chapter an exemplar-based random-walk model...

C H A P T E R

7 An Exemplar-Based Random-Walk Modelof Categorization and Recognition

Robert M. Nosofsky and Thomas J. Palmeri

Abstract

In this chapter, we provide a review of a process-oriented mathematical model ofcategorization known as the exemplar-based random-walk (EBRW) model (Nosofsky & Palmeri,1997a). The EBRW model is a member of the class of exemplar models. According to suchmodels, people represent categories by storing individual exemplars of the categories inmemory, and classify objects on the basis of their similarity to the stored exemplars. TheEBRW model combines ideas ranging from the fields of choice and similarity, to thedevelopment of automaticity, to response-time models of evidence accumulation anddecision-making. This integrated model explains relations between categorization and otherfundamental cognitive processes, including individual-object identification, the developmentof expertise in tasks of skilled performance, and old-new recognition memory. Furthermore, itprovides an account of how categorization and recognition decision-making unfold throughtime. We also provide comparisons with some other process models of categorization.

Key Words: categorization, recognition, exemplar model, response times, automaticity,random walk, memory search, expertise, similarity, practice effects

IntroductionA fundamental issue in cognitive psychology and

cognitive science concerns the manner in whichpeople represent categories and make classifica-tion decisions (Estes, 1994; Smith & Medin,1981). There is a wide variety of process-orientedmathematical models of categorization that havebeen proposed in the field. For example, accord-ing to prototype models (e.g., Posner & Keele,1968; Smith & Minda, 1998), people representcategories by storing a summary representation,usually presumed to be the central tendency ofthe category distribution. Classification decisionsare based on the similarity of a test item to theprototypes of alternative categories. According todecision-boundary models (e.g., Ashby & Maddox,1993; McKinley & Nosofsky, 1995), peopleconstruct boundaries, usually assumed to be linearor quadratic in form, to divide a stimulus space

into category response regions. If an object isperceived to lie in Region A of the space, then theobserver emits a Category-A response. Accordingto rule-plus-exception models (e.g., Davis, Love, &Preston, 2012; Erickson & Kruschke, 1998;Nosofsky, Palmeri, & McKinley, 1994), peopleconstruct low-dimensional logical rules for sum-marizing categories, and they remember occasionalexceptions that may be needed to patch those rules.

In this chapter, however, our central focus ison exemplar models of classification. Accordingto exemplar models, people represent categoriesby storing individual exemplars in memory, andclassify objects on the basis of their similarity tothe stored exemplars (Hintzman, 1986; Medin &Schaffer, 1978; Nosofsky, 1986). For instance, suchmodels would assume that people represent thecategory of “birds” by storing in memory the vastcollection of different robins, sparrows, eagles (and

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so forth) that they have experienced during theirlifetimes. If a novel object were similar to some ofthese bird exemplars, then a person would tend toclassify it as a bird.

Although alternative classification strategies arelikely to operate across different experimentalcontexts, there are several reasons why we choseto focus on exemplar models in this chapter. Onereason is that models from that class have provided asuccessful route to explaining relations between cat-egorization and a wide variety of other fundamentalcognitive processes, including individual-objectidentification (Nosofsky, 1986, 1987), the develop-ment of automaticity in tasks of skilled performance(Logan, 1988; Palmeri, 1997), and old-newrecognition memory (Estes, 1994; Hintzman, 1988;Nosofsky, 1988, 1991). A second reason is that,in our view, for most “natural” category structures(Rosch, 1978) that cannot be described in termsof simple logical rules, exemplar models seem toprovide the best-developed account for explaininghow categorization decision-making unfolds overtime. Thus, beyond predicting classification choiceprobabilities, exemplar models provide detailedquantitative accounts of classification responsetimes (Nosofsky & Palmeri, 1997a). We now brieflyexpand these themes before turning to the mainbody of our chapter.

One of the central goals of exemplar mod-els has been to explain relations between cate-gorization and other fundamental cognitive pro-cesses, including old-new recognition memory (Estes,1994; Hintzman, 1988; Nosofsky, 1988, 1991;Nosofsky & Zaki, 1998). Whereas in categoriza-tion people organize distinct objects into groups, inrecognition the goal is to determine if some indi-vidual object is “old” (previously studied) or “new.”Presumably, when people make recognition judg-ments, they evaluate the similarity of test objects tothe individual previously studied items (i.e., exem-plars). If categorization decisions are also based onsimilarity comparisons to previously stored exem-plars, then there should be close relations betweenthe processes of recognition and categorization.

A well-known model that formalizes these ideasis the generalized context model (GCM; Nosofsky,1984, 1986, 1991). In the GCM, individualexemplars are represented as points in a mul-tidimensional psychological space, and similaritybetween exemplars is a decreasing function of thedistance between objects in the space (Shepard,1987). The model presumes that both classifi-cation and recognition decisions are based on

the “summed similarity” of a test object to theexemplars in the space. By conducting similarity-scaling studies, one can derive multidimensionalscaling (MDS) solutions in which the locations ofthe exemplars in the similarity space are preciselylocated (Nosofsky, 1992). By using the GCM incombination with these MDS solutions, one canthen achieve successful fine-grained predictions ofclassification and recognition choice probabilitiesfor individual items (Nosofsky, 1986, 1987, 1991;Shin & Nosofsky, 1992).

A significant development in the application ofthe GCM has involved extensions of the model toexplaining how the categorization process unfoldsthrough time. So, for example, the exemplar modelnot only predicts choice probabilities, but alsopredicts categorization and recognition responsetimes (RTs). This development is important becauseRT data often provide insights into cognitiveprocesses that would not be evident based on exam-ination of choice-probability data alone. Nosofskyand Palmeri’s (1997a,b) exemplar-based random-walk (EBRW) model adopts the same fundamentalrepresentational assumptions as does the GCM.However, it extends that earlier model by assumingthat retrieved exemplars drive a random walk process(e.g., Busemeyer, 1985; Link, 1992; Ratcliff, 1978).This exemplar-based random-walk model allowsone to predict the time course of categorization andrecognition decision making.

In this chapter, we provide a review of thisEBRW model and illustrate its applications tovarieties of categorization and recognition choice-probability and RT data. In the section on TheFormal EBRW Model of Categorization RTs weprovide a statement of the formal model as appliedto categorization. As will be seen, the EBRWmodel combines ideas ranging from the fields ofchoice and similarity, to the development of auto-maticity, to RT models of evidence accumulationand decision making. In the section Effects ofSimilarity and Practice on Speeded Classification,we describe applications of the model to speededperceptual classification, and illustrate how it cap-tures fundamental phenomena including effects ofsimilarity and practice. In the section Automaticityand Perceptual Expertise we describe how theEBRW accounts for the development of automaticcategorization and perceptual expertise. In thesection Using Probabilistic Feedback Manipulationsto Contrast the Predictions From the EBRW Modeland Alternative Models we describe experimentalmanipulations that have been used to try to

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Fig. 7.1 Schematic illustration of the ideas behind the exemplar-based random-walk model. Panel a: Exemplars are represented aspoints in a multidimensional psychological space. Similarity is a decreasing function of the distance between objects in the space.Presentation of a test probe causes the exemplars to be “activated” in accord with how similar they are to that probe. The activatedexemplars “race” to be retrieved. Panel b: The retrieved exemplars drive a random-walk process for making categorization decisions.Each time that an exemplar from Category A is retrieved, the random walk takes unit step towards the Category-A response threshold,and likewise for Category B. The retrieval process continues until one of the response thresholds is reached.

distinguish between the predictions of the EBRWmodel and some other major models of speededcategorization, including decision-boundary andprototype models. Finally, in the section Extendingthe EBRW Model to Predict Old-New RecognitionRTs, we describe recent developments in which theEBRW model has been used to account for old-new recognition RTs. We then provide conclusionsand questions for future research in the sectionConclusions and New Research Goals.

