Generalisation: mechanistic and functional explanations

An overview of mechanistic and functional accounts of stimulus generalisation is given. Mechanistic accounts rely on the process of spreading activation across units representing stimuli. Different models implement the spread in different ways, ranging from diffusion to connectionist networks. A functional account proposed by Shepard analyses the probabilistic structure of the world for invariants. A universal law based on one such invariant claims that under a suitable scaling of the stimulus dimension, generalisation gradients should be approximately exponential in shape. Data from both vertebrates and invertebrates so far uphold Shepard's law. Some data on spatial generalisation in honeybees are presented to illustrate how Shepard's law can be used to determine the metric for combining discrepancies in different stimulus dimensions. The phenomenon of peak shift is discussed. Comments on mechanistic and functional approaches to generalisation are given. Accepted after revision: 10 September 2001 Electronic Publication     

Shepard's universal law supported by honeybees in spatial generalization

An animal that is rewarded for a response in one situation (the S+) is likely to respond to similar but recognizably different stimuli, the ubiquitous phenomenon of stimulus generalization. On the basis of functional analyses of the probabilistic structure of the world, Shepard formulated a universal law of generalization, claiming that generalization gradients, as a function of the appropriately scaled distance of a stimulus from S+, should be exponential in shape. This law was tested in spatial generalization in honeybees. Based on theoretically derived scales, generalization along both the dimensions of the distance from a landmark and the direction to a landmark followed Shepard's law. Support in an invertebrate animal increases the scope of the law, and suggests that the ecological structure of the world may have driven the evolution of cognitive structures in diverse animals.     

Similarity, kernels, and the triangle inequality

Similarity is used as an explanatory construct throughout psychology and multidimensional scaling (MDS) is the most popular way to assess similarity. In MDS, similarity is intimately connected to the idea of a geometric representation of stimuli in a perceptual space. Whilst connecting similarity and closeness of stimuli in a geometric representation may be intuitively plausible, Tversky and Gati [Tversky, A., & Gati, I. (1982). Similarity, separability, and the triangle inequality. Psychological Review, 89(2), 123-154] have reported data which are inconsistent with the usual geometric representations that are based on segmental additivity. We show that similarity measures based on Shepard’s universal law of generalization [Shepard, R. N. (1987). Toward a universal law of generalization for psychologica science. Science, 237(4820), 1317-1323] lead to an inner product representation in a reproducing kernel Hilbert space. In such a space stimuli are represented by their similarity to all other stimuli. This representation, based on Shepard’s law, has a natural metric that does not have additive segments whilst still retaining the intuitive notion of connecting similarity and distance between stimuli. Furthermore, this representation has the psychologically appealing property that the distance between stimuli is bounded.     

Attention, similarity, and the identification-categorization relationship

A unified quantitative approach to modeling subjects' identification and categorization of multidimensional perceptual stimuli is proposed and tested. Two subjects identified and categorized the same set of perceptually confusable stimuli varying on separable dimensions. The identification data were modeled using Shepard's (1957) multidimensional scaling-choice framework. This framework was then extended to model the subjects' categorization performance. The categorization model, which generalizes the context theory of classification developed by Medin and Schaffer (1978), assumes that subjects store category exemplars in memory. Classification decisions are based on the similarity of stimuli to the stored exemplars. It is assumed that the same multidimensional perceptual representation underlies performance in both the identification and categorization paradigms. However, because of the influence of selective attention, similarity relationships change systematically across the two paradigms. Some support was gained for the hypothesis that subjects distribute attention among component dimensions so as to optimize categorization performance. Evidence was also obtained that subjects may have augmented their category representations with inferred exemplars. Implications of the results for theories of multidimensional scaling and categorization are discussed.     

