Identification Model Based on the Maximum Information Entropy Principle

A new theoretical approach to stimulus identification is proposed through a probabilistic multidimensional model based on the maximum information entropy principle. The approach enables us to derive the multidimensional scaling (MDS) choice model, without appealing to Luce's choice rule and without defining a similarity function. It also clarifies the relationship between the MDS choice model and the optimal version of the identification model based on Ashby's general recognition theory; it is shown theoretically that the identification model derived from the new approach includes these two models as special cases. Finally, as an application of our approach, a model of similarity judgment is proposed and compared with Ashby's extended similarity model. Copyright 2001 Academic Press.     

Tests of an exemplar model for relating perceptual classification and recognition memory

Experiments were conducted in which Ss made classification, recognition, and similarity judgments for 34 schematic faces. A multidimensional scaling (MDS) solution for the faces was derived on the basis of the similarity judgments. This MDS solution was then used in conjunction with an exemplar-similarity model to accurately predict Ss' classification and recognition judgments. Evidence was provided that Ss allocated attention to the psychological dimensions differentially for classification and recognition. The distribution of attention came close to the ideal-observer distribution for classification, and some tendencies in that direction were observed for recognition. Evidence was also provided for interactive effects of individual exemplar frequencies and similarities on classification and recognition, in accord with the predictions of the exemplar model. Unexpectedly, however, the frequency effects appeared to be larger for classification than for recognition.     

Further tests of an exemplar-similarity approach to relating identification and categorization

Further tests were provided of an exemplar-similarity model for relating the identification and categorization of separable-dimension stimuli (Nosofsky, 1986). On the basis of confusion errors in an identification paradigm, a multidimensional scaling (MDS) solution was derived for a set of 16 separable-dimension stimuli. This MDS solution was then used in conjunction with the exemplar-similarity model to accurately predict performance in four separate categorization paradigms with the same stimuli. A key to achieving the accurate quantitative fits was the assumption that a selective attention process systematically modifies similarities among exemplars across different category structures. The tests reported go well beyond earlier ones (Nosofsky, 1986) in demonstrating the generalizability and utility of the theoretical approach. Implications of the results for alternative quantitative models of classification performance, including Ashby and Perrin's (1988) general recognition theory, were also considered.     

Division by multiplication

In two experiments, item-specific transfer was examined in simple multiplication and division with prime and probe problems separated by four to six trials. As was predicted by Rickard and Bourne's (1996) identical-elements model, response time (RT) savings were larger with identical (e.g., prime 63 divided by 7, probe 63 divided by 7) than with inverted (63 divided by 9 and 63 divided by 7) division problems, whereas identical (7 x 9 and 7 x 9) and inverted (9 x 7 and 7 x 9) multiplication problems produced equivalent transfer. Nonetheless, there was statistically significant transfer between inverted division problems. Furthermore, RT savings in the multiplication-to-division transfer conditions (e.g., prime 7 x 9, probe 63 divided by 7) indicated that multiplication mediated large-number division problems. These latter effects are not predicted by the identical-elements model but may be reconciled with the model by distinguishing associative transfer (facilitation owing to strengthening of a common problem node in memory) from mediated transfer (facilitation owing to mediation by a strengthened, related problem). Skilled adults can exploit the conceptual correspondences between multiplication and division facts in a highly efficient way to facilitate performance.     

Adults’ strategy choices for simple addition: Effects of retrieval interference

Simple addition (e.g., 3 + 2, 7 + 9) may be performed by direct memory retrieval or by such procedures as counting or transformation. The distribution of associations (DOA) model of strategy choice (Siegler, 1988) predicts that procedure use should increase as retrieval interference increases. To test this, 100 undergraduates performed simple addition problems, either after blocks of simple multiplication (high-interference context) or after blocks of simple division problems (low-interference context). Addition took longer and was more error prone after multiplication; in particular, there were more multiplication confusion errors on the relatively easy, small-number addition problems (e.g., 3 + 2 = 6, 4 + 3 = 12), but not on the more difficult, large-number additions. Consistent with the DOA, participants reported greater use of procedures for addition after multiplication, but more so for small addition problems. The findings demonstrate that adults’ use of procedural strategies for simple addition is substantially influenced by retrieval interference.     