The Formal EBRW Model ofCategorization RTs

The EBRW model and the GCM build uponclassic theories in the areas of choice and similarity(Shepard, 1957, 1987). As described in the intro-duction, in the model, exemplars are representedas points in a multidimensional psychological space(for an illustration, see Figure 7.1a). The distancebetween exemplars i and j (dij) is given by theMinkowski power model,

dij =[

K∑k=1

wk∣∣xik− xjk

∣∣ρ] 1ρ

, (1)

where xik is the value of exemplar i on psychologicaldimension k; K is the number of dimensions thatdefine the space; ρ defines the distance metric ofthe space; and wk (0 < wk,

∑wk = 1) is the

weight given to dimension k in computing distance.In situations involving the classification of holistic

or integral-dimension stimuli (Garner, 1974), whichwill be the main focus of the present chapter, ρ isset equal to 2, which yields the familiar Euclideandistance metric. The dimension weights wk are freeparameters that reflect the degree of “attention”that subjects give to each dimension in makingtheir classification judgments (Carroll & Wish,1974). In situations in which some dimensions arefar more relevant than others in allowing subjectsto discriminate between members of contrastingcategories, these attention-weight parameters mayplay a significant role (e.g., Nosofsky, 1986, 1987).In the experimental situations considered in thischapter, however, all dimensions tend to be relevantand the attention weights will turn out to play arelatively minor role.

The similarity of test item i to exemplar jis an exponentially decreasing function of theirpsychological distance (Shepard, 1987),

sij = exp(− c ·dij), (2)

where c is an overall sensitivity parameter that mea-sures overall discriminability in the psychologicalspace. The sensitivity governs the rate at whichsimilarity declines with distance. When sensitivity ishigh, the similarity gradient is steep, so even objectsthat are close together in the space may be highlydiscriminable. By contrast, when sensitivity is low,the similarity gradient is shallow, and objects arehard to discriminate.

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Each exemplar j is stored in memory (along withits associated category feedback) with memory-strength mj . As will be seen, the precise assumptionsinvolving the memory strengths vary with the specificexperimental paradigm that is tested. For example,in some paradigms, we attempt to model perfor-mance on a trial-by-trial basis, and assume that thememory strengths of individual exemplars decreasesystematically with their lag of presentation. Inother paradigms, we attempt to model performancein a transfer phase that follows a period of extendedtraining. In that situation, we typically assumethat the memory strengths are proportional to thefrequency with which each individual exemplar waspresented in common with given category feedbackduring the initial training phase.

When a test item is presented, it causes allexemplars to be “activated” (Figure 7.1a). Theactivation for exemplar j, given presentation of itemi, is given by

aij =mjsij. (3)

Thus, the exemplars that are most highly activatedare those that have the greatest memory strengthsand are highly similar to the test item. In modelingearly-learning behavior, we also presume that“background” elements exist in memory at the startof training that are not associated with any of thecategories. These background elements are presumedto have fixed activation b, independent of the testitem that is presented.

Borrowing from Logan’s (1988) highly influen-tial instance theory of automaticity, the EBRWassumes that presentation of a test item causesthe activated stored exemplars and backgroundelements to “race” to be retrieved. For mathematicalsimplicity, the race times are presumed to be inde-pendent exponentially distributed random variableswith rates proportional to the degree to whichexemplar j is activated by item i (Bundesen, 1990;Logan, 1997; Marley, 1992). Thus, the probabilitydensity that exemplar j completes its race at time t,given presentation of item i, is given by

f (t)= aij · exp(− aij · t). (4)

This assumption formalizes the idea that althoughthe retrieval process is stochastic, the exemplars thattend to race most quickly are those that are mosthighly activated by the test item. The exemplar(or background element) that “wins” the race isretrieved.

Whereas in Logan’s (1988) model the response isbased on only the first retrieved exemplar, in theEBRW model exemplars from multiple retrievals

drive a random-walk evidence accumulation process(e.g., Busemeyer, 1982; Ratcliff, 1978). In a two-category situation, the process operates as follows(for an illustration, see Figure 7.1b): First, there is arandom-walk counter with initial setting zero. Theobserver establishes response thresholds representingthe amount of evidence needed to make either aCategory-A response (+A) or a Category-B response(–B). Suppose that exemplar x wins the race ona given step. If x received Category-A feedbackduring training, then the random-walk counter isincreased by unit value in the direction of +A,whereas if x received Category-B feedback, thecounter is decreased by unit value in the directionof –B. (If a background element wins the race, thecounter’s direction of change is chosen randomly.)If the counter reaches either threshold +A or –B,the corresponding categorization response is made.Otherwise, the races continue, another exemplar isretrieved (possibly the same one as on the previousstep), and the random walk takes its next step.

Given the processing assumptions outlined ear-lier, Nosofsky and Palmeri (1997a) showed that,on each step of the random walk, the probability(pi) that the counter is increased in the direction ofthreshold +A is given by

pi = (SiA+b)/(SiA+SiB+2b), (5)

where SiA denotes the summed activation of allcurrently stored Category-A exemplars given pre-sentation of item i (and likewise for SiB). (Theprobability that the counter is decreased in thedirection of Category B is given by qi = 1-pi.) Thus,as the summed activation of Category-A exemplarsincreases, the probability of retrieving Category-Aexemplars, and thereby the probability of movingthe counter in the direction of +A, increases. Thecategorization decision time is determined jointlyby the total number of steps required to completethe random walk and by the speed with which thoseindividual steps are made.

Given these random-walk processing assump-tions, it is straightforward to derive analyticpredictions of classification choice probabilities andmean RTs for each stimulus at any given stage of thelearning process (Busemeyer, 1982). The relevantequations are summarized by Nosofsky and Palmeri(1997a, pp. 269–270, 291–292). Here, we focus onsome key conceptual predictions that follow fromthe model.

A first prediction is that rapid classificationdecisions should be made for items that arehighly similar to exemplars from their own target


category and that are dissimilar to exemplars fromthe contrast category. Under such conditions, allretrieved exemplars will tend to come from thetarget category, so the random walk will marchconsistently toward that category’s response thresh-old. For example, if an item is highly similar tothe exemplars from Category A, and dissimilar tothe exemplars from Category B, then the value piin Eq. 5 will be large, so almost all steps in therandom walk will move in the direction of +A. Bycontrast, if an item is similar to the exemplars ofboth Categories A and B, then exemplars from bothcategories will be retrieved; in that case, the randomwalk will meander back and forth, leading to aslow RT.

A second prediction is that practice in theclassification task will lead to more accurate re-sponding and faster RTs. Early in training, beforeany exemplars have been stored in memory, thebackground-element activations are large relative tothe summed-activations of the stored exemplars (seeEq. 5). Thus, the random-walk step probabilities(pi) hover around .5, so responding is slow andprone to error. As the observer accumulates morecategory exemplars in memory, the summed activa-tions SiA and SiB grow in magnitude, so respondingis governed more by the stored exemplars. A secondreason that responding speeds up is that the greaterthe number of exemplars stored in memory, thefaster the “winning” retrieval times tend to be (cf.Logan, 1988). The intuition is that the greater thenumber of exemplars that participate in the race,the higher is the probability that some particularexemplar will finish quickly. Therefore, as moreexemplars are stored, the speed of the individualsteps in the random walk increases.

As discussed later in this chapter, many morefine-grained predictions follow from the model,some of which are highly diagnostic for distinguish-ing between the predictions of the EBRW modeland alternative models. We describe these predic-tions in the context of the specific experimentalparadigms in which they are tested.

One other important formal aspect of the modelinvolves its predictions of classification choiceprobabilities. In particular, in the special case inwhich the response thresholds +A and −B areset an equal magnitude γ from the starting pointof the random walk, the model predicts that theprobability that item i is classified in Category Ais given by

P(A|i)= (SiA+b)γ /[(SiA+b)γ +[(SiB+b)γ ]. (6)

This equation is the descriptive equation of choiceprobability that is provided by the GCM, one ofthe most successful formal models of perceptualclassification (for discussion, see, e.g., Nosofsky,1986; Wills & Pothos, 2012).1 Thus, besidesproviding a formal account of classification RTs, theEBRW model provides a processing interpretationfor the emergence of the GCM response rule.

