On minima of discrimination functions

A discrimination function ψ(x,y) assigns a measure of discriminability to stimulus pairs x,y (e.g., the probability with which they are judged to be different in a same-different judgment scheme). If for every x there is a single y least discriminable from x, then this y is called the point of subjective equality (PSE) for x, and the dependence h(x) of the PSE for x on x is called a PSE function. The PSE function g(y) is defined in a symmetrically opposite way. If the graphs of the two PSE functions coincide (i.e., g≡h⁻¹), the function is said to satisfy the Regular Minimality law. The minimum level functions are restrictions of ψ to the graphs of the PSE functions. The conjunction of two characteristics of ψ, (1) whether it complies with Regular Minimality, and (2) whether the minimum level functions are constant, has consequences for possible models of perceptual discrimination. By a series of simple theorems and counterexamples, we establish set-theoretic, topological, and analytic properties of ψ which allow one to relate to each other these two characteristics of ψ.     

Resolution time and the coding of arithmetic relations on supraliminally different visual extents

Comparison time for pairs of vertical-line stimuli, sufficiently different that they can be errorlessly discriminated with respect to visual extent, was examined as a function of arithmetic relations (physical ratio and difference) on members of the pair. Arithmetic relations are coded very precisely by judgment time: Responses slow as stimulus ratios approach one with difference fixed, and as stimulus differences approach zero with ratio fixed. Most models which assume a simple (Difference or Ratio) resolution rule operating on independent sensations require judgment time to depend on either ratios or on differences but not on both. Further tests showed both an index based on median judgment times and a confusion index based on pairs of observed judgment times, satisfied the requirements for a Positive Difference Structure. One representation of these data, which remains acceptable through all analyses, is a Difference resolution rule operating on sensations determined by a power psychophysical function with β < 1. Specifically, L(x, y) = F{ψ(x) ? ψ(y)} + R, where L(x, y) is the judgment time with the stimulus pair x and y, ψ(x) = Ax^β + C, R is a positive constant, and F is a continuous monotone decreasing function.     

Dissimilarity cumulation theory in smoothly connected spaces

This is the third paper in the series introducing the Dissimilarity Cumulation theory and its main psychological application, Universal Fechnerian Scaling. The previously developed dissimilarity-based theory of path length is used to construct the notion of a smooth path, defined by the property that the ratio of the dissimilarity between its points to the length of the subtended fragment of the path tends to unity as the points get closer to each other. We consider a class of stimulus spaces in which for every path there is a series of piecewise smooth paths converging to it pointwise and in length; and a subclass of such spaces where any two sufficiently close points can be connected by a smooth “geodesic in the small”. These notions are used to construct a broadly understood Finslerian geometry of stimulus spaces representable by regions of Euclidean n-spaces. With an additional assumption of comeasurability in the small between the canonical psychometric increments of the first and second kind, this establishes a link between Universal Fechnerian Scaling and Multidimensional Fechnerian Scaling in Euclidean n-spaces. The latter was a starting point for our theoretical program generalizing Fechner’s idea that sensation magnitudes can be computed by integration of a local discriminability measure.     

Affine representations in psychophysics and the near-miss to Weber’s law

A theoretical account for the near-miss to Weber’s law in the form of a power function, with a special emphasis on the interpretation of the exponent, was proposed by Falmagne [Falmagne, J.-C. (1985). Elements of psychophysical theory. New York: Oxford University Press] within the framework of a subtractive representation, P(x,y)=F(u(x)−g(y)). In this paper, we examine a more general affine representation, P(x,y)=F(u(x)h(y)+g(y)). We first obtain a uniqueness theorem for the affine representation. We then study the conditions that force an affine representation to degenerate to a subtractive one. Part of that study involves the case for which two different affine representations co-exist for the same data. We also show that the balance condition P(x,y)+P(y,x)=1 constrains an affine representation to be a special kind of subtractive representation, a Fechnerian one. We further show that Falmagne’s power law takes on a special form for a so-called weakly balanced system of probabilities, in which case the affine representation is Fechnerian. Finally, following Iverson [Iverson, G.J. (2006a). Analytical methods in the theory of psychophysical discrimination I: Inequalities, convexity and integration of just noticeable differences. Journal of Mathematical Psychology, 50, 271-282], we generalize the Fechner method to construct the sensory scales in a weakly balanced affine representation by integrating (derivatives of) just noticeable differences.     

