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.     

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.     

Multidimensional Fechnerian Scaling: Probability-Distance Hypothesis

The probability-distance hypothesis states that the probability with which one stimulus is discriminated from another is a function of some subjective distance between these stimuli. The analysis of this hypothesis within the framework of multidimensional Fechnerian scaling yields the following results. If the hypothetical subjective metric is internal (which means, roughly, that the distance between two stimuli equals the infimum of the lengths of all paths connecting them), then the underlying assumptions of Fechnerian scaling are satisfied and the metric in question coincides with the Fechnerian metric. Under the probability-distance hypothesis, the Fechnerian metric exists (i.e., the underlying assumptions of Fechnerian scaling are satisfied) if and only if the hypothetical subjective metric is internalizable, which means, roughly, that by a certain transformation it can be made to coincide in the small with an internal metric; and then this internal metric is the Fechnerian metric. The specialization of these results to unidimensional stimulus continua is closely related to the so-called Fechner problem proposed in 1960's as a substitute for Fechner's original theory.     

Multidimensional Fechnerian Scaling: Regular Variation Version

The underlying assumptions of Fechnerian scaling are complemented by an assumption that ensures that any psychometric differential (the rise in the value of a discrimination probability function as one moves away from its minimum in a given direction) regularly varies at the origin with a positive exponent. This is equivalent to the following intuitively plausible property: any two psychometric differentials are comeasurable in the small (i.e., asymptotically proportional at the origin), without, however, being asymptotically equal to each other unless the corresponding values of the Fechner-Finsler metric function are equal. The regular variation version of Fechnerian scaling generalizes the previously proposed power function version while retaining its computational and conceptual simplicity.     

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.     

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.     

Psychophysics without physics: extension of Fechnerian scaling from continuous to discrete and discrete-continuous stimulus spaces

The computation of subjective (Fechnerian) distances from discrimination probabilities involves cumulation of appropriately transformed psychometric increments along smooth arcs (in continuous stimulus spaces) or chains of stimuli (in discrete spaces). In a space where any two stimuli that are each other's points of subjective equality are given identical physical labels, psychometric increments are positive differences ψ(x,y)-ψ(x,x) and ψ(y,x)-ψ(x,x), where x≠y and ψ is the probability of judging two stimuli different. In continuous stimulus spaces the appropriate monotone transformation of these increments (called overall psychometric transformation) is determined uniquely in the vicinity of zero, and its extension to larger values of its argument is immaterial. In discrete stimulus spaces, however, Fechnerian distances critically depend on this extension. We show that if overall psychometric transformation is assumed (A) to be the same for a sufficiently rich class of discrete stimulus spaces, (B) to ensure the validity of the Second Main Theorem of Fechnerian Scaling in this class of spaces, and (C) to agree in the vicinity of zero with one of the possible transformations in continuous spaces, then this transformation can only be identity. This result is generalized to the broad class of “discrete-continuous” stimulus spaces, of which continuous and discrete spaces are proper subclasses.     

Sources and influence of perceptual variance: Comment on Dzhafarov's Regular Minimality Principle

Dzhafarov [(2002). Multidimensional Fechnerian scaling: Pairwise comparisons, regular minimality, and nonconstant self-similarity. Journal of Mathematical Psychology, 46, 583-608] claims that Regular Minimality (RM) is a fundamental property of “same-different” discrimination probabilities and supports his claim with some empirical evidence. The key feature of RM is that the mapping, h, between two observation areas based on minimum discrimination probability is invertible. Dzhafarov [(2003a). Thurstonian-type representations for “same-different” discriminations: Deterministic decisions and independent images. Journal of Mathematical Psychology, 47, 184-204; (2003b). Thurstonian-type representations for “same-different” discriminations: Probabilistic decisions and interdependent images. Journal of Mathematical Psychology, 47, 229-243] also demonstrates that well-behaved Thurstonian models of “same-different” judgments are incompatible with RM and Nonconstant Self-Similarity (NCSS). There is extensive empirical support for the latter. Stimulus and neural sources of perceptual noise are discussed and two points are made:

Point 1: Models that require discrimination probabilities for noisy stimuli to possess the property that h is invertible would be too restrictive.