Effects of Similarity and Practice onSpeeded Classification

In their initial tests of the EBRW model,Nosofsky and Palmeri (1997a) conducted a speededclassification task using the category structureshown in Figure 7.2. The stimuli were a set of 12Munsell colors of a constant red hue varying intheir brightness and saturation2. As illustrated inthe figure, half the exemplars were assigned by theexperimenter to Category A (squares) and half toCategory B (circles). On each trial, a single colorwas presented, the observer classified it into one ofthe two categories, and corrective feedback was thenprovided. Testing was organized into 150 blocksof 12 trials (30 blocks per day), with each colorpresented once in a random order in each block.(See Nosofsky and Palmeri, 1997a, pp. 273–274for further methodological details.)

The mean RTs observed for one of the par-ticipants are shown in Figure 7.3. The top panelillustrates the mean RT for each individual color,averaged across the final 120 blocks. The diameterof the circle enclosing each stimulus is linearlyrelated to the stimulus’s mean RT. To help interpretthese data, the figure also displays a dotted bound-ary of equal summed-similarity to the exemplars

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Fig. 7.3 Data from Experiment 1 of Nosofsky and Palmeri(1997a). (Top) Mean RTs for individual colors (RT proportionalto the size of the circle). (Bottom) Mean RTs averaged across allcolors as a function of grouped blocks of practice.

of Category A and Category B. Points falling tothe upper right have greater summed similarity toCategory A, and points falling to the lower left havegreater summed similarity to Category B. As canbe seen, the mean RTs tend to get faster as onemoves farther away from the boundary of equalsummed similarity. The bottom panel of Figure7.3 provides another perspective on the data. Thispanel plots the overall mean RT for all 12 colors foreach “grouped-block” of practice in the task, wherea grouped-block corresponds to five consecutiveindividual blocks. It is evident that there is aspeed-up with practice, with the lion’s share of thespeed-up occurring during the early blocks.

To apply the EBRW model to these data, we firstderived a multidimensional scaling (MDS) solutionfor the colors by having the participant provideextensive similarity ratings between all pairs of thecolors. A two-dimensional scaling solution yielded agood account of the similarity ratings. The solutionis depicted in the top panel of Figure 7.3, where thecenter of each circle gives the MDS coordinates ofthe color. We then used the EBRW model in combi-nation with the MDS solution to simultaneously fitthe mean RTs associated with the individual colors(Figure 7.3, top panel) as well as the aggregatedmean RTs observed as a function of grouped-blocksof practice (Figure 7.3, bottom panel). Specifically,the MDS solution provides the coordinate valuesxik that enter into the EBRW model’s distancefunction (Equation 1). For simplicity, in fitting themodel, we assumed that on each individual blockof practice, an additional token of each individualexemplar was stored in memory with strength equalto one. (To reduce the number of free parameters,we set the background-element activation b equalto zero.) The first step in the analysis was touse the model to predict the mean RT for eachindividual color in each individual block. Then,for each individual color, we averaged the predic-tions across Blocks 31–150 to predict the overallindividual-color mean RTs. Likewise, we averagedthe predicted mean RTs over all colors in eachgrouped-block of practice to predict the speed-upcurves. We fitted the model by searching for the freeparameters that minimized the total sum of squareddeviations between predicted and observed meanRTs across both data sets. These free parametersincluded the overall sensitivity parameter c; anattention-weight parameter w1(with w2 = 1 – w1);and a response-threshold parameter +A (with +A =|–B|). In addition, we estimated a mean residual-time parameter μR; a scaling constant v fortransforming the number of steps in the randomwalk into ms; and an individual step-time constantα (see Nosofsky & Palmeri, 1997a, pp. 268–270 fordetails).

The model-fitting results for the individual-color RTs are displayed in Figure 7.4, which plotsobserved against predicted mean RTs for eachindividual color. The model provides a reasonablygood fit (r = 0.89), especially considering that itis constrained to simultaneously fit the speed-upcurves. The model predicts these results because col-ors far from the exemplar-based boundary (e.g., 2,4, 7, and 10—see top panel of Figure 7.3) tend to besimilar only to exemplars from their own category.


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Fig. 7.4 Scatterplot of observed individual-color mean RTsagainst the predictions from the EBRW model.

Thus, only exemplars from a single category areretrieved, and the random walk marches in rapidfashion to the appropriate response threshold. Bycontrast, colors close to the boundary (e.g., 3, 9,and 12) are similar both to exemplars from theirown category and to the contrast category. Thus,exemplars from both categories are retrieved and therandom walk meanders back and forth, leading toslow mean RTs.

The fits to the speed-up curves are shown alongwith the observed data in the bottom panel ofFigure 7.3. Again, the model captures these datareasonably well (r = 0.931). It predicts the speed-upfor essentially the same reason as in Logan’s (1988)model: As practice continues, an increased numberof exemplars are stored in memory. The greaterthe number of exemplars that race to be retrieved,the faster are the winning retrieval times, so theindividual steps in the random walk take place morequickly.3

Automaticity and Perceptual ExpertiseThe EBRW provides a general theory of cate-

gorization, automaticity, and the development ofperceptual expertise (Nosofsky & Palmeri, 1997a;Palmeri, 1997; Palmeri, Wong, & Gauthier, 2004).In some real-world domains, novices learn tocategorize objects by first learning to use a setof explicit verbal rules. This novice categorizationcan be a relatively slow, deliberate, attention-demanding process. With experience, as peopledevelop perceptual expertise, categorization often

becomes fast and automatic. Logan (1988) gen-erally attributed the development of automaticityin cognitive skills to a shift from strategic al-gorithmic processes to exemplar-based memoryretrieval. Palmeri (1997) specifically examined thedevelopment of automaticity in categorization as ashift from a rule-based process to an exemplar-basedprocess assumed by EBRW (see also Johansen &Palmeri, 2002).

In Palmeri (1997, 1999), subjects were told touse an explicit rule in a dot-counting categorizationtask.They were asked to categorize random patternscontaining between six and eleven dots according tonumerosity and did so over multiple sessions. RTsobserved in the first session increased linearly withthe number of dots, as shown in Figure 7.5A. Thedot patternswere repeated across multiple sessions,giving subjects an opportunity to develop auto-maticity in the task. Indeed, RTs observed in the13th session were flat with numerosity, a signatureof automaticity. In a subsequent transfer test, shownin Figure 7.5B, new unrelated test patterns hadcategorization RTs that were a linear function ofnumerosity, just like the first session, and oldtraining patterns continued to be categorized withthe same RTs irrespective of numerosity, just likethe last session. Dot patterns of low or moderatesimilarity to the training patterns were categorizedwith RTs intermediate to the new unrelated and oldtraining patterns.

Categorization RTs over sessions and numerositywere successfully modeled as a horse race betweenan explicit dot-counting process, whose stochasticfinishing time increased linearly with the numberof dots in the pattern; and an exemplar-basedcategorization process determined by the similarityof a dot pattern to stored exemplars of patternspreviously categorized—an elaboration of EBRW.4

When few patterns have been experienced, catego-rization is based entirely on the finishing times ofthe explicit counting process, predicting increasedRTs with increased numerosity (Figure 7.5C). Withexperience, the memory strength mj (Eq. 3) ofexemplars in EBRW increases, causing a fasteraccumulation of perceptual evidence to a responsethreshold. Faster categorizations based on exemplarretrieval imply increased likelihood that thosecategorizations finish more quickly than explicitcounting. With sufficient experience, EBRW fin-ishes before counting for nearly all categorizations,predicting flat RTs with numerosity. The similarity-based retrieval in EBRW also predicts the transferresults, as shown in Figure 7.5D.


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Fig. 7.5 Data from Experiment 1 of Palmeri (1997). Subjects categorized dot patterns varying in numerosity over thirteen trainingsessions and these were followed by a transfer test. (A) Response time (ms) as a function of numerosity in each training session (1-13).(B) Response time (ms) as a function of numerosity during transfer for old patterns, new unrelated patterns, and patterns of low andmoderate similarity to old patterns. (C and D) EBRW predictions.