Dual scaling of comparison and reference stimuli in multi-dimensional psychological space

Dzhafarov and Colonius (Psychol. Bull. Rev. 6 (1999)239; J. Math. Psychol. 45(2001)670) proposed a theory of Fechnerian scaling of the stimulus space based on the psychometric (discrimination probability) function of a human subject in a same-different comparison task. Here, we investigate a related but different paradigm, namely, referent-probe comparison task, in which the pair of stimuli (x and y) under comparison assumes substantively different psychological status, one serving as a referent and the other as a probe. The duality between a pair of psychometric functions, arising from assigning either x or y to be the fixed reference stimulus and the other to be the varying comparison stimulus, and the 1-to-1 mapping between the two stimulus spaces X and Y under either assignment are analyzed. Following Dzhafarov and Colonius, we investigate two properties characteristic of a referent-probe comparison task, namely, (i) Regular cross-minimality—for the pair of stimulus values involved in referent-probe comparison, each minimizes a discrimination probability function where the other is treated as the fixed reference stimulus; (ii) Nonconstant self-similarity—the value of the discrimination probability function at its minima is a nonconstant function of the reference stimulus value. For the particular form of psychometric functions investigated, it is shown that imposing the condition of regular cross-minimality on the pair of psychometric functions forces a consistent (but otherwise still arbitrary) mapping between X and Y, such that it is independent of the assignment of reference/comparison status to x and to y. The resulting psychometric differentials under both assignments are equal, and take an asymmetric, dualistic form reminiscent of the so-called divergence measure that appeared in the context of differential geometry of the probability manifold with dually flat connections (Differential Geometric Methods in Statistics, Lecture Notes in Statistics, Vol. 28, Springer, New York, 1985). The pair of divergence functions on X and on Y, respectively, induce a Riemannian metric in the small, with psychometric order (defined in Dzhafarov & Colonius, 1999) equal to 2. The difference between the Finsler-Riemann geometric approach to the stimulus space (Dzhafarov & Colonius, 1999) and this dually-affine Riemannian geometric approach to the dual scaling of the comparison and the reference stimuli is discussed.     

Dissimilarity cumulation theory and subjective metrics

We present a new mathematical notion, dissimilarity function, and based on it, a radical extension of Fechnerian Scaling, a theory dealing with the computation of subjective distances from pairwise discrimination probabilities. The new theory is applicable to all possible stimulus spaces subject to the following two assumptions: (A) that discrimination probabilities satisfy the Regular Minimality law and (B) that the canonical psychometric increments of the first and second kind are dissimilarity functions. A dissimilarity function Dab for pairs of stimuli in a canonical representation is defined by the following properties: (1) a≠b?Dab>0; (2) Daa=0; (3) If and , then ; and (4) for any sequence {a_nX_nb_n}_n∈N, where X_n is a chain of stimuli, Da_nX_nb_n→0?Da_nb_n→0. The expression DaXb refers to the dissimilarity value cumulated along successive links of the chain aXb. The subjective (Fechnerian) distance between a and b is defined as the infimum of DaXb+DbYa across all possible chains X and Y inserted between a and b.     

Fechnerian metrics in unidimensional and multidimensional stimulus spaces

A new theory is proposed for subjective (Fechnerian) distances among stimuli in a continuous stimulus space of arbitrary dimensionality. Each stimulus in such a space is associated with a psychometric function that determines probabilities with which it is discriminated from other stimuli, and a certain measure of its discriminability from its infinitesimally close neighboring stimuli is computed from the shape of this psychometric function in the vicinity of its minimum. This measure of discriminability can be integrated along any path connecting any two points in the stimulus space, yielding the psychometric length of this path. The Fechnerian distance between two stimuli is defined as the infimum of the psychometric lengths of all paths connecting the two stimuli. For a broad class of models defining the dichotomy of response bias versus discriminability, the Fechnerian distances are invariant under response bias changes. In the case in which physically multidimensional stimuli are discriminated along some unidimensional subjective attribute, a systematic construction of the Fechnerian metric leads to a resolution of the long-standing controversy related to the numbers of just-noticeable differences between isosensitivity curves. It is argued that for unidimensional stimulus continua, the proposed theory is close to the intended meaning of Fechner's original theory.     