Point 2: In the absence of stimulus noise, violations of RM may be so subtle that their detection would be unlikely.

     

Regular Minimality and Thurstonian-type modeling

A Thurstonian-type model for pairwise comparisons is any model in which the response (e.g., “they are the same” or “they are different”) to two stimuli being compared depends, deterministically or probabilistically, on the realizations of two randomly varying representations (perceptual images) of these stimuli. The two perceptual images in such a model may be stochastically interdependent but each has to be selectively dependent on its stimulus. It has been previously shown that all possible discrimination probability functions for same–different comparisons can be generated by Thurstonian-type models of the simplest variety, with independent percepts and deterministic decision rules. It has also been shown, however, that a broad class of Thurstonian-type models, called “well-behaved” (and including, e.g., models with multivariate normal perceptual representations whose parameters are smooth functions of stimuli) cannot simultaneously account for two empirically plausible properties of same–different comparisons, Regular Minimality (which essentially says that “being least discriminable from” is a symmetric relation) and nonconstancy of the minima of discrimination probabilities (the fact that different pairs of least discriminable stimuli are discriminated with different probabilities). These results have been obtained for stimulus spaces represented by regions of Euclidean spaces. In this paper, the impossibility for well-behaved Thurstonian-type models to simultaneously account for Regular Minimality and nonconstancy of minima is established for a much broader notion of well-behavedness applied to a much broader class of stimulus spaces (any Hausdorff arc-connected ones). The universality of Thurstonian-type models with independent perceptual images and deterministic decision rules is shown (by a simpler proof than before) to hold for arbitrary stimulus spaces.     

Brief presentations are sufficient for pigeons to discriminate arrays of same and different stimuli

Four pigeons first learned to discriminate 16-item arrays of same from different pictorial stimuli. They were then tested with reduced exposure to the pictorial arrays, brought about by changes in the stimulus viewing requirement under fixed-ratio (FR) and fixed-interval (FI) schedules. Increasing the FR requirement enhanced discriminative performance up to 10 pecks; increasing the FI requirement enhanced discriminative performance up to 5 s. Exposures to the stimulus arrays averaging only 2 s supported reliable discrimination. Pigeons thus discriminate same from different stimuli with considerable speed, suggesting that same-different discrimination behavior is of substantial adaptive significance.     

A relational differential outcomes effect: pigeons can classify outcomes as “good” and “better”

In a conditional discrimination each of two sample stimuli indicates which of two comparison stimuli is correct. When correct choice following each conditional stimulus is followed by a different outcome (one kind of food following one, a different kind of food following the other) it often facilitates acquisition and improves memory. In transfer designs, in which two different conditional discriminations are followed by the same two differential outcomes, outcome expectation can be shown to be sufficient for comparison choice. That is, the samples from one conditional discrimination are matched to comparisons from the other conditional discrimination based on the common outcomes alone. In the present study we asked if for pigeons the relative value of the differential outcomes (higher versus lower value) can serve as the basis for comparison choice, independent of other characteristics of the outcomes and of differential sample responding. That is, would different outcomes that could be described as “good” and “better” form two stimulus classes. For one conditional discrimination, the differential outcomes involved differential probability of reinforcement for choice of the correct comparison stimulus (0.80 vs. 0.20 for correct choice of the two comparisons, respectively). For the other conditional discrimination, the differential outcomes involved differential responding to the two comparison stimuli (5 pecks vs. 20 pecks to the correct comparisons, respectively). On test trials, when conditional stimuli from the two conditional discriminations were interchanged and the relative value of the differential outcomes could serve as the only basis for comparison choice, we found positive transfer. The results indicate that relational attributes of outcomes can serve as effective cues for comparison choice.     