A shift from rules to exemplars is just onecommon characteristic of the development ofperceptual expertise that EBRW can help explain(see Palmeri et al., 2004; Palmeri & Cottrell,2009). As one other brief example, considerthe well-known basic-level advantage. The seminalwork of Rosch, Morris, Gray, Johnson, & Boyes-Braem (1976) showed that people are faster toverify the category of an object at an intermediatebasic level of abstraction (“bird”) than at moresuperordinate (“animal”) or subordinate (“NorthernCardinal”) levels. With expertise, subordinate-levelcategorizations are verified as quickly as basic-levelcategorizations (Tanaka & Taylor, 1991). Oneexplanation for the basic-level advantage and relatedfindings in novices is that basic-level categorizations

reflect an initial stage of visual processing, perhapsas early as object segmentation (Grill-Spector &Kanwisher, 2005; see also Jolicoeur, Gluck, & Koss-lyn, 1984). For novices, subordinate categorizationsare slow because basic-level categorizations mustbe made first. For experts, the stage of basic-levelcategorization is somehow bypassed.

EBRW provides an alternative account (seeMack & Palmeri, 2011; Mack, Wong, Gauthier,Tanoka, & Palmeri, 2007). Faster or slowercategorization at different levels of abstractionneed not reflect stages of processing but mayinstead reflect differences in the speed of evidenceaccumulation in the random walk. Mack et al.(2007) simulated basic-level and subordinate-levelcategorization by novices and experts. For these


simulations, subordinate categories were simplyassumed to be individual identities of objects withinclusters that were the basic-level categories. Formoderate levels of the sensitivity parameter, c,which reflects the overall discriminability in thepsychological space (Eq. 2), a basic-level advantagewas predicted. But for high levels of sensitivity, re-flecting the greater discriminability that may comewith perceptual expertise, equivalent RTs for thesubordinate and basic levels were predicted. Furtherempirical evidence has supported the hypothesisthatdifferences in RTs at different levels of abstractionreflect how quickly perceptual evidence accumulatesrather than differences in discrete visual processingstages (e.g., Mack, Gauthier, Sadr, & Palmeri, 2008;et al., 2009; Mack & Palmeri, 2010).

Using Probabilistic FeedbackManipulations to Contrast the Predictionsfrom the EBRW Model and AlternativeModels

One the major alternative modeling approachesto predicting multidimensional classification RTsarises from a general framework known as “decision-boundary theory” (e.g., Ashby & Townsend, 1986).According to this framework, people constructdecision boundaries for dividing a perceptual spaceinto category response regions. Test items areassumed to give rise to noisy representations in themultidimensional perceptual space. If a perceivedrepresentation falls in Region A of the space, thenthe observer classifies the item into Category A.

Most past approaches to generating RT predic-tions from decision-boundary theory have involvedapplications of the RT-distance hypothesis (Ashby,Boynton, & Lee, 1994). According to this hypoth-esis, RT is a decreasing function of the distance of astimulus from the decision boundary. More recentmodels that formalize the operation of decisionboundaries posit perceptual-sampling mechanismsthat drive random-walk or diffusion processes,similar to those of the EBRW model (e.g.,Ashby, 2000; Fific, Little, & Nosofsky, 2010;Nosofsky & Stanton, 2005). For example, inNosofsky and Stanton’s (2005, p. 625) approach,on each step of a random walk, a percept issampled randomly from the perceptual distributionassociated with a presented stimulus. If the perceptfalls in Region A (as defined by the decisionboundary), then the random walk steps towardthreshold A, otherwise it steps toward threshold –B. The perceptual-sampling process continues untileither threshold is reached. Such models provide

process interpretations for why RT should get faster(and accuracy should increase) as distance of astimulus from a decision boundary increases.

According to the EBRW model and decision-boundary models, the nature of the memoryrepresentations that are presumed to underlie cat-egorization are dramatically different (i.e., storedexemplars versus boundary lines). Despite this dra-matic difference, it is often difficult to distinguishbetween the predictions from the models. Thereason is that distance-from-boundary and relativesummed-similarity tend to be highly correlated inmost experimental designs. For example, as wealready described with respect to the Figure 7.3(top panel) structure, items far from the boundarytend to be highly similar to exemplars from theirown category and dissimilar to exemplars from thecontrast category. Thus, both the distance-from-boundary model and the EBRW model tend topredict faster RTs and more accurate responding asdistance from the boundary increases.

In one attempt to discriminate between the RTpredictions of the models, Nosofsky and Stanton(2005) conducted a design that aimed to decoupledistance-from-boundary and relative summed simi-larity (for a related approach, see Rouder & Ratcliff,2004). The key idea in the design was to make useof probabilistic feedback manipulations associatedwith individual stimuli in the space. The designis illustrated in Figure 7.6. The stimuli wereagain a set of 12 Munsell colors of a constanthue, varying in their brightness and saturation.As illustrated in the figure, Colors 1–6 belongedto Category A, whereas Colors 7–12 belonged toCategory B. To help motivate the predictions, wehave drawn a diagonal linear decision boundaryfor separating the two categories of colors intoresponse regions. Given reasonable assumptions(for details, see Nosofsky & Stanton, 2005), thisboundary is the optimal (ideal-observer) boundaryaccording to decision-boundary theory. That is, itis the boundary that would maximize an observer’spercentage of correct categorization decisions. Ingenerating predictions, decision-bound theoristsoften assume that observers will use a boundarywith an optimal form. However, we will considermore general possibilities later in this section.

The key experimental manipulation was that,across conditions, either Stimulus Pair 4/8 orStimulus Pair 5/9 received probabilistic feedback,whereas all other stimuli received deterministicfeedback. Specifically, in Condition 4/8, Stimulus 4received Category-A feedback with probability 0.75


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but received Category- B feedback with probability0.25; whereas Stimulus 8 received Category- Bfeedback with probability 0.75 and Category- Afeedback with probability 0.25. Analogous proba-bilistic feedback assignments were given to Stimuli5 and 9 in Condition 5/9. In each condition, werefer to the pair that received probabilistic feedbackas the probabilistic pair and to the pair that receiveddeterministic feedback as the deterministic pair.

The key conceptual point is that, because ofthe symmetric probabilistic assignment of stimulito categories, the optimal boundary for partitioningthe space into response regions is the same diagonallinear decision boundary that we have alreadyillustrated in Figure 7.6. There is no way toadjust the boundary to achieve more accurateresponding in the face of the probabilistic feedbackassignments. Furthermore, because the probabilisticand deterministic pairs are an equal distance fromthe decision boundary, the most natural predictionfrom that modeling approach is that RTs forthe probabilistic and deterministic pairs should bethe same.

By contrast, the EBRW model predicts that theprobabilistic pair should be classified more slowly(and with lower accuracy) than the deterministicpair. For example, in Condition 4/8, in casesin which stimulus 4 is presented and tokens ofexemplar 4 are retrieved from memory, 0.75 ofthe steps in the random walk will move in thedirection of threshold +A, but 0.25 of the steps will

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move in the direction of threshold –B. By contrast,presentations of the deterministic pair will result inmore consistent steps of the random walk, leadingto faster RTs and more accurate responding.

Across two experiments (in which the instruc-tions varied the relative emphasis on speed versusaccuracy in responding), the qualitative patternof results strongly favored the predictions fromthe EBRW model compared to the distance-from-boundary model. In particular, in both experi-ments, observers responded more slowly and withlower accuracy to the probabilistic pair than tothe deterministic pair. These results were observedat both early and late stages of testing and wereconsistent across the majority of the individualparticipants. As a converging source of evidence,the EBRW model also provided better overallquantitative fits to the individual-subject choice-probability and RT data than did the distance-from-boundary model, including versions of thelatter model in which the slope and y-intercept ofthe linear boundary were allowed to vary as freeparameters.