Similarity of rectangles

Krantz and Tversky found that neither (log-) height (y) and width (x), nor area (x + y) and shape (x ? y) qualify as “subjective dimensions of rectangles” because both pairs violate the decomposability condition for their dissimilarity data. However, the data suggest a nonlinear transformation of x, y into a pair of subjective dimensions u(x, y), v(x, y) for which decomposability should be approximately satisfied. An explicit statement of this mapping is given.     

Dissimilarity cumulation theory in arc-connected spaces

This paper continues the development of the Dissimilarity Cumulation theory and its main psychological application, Universal Fechnerian Scaling [Dzhafarov, E.N and Colonius, H. (2007). Dissimilarity Cumulation theory and subjective metrics. Journal of Mathematical Psychology, 51, 290-304]. In arc-connected spaces the notion of a chain length (the sum of the dissimilarities between the chain’s successive elements) can be used to define the notion of a path length, as the limit inferior of the lengths of chains converging to the path in some well-defined sense. The class of converging chains is broader than that of converging inscribed chains. Most of the fundamental results of the metric-based path length theory (additivity, lower semicontinuity, etc.) turn out to hold in the general dissimilarity-based path length theory. This shows that the triangle inequality and symmetry are not essential for these results, provided one goes beyond the traditional scheme of approximating paths by inscribed chains. We introduce the notion of a space with intermediate points which generalizes (and specializes to when the dissimilarity is a metric) the notion of a convex space in the sense of Menger. A space is with intermediate points if for any distinct there is a different point such that (where D is dissimilarity). In such spaces the metric G induced by D is intrinsic: coincides with the infimum of lengths of all arcs connecting to In Universal Fechnerian Scaling D stands for either of the two canonical psychometric increments and (ψ denoting discrimination probability). The choice between the two makes no difference for the notions of arc-connectedness, convergence of chains and paths, intermediate points, and other notions of the Dissimilarity Cumulation theory.     

Conditions under which thurstone case III representations for binary choice probabilities are also Fechnerian

By a Thurstone Case III representation for binary symmetric choice probabilities P_x,y we mean that there exist functions F, μ, σ > 0 such that $\text{P}{}_{\mathrm{x,y}}\text{= F[}\text{(μ(x) ? μ(y))}\text{(σ}{}^2\text{(x) + σ}{}^2\text{(y))}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{]}$. We show that the constraint σ = constant, or μ = ασ + β, α ≠ 0, is both necessary and sufficient for a Thurstone Case III representation to be Fechnerian, i.e., to be reexpressable as as P_x,y = G(u(x) ? u(y)) for some suitably chosen functions G, u.     

Multidimensional Fechnerian Scaling: Basics

Fechnerian scaling is a theory of how a certain (Fechnerian) metric can be computed in a continuous stimulus space of arbitrary dimensionality from the shapes of psychometric (discrimination probability) functions taken in small vicinities of stimuli at which these functions reach their minima. This theory is rigorously derived in this paper from three assumptions about psychometric functions: (1) that they are continuous and have single minima around which they increase in all directions; (2) that any two stimulus differences from these minimum points that correspond to equal rises in discrimination probabilities are comeasurable in the small (i.e., asymptotically proportional), with a continuous coefficient of proportionality; and (3) that oppositely directed stimulus differences from a minimum point that correspond to equal rises in discrimination probabilities are equal in the small. A Fechnerian metric derived from these assumptions is an internal (or generalized Finsler) metric whose indicatrices are asymptotically similar to the horizontal cross-sections of the psychometric functions made just above their minima. Copyright 2001 Academic Press.     

