Resonating memory traces account for the perceptual magnet effect

Transcription

1 Resonating memory traces account for the perceptual magnet effect Gerhard Jäger Dept. of Linguistics, University of Tübingen, Germany Introduction In a series of experiments, atricia Kuhl and co-workers investigated whether categories of speech sounds vowels, to be precise display an internal structure. To this end, they conducted experiments where participants had to discriminate between various pairs of perceptual stimuli of varying psycho-physical distance, which all belonged to the same vowel category. Their main finding was that the ability of participants to discriminate between stimuli was lowest in the neighborhood of the prototypical instance of the category, and increased monotonically with the distance from the prototype. The effect was strongest with adults, weaker but still present with infants, and absent with monkeys (cf. (Kuhl, 99)). In a later study (Kuhl, Williams, Lacerda, Stevens, & Lindblom, 992), it was established that the location of the prototype depends on the native language of the participants, and that this effect is already present in infants of six months age. Lacerda (995) pointed out that these findings are not necessarily evidence for prototypicality effects. He gives an alternative interpretation of Kuhl et al. s results, using exemplar theory. According to exemplar theory, every perceptual stimulus leaves a trace in memory. Newly incoming stimuli are processed in comparison to similar exemplars in memory. Lacerda assumes that exemplars are tagged with a category label, and that the perceptual magnet effect is a consequence of the presence of competing categories. Briefly put, the discrimination measure of a stimulus position is inversely related to the homogeneity of categorization of the exemplars in the neighborhood of the stimulus. At the center of a category (i.e. in an area of high density of exemplars of that category, and low density of exemplars from other categories), homogeneity is high and discrimination thus low. At the boundary regions between categories, homogeneity is low and discrimination thus high. This, according to Lacerda, accounts for the ME (perceptual magnet effect). Lacerda s proposal takes category membership of exemplars as basic and derives notions like discrimination or similarity (which is inversely related to discrimination) from categorization. Since (Nosofsky, 986), a lot of work in exemplar theory, however, has taken a notion of similarity between exemplars as basic and has attempted to derive categorization from it. In this paper, I will propose another way to derive the ME from an exemplar model. Unlike Lacerda, I will assume a notion of similarity between exemplars and stimuli as basic, and I will make no assumptions about categorization. The crucial inspiration for my model comes from (Hintzman, 986) and (Goldinger, 998). I assume that a perceptual stimuli lets all exemplars in memory resonate, where the strength of the resonance of an exemplar is proportional to its similarity to the stimulus. The perceived location of the stimulus is the result of superimposing the stimulus with all resonating exemplars according to their resonance strength. As a consequence, stimuli in the neighborhood of regions with high exemplar density will be pulled strongly towards the center of this region, while this effect is weak in greater distance from the center of gravitation. If we assume that exemplars of a category are concentrated near the center of this category (its prototype, if you like), the ME falls out as a consequence. The erceptual Magnet Effect In this section I will briefly recapitulate the main points of (Kuhl, 99), where the ME was first established. In the crucial experiment (experiment 2), participants were presented with pairs of vowel stimuli, consisting of a reference vowel and the comparison vowel. They had to decide whether or not the two sounds were identical. The reference sounds were either a prototypical instance of the vowel category /i/ ( condition) or a specific non-prototypical instance of the same category ( condition). The comparison vowel was either identical to the referent vowel, or it was one out of 32 variants that are located around the referent sound in four concentric orbits of increasing diameter (in the two-dimensional F/F2 acoustic space). The location of the stimuli is depicted in figure. Figure. The prototype /i/ vowel () and variants on four orbits surrounding it (open circles) and the nonprototype /i/ vowel () and variants surrounding it (closed circles). The stimuli on one vector were common to both sets. (taken from Kuhl, 99)