As suggested by Nosofsky and Stanton (2005,p. 623), the results were particularly intriguingbecause they pointed toward a stubborn formof suboptimality in human performance: In theFigure 7.6 design, subjects would perform optimallyby simply ignoring the probabilistic feedback andclassifying objects based on the linear decisionboundary. Nevertheless, despite being providedwith monetary payoffs for correct responses, sub-jects’ behavior departed from that optimal strategyin a manner that was well predicted by theEBRW model. Similar forms of evidence that favorthe predictions from the EBRW model have beenreported in related studies that manipulated theoverall familiarity of individual study exemplarsrather than probabilistic feedback assignments (e.g.,Nosofsky & Palmeri, 1997a; Verguts, Storms, andTuerlinckx, 2003).

Although the focus of Nosofsky and Stanton’s(2005) study was to contrast predictions fromthe EBRW model and decision-bound models, thedesigns also yielded sharp contrasts between theEBRW model and prototype models.5 Specifically,Nosofsky and Stanton (2005, p. 610) formulated aprototype-based random-walk (PBRW) model, anal-ogous in all respects to the EBRW model, exceptthat the category representation was presumed tocorrespond to the central tendency of each categorydistribution rather than to the individual exemplars.It turns out that for the stimulus spacings and


probabilistic stimulus-category assignments used inthe Figure 7.6 design, the central tendency of eachcategory is equidistant to the probabilistic anddeterministic stimulus pairs. Thus, the PBRWmodel predicted incorrectly that those pairs wouldshow identical choice probabilities and RTs. Notsurprisingly, therefore, the PBRW yielded far worsequantitative fits to the full sets of choice-probabilityand RT data than did the EBRW model.

Extending the EBRW Model to PredictOld-New Recognition RTsOverview

As noted in the introduction, a central goal of ex-emplar models is to explain not only categorization,but other fundamental cognitive processes suchas old-new recognition. The GCM has providedsuccessful accounts of old-new recognition choiceprobabilities in wide varieties of experimentalsituations. When applied to recognition, the GCMassumes that each item from a study list is storedas a distinct exemplar in memory. The observer ispresumed to sum the similarity of a test item tothese stored exemplars. The greater the summedsimilarity, the more familiar is the test item,so the greater is the probability with which theobserver responds “old.” Indeed, the GCM canbe considered a member of the general class of“global matching” models of old-new recognition(e.g., Clark & Gronlund, 1996; Gillund & Shiffrin,1984; Hintzman, 1988; Murdock, 1982). Withinthis broad class, an important achievement of themodel is that it predicts fine-grained differencesin recognition probabilities for individual itemsbased on their fine-grained similarities to individualexemplars in the study set (e.g., Nosofsky, 1988,1991; Nosofsky & Zaki, 2003).

Just as has been the case for categorization, amajor development in recent years has involvedextensions of the GCM in terms of the EBRWmodel to allow it to account for recognition RTs(Donkin & Nosofsky, 2012a,b; Nosofsky, Little,Donkin, & Fific, 2011; Nosofsky & Stanton,2006). In this section we describe these formaldevelopments and illustrate applications to variantsand extensions of the classic Sternberg (1966) short-term probe-recognition paradigm. In this paradigm,subjects are presented on each trial with a short listof items (the memory set) and are then presentedwith a test probe. The subjects judge, as rapidlyas possible, while minimizing errors, whether theprobe occurred in the memory set. Fundamentalvariables that are manipulated in the paradigm

include the size of the memory set; whether the testprobe is old (a “positive” probe) or new (a “negative”probe); and, if old, the serial position with whichthe positive probe occurred in the memory set.

Whereas in the standard version of the Sternbergparadigm the to-be-recognized items are generallyhighly distinct entities, such as alphanumeric char-acters, recent extensions have examined recognitionperformance in cases involving confusable stimuliembedded in a continuous-dimension similarityspace (e.g., Kahana & Sekuler, 2002). We focuson this type of extended similarity-based paradigmin the initial part of this section; however, we willillustrate applications of the EBRW model to themore standard paradigm as well.

The Formal ModelThe EBRW-recognition model makes the same

representational assumptions as does the categoriza-tion model: (a) Exemplars are conceptualized asoccupying points in a multidimensional similarityspace; (b) similarity is a decreasing function ofdistance in the space (Eqs. 1 and 2); (c) activationof the exemplars is a joint function of their memorystrength and their similarity to the test probe(Equation 3); and (d) the stored exemplars raceto be retrieved with rates proportional to theiractivations (Eq. 4).

In our previous applications to categorization,a single global level of sensitivity (c in Eq. 2)was assumed that applied to all exemplar tracesstored in long-term memory. In application toshort-term recognition paradigms involving high-similarity lures, however, allowance is made for aform of exemplar-specific sensitivity. In particular,an observer’s ability to discriminate between testitem i and a nonmatching exemplar-trace j willalmost certainly depend on the recency withwhich exemplar j was presented: Discriminationis presumably much easier if an exemplar was justpresented than if it was presented in the distant past.In the version of the model applied by Nosofskyet al. (2011), a separate sensitivity parameter cjwas estimated for each individual lag j on thestudy list, where lag is counted backward fromthe presentation of the test probe to the memory-set exemplar. For example, for the case in whichmemory set-size is 4, the exemplar in the fourthserial position has Lag 1, the exemplar in the thirdserial position has Lag 2, and so forth.

Likewise, the memory strengths of the individualexemplars (mj) were also assumed to depend onlag j: Presumably, the more recently an exemplar


was presented on the study list, the greater itsmemory strength (e.g., McElree & Dosher, 1989;Monsell, 1978). (Although the effects are smaller, inmodeling short-term recognition with the EBRW,allowance is also typically made for a primacy effecton memory strength. The memory strength of theitem in the first serial position of the memory setis given by PM · mj , where mj is the memorystrength for an item with lag j, and PM is aprimacy-multiplier parameter.)

To adapt the EBRW model to the domain ofold-new recognition, it is assumed that the observerestablishes what are termed “criterion elements” inthe memory system. These elements are similar tothe “background elements” used for modeling theearly stages of category learning. Just as is the casefor the stored exemplars, upon presentation of a testprobe, the criterion elements race to be retrieved.However, whereas the retrieval rates of the storedexemplars vary with their lag-dependent memorystrengths and their similarity to the test probe,the retrieval rates of the criterion elements areindependent of these factors. Instead, the criterionelements race with some fixed rate β, independentof the test probe that is presented. The setting of βis presumed to be, at least in part, under the controlof the observer.

Finally, the retrieved exemplars and criterionelements drive a random-walk process that governsold-new recognition decisions. The observer setsresponse thresholds +OLD and –NEW that estab-lish the amount of evidence needed for making an“old” or a “new” response. On each step of therandom walk, if an old exemplar wins the retrievalrace, then the random-walk counter takes a stepin the direction of the +OLD response threshold;whereas if a criterion element wins the race, then thecounter takes a step in the direction of the –NEWthreshold. The retrieval process continues until oneof the thresholds is reached.

Given the processing assumptions outlined ear-lier, then on each step of the random walk, theprobability that the counter steps in the directionof the +OLD threshold is given by

pi = F i/(F i+β), (7)

where Fi gives the summed activation (“familiarity”)of the test probe to all old exemplars on the studylist (and β is the fixed setting of criterion-elementactivation). Note that test probes that matchrecently presented exemplars (with high memorystrengths) will cause high summed familiarity (Fi),leading the random walk to march quickly to the

+OLD threshold and resulting in fast old RTs.By contrast, test probes that are highly dissimilarto the memory-set items will not activate thestored exemplars, so only criterion elements willbe retrieved. In this case, the random walk willmarch quickly to the –NEW threshold, resulting infast new RTs. Through experience in the task, theobserver is presumed to learn an appropriate settingof the criterion-element activation β, such thatsummed activation (Fi) tends to exceed β when thetest probe is old, but tends to be less than β whenthe test probe is new. In this way, the random walkwill tend to step toward the appropriate responsethreshold on trials in which old versus new probesare presented.

Experimental TestsIn Nosofsky et al.’s (2011) initial experiment for

testing the model, the stimuli were a set of 27Munsell colors that varied along the dimensions ofhue, brightness, and saturation. Similarity-scalingprocedures were used to derive a precise MDSsolution for the colors.