Reconstructing Distances among Objects from Their Discriminability

We describe a principled way of imposing a metric representing dissimilarities on any discrete set of stimuli (symbols, handwritings, consumer products, X-ray films, etc.), given the probabilities with which they are discriminated from each other by a perceiving system, such as an organism, person, group of experts, neuronal structure, technical device, or even an abstract computational algorithm. In this procedure one does not have to assume that discrimination probabilities are monotonically related to distances, or that the distances belong to a predefined class of metrics, such as Minkowski. Discrimination probabilities do not have to be symmetric, the probability of discriminating an object from itself need not be a constant, and discrimination probabilities are allowed to be 0’s and 1’s. The only requirement that has to be satisfied is Regular Minimality, a principle we consider the defining property of discrimination: for ordered stimulus pairs (a,b), b is least frequently discriminated from a if and only if a is least frequently discriminated from b. Regular Minimality generalizes one of the weak consequences of the assumption that discrimination probabilities are monotonically related to distances: the probability of discriminating a from a should be less than that of discriminating a from any other object. This special form of Regular Minimality also underlies such traditional analyses of discrimination probabilities as Multidimensional Scaling and Cluster Analysis. This research was supported by the NSF grant SES 0318010 (E.D.), Humboldt Research Award (E.D.), Humboldt Foundation grant DEU/1038348 (H.C. & E.D.), and DFG grant Co 94/5 (H.C.).     

Nondecomposable conjoint measurement for bisymmetric structures

This paper discusses two “nondecomposable” conjoint measurement representations for an asymmetric binary relation ? on a product set A × X, namely (a, x) ? (b, y) iff f₁(a) + g₁(a)g₂(x) > f₁(b) + g₁(b)g₂(y), and (a, x) ? (b, y) iff f₁(a) + f₂(x) + g₁(a)g₂(x) > f₁(b) + f₂(y) + g₁(b)g₂(y). Difficulties in developing axioms for ? on A × X which imply these representations in a general formulation have led to their examination from the standpoint of bisymmetric structures based on applications of a binary operation to A × X. Depending on context, the binary operation may refer to concatenation, extensive or intensive averaging, gambles based on an uncertain chance event, or to some other interpretable process. Independence axioms which are necessary and sufficient for the special representations within the context of bisymmetric structures are presented.     

Multidimensional Fechnerian Scaling: Perceptual Separability

A new definition of the perceptual separability of stimulus dimensions is given in terms of discrimination probabilities. Omitting technical details, stimulus dimensions are considered separable if the following two conditions are met: (a) the probability of discriminating two sufficiently close stimuli is computable from the probabilities with which one discriminates the projections of these stimuli on the coordinate axes; (b) the psychometric differential for discriminating two sufficiently close stimuli that differ in one coordinate only does not depend on the value of their matched coordinates (the psychometric differential is the difference between the probability of discriminating a comparison stimulus from a reference stimulus and the probability with which the reference is discriminated from itself). Thus defined perceptual separability is analyzed within the framework of the regular variation version of multidimensional Fechnerian scaling. The result of this analysis is that the Fechnerian metric of a stimulus space with perceptually separable dimensions has the structure of a Minkowski power metric with respect to these dimensions. The exponent of this metric equals the psychometric order of the stimulus space, or 1, whichever is greater.     

The parameters in the near-miss-to-Weber's law

Many empirical data support the hypothesis that the sensitivity function grows as a power function of the stimulus intensity. This is usually referred to as the near-miss-to-Weber's law. The aim of the paper is to examine the near-miss-to-Weber's law in the context of psychometric models of discrimination. We study two types of psychometric functions, characterized by the representations P_a(x)=F(ρ(a)x^γ(a)) (type A), and P_a(x)=F(γ(a)+ρ(a)x) (type B). A central result shows that both types of psychometric functions are compatible with the near-miss-to-Weber's law. If a representation of type B exists, then the exponent in the near-miss is necessarily a constant function, that is, does not depend on the criterion value used to define “just noticeably different”. If, on the other hand, a representation of type A exists, then the exponent in the near-miss-to-Weber's law can vary with the criterion value. In that case, the parameters in the near-miss co-vary systematically.     