2 2 GERHARD JÄGER Not surprisingly, the generalization the likelihood that participants wrongly classify the comparison stimulus as identical to the reference stimulus was strongest for the innermost orbits and decreased with increasing distance from the referent vowel. However, it turned out that for all orbits, the generalization was significantly higher in the condition than in the condition. The results displayed in figure 2. Qualitively similar results were obtained with infants of 6-7 Generalization (%) Figure 2. Average generalization scores shown for stimuli surrounding the prototype and the nonprototype. (data taken from Kuhl, 99) months of age. There the generalization scores were generally higher, but the difference between the and the condition showed up as well. No significant difference between the two conditions were found when the experiment was replicated with monkeys (Rhesus macaques). (Kuhl et al., 992) reports a comparative study with American and Swedish infants. American English and Swedish differ with respect to the location of prototype of the vowel category that was investigated. It turned out that the ME is language dependent; even with infants the effect is strongest in the neighborhood of their respective native language prototype. An exemplar-based account of the ME Exemplar theories were developed in psychology as a model of perception and categorization (see for instance (Nosofsky, 986)). In these models, it is assumed that each experience leaves a trace in the long term memory. These traces contain very specific information about the properties of the stimuli that generated them. For instance, an exemplar (or trace) of a vowel stimulus contains specific information about the formant frequencies of that stimulus, not just about categorical features like high or fronted. When a newly incoming stimulus is processed, the properties of similar exemplars in memory are used. Models differ regarding the precise role of memory exemplars in stimulus processing. My proposal is crucially inspired by Hintzman s (986) MINERVA 2 model (which is also used in (Goldinger, 998)). Here, if a new stimulus has to be processed, all exemplars in memory are activated to a degree that is monotonically related to their similarity to the stimulus. The superposition of all memory exemplars (weighted according to their activation strength) constitutes the echo of the stimulus. Crucially, the echo may possess features that are absent in the stimulus but are inherited from the exemplars in long term memory. The echo is used to model subsequent processes like categorization. In Hintzman s model, exemplars and stimuli are highdimensional vectors of discrete values (-,, or +). Similarity between vectors is defined as their inner product. To apply this conception to the vowel space, I assume a continuous two-dimensional vector space (representing the first and second formant frequency in mel). Similarity between vectors is defined as a Gaussian function (with standard deviation σ) of the Euclidean distance between the vectors: x y 2 sim(x,y) = exp( 2σ 2 ) () The long term memory consists of a collection of stored exemplar vectors. I assume that exemplar strength exponentially decays over time. Within one time unit, the activation strength of each exemplar is divided by some constant a >. If exemplars are added at constant intervals, the strength of the ith of N exemplars is a i N. The echo of a stimulus x is the weighted sum of all exemplars in memory weighted by their activation strength and their similarity to the stimulus and the stimulus itself. If the longterm memory is represented by a sequence of exemplars v = v,...,v N, we have echo(x, v) = x + N i= ai N sim(x,v i )v i + N i= ai N sim(x,v i ) The perceived similarity (or generalization) between two stimuli is modeled as the similarity between their echoes. (2) generalization(x, y, v) = sim(echo(x, v), echo(y, v))(3) For reasons of analytic tractability, I assume that we are dealing with only one vowel category (/i/, say), and that the exemplars of this category are normally distributed. The coordinate system is normalized so that this distribution is standard, i.e. it has mean and standard deviation. If the exemplar population is sufficiently large, it can be approximated by a density function f which is given by f (v) = 2π exp( v 2 2 ) (4) Note that I assume a stationary distribution here, so the decay parameter a does not play a role under this assumption. The echo function now becomes: echo(x) = x + R2 sim(x,y) f (y)ydy + R2 sim(x,y) f (y)dy Some calculations reveal that this can be simplified to σ ri(x) echo(x) = x (σ 2 + )( + ri(x)), To keep things simple, I assume that the long term memory remains constant during the experiment.