Recognition memory for realistic synthetic faces

A series of experiments examined short-term recognition memory for trios of briefly presented, synthetic human faces derived from three real human faces. The stimuli were a graded series of faces, which differed by varying known amounts from the face of the average female. Faces based on each of the three real faces were transformed so as to lie along orthogonal axes in a 3-D face space. Experiment 1 showed that the synthetic faces' perceptual similarity structure strongly influenced recognition memory. Results were fit by a noisy exemplar model (NEMO) of perceptual recognition memory. The fits revealed thatrecognition memory was influenced both by the similarity of the probe to the series items and by the similarities among the series items themselves. Nonmetric multidimensional scaling (MDS) showed that the faces' perceptual representations largely preserved the 3-D space in which the face stimuli were arrayed. NEMO gave a better account of the results when similarity was defined as perceptual MDS similarity, rather than as the physical proximity of one face to another. Experiment 2 confirmed the importance of within-list homogeneity directly, without mediation of a model. We discuss the affinities and differences between visual memory for synthetic faces and memory for simpler stimuli.     

Perceptual representation of spam and phishing emails

Understanding how computer users allocate attention to features of potentially dangerous emails could help mitigate costly errors. Which features are salient? How stable is attention allocation across variation in email features? We attempted to measure the mental salience of several email features common in spam and/or phishing emails. We created two email sets: one in which messages contained company logos and urgent actionable links and one without these features. Participants rated pairwise similarity of emails within each set. Multidimensional scaling (MDS) analysis was conducted to quantify psychological similarity between emails. A separate group rated the same emails for presence of five other features: important downloadable content, collecting personal information, account deletion or suspension, advertisement, and large images with clickable content. Regressing feature ratings onto the MDS coordinates revealed that similarity judgments were influenced mostly by advertisement/large images and collecting personal information, regardless of presence or absence of company logos and urgent actionable links.     

Cognitive arithmetic: comparison of operations

Adults' performance of simple arithmetic calculations (addition, multiplication, and numerical comparison) was examined to test predictions of digital (counting), analog, and network models. Although all of these models have been supported by studies of mental addition, each leads to a different prediction concerning relations between the times required for addition, multiplication, and numerical comparison. Pairs of single-digit integers were presented and reaction times (RTs) for adding, multiplying, and comparing the stimuli were collected. A high correlation between RT for addition and multiplication of the same digits was obtained. This result is consistent with a network model, but presents difficulties for both analog and counting models. A "ties" effect of no increase in RT with increases in problem size for doubles such as 2 + 2 has been found in previous studies of addition using verification procedures, but was not found with the production task employed in the present study. Instead, a different kind of ties effect was found. Reaction time for both addition and multiplication of ties increased more slowly with problem size than did RT for non-tie problems. This ties effect, and the finding that probability of making errors contributes independently of problem size to RT support a distinction between location and accessibility of information in a network.     

Adults' representations of division facts: a consequence of learning history?

There has been a recent increase in the study of adults' performance on simple division problems. Researchers up to now have focused on the relationship between multiplication and division and have found that multiplication often has a mediating role in the solution of division problems (Campbell, 1997, 1999; LeFevre & Morris, 1999; Mauro, LeFevre, & Morris, 2002). In this study, division was exclusively examined to determine the strategies that are used to solve these problems and to identify factors relating to particular strategy use. Thirty-two participants were asked to solve two sets of 64 simple division problems (from 4 divided by 2 to 81 divided by 9) and error, latency, and strategy report data were collected. Fewer errors were made on easy problems, which were also solved more quickly than difficult problems. Participants used retrieval, multiplication, and other strategies to solve the problems and tended to use retrieval more on easy than difficult problems and used multiplication more on difficult problems than easy problems. Unexpected age differences in strategy use were also found. Older participants tended to rely more heavily on retrieval than younger participants. These results suggest that older participants may have stronger representations for simple division problems than younger participants.     