Multidimensional Fechnerian Scaling: Pairwise Comparisons, Regular Minimality, and Nonconstant Self-Similarity

Stimuli presented pairwise for same-different judgments belong to two distinct observation areas (different time intervals and/or locations). To reflect this fact the underlying assumptions of multidimensional Fechnerian scaling (MDFS) have to be modified, the most important modification being the inclusion of the requirement that the discrimination probability functions possess regular minima. This means that the probability with which a fixed stimulus in one observation area (a reference) is discriminated from stimuli belonging to another observation area reaches its minimum when the two stimuli are identical (following, if necessary, an appropriate transformation of the stimulus measurements in one of the two observation areas). The remaining modifications of the underlying assumptions are rather straightforward, their main outcome being that each of the two observation areas has its own Fechnerian metric induced by a metric function obtained in accordance with the regular variation version of MDFS. It turns out that the regular minimality requirement, when combined with the empirical fact of nonconstant self-similarity (which means that the minimum level of the discrimination probability function for a fixed reference stimulus is generally different for different reference stimuli), imposes rigid constraints on the interdependence between discrimination probabilities and metric functions within each of the observation areas and on the interdependence between Fechnerian metrics and metric functions belonging to different observation areas. In particular, it turns out that the psychometric order of the stimulus space cannot exceed 1.     

Some counterexamples in multidimensional scaling

Given a set X with elements x, y,… which has a partial order < on the pairs of the Cartesian product X², one may seek a distance function ? on such pairs (x, y) which satisfies ?(x₁, y₁) < ?(x₂, y₂) precisely when (x₁, y₁) < (x₂, y₂), and even demand a metric space (X, ?) with some such compatible ? which has an isometric imbedding into a finite-dimensional Euclidean space or a separable Hilbert space. We exhibit here systems (X, <) which cannot meet the latter demand. The space of real m-tuples (ξ₁,…,ξ_m) with either the “city-block” norm Σ_i ∥ξ_i∥ or the “dominance” norm max_i, ∥ξ_i∥ cannot possibly become a subset of any finite-dimensional Euclidean space. The set of real sequences (ξ₁, ξ₂,…) with finitely many nonzero elements and the supremum norm sup_i, ∥ξ_i∥ cannot even become a subset of any separable Hilbert space.     

Occasion setting is specific to the CS-US association

In Experiment 1, rats were trained on a discrimination in which one occasion setter, A, signaled that one cue (conditioned stimulus, CS), x, would be followed by one outcome, p (unconditioned stimulus, US), and a second CS, y, by a different outcome, q (x → p and y → q); a second occasion setter, B signalled the reverse CS-outcome relations (x → q and y → p). In a subsequent stage, the animals were divided into two groups, and trained as before, except that both A and B were presented in compound with a novel occasion setter, C. For Group S (same) the CS-outcome relations following A and B were identical to those in the pretraining stage, whereas in Group D (different) they were reversed. In a subsequent test, stimulus C was shown to be a more effective occasion setter in Group D than in Group S. In Experiment 2, rats were trained on a negative occasion-setting discrimination in which CS x signaled outcome p, and y outcome q; when x and y were signaled by the occasion setter A then no outcome followed (x → p, y → q, A:x-, and A:y-). In a subsequent stage, A was now trained as a positive occasion setter, signaling reinforcement of x and y. In Group S, x and y signaled the same outcomes as in the prior training stage (x-, y-, A:x → p, and A:y → q), whereas in Group D they signaled the opposite outcomes (x, y, A:x → q, and A:y → p); more efficient test performance was seen in the latter group. These results suggest that the each occasion setter conveyed information about the specific combination of CS and US paired in its presence (i.e., x → p and y → q, or x → no p and y → no q). These results are consistent with the suggestion that occasion setters operate, at least in part, on a specific CS-US association.