A derived transfer of simple discrimination and self-reported arousal functions in spider fearful and non-spider-fearful participants

Two experiments investigated the derived transfer of functions through equivalence relations established using a stimulus pairing observation procedure. In Experiment 1, participants were trained on a simple discrimination (A1+/A2-) and then a stimulus pairing observation procedure was used to establish 4 stimulus pairings (A1-B1, A2-B2, B1-C1, B2-C2). Subsequently, a transfer of the simple discrimination functions through equivalence relations was observed (e.g., C1+/C2-). These procedures were modified in Experiment 2, which demonstrated that spider-fearful and non-spider-fearful participants show differing levels of a transfer of self-reported arousal functions for stimuli used in equivalence relations with video-based material depicting scenes with spiders. The results demonstrate that the stimulus pairing observation procedure provides a viable alternative to matching-to-sample, and also offer tentative support for a derived-relations model of the acquisition of anxiety responses in at least one sub-clinical population.     

Observing responses maintained by conditional discriminative stimuli.

In a conditional discrimination procedure, pigeons' observing responses were analyzed to examine whether two color stimuli (blue or red), conditionally related to whether each of two line stimuli (vertical or horizontal) accompanied reinforcement or nonreinforcement, functioned as conditioned reinforcers. If a variable-interval (VI) 10-s requirement was fulfilled, an observing response produced onset of a color stimulus. A little later, a line stimulus was presented independently of responding, added to the color stimulus to form a compound stimulus. If 55 s elapsed with a response not having occurred either through 55 s or after the variable-interval 10-s had timed out, one of the color-line compound stimuli was presented independently of responding. To control for sensory reinforcement effects and for earlier entrance to the later link, a simple discrimination procedure also was conducted in which reinforcement was not correlated with the color stimuli but with the line stimuli only. As in the conditional discrimination, the observing response also could produce earlier presentation of blue or red. The observing response occurred more frequently during the conditional discrimination than during the simple discrimination. The results were related to different theoretical accounts of conditioned reinforcement, particularly the information hypothesis.     

Categorical perception of stereoscopic stimuli

Two studies of the stereopsis model of Richards were conducted using a between-subjects balanced design and a repeated trials procedure rather than the within-subjects procedure of Richards. Measures of the accuracy of discrimination between classes of retinal disparity were defined by the method of successive intervals in place of the d’ index of SDT. The results support the predictions (1) of symmetric discrimination functions for convergent and divergent stimuli, (2) of greater discrimination for stimuli of 45 min of retinal disparity than for the stimuli of 15 min disparity, (3) of greater variability for the stimuli of greater disparity, and (4) of the use by nonanomalous subjects of at least two different processing mechanisms for stimuli of different disparity classes. The results do not support the prediction (1) of different processing mechanisms for foveal stimuli differing in figure-ground contrast or (2) of greater usefulness for monocular stimuli as a null disparity reference condition, as compared with binocular null disparity stimuli. Similar results were found for two stimulus shapes, a bar and a disk of equal area. Unreliability of subjects’ response systems was suggested as a factor in the prediction of figure-ground contrast changes and as a factor in the classification of subjects as stereo anomalous. The pattern of accuracy of discrimination between classes of disparity stimuli also raises questions of the adequacy of the simple stereopsis model.     

Generalization,similarity, and Bayesian inference

Shepard has argued that a universal law should govern generalization across different domains of perception and cognition, as well as across organisms from different species or even different planets. Starting with some basic assumptions about natural kinds, he derived an exponential decay function as the form of the universal generalization gradient, which accords strikingly well with a wide range of empirical data. However, his original formulation applied only to the ideal case of generalization from a single encountered stimulus to a single novel stimulus, and for stimuli that can be represented as points in a continuous metric psychological space. Here we recast Shepard's theory in a more general Bayesian framework and show how this naturally extends his approach to the more realistic situation of generalizing from multiple consequential stimuli with arbitrary representational structure. Our framework also subsumes a version of Tversky's set-theoretic model of similarity, which is conventionally thought of as the primary alternative to Shepard's continuous metric space model of similarity and generalization. This unification allows us not only to draw deep parallels between the set-theoretic and spatial approaches, but also to significantly advance the explanatory power of set-theoretic models.     