The design of the probe-recognition experimentinvolved a broad sampling of different list structuresto provide a comprehensive test of the model. Therewere 360 lists in total. The size of the memoryset on each trial was either 1, 2, 3 or 4 items,with an equal number of lists at each set size. Foreach set size, half the test probes were old and halfwere new. In the case of old probes, the matchingitem from the memory set occupied each serialposition equally often. To create the lists, itemswere randomly sampled from the full set of stimuli,subject to the constraints described earlier. Thus, ahighly diverse set of lists was constructed, varyingnot only in set size, old/new status of the probe,and serial position of old probes, but also in thesimilarity structure of the lists.

Because the goal was to predict performance atthe individual-subject level, three subjects were eachtested for approximately 20 one-hour sessions, witheach of the 360 lists presented once per session.As it turned out, each subject showed extremelysimilar patterns of performance, and the fits of theEBRW model yielded similar parameter estimatesfor the three subjects. Therefore, for simplicity, andto reduce noise in the data, we report the resultsfrom the analysis of the averaged subject data.

In the top panels of Figure 7.7 we reportsummary results from the experiment. The top-right panel reports the observed mean RTs plottedas a function of: (a) set size, (b) whether the probe


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was old or new (i.e., a lure), and (c) the lag withwhich old probes appeared in the memory set. Forold probes, there was a big effect of lag: In general,the more recently a probe appeared on the study list,the shorter was the mean RT. Indeed, once one takeslag into account, there is little remaining effect of setsize on the RTs for the old probes. That is, as can beseen, the different set size functions are nearly over-lapping. The main exception is a persistent primacyeffect, in which the mean RT for the item at thelongest lag for each set size is “pulled down.” (Theitem at the longest lag occupies the first serial posi-tion of the list.) By contrast, for the lures, there is abig effect of set size, with longer mean RTs as set sizeincreases. The mean proportions of errors for thedifferent types of lists, shown in the top-left panel ofFigure 7.7, mirror the mean RT data just described.

The goal of the EBRW modeling, however,was not simply to account for these summarytrends. Instead, the goal was to predict the choiceprobabilities and mean RTs observed for eachof the individual lists. Because there were 360unique lists in the experiment, this goal entailedsimultaneously predicting 360 choice probabilitiesand 360 mean RTs. The results of that model-fittinggoal are shown in the top and bottom panels of

Figure 7.8. The top panel plots, for each individuallist, the observed probability that the subjectsjudged the probe to be “old” against the predictedprobability from the model. The bottom panel doesthe same for the mean RTs. Although there are afew outliers in the plots, overall the model achievesa good fit to both data sets, accounting for 96.5%of the variance in the choice probabilities and for83.4% of the variance in the mean RTs.

The summary-trend predictions that result fromthese global fits are shown in the bottom panelsof Figure 7.7. It is evident from inspection thatthe EBRW does a good job of capturing thesesummary results. For the old probes, it predictsthe big effect of lag on the mean RTs, the nearlyoverlapping set-size functions, and the facilitationin RT with primacy. Likewise, it predicts with goodquantitative accuracy the big effect of set size on thelure RTs. The error-proportion data (left panels ofFigure 7.7) are also well predicted, with the mainexception that a primacy effect was predicted butnot observed for the size-2 lists.

The explanation of these results in terms ofthe EBRW model is straightforward. Accordingto the best-fitting parameters from the model (seeNosofsky et al., 2011, Table 2), more recently


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presented exemplars had greater memory strengthsand sensitivities than did less recently presentedexemplars. From a psychological perspective, thispattern seems highly plausible. For example,presumably, the more recently an exemplar waspresented, the greater should be its strength inmemory. Thus, if an old test probe matches therecently presented exemplar, it will give rise togreater overall activation, leading to faster meanold RTs. In the case of a lure, as set size increases,the overall summed activation yielded by the lurewill also tend to increase. This pattern arisesboth because a greater number of exemplars willcontribute to the sum, and because the greater theset size, the higher is the probability that a leastone exemplar from the memory set will be highlysimilar to the lure. As summed activation yieldedby the lures increases, the probability that therandom walk takes correct steps toward the –NEWthreshold decreases, so mean RTs for the lures getlonger.

Beyond accounting well for these summarytrends, inspection of the detailed scatterplots inFigure 7.8 reveals that the model accounts forfine-grained changes in choice probabilities andmean RTs depending on the fine-grained similaritystructure of the lists. For example, consider thechoice-probability plot (Figure 7.8, top panel) andthe Lure-Size-4 items (open diamonds). Whereasperformance for those items is summarized by asingle point on the summary-trend figure (Figure7.7), the full scatterplot reveals extreme variabilityin results across different tokens of the Lure-Size-4 lists. In some cases the false-alarm rates associatedwith these lists are very low, in other cases moderate,and in still other cases the false-alarm rates exceedthe hit rates associated with old lists. The EBRWcaptures well this variability in false-alarm rates. Insome cases, the lure might not be similar to any ofthe memory-set items, resulting in a low false-alarmrate; whereas in other cases the lure might be highlysimilar to some of the memory-set items, resultingin a high false-alarm rate.

The application reviewed earlier involved aversion of a short-term probe-recognition paradigmthat used confusable stimuli embedded in acontinuous-dimension space. However, the EBRWmodel has also been applied successfully to morestandard versions of such paradigms that involveeasy-to-discriminate stimuli such as alphanumericcharacters. In those applications, instead of adopt-ing MDS approaches, a highly simplified modelof similarity is used: The similarity of a probe to

itself is set equal to one, whereas the similaritybetween a probe and any nonmatching item isset equal to a free parameter s. Not only hasthis simple version of the EBRW model capturedmany of the classic patterns of results involvingmean RTs in such paradigms, it also accountssuccessfully for the detailed RT-distribution datathat have been observed (Nosofsky et al., 2011;Donkin & Nosofsky, 2012a,b).

Furthermore, applications of the EBRW modelto both continuous-similarity and discrete versionsof the probe-recognition paradigm have led tothe discovery of an interesting regularity involvingmemory strength. As noted earlier, in the initialtests of the model, separate-memory strengthparameters were estimated corresponding to eachindividual lag on the study list. It turns out,however, that the estimated memory strengthsfollow almost a perfect power function of this lag.For example, in an experiment reported by Donkinand Nosofsky (2012a), participants studied 12-itemlists consisting of either letters or words, followed bya test probe. Separate RT-distribution data for hitsand misses for positive probes were collected at eachstudy lag. (RT-distribution data for false alarms andcorrect rejections for negative probes were collectedas well.) The EBRW model provided an excellentquantitative account of this complete set of detailedRT-distribution and choice-probability data.6 Thediscovery that resulted from the application of themodel is illustrated graphically in Figure 7.9. Thefigure plots, for each of four individual participantswho were tested, the estimated memory-strengthparameters against lag. As shown in the figure, themagnitudes of the memory strengths are extremelywell captured by a simple power function. Inter-estingly, other researchers have previously reportedthat a variety of empirical forgetting curves arewell described as power functions (e.g., Anderson& Schooler, 1991; Wickelgren, 1974; Wixted &Ebbesen, 1991). For example, Wixted and Ebbesen(1991) reported that diverse measures of forgetting,including proportion correct of free recall of wordlists, recognition judgments of faces, and savingsin relearning lists of nonsense syllables, were welldescribed as power functions of the retention inter-val. Wixted (2004) considered a variety of possiblereasons for the emergence of these empirical power-function relations and concluded that the bestexplanation was that the strength of the mem-ory traces themselves may exhibit power-functiondecay. The model-based results from Donkinand Nosofsky (2012a,b) lend support to Wixted’s


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Science, 23, 625–634. Adapted with permission of SAGE Publications.

suggestion and now motivate the new researchgoal of unpacking the detailed psychological andneurological mechanisms that give rise to thisdiscovered power law of memory strength.