Representation is representation of similarities

Advanced perceptual systems are faced with the problem of securing a principled (ideally, veridical) relationship between the world and its internal representation. I propose a unified approach to visual representation, addressing the need for superordinate and basic-level categorization and for the identification of specific instances of familiar categories. According to the proposed theory, a shape is represented internally by the responses of a small number of tuned modules, each broadly selective for some reference shape, whose similarity to the stimulus it measures. This amounts to embedding the stimulus in a low-dimensional proximal shape space spanned by the outputs of the active modules. This shape space supports representations of distal shape similarities that are veridical as Shepard's (1968) second-order isomorphisms (i.e., correspondence between distal and proximal similarities among shapes, rather than between distal shapes and their proximal representations). Representation in terms of similarities to reference shapes supports processing (e.g., discrimination) of shapes that are radically different from the reference ones, without the need for the computationally problematic decomposition into parts required by other theories. Furthermore, a general expression for similarity between two stimuli, based on comparisons to reference shapes, can be used to derive models of perceived similarity ranging from continuous, symmetric, and hierarchical ones, as in multidimensional scaling (Shepard 1980), to discrete and nonhierarchical ones, as in the general contrast models (Shepard & Arabie 1979; Tversky 1977).     

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.     

Stimulus dimensions or stimulus aspects

Summary The questionableness of geometric models of stimulus similarity has led to the development of an alternative approach by Tversky which makes no dimensional or metric assumptions. Rather, stimuli are described as sets of qualitative stimulus aspects and stimulus similarity as a function of common and non-common aspects. According to Restle's model of stimulus similarity, however, the perception of stimuli of a categorial nature can be organized along dimensions because stimulus aspects form dimensions under certain conditions. The present study supports the empirical validity of this assumption. Further it is suggested that contrary to the present opinion quantitative stimulus characteristics are probably not perceived as dimensions in the sense of the geometric models but only dimensions as described in the Restle model.     

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.     

Dissimilarity, Quasimetric, Metric

Dissimilarity is a function that assigns to every pair of stimuli a nonnegative number vanishing if and only if two stimuli are identical, and that satisfies the following two conditions called the intrinsic uniform continuity and the chain property, respectively: it is uniformly continuous with respect to the uniformity it induces, and, given a set of stimulus chains (finite sequences of stimuli), the dissimilarity between their initial and terminal elements converges to zero if the chains’ length (the sum of the dissimilarities between their successive elements) converges to zero. The four properties axiomatizing this notion are shown to be mutually independent. Any conventional, symmetric metric is a dissimilarity function. A quasimetric (satisfying all metric axioms except for symmetry) is a dissimilarity function if and only if it is symmetric in the small. It is proposed to reserve the term metric (not necessarily symmetric) for such quasimetrics. A real-valued binary function satisfies the chain property if and only if whenever its value is sufficiently small it majorates some quasimetric and converges to zero whenever this quasimetric does. The function is a dissimilarity function if, in addition, this quasimetric is a metric with respect to which the function is uniformly continuous.     

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.     

Contextual control of stimulus generalization and stimulus equivalence in hierarchical categorization

The purpose of this study was to determine whether hierarchical categorization would result from a combination of contextually controlled conditional discrimination training, stimulus generalization, and stimulus equivalence. First, differential selection responses to a specific stimulus feature were brought under contextual control. This contextual control was hierarchical in that stimuli at the top of the hierarchy all evoked one response, whereas those at the bottom each evoked different responses. The evocative functions of these stimuli generalized in predictable ways along a dimension of physical similarity. Then, these functions were indirectly acquired by a set of nonsense syllables that were related via transitivity relations to the originally trained stimuli. These nonsense syllables effectively served as names for the different stimulus classes within each level of the hierarchy.     

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.     

Superordinate categorization via learned stimulus equivalence: quantity of reinforcement, hedonic value, and the nature of the mediator

Three experiments examined superordinate categorization via stimulus equivalence training in pigeons. Experiment 1 established superordinate categories by association with a common number of food pellet reinforcers, plus it established generalization to novel photographic stimuli. Experiment 2 documented generalization of choice responding from stimuli signaling different numbers of food pellets to stimuli signaling different delays to food reinforcement. Experiment 3 indicated that different numbers of food pellets did not substitute as discriminative stimuli for the photographic stimuli with which the food pellets had been paired. The collective results suggest that the effective mediator of superordinate categories that are established via learned stimulus equivalence is not likely to be an accurate representation of the reinforcer, neither is it likely to be a distinctive response that is made to the discriminative stimulus. Motivational or emotional mediation is a more likely account.     