Visible persistence of moving objects

A single line was presented in a succession of orientations, each orientation separated by a fixed angle and by a fixed interval of time, and subjects reported the number of successive lines that appeared to rotate together. The perceived number of rotating lines increased linearly with the rate of stimulus presentation, with a slope that was proportional to the spatial separation. The linear functions obtained in this first experiment predicted the results of a second experiment in which subjects adjusted the spatial and temporal variables to a discrimination threshold for seeing two rotating lines. If the slope of the linear functions is considered to be an estimate of the duration of visible persistence, then these results suggest that the visible persistence of a briefly presented stimulus increases with the distance separating that stimulus from other stimuli.     

The ABCs of categorical perception

Studies of speech perception first revealed a surprising discontinuity in the way in which stimulus values on a physical continuum are perceived. Data which demonstrate the effect in nonspeech modes have challenged the contention that categorical perception is a hallmark of the speech mode, but the psychophysical models that have been proposed have not resolved the issues raised by empirical findings. This study provides data from judgments of four sensory continua, two visual and two tactual-kinesthetic, which show that the adaptation level for a set of stimuli serves as a category boundary whether stimuli on the continuum differ by linear or logarithmic increments. For all sensory continua studied, discrimination of stimuli belonging to different perceptual categories was more accurate than discrimination of stimuli belonging to the same perceptual category. Moreover, shifts in the adaptation level produced shifts in the location of the category boundary. The concept of Adaptation-level Based Categorization (ABC) provides a unified account of judgmental processes in categorical perception without recourse to post hoc constructs such as implicit anchors or external referents.     

On the law of Regular Minimality: Reply to Ennis

Ennis's critique touches on issues important for psychophysics, but the points he makes against the hypothesis that Regular Minimality is a basic property of sensory discrimination are not tenable.(1) Stimulus variability means that one and the same apparent stimulus value (as measured by experimenter) is a probabilistic mixture of true stimulus values. The notion of a true stimulus value is a logical necessity: variability and distribution presuppose the values that vary and are distributed (even if these values are represented by processes or sets rather than real numbers). Regular Minimality is formulated for true stimulus values. That a mixture of probabilities satisfying Regular Minimality does not satisfy this principle (unless it also satisfies Constant Self-Similarity) is an immediate consequence of my 2003 analysis. Stimulus variability can be controlled or estimated: the cases when observed violations of Regular Minimality can be accounted for by stimulus variability corroborate rather than falsify this principle. In this respect stimulus variability is no different from fatigue, perceptual learning, and other factors creating mixtures of discrimination probabilities in an experiment.(2) Could it be that well-behaved Thurstonian-type models are true models of discrimination but their parameters are so adjusted that the violations of Regular Minimality they lead to (due to my 2003 theorems) are too small to be detected experimentally? This is possible, but this amounts to admitting that Regular Minimality is a law after all, albeit only approximate: nothing in the logic of the Thurstonian-type representations per se prevents them from violating Regular Minimality grossly rather than slightly. Moreover, even very small violations predicted by a given class of Thurstonian-type models can be tested in specially designed experiments (perhaps under additional, independently testable assumptions). The results of one such experiment, in which observers were asked to alternately adjust to each other the values of stimuli in two observation areas, indicate that violations of Regular Minimality, if any, are far below limits of plausible interpretability.     

Serial conditioning as a function of stimulus, response, and temporal dependencies

Six experiments were used to examine the effects of explicit response, stimulus, and temporal dependencies on responding in an interfood interval. The first two experiments demonstrated that 10-segment 60-s interfood clocks controlled similar distributions of key pecking in pigeons regardless of whether response–reinforcement contiguity was required, allowed, or precluded. The third and fourth experiments found that in the absence of an explicit response–reinforcement dependency, systematic explicit stimuli in an interfood interval were sufficient to establish and maintain the characteristic distribution of key pecking and that an interval without an explicit clock failed to establish or maintain key pecking. The last two experiments demonstrated that the interfood interval need not be of fixed length, and that a simple correlation of stimuli with increments from either a minimum to a maximum imminency or probability of food presentation controlled behavior in a similar manner. Successively higher rates generally occurred to successively later stimuli in the upper half of the range.