic assumptions. The possible impact of these results on statistical models is discussed.

Distance correlation: Discovering meta-analytic relationships between variables when other correlation coefficients fail

Assessing nonlinear relationships  Comparison of four different coefficients • Distance correlat... more Assessing nonlinear relationships  Comparison of four different coefficients • Distance correlation, Pearson's ρ, Kendall's τ, and Spearman's ρ Linear relationship Inverted-U relationship Distance correlation in meta-analysis Lukasz Stasielowicz & Reinhard Suck Research Synthesis, Dubrovnik Methods in meta-analysis (29.05.2019

Probabilistic Biclassification and Random Variable Representations

Journal of Mathematical Psychology, Mar 1, 1997

Random utility theory has been generalized by various authors to a large variety of relations. A ... more Random utility theory has been generalized by various authors to a large variety of relations. A fairly general description of this theory is given and its combinatorial, geometric, and probabilistic ingredients are outlined. Subsequently, the technique is applied to the class of biclassification relations. Furthermore, a modification is presented which is applicable in cases in which the standard theory breaks down. The principles of applying the modified version to the problem of interrater reliability are outlined. The paper closes with considering statistical questions both in the standard and in the modified version.

Factorization and Additive Decomposition of A Weak Order

Advances in psychology, 1984

Abstract Possibilities of deriving a Cartesian nroduct structure from a weak order are investigat... more Abstract Possibilities of deriving a Cartesian nroduct structure from a weak order are investigated. As an intermediate step a “comoensatory structure” is introduced and a renresentation theorem for this special type proved. Conditions are formulated that guarantee the desired factorization and yield an additive reoresentation. The naner closes with some remarks on possible applications.

Algebraic Representation of Additive Structures with an Infinite Number of Components

Journal of Mathematical Psychology, Dec 1, 1993

Representation theorems for additive conjoint structures with an in!inite number of components we... more Representation theorems for additive conjoint structures with an in!inite number of components were !irst developed by T. C. Koopmans (1960, Econometrica 28, 287-309) for modelling dynamic decision behavior. However, he proved his theorems within a topological framework. Here, the theorems are adapted to and proved within a more general algebraic framework. Moreover, while Koopmans provided a representation only for bounded component sequences, here the representation is extended to unbounded sequences satisfying polynomial growth. Additive conJomt structures with an infinte number of components were intro duced by Koopmans (1960) to model dynamic decision behavior. He derived representations for preference orderings of consumption programs (see also Koopmans, Diamond, and Williamson, 1964; Diamond, 1965; Koopmans, 1972). Recently, Hubner (l989a) showed that such structures are not only useful in modelling other dynamic behavior as well, but that they are also valuable in combining axiomatic measurement theory and linear system theory. By successfully modelling loudness adaption he demonstrated that his ideas are even suitable for system identification (Hubner, 1989b). A disadvantage of Koopmans' results is that they are developed within the topological framework of Debreu (1960). It is not only that this formalism is uncommon for psychologically motivated measurement theories, but, as has been mentioned by Krantz, Luce, Suppes, and Tversky (1971) and recently emphasized by Wakker (1988), the algebraic approach of Krantz e/ al. (1971) is more general than the topological; i.e., the algebraic approach is applicable to more cases.

Independent random utility representations

Mathematical Social Sciences, Jul 1, 2002

ABSTRACT The question of the existence of independent random variables which represent a given se... more ABSTRACT The question of the existence of independent random variables which represent a given set of binary choice data is investigated. It is related to the linear ordering polytope. The subset of it which is independently representable, denoted by ILOn, is characterized for three elements. For the general case, necessary conditions are given and their geometric meaning is discussed. A procedure, called mixture technique, is developed which allows one to construct a new point in ILOn and its independent representation from known points in ILOn. Finally, a few results on parametric representations are derived.