Perceptual organization of acoustic stimuli by budgerigars (Melopsittacus undulatus): III. Contact calls

We trained budgerigars by operant conditioning to discriminate among a set of contact calls in a same-different task and analyzed response latencies from this task by using multidimensional-scaling (MDS) and cluster-analysis procedures. Humans listened to the same calls and indicated the similarity between pairs of calls by a direct rating procedure. An MDS program (sindscal) was used to arrange these complex acoustic stimuli in multidimensional space reflecting perceptual organization. Multiple regression techniques were used to identify the acoustic characteristics of contact calls that were correlated with the perceptual dimensions obtained from MDS. A number of spectral characteristics (e.g., peak frequency, rate of frequency modulation, and concentration of spectral energy) emerged as important for both budgerigars and humans, but the relative salience of these cues differed for the two species. Additional tests with two groups of budgerigars--cagemates and noncagemates--showed that experience with calls can change the salience of various acoustic characteristics used for perceptual organization and individual recognition.     

Conditions of error priming in number-fact retrieval

Analysis of errors in simple multiplication has shown that answers retrieved on previous trials are initially inhibited (negative error priming) but later are promoted as errors to subsequent problems (positive error priming). Two experiments investigated whether error priming is associated either with problem-specific retrieval processes or with representations of answers that can be manipulated independently of problems. In Experiment 1, answers were primed by visually presenting products for 200 msec prior to problems. Correct-answer primes facilitated retrieval, related-incorrect primes interfered with retrieval more than unrelated primes, and both effects were greater for more difficult problems. Primes affected only the trial on which they were presented, however, whereas both negative and positive error priming from previous problems were observed across trials. In Experiment 2, subjects named and retrieved multiplication products on alternating trials. Just-named products were inhibited as errors to the following multiplication problem (i.e., negative error priming), but, compared to positive priming from previous retrieved products, positive error priming from previously named numbers was weak. The results indicate that positive error priming is due mainly to an encoding or retrieval bias produced by previous problems, whereas negative error priming entails suppression, or de-selection, of answer representations.     

A multidimensional scaling analysis of own- and cross-race face spaces

We examined predictions derived from Valentine’s (1991) Multidimensional Space (MDS) framework for own- and other-race face processing. A set of 20 computerized faces was generated from a single prototype. Each face was saved as Black and White, changing only skin tone, such that structurally identical faces were represented in both race categories. Participants made speeded “same-different” judgments to all possible combinations of faces, from which we generated psychological spaces, with “different” RTs as the measure of similarity. Consistent with the MDS framework, all faces were pseudo-normally distributed around the (unseen) prototype. The distribution of faces was consistent with Valentine’s (1991) predictions: despite their physical identity to the White faces, Black faces had lower mean inter-object distances in psychological space. Other-race faces are more densely clustered in psychological space, which could underlie well-known recognition deficits.     

Physical and Psychological Representations of Faces: Evidence FromMorphing

Models of face recognition and classification often adopt a framework in which faces are represented as points in a multi-dimensional space. This psychological face space organizes the faces according to similarity and makes predictions for representational theories of faces. A variety of image-processing techniques have been used to create novel stimuli in this space that represent the average of a population or make a face appear more distinctive. The current research examined the relation between the stimuli created by these image-processing techniques and the underlying psychological representation as measured by multidimensional scaling (MDS) procedures. Morphing procedures were used to create 16 faces that were embedded in a set of 84 other faces. Similarity ratings between all possible pairs of faces were collected, and the data were analyzed using MDS procedures. Dimensions that emerged from the MDS solution included age, race, adiposity, and facial hair. In the MDS space, the morphs appeared more typical than the parents, as predicted by the geometric model. A number of biases were examined, including the tendency of the morphs to be less typical than predicted, which may be attributed to the effects of density near the center of face space. In addition, age and facial-adiposity biases were found. The results support the use of the face-space framework for models of face recognition, although image-processing techniques that are designed to create novel stimuli in this space may introduce systematic biases.     