Stimulus-specific processing consequences of pattern goodness

Two issues concerning the effects of visual pattern goodness on information processing time were investigated: the role of memory vs. encoding and the role of individual stimulus goodness vs. stimulus similarity. A sequential “same-different” task was used to provide differentiation of target item or memory effects from display item or encoding effects. Experiment 1 used four alternative stimuli in each block of trials. The results showed that good patterns were processed faster than poor patterns for both “same” and “different” responses. Furthermore, the goodness of the target item had a greater effect on reaction time than did the goodness of the display item, indicating that memory is more important than encoding in producing faster processing of good stimuli. Effects of interstimulus similarity on processing time were minimal, although isolation of good stimuli in a similarity space could explain many of the results. Experiment 2 replicated the results of Experiment 1, despite the fact that differences in similarity space had been minimized by using only two alternative stimuli in each block. In addition, the speed of processing a “same” pair was essentially independent of the particular alternative stimulus in a block. These results suggest that in this task, there is a processing advantage for good stimuli that is stimulus specific, with the effect operating primarily in memory.     

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.     

Free classification: Element-level and subgroup-level similarity

Subjects classified sets of multidimensional stimuli into two groups in any way they wished. The sets were composed of 6 or 12 stimuli: 2 or 4 instances of 3 different stimuli (e.g., 2 blue circles, 2 green circles, 2 red circles). There were striking individual differences in the preferred classification. Some subjects maximized the similarity between subgroups by matching the composition of the subgroups--one instance of each stimulus was placed in each group. The other subjects maximized the similarity among stimuli within each subgroup by placing similar stimuli in each group (the blues and greens in one group, the reds in the other). The nature of the stimuli as well as the relationships among the three stimuli had little effect on classification. In this case, cognitive styles specific to individuals but general across diverse dimensions and stimulus sets determined classification.     

Bayesian generalization with circular consequential regions

Generalization–deciding whether to extend a property from one stimulus to another stimulus–is a fundamental problem faced by cognitive agents in many different settings. Shepard (1987) provided a mathematical analysis of generalization in terms of Bayesian inference over the regions of psychological space that might correspond to a given property. He proved that in the unidimensional case, where regions are intervals of the real line, generalization will be a negatively accelerated function of the distance between stimuli, such as an exponential function. These results have been extended to rectangular consequential regions in multiple dimensions, but not for circular consequential regions, which play an important role in explaining generalization for stimuli that are not represented in terms of separable dimensions. We analyze Bayesian generalization with circular consequential regions, providing bounds on the generalization function and proving that this function is negatively accelerated.     

Control of a continuous response dimension by a continuous stimulus dimension

Pigeons were trained to respond to stimuli from a continuous stimulus dimension (tonal frequency) with response values from a continuous response dimension. Both the number of points of correspondence and problem difficulty were varied. After training, subjects were tested with stimulus values intermediate to those trained. During these test tones, subjects emitted only those response values reinforced during training. The study suggested that if there are fast and efficient methods to obtain control of a continuous response dimension by a continuous stimulus dimension, these methods must depend on factors other than simple generalization.     

Theoretical analysis of an alphabetic confusion matrix

A study was undertaken to acquire a confusion matrix of the entire upper-case English alphabet with a simple nonserifed font under tachistoscopic conditions. This was accomplished with two experimental conditions, one with blank poststimulus field and one with noisy poststimulus field, for six Ss run 650 trials each. Three mathematical models of recognition, two based on the concept of a finite number of sensory states and one being the choice model, were compared in their ability to predict the confusion matrix after their parameters were estimated from functions of the data. In order to ascertain the facility with which estimates of similarity among the letters could lead to a psychological space containing the letters, η_ij, the similarity parameter of the choice model was input to an ordinally based multidimensional scaling program. Finally, correlation coefficients were computed among parameters of the models, the scaled space, and a crude measure of physical similarity. Briefly, the results were: (1) the finite-state model that assumed stimulus similarity (the overlap activation model) and the choice model predicted the confusion-matrix entries about equally well in terms of a sum-of-squared deviations criterion and better than the all-or-none activation model, which assumed only a perfect perception or random-guessing state following a stimulus presentation; (2) the parts of the confusion matrix that fit best varied with the particular model, and this finding was related to the models; (3) the best scaling result in terms of a goodness-of-fit measure was obtained with the blank poststimulus field condition, with a technique allowing different distances for tied similarity values, and with the Euclidean as opposed to the city-block metric; and (4) there was agreement among the models in terms of the way in which the models reflected sensory and response bias structure in the data, and in the way in which a single model measured these attributes across experimental conditions, as well as agreement among similarity ami distance measures with physical Similarity.