Stress Caused by Waiting: A Theoretical Evaluation of a Mathematical Model

Journal of Mathematical Psychology, Sep 1, 1997

According to cognitive stress theories, stress caused by waiting is influenced by two components,... more According to cognitive stress theories, stress caused by waiting is influenced by two components, the (psychological) cost of waiting, that is, a function C of time, and a probability distribution over waiting times. Osuna (1985a) suggested a model by which stress as a function of time could be calculated from these constituents. The aim of this paper is (1) to generalize the model, (2) to investigate its mathematical properties, (3) to derive predictions for experimental tests of the model, and (4) to give a precise meaning to the variability and duration hypothesis discussed in the experimental literature. In particular, several theorems are derived which specify situations in which the model enables the user to compare conditions for their stress inducing potential. Such conditions are the duration of the waiting time and the predictability of the length of a waiting period. The core of the mathematical derivations is an identity for expected stress which simplifies the calculations of the Osuna model considerably.

Conjointness as a derived property

Journal of Mathematical Psychology, Mar 1, 1990

Abstract In this paper we investigate the question of which conditions imposed on a weak order ( ... more Abstract In this paper we investigate the question of which conditions imposed on a weak order ( A , ≾) allow for a representation of ( A , ≾) as a Cartesian product structure ( X 1 , X 2 , ≾′) with independent components X 1 , X 2 and a weak order ≾′ on X 1 × X 2 . Two possibilities are analyzed: first, two weak orders on A are supposed to exist in addition to ≾ and a compatibility condition between all three is shown to yield the decomposition; second, one additional weak order, together with a relation which reflects the compensatory trade-off components typically exhibit in an additive representation, is demonstrated to give essentially the same result. Furthermore, the paper deals with the automorphism groups of these structures and relates them to the important concept of a factorizable automorphism of a conjoint structure. Next, a generalization to more than two components is presented and it is shown that the compatibility between the weak orders gives rise to a partial order or, if additional assumptions are met, even to a lattice. The final section of the paper discusses an application of one of our theorems to the theory of psychological tests.

Parsimonious set representations of orders, a generalization of the interval order concept, and knowledge spaces

Discrete Applied Mathematics, Apr 1, 2003

Partial orders can be represented by subsets of a given set ordered by inclusion. Special kinds o... more Partial orders can be represented by subsets of a given set ordered by inclusion. Special kinds of such set representations are investigated because they facilitate the exploration of properties of knowledge spaces. A new type of order relation is deÿned which is closely related to interval orders. These orders and the interval orders are investigated with respect to their parsimonious set representations. The theorems are applied to knowledge space theory.

Partial Orders

Elsevier eBooks, 2015

Erratum to: “Media: A mathematical structure, its development, forerunners and applications” [J. Math. Psychol. 52 (2008) 202–205]

Journal of Mathematical Psychology, Oct 1, 2008

The publisher regrets that the following corrections requested by the author at the proof stage w... more The publisher regrets that the following corrections requested by the author at the proof stage were not incorporated. On page 202, the book author’s name should read ‘‘David Eppstein’’ instead of ‘‘David Epstein’’. On page 203, in line 6 of the third full paragraph in the right column, the extraneous ‘‘the’’ before ‘‘these sets’’ should be deleted. On page 203, in lines 2 and 3 from the bottom of the page, ‘‘twodimensional’’ should read ‘‘two dimension’’. On page 204, in line 5 from the top, ‘‘their close relation to graphs’’ should read ‘‘the close relation between media and graphs’’. On page 204, in line 13 of the second full paragraph in the right column, ‘‘space is a union of a closed collection of subsets’’ should read ‘‘space is a union closed collection of subsets’’. On page 204, in the fourth paragraph in the right column, the

Media: A mathematical structure, its development, forerunners and applications

Journal of Mathematical Psychology, May 1, 2008

An ordinal evaluation of categorical judgement data by random utilities and a corresponding correlation analysis