Calculation,culture, and the repeated operand effect

A basic phenomenon of cognitive arithmetic is that problems composed of a repeated operand, so-called "ties" (e.g. 6+6, 7 x 7), typically are solved more quickly and accurately than comparable non-tie problems (e.g. 6+5, 7 x 8). In Experiment 1, we present evidence that the tie effect is due to more efficient memory for ties than for non-ties, which participants reported solving more often using calculation strategies. The memory/strategy hypothesis accounts for differences in the tie effect as a function of culture (Asian Chinese vs. non-Asian Canadian university students), operation (addition, multiplication, subtraction, and division), and problem size (numerically small vs. large problems). Nonetheless, Blankenberger (Cognition 82 (2001) B15) eliminated the tie response time (RT) advantage by presenting problems in mixed formats (e.g. 4 x four), which suggests that the tie effect with homogenous formats (4 x 4 or four x four) is due to encoding. In Experiment 2, using simple multiplication problems, we replicated elimination of the tie effect with mixed formats, but also demonstrated an interference effect for mixed-format ties that slowed RTs and increased errors relative to non-tie problems. Additionally, practicing non-tie problems in both orders (e.g. 3 x 4 and 4 x 3) each time ties were tested once (cf. Cognition 82 (2001) B15) reduced the tie effect. The format-mismatch effect on ties, combined with a reduced tie advantage because of extra practice of non-ties, eliminated the tie effect. Rather than an encoding advantage, the results indicate that memory access for ties was better than for non-ties.     

Cognitive addition and multiplication: Evidence for a single memory network

In an experiment using verification task procedures, 100 subjects responded to simple and complex problems of addition and multiplication. Identical structural parameters were found to model reaction time accurately to both addition and multiplication problems. Slope estimates for a memory network parameter did not differ significantly between simple and complex problems within an operation or between addition and multiplication problems. Both complex addition and complex multiplication problems were processed columnwise, with column sums or products being retrieved from an interrated memory network. The two types of complex problems included similar processes for carrying and for encoding of single digits, and both were self-terminated when an error in the units column was encountered. Addition and multiplication facts appear to be retrieved from a single interrelated memory network. A conceptual model for this interrelated network is discussed.     

Avoiding the dangers of averaging across subjects when using multidimensional scaling

Ashby, Maddox and Lee (Psychological Science, 5 (3) 144) argue that it can be inappropriate to fit multidimensional scaling (MDS) models to similarity or dissimilarity data that have been averaged across subjects. They demonstrate that the averaging process tends to make dissimilarity data more amenable to metric representations, and conduct a simulation study showing that noisy data generated using one distance metric, when averaged, may be better fit using a different distance metric. This paper argues that a Bayesian measure of MDS models has the potential to address these difficulties, because it takes into account data-fit, the number of dimensions used by an MDS representation, and the precision of the data. A method of analysis based on the Bayesian measure is demonstrated through two simulation studies with accompanying theoretical analysis. In the first study, it is shown that the Bayesian analysis rejects those MDS models showing better fit to averaged data using the incorrect distance metric, while accepting those that use the correct metric. In the second study, different groups of simulated ‘subjects’ are assumed to use different underlying configurations. In this case, the Bayesian analysis rejects MDS representations where a significant proportion of subjects use different configurations, or when their dissimilarity judgments contain significant amounts of noise. It is concluded that the Bayesian analysis provides a simple and principled means for systematically accepting and rejecting MDS models derived from averaged data.     

Effects of problem format on division and multiplication performance: division facts are mediated via multiplication-based representations

In 2 experiments participants solved division problems presented in multiplication-based formats (e.g., 8 x _ = 72) more quickly than division problems presented in division-based formats (e.g., 72 / 8 = _). In contrast, participants solved multiplication problems presented in a division-based format (e.g., _ / 8 = 9) slowly and made many errors. In both experiments, the advantage for multiplication-based formats on division problems was found only for large problems (i.e., those with products or dividends greater than 25). These findings provide support for the view that large single-digit division facts are mediated via multiplication-based representations and that multiplication is the primary mode of representation for both division and multiplication facts.     

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.