Conclusions and New Research GoalsIn sum, the EBRW is an important candidate

model for explaining how the process of catego-rization unfolds over time. The model combinesassumptions involving exemplar-based categoryrepresentation and processes of evidence accumu-lation within a unified framework to account forcategorization and recognition choice-probabilityand RT data. As reviewed in this chapter, it accountssuccessfully for a wide variety of fundamental effectsin these domains including effects of similarity,distance-from-boundary, familiarity, probabilisticfeedback, practice, expertise, set size, and lag.Although we were able to sample only a limitednumber of example applications in this singlechapter, we should clarify that the exemplar modelhas been applied in a wide variety of stimulusdomains and to varied category and study-liststructures. The stimulus domains include colors,dot patterns, multidimensional cartoon drawings,geometric forms, schematic faces, photographs of

faces, alphanumeric characters, words, and picturesof real-world objects. The category structuresinclude small collections of continuous-dimensionstimuli separated by boundaries of varying de-grees of complexity, normally distributed categorystructures, high-dimensional stimuli composed ofdiscrete dimensions, and categories generated fromstatistical distortions of prototypes (e.g., see Richler& Palmeri, 2014). Furthermore, growing neuralevidence ranging from single-unit records to func-tional brain imaging supports a number of theprocessing assumptions embodied in models likeEBRW (see Box 1). Finally, as illustrated in ourchapter, a key theme of the theoretical approach isthat, despite the dramatically different task goals,the processes of categorization and old-new recog-nition may be closely related (but see Box 2 for dis-cussion of a major theoretical debate regarding thisissue in the cognitive-neuroscience literature). Alikely reason for the model’s success is that it buildson the strengths of classic previous approaches forunderstanding processes of choice and similarity,the development of automaticity, and evidenceaccumulation in decision-making and memory.

Beyond accounting for categorization and recog-nition, we believe that the EBRW model can serve


Box 1 Neural Evidence SupportsMechanistic Assumptions in EBRW.EBRW proposes many mechanistic assump-tions, such as exemplar representations, atten-tion weights along relevant dimensions, andaccumulation of perceptual evidence. In mostmodeling, these are supported by evaluatingpredictions of behavioral data like accuracy andRTs. But we can now evaluate particular mech-anistic assumptions using relevant neural datafrom brain regions hypothesized to instantiatethose mechanisms.

For example, Mack, Preston and Love,(2013; see also Palmeri, 2014) turned topatterns of brain activity measured with fMRIto evaluate whether the brain represents cat-egories using exemplars or prototypes. Sub-jects learned to classify objects into oneof two categories and then in the scannerwere tested on training and transfer objectswithout feedback (Medin & Schaffer, 1978).Typical fMRI analyses would correlate brainactivity with stimuli or responses, for exam-ple highlighting regions that modulate withcategorization difficulty. Instead, Mack andcolleagues first fitted exemplar and prototypemodels to individual subjects’ categorizationresponses; despite the fact that these modelsmade fairly similar behavioral predictions, theydiffered in patterns of summed similarity totheir respective exemplar or prototype repre-sentations. Mack and colleagues showed thatpatterns of individual subjects’ brain activ-ity were more consistent with patterns ofsummed similarity predicted by an exemplarmodel than those predicted by a prototypemodel. According to exemplar models, learn-ing to categorize objects can cause selectiveattention to relevant psychological dimensions,stretching psychological space to better allowsubjects to discriminate between members ofcontrasting categories. Neurophysiology andfMRI have suggested that category-relevantdimensions can be emphasized in visual cortex.After monkeys learned to categorize multidi-mensional objects, neurons in inferotemporalcortex were more sensitive to variations alonga relevant dimension than an irrelevant dimen-sion (De Baene, Ons, Wagemans, & Vogels,2008; Sigala and Logothetis 2002; see alsoGauthier & Palmeri, 2002). Similarly, afterpeople learned object categories, psychological

stretching of relevant dimensions was accompa-nied by neural stretching of relevant dimensionsmeasured by fMRI (Folstein, Palmeri, &Gauthier, 2013; see also Folstein, Gauthier, &Palmeri, 2012).

Finally, mathematical psychology and systemsneuroscience have converged on accumulationof perceptual evidence as a general theoret-ical framework to explain the time courseof decision making (see Palmeri, Schall, &Logan, this volume). Some neurons showdynamics predicted by accumulator models,other neurons show activity consistent withencoded perceptual evidence to be accumulatedover time, and an ensemble of neurons predictsthe time course of decisions made by awakebehaving monkeys (e.g., Purcell, Schall, Logan,& Palmeri, 2012; Zandbelt, Purcell, Palmeri,Logan, Schall, 2014).

Box 2 The Exemplar Model Accountsfor Dissociations BetweenCategorization and RecognitionDemonstrated in theCognitive-Neuroscience Literature.Interestingly, in contrast to the theme empha-sized in this chapter, the prevailing view in thecognitive neuroscience literature is that separatecognitive/neural systems mediate categorizationand recognition (Smith, 2008). The mainsource of evidence involves the demonstrationof intriguing dissociations between categoriza-tion and recognition. For example, studieshave demonstrated that amnesics with poorrecognition memory perform at normal levelsin categorization tasks involving the sametypes of stimuli (e.g., Knowlton & Squire,1993). Nevertheless, formal modeling analyseshave indicated that even these dissociationsare consistent with the predictions from theexemplar model (e.g., Nosofsky, Denton,Zaki, Murphy-Knudson, & Unverzagt 2012;Nosofsky & Zaki, 1998; Palmeri & Flan-ery, 1999, 2002; Zaki, Nosofsky, Jessup,& Unverzagt, 2003). The general approachin the modeling was to assume that am-nesics have reduced ability to discriminateamong distinct exemplars in memory. Thisreduced discriminability is particularly detri-mental to old-new recognition, which may


Box 2 Continuedrequire the observer to make fine-grained dis-tinctions between old versus new items. How-ever, the reduced discriminability is not verydetrimental in typical tasks of categorization,which may require only gross-level assessmentsof similarity to be made. A more direct chal-lenge to the exemplar-model hypothesis comesfrom brain-imaging studies that show that dis-tinct brain regions are activated when observersengage in recognition vs. categorization tasks(Reber et al., 1998a,b). Exemplar theorists haveresponded, however, by providing evidencethat these brain-imaging dissociations may notreflect the operation of separate neural systemsdevoted to categorization versus recognition perse. Instead, the brain-imaging dissociations mayreflect changes in stimulus-encoding strategiesacross task situations (Gureckis, James, &Nosofsky, 2011), differences in the precisestimuli that are tested (Nosofsky, Little, &James, 2012; Reber et al., 2003), as wellas adaptive changes in parameter settingsthat allow observers to meet the competingtask goals of categorization versus recognition(Nosofsky et al., 2012).

as a useful analytic device for assessing humanperformance. For example, note that Ratcliff ’s(1978) diffusion model has been applied to analyzechoice behavior in various special populations,including elderly adults, sleep-deprived subjects,and so forth (see Chapter 2 of this volume).The model-based analyses provide a deeper under-standing of the locus of the cognitive/perceptualdeficits in such populations by tracing them tochanges in diffusion-model drift rates, response-threshold settings, or residual times. The EBRWmodel has potential to reveal even more fine-grainedinformation along these lines. For example, inthat model, the random-walk step probabilities(i.e., drift rates) emerge from cognitive/perceptualfactors such as overall sensitivity, attention-weightsettings, and memory strengths of stored exemplars,each of which can be measured by fitting themodel to data obtained in suitable categorizationand recognition paradigms. Although exemplarmodels have been applied to help interpret thebehavior of amnesic subjects and patients withmild memory-based cognitive impairment (seeBox 2), we have only scratched the surface of suchpotential applications to many more clinical groups.