Journal of Mathematical Psychology, Dec 1, 2005

ABSTRACT Categorical judgement data are analyzed along the lines of random utility theory. A clas... more ABSTRACT Categorical judgement data are analyzed along the lines of random utility theory. A class of orders is introduced (categorical weak orders); their characteristic vectors are regarded as points in a Euclidean space; their convex hull forms a polytope whose facets are fully characterized. This polytope is shown to correspond to an order polytope. Furthermore, its relation to the biorder polytope is pointed out. The convex representations of a given point of the polytope are discussed. The impact of these results on the methods of analyzing data arising from a categorical judgement procedure is outlined. In particular, some consequences are drawn with respect to the usual evaluation of correlations of such data.

A Note on the Eigensystem of the Covariance Matrix of Dichotomous Guttman Items

Frontiers in Psychology, Dec 1, 2015

We consider the covariance matrix for dichotomous Guttman items under a set of uniformity conditi... more We consider the covariance matrix for dichotomous Guttman items under a set of uniformity conditions, and obtain closed-form expressions for the eigenvalues and eigenvectors of the matrix. In particular, we describe the eigenvalues and eigenvectors of the matrix in terms of trigonometric functions of the number of items. Our results parallel those of Zwick (1987) for the correlation matrix under the same uniformity conditions. We provide an explanation for certain properties of principal components under Guttman scalability which have been first reported by Guttman (1950).

Ordered Relational Structures

Elsevier eBooks, 2001

Abstract The concepts of a relation, a mapping, and an operation are introduced. An ordered relat... more Abstract The concepts of a relation, a mapping, and an operation are introduced. An ordered relational structure consists of a set and a few relations defined on it. The formal definition is presented and illustrated by numerous examples from social sciences and from mathematics. Homomorphisms and isomorphisms between structures are defined. These are mappings of one structure into another which, in a way, keep its structural properties, while possibly changing the meaning of its primitives. Automorphisms are mappings of the structure onto itself. They reveal most of its important properties. The group of automorphisms of a structure is described leading to the concept of homogeneity and uniqueness of a structure. The Alper/Narens theorem characterizes structures on the reals with specified homogeneity and uniqueness properties of their automorphism group. This famous theorem is discussed along with a recent generalization. The order topology and some results in connection with it are presented. Finally, the problems which arise when a structure is embedded into a Dedekind complete ordered relational structure, are briefly mentioned. This article is closely related to the article on representational measurement theory. Furthermore, the section on the completion of structures draws on Section Order Topology of the article Partial Orders .

A Qualitative Characterization of the Exponential Distribution

Journal of Mathematical Psychology, Dec 1, 1998

An axiomization of a random variable representation is developed. The existence of the probabilit... more An axiomization of a random variable representation is developed. The existence of the probability measure with respect to which the representations are measurable is derived from qualitative conditions in the sense of measurement theory. The structure is based on a combination of a difference with an extensive measurement structure. Furthermore, a special condition, qualitative as the others, is shown to yield the result that the real valued representations of the structure are random variables with an exponential distribution. The key property is a condition relating the difference order and the concatenation in such a way that shifting two intervals by concatenating their endpoints with the same element does not affect their order relation.

Progress in mathematical psychology

Elsevier eBooks, 1987

Measurement Theory. Cognitive Processes. Performance and Timing. Perception and Psychophysics. Ps... more

The Empirical Claim of Probability Statements, Idealized Bernoulli Experiments and their Approximate Version

BRILL eBooks, 1995

Ordering orders

Mathematical Social Sciences, Sep 1, 1998

This paper is concerned with structures of the form (A, O) where A is a nonempty set and O a set ... more This paper is concerned with structures of the form (A, O) where A is a nonempty set and O a set of order relations on A. In particular we investigate their representability as a product structure with a weak order defined on a Cartesian product satisfying independence in the Conjoint Measurement sense. It is shown by a number of examples