3 RESONATING MEMORY TRACES ACCOUNT FOR THE ERCETUAL MAGNET EFFECT 3 where ri(x) is the resonance intensity of the exemplar population f when responding to x, which is given by ri(x) = sim(x,y) f (y)dy R2 = x 2 σ 2 exp( 2(σ 2 +) ) (σ 2 + ) It can be seen from these formulas that (a) the echo of a stimulus always lies on a straight line between the stimulus and the mean of the exemplar population, and (b) that the distance between the echo and the mean ( echo(x) ) only depends on the distance of the stimulus from the memory mean x, not on its absolute position. If x is small, the echo of x is pulled towards the memory mean. At larger distances from the mean, this effect vanishes. Figure 3 gives a visualization of this effect. It gives the factor by which the distance of the echo from the mean and of the stimulus from the mean differ, as a function of stimulus distance. Here it is assumed that σ = ; for other parameter values the effect is qualitatively similar. and the non-prototypical vowel. The standard deviation of the similarity function was set to. The average perceived distances of the stimuli on the orbits from the center of the orbit are displayed in figure 4. For the conditions, the precise values for the four orbits are.86,.84, 2.92, and For the condition, the values are.4, 2.2, 3.2, and erceived distance Figure 4. Average perceived distances between reference stimulus and comparison stimulus: numerical simulation echo(x) x So in the neighborhood of the vowel which is located at the mean of the exemplar population all stimuli are pulled towards the center, thus diminishing perceived distances. In the neighborhood of the vowel, the orbits are distorted. Comparison vowels near the mean are pulled towards the mean, while peripheral elements barely change their position. Therefore the average distances are even slightly enhanced. As argued for above, the generalization, i.e. the likelihood that two stimuli are perceived as identical, is inversely related to their perceived distance. The similarity of the echoes of two stimuli might serve as an approximation (which is not entirely correct because it does not take false discrimination between identical stimuli into account). Figure 5 gives the orbit-wise average generalization between the reference vowels and the comparison vowels. As in Kuhl s experi- Figure 3. 5 x Relative echo position as a function of stimulus position The perceived distance between two stimuli can straightforwardly be defined as the distance between their echoes. Since stimuli near the center of the exemplar population are pulled strongly towards the center, perceived distances are lower than objective distances. In the peripheral region, the magnet effect is negligible. This essentially reconstructs Kuhl s experimental findings. In a numerical simulation, the stimuli were arranged in the same pattern as in Kuhl s experiments, with a radius of, 2, 3, and 4 units for each orbit (where a unit is defined as the standard deviation of the exemplar population), and a distance of 4 standard deviation between the prototypical erceived similarity,7,6,5,4,3,2, Figure 5. Average perceived similarities between reference stimulus and comparison stimulus: numerical simulation ments, the generalization between the reference stimuli and

4 4 GERHARD JÄGER the comparison stimuli is generally higher in the condition than in the condition. Comparison to Lacerda s model An exemplar based model of the ME has been proposed already in (Lacerda, 995). In this section I will briefly compare the two models. Lacerda assumes that exemplars are tagged with a category label, and that the ME is essentially a side effect of competition between categories. To illustrate his model, he presents an example with two categories, A and B, within a one-dimensional exemplar space. Exemplars of the categories are normally distributed around the means µ A and µ B with a standard deviation of. 2 The number of exemplars in category A and B is w A and w B respectively. Based on this, Lacerda defines a notion of similarity of a stimulus to a category. Simplifying his model somewhat for the purpose of exposition 3, this can be defined as sim(x, A) = w A N(x;µ A ) w A N(x;µ A ) + w B N(x;µ B ) The discrimination measure of a stimulus measures measures the local variation of the category membership of the exemplars in the immediate environment of the stimulus. It is defined as (5) Generalization discr(x),95,9, x Figure 6. Lacerda s discrimination measure discr(x) = dx d sim(x,a) + dx d sim(x,b), (6) 2 Const where Const is a normalization constant that ensures that the discrimination measure