Finally, an important theme in the categorizationliterature is that there may be multiple systems ofcategorization (e.g., Ashby, Alfonso-Reese, Turken,& Waldron, 1998; Erickson & Kruschke, 1998;Johansen & Palmeri, 2002; Nosofsky, Palmeri, &McKinley, 1994). A classic idea, for example, is thatmany categories may be represented and processedby forming and evaluating logical rules. In somemodern work that pursues this avenue, researchershave considered the RT predictions from logical-rule models of classification. Furthermore, suchapproaches have been used to develop sharp con-trasts between the predictions of the ERBW modeland rule-based forms of category representation andprocessing (Fific, Little, & Nosofsky, 2010; Lafond,Lacouture, & Cohen, 2009; Little, Nosofsky,& Denton, 2011). In domains involving highlyseparable-dimension stimuli in which the categorystructures can be described in terms of exceedinglysimple logical rules, evidence has been mountingthat the logical-rule models provide better accountsof the detailed patterns of RT data than doesthe EBRW model. An important target for futureresearch is to develop a deeper understanding ofthese multiple forms of categorization, to learnabout the experimental factors that promote theuse of each strategy, and to explain the manner inwhich exemplar-based and rule-based systems mayinteract.

AcknowledgmentsThis work was partially supported by NSF grants

SMA-1041755 and SBE-1257098 and by AFOSRgrant FA9550-14-1-0307.

Notes1. In the context of the GCM, the parameter γ is referred to

as a response-scaling parameter. When γ =1, observers respond by“probability-matching” to the relative summed similarities. As γgrows greater than 1, observers respond more deterministicallywith the category that yields the greater summed similarity(Ashby & Maddox, 1993; McKinley & Nosofsky, 1995).

2. The version of the EBRW model described in this chapteris applicable to “integral-dimension” stimuli, which are encodedand perceived holistically. A common example of suchintegral-dimension stimuli are colors varying in hue, brightness,and saturation. Because there has been extensive previousscaling work indicating that similarity relations among thesestimuli are extremely well described in terms of thesedimensions, we often use these stimuli in our tests of the EBRWmodel. An extended version of the EBRW model has also beendeveloped that is applicable to separable-dimension stimuli(Cohen & Nosofsky, 2003). In this version, rather thanencoding stimuli in holistic fashion, the encoding of individualstimulus dimensions is a stochastic process, and similarity


relations between a test item and the stored exemplars changedynamically during the time course of processing (see alsoLamberts, 2000).

3. Because accuracies were near ceiling in the presentexperiment, we focused our analysis primarily on the patterns ofRT data. However, in a follow-up study, Nosofsky andAlfonso-Reese (1999) tested conditions that allowedexamination of how both speed and accuracy changed duringthe early stages of learning. By including thebackground-activation parameter b in its arsenal, the EBRWmodel provided good quantitative fits to not only the speed-upin mean RTs, but to the improvements in choice-accuracy dataas well. (As noted by Nosofsky and Palmeri, 1997a, p. 291, withb=0, the EBRW model does not predict changes in responseaccuracy.)

4. The original EBRW model (Nosofsky & Palmeri, 1997a)applies to two-alternative forced- choice responses. There is asingle accumulator whose value increases or decreases asevidence accumulates in the random walk until an upper orlower response threshold is reached. Numerosity judgments inPalmeri (1997) permitted six possible responses, so EBRW wasextended to allow multiple alternatives. Each responsealternative was associated with its own counter, so with sixnumerosity responses there were six counters. Whenever anexemplar was retrieved with the label associated with a particularcounter, the value of that counter was incremented. A responsewas made whenever the value of one of the counters exceeded allthe rest by some relative amount. With only two alternatives,this multiple counter model with a relative threshold responserule generally mimics a standard random-walk model with oneaccumulator with a positive and negative threshold.

5. There is a long history of debate, too extensive to bereviewed in this chapter, between the proponents of exemplarand prototype models. For examples of recent research that hasargued in favor of the prototype view, see Minda and Smith(2001), Smith and Minda (1998, 2000) and Smith (2002). Forexamples from the exemplar perspective, seeNosofsky (2000), Nosofsky and Zaki (2002), Palmeri andFlanery (2002), and Zaki and Nosofsky (2007).

6. The version of the exemplar-recognition model reportedby Donkin and Nosofsky (2012a) assumed a linear-ballisticaccumulation process (Brown & Heathcote, 2008) instead of arandom-walk accumulation process. However, the sameevidence for a power-law relation between memory strength andlag was obtained regardless of the specific accumulation processthat was assumed. We should note as well that, in fittingcomplete RT distributions for correct and error responses, suchas occurred in the Donkin and Nosofsky (2012a) experiment,the exemplar model makes provision for drift-rate variabilityand response-threshold variability across trials (e.g, Donkin &Nosofsky, 2012a; Nosofsky & Stanton, 2006), in a manneranalogous to the approach used in Ratcliff ’s diffusion model(e.g, Ratcliff, Van Zandt, & McKoon, 1999).

GlossaryAttention-weight parameters: a set of parameters in theGCM and EBRW models that describe the extent to whicheach dimension is weighted when computing distancesamong objects.

Automaticity: ability to perform some task at a satisfactorylevel without requiring conscious attention or effort andwithout limits in capacity.

Background element: a hypothetical construct in theGCM and EBRW models that describes initial backgroundnoise in people’s memories for members of alternativecategories.

Basic-level of categorization: an intermediate level of acategory hierarchy that is hypothesized to lead to privilegedforms of cognitive processing.

Categorization: process in which observers classify distinctobjects into groups.

Criterion element: a hypothetical entity in the EBRWrecognition model. Retrieval of criterion elements leads therandom walk to step in the direction of the NEW responsethreshold. Biases in the random-walk step probabilities aredetermined by the strength of the criterion elements.

Decision-boundary model: model of categorization thatassumes that people form boundaries to divide a stimulusspace into category-response regions.

Exemplar model: model of categorization that assumesthat observers store individual examples of categories inmemory.

Exemplar-based random-walk (EBRW) model: an exten-sion of the generalized context model that explains how theprocesses of categorization and recognition unfold over time.Exemplars stored in memory “race” to be retrieved, and theretrieved exemplars drive a random-walk decision-makingprocess.

Exponential distribution: a probability distribution thatdescribes the time between events, in which the eventsoccur continuously and independently at a constant averagerate.

Generalized context model (GCM): a member of the classof exemplar models. In the GCM, exemplars are representedas points in a multidimensional psychological space, andsimilarity is a decreasing function of distance in the space.

Integral-dimension stimuli: stimuli composed of individ-ual dimensions that combine into unitary, integral wholes.

Memory strength parameters: parameters in the GCMand EBRW models that describe the strength with whichthe exemplars are stored in memory.

Minkowski power model: a model for computing dis-tances between points in a space.

Multidimensional scaling: a modeling technique for rep-resenting similarity relations among objects. The objects arerepresented as points in a multidimensional psychologicalspace and similarity is a decreasing function of the distancebetween the points in the space.

Prototype model: model of categorization that assumesthat observers represent categories by forming a summaryrepresentation, usually assumed to be the central-tendencyof the category distribution.


GlossaryPrototype-based random-walk (PBRW) model: a modelthat is analogous in all respects to the EBRW model, exceptthat the category representation corresponds to the centraltendency of each category distribution rather than to theindividual exemplars.

Random walk: a mathematical model that describes a pathof outcomes consisting of a sequence of random steps.

Response thresholds: parameters in evidence-accumulation models that determine how muchevidence is required before a response is initiated.

Recognition memory: process in which observers decidewhether objects are “old” (previously experienced) or“new.”

Response-scaling parameter: a parameter in the GCMthat describes the extent to which observers respond usingprobabilistic versus deterministic response rules.

RT-distance hypothesis: hypothesis that categorizationRT is a decreasing function of the distance of a stimulusfrom the decision boundary.

Rule-plus-exception model: model of categorization thatassumes that people classify objects by forming simplelogical rules and remembering occasional exceptions tothose rules.

Sensitivity parameter: a parameter in the GCMand EBRW models that describes overall discriminabilityamong distinct items in the multidimensional psychologicalspace.

Short-term probe-recognition task: task in whichobservers are presented with a short list of to-be-remembered items followed by a test probe. The observersjudge as rapidly as possible, while trying to minimizeerrors, whether the probe is old or new.

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