is always between and. Figure 6 graphically displays the discrimination measure for a choice of parameters that is also discussed in Lacerda s paper (µ A =, µ B = 3, w A =, w B = ). The two vertical lines at and 3 indicate the centers of the two categories. It can be seen that discrimination is highest in the area between the categories. The maximum is closer to µ B than to µ A because category A is more densely populated than B. Discrimination is lowest in regions that clearly belong to one category rather than the other. This is not at the mean of the category population but, in the absence of further competing categories, at the extreme position of the exemplar space. Lacerda defines the generalization of a comparison stimulus as minus the discrimination measure of the stimulus. Figure 7 gives the results of a numerical simulation that used Lacerda s model to account for Kuhl s findings. The comparison stimuli were arranged at orbits with radiuses of.5,.3,.45 and.6 units around the reference stimuli (where units are defined as standard deviations of the exemplar populations), with the stimulus at a distance of.6 units from the stimulus (thus again replicating the configuration from Kuhl s experiments). The columns in the figure give the average generalization for each orbit. Again, the generalization is generally higher in the vicinity of the prototype vowel than around the non-prototype vowel.,8 Figure 7. Average generalization between reference stimulus and comparison stimulus in Lacerda s model: numerical simulation So far, both Lacerda s and my model qualitatively predict the ME. Also, both models account for the native language tuning effects that were uncovered in (Kuhl et al., 992). According to either model, the ME is not induced by some abstract prototype. Rather, each exemplar in memory acts as a tiny little perceptual magnet, and the macroscopic ME is a result of the cumulative microscopic effects. So the targets of the ME are regions of high exemplar density, and this of course depends on the perceptual history of the participants. Despite this general similarity, the two models differ conceptually. Lacerda takes category membership of exemplars to be basic, and he defines generalization ( similarity) in terms of category membership, as a derived notion. However, since (Nosofsky, 986) much work in exemplar theory attempts to explain categorization via exemplar models, assuming some basic similarity measure between exemplars as given. The present model is in line with these assumption. It makes no assumptions whatsoever about categorization 2 The coordinate system can always be normalized such that the standard deviation is, so this assumption does not restrict generality. 3 Rather than taking his parameter ε into account, I consider the limiting case were ε converges to.

5 RESONATING MEMORY TRACES ACCOUNT FOR THE ERCETUAL MAGNET EFFECT 5 the model only makes reference to a similarity measure and exemplar densities. There are various ways how the two models can and should be compared in future research. First and foremost, it has to be tested which model provides a better fit for the quantitative data from Kuhl s experiments and similar studies. Besides, the models make significantly different predictions in other settings. For instance, if participants are repeatedly confronted with (normally distributed) stimuli from a domain they are unfamiliar with (for instance unusual graphical or musical patterns), the present model predicts that the ME will eventually emerge. Lacerda s model predicts that the ME will not arise because generalization is constant if there is no category competition. References Goldinger, S. D. (998). Echoes of echoes? An episodic theory of lexical access. sychological Review, 5, Hintzman, D. L. (986). Schema abstraction in a multiple-trace memory model. sychological Review, 93, Kuhl,. K. (99). Human adults and human infants show a perceptual magnet effect for the prototypes of speech categories, monkeys do not. erception & psychophysics, 5(2), Kuhl,. K., Williams, K. A., Lacerda, F., Stevens, K. N., & Lindblom, B. (992). Linguistic experience alters phonetic perception in infants by 6 months of age. Science, 255(544), Lacerda, F. (995). The perceptual magnet effect: An emergent consequence of exemplar-based phonetic memory. In K. Elenius &. Branderyd (Eds.), roceedings of the XIIIth international congress of phonetic sciences (Vol. 2, p. 4-47). Stockholm: KTH and Stockholm University. Nosofsky, R. M. (986). Attention, similarity, and the identification-categorization relationship. Journal of Experimental sychology: General, 5,