Mixture Models of Categorization

Many currently popular models of categorization are either strictly parametric (e.g., prototype models, decision bound models) or strictly nonparametric (e.g., exemplar models) (F. G. Ashby & L. A. Alfonso-Reese, 1995, Journal of Mathematical Psychology, 39, 216-233). In this article, a family of semiparametric classifiers is investigated where categories are represented by a finite mixture distribution. The advantage of these mixture models of categorization is that they contain several parametric models and nonparametric models as a special case. Specifically, it is shown that both decision bound models (F. G. Ashby & W. T. Maddox, 1992, Journal of Experimental Psychology: Human Perception and Performance, 16, 598-612; 1993, Journal of Mathematical Psychology, 37, 372-400) and the generalized context model (R. M. Nosofsky, 1986, Journal of Experimental Psychology: General, 115, 39-57) can be interpreted as two extreme cases of a common mixture model. Furthermore, many other (semiparametric) models of categorization can be derived from the same generic mixture framework. In this article, several examples are discussed and a parameter estimation procedure for fitting these models is outlined. To illustrate the approach, several specific models are fitted to a data set collected by S. C. McKinley and R. M. Nosofsky (1995, Journal of Experimental Psychology: Human Perception and Performance, 21, 128-148). The results suggest that semi-parametric models are a promising alternative for future model development.     

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.     

知道与不知道要学多少：类别学习中样例量的预期作用

采用“5/4模型”类别结构探讨了类别学习中样例量的预期作用。设置了两种学习条件(“知道样例量”和“不知道样例量”), 分别探讨两种学习条件下的学习效率、学习策略以及所形成的类别表征。106名大学生参加了实验, 结果表明：在类别学习中, 样例量的预期作用显著, 知道样例量组的学习效率高于不知道样例量组; 样例量的预期作用对类别学习效率的影响是通过影响学习过程中使用的策略来实现的; 样例量的预期作用不影响两种学习条件的学习后形成的类别表征, 且两种学习条件的被试自始至终表现出样例学习的表征模式。     

Overestimation of base-rate differences in complex perceptual categories

The optimality of multidimensional perceptual categorization performance was examined for several base-rate ratios, for both integral and separable dimension stimuli, and for complex category structures. In all cases, the optimal decision bound was highly nonlinear. Observers completed several experimental sessions, and all analyses were performed at the single-observer level using a series of nested models derived from decision-bound theory (Maddox, 1995; Maddox & Ashby, 1993). In every condition, all observers were found to be sensitive to the base-rate manipulations, but the majority of observers appeared to overestimate the base-rate difference. These findings converge with those for cases in which the optimal decision bound was linear (Maddox, 1995) and suggest that base-rates are learned in a similar fashion regardless of the complexity of the optimal decision bound. Possible explanations for the consistent overestimate of the base-rate difference are discussed. Several continuous-valued analogues of Kruschke’s (1996) theory of base-rate learning with discrete-valued stimuli were tested. These models found some support, but in all cases were outperformed by a version of decision-bound theory that assumed accurate knowledge of the category structure and an overestimate of the base-rate difference.     

Journey to the center of the category: the dissociation in amnesia between categorization and recognition

The authors' theoretical analysis of the dissociation in amnesia between categorization and recognition suggests these conclusions: (a) Comparing to-be-categorized items to a category center or prototype produces strong prototype advantages and steep typicality gradients, whereas comparing to-be-categorized items to the training exemplars that surround the prototype produces weak prototype advantages and flat typicality gradients; (b) participants often show the former pattern, suggesting their use of prototypes; (c) exemplar models account poorly for these categorization data, but prototype models account well for them; and (d) the recognition data suggest that controls use a single-comparison exemplar-memorization process more powerfully than amnesics. By pairing categorization based in prototypes with recognition based in exemplar memorization, the authors support and extend other recent accounts of cognitive performance that intermix prototypes and exemplars, and the authors reinforce traditional interpretations of the categorization-recognition dissociation in amnesia.     

How many exemplars are used? Explorations with the Rex Leopold I model

The goal of this research is to test the hypothesis that a category is not necessarily represented by all observed exemplars, but by a reduced subset of these exemplars. To test this hypothesis, we made use of a study reported by Nosofsky, Clark, and Shin (1989), and replicated their Experiment 1 in order to gather individual-participant data. Both a full exemplar model and a reduced exemplar model were fit to the data. In general, the fits of the reduced exemplar model were superior to those of the full exemplar model. The results suggest that only a subset of exemplars may be sufficient for category representation.     

Memory storage and retrieval processes in category learning

The detailed course of learning is studied for categorization tasks defined by independent or contingent probability distributions over the features of category exemplars. College-age subjects viewed sequences of bar charts that simulated symptom patterns and responded to each chart with a recognition and a categorization judgment. Fuzzy, probabilistically defined categories were learned relatively rapidly when individual features were correlated with category assignment, more slowly when only patterns carried category information. Limits of performance were suboptimal, evidently because of capacity limitations on judgmental processes as well as limitations on memory. Categorization proved systematically related to feature and exemplar probabilities, under different circumstances, and to similarity among exemplars of categories. Unique retrieval cues for exemplar patterns facilitated recognition but entered into categorization only at retention intervals within the range of short-term memory. The findings are interpreted within the framework of a general array model that yields both exemplar-similarity and feature-frequency models as special cases and provides quantitative accounts of the course of learning in each of the categorization tasks studied.     

Comparing exemplar-retrieval and decision-bound models of speeded perceptual classification

The authors compared the exemplar-based random-walk (EBRW) model of Nosofsky and Palmeri (1997) and the decision-bound model (DBM) of Ashby and Maddox (1994; Maddox & Ashby, 1996) on their ability to predict performance in Garner’s (1974) speeded classification tasks. A key question was the extent to which the models could predict facilitation in the correlated task and interference in the filtering task, in situations involving integral-dimension stimuli. To obtain rigorous constraints for model evaluation, the goal was to fit the detailed structure of the response time (RT) distribution data associated with each individual stimulus in each task. Both models yielded reasonably good global quantitative fits to the RT distribution and accuracy data. However, the DBM failed to properly characterize the interference effects in the filtering task. Apparently, a fundamental limitation of the DBM is that it predicts that the fastest RTs in the filtering task should be faster than the fastest RTs in the control task, whereas the opposite pattern was observed in our data.     

6岁儿童的类别学习能力:类别表征、注意和分类策略

探讨了6岁儿童的类别学习能力、类别表征和分类策略。62名儿童参加了实验,实验1采用了"5/4模型"类别结构,实验2采用了"3/3类别结构"。结果发现:6岁儿童已经具备了一定的类别学习能力;相对于原型表征,6岁儿童更倾向于进行样例表征;6岁儿童在注意上具有定位在高典型性特征维度上的能力,但不具备定位在区分性特征维度上的能力;在类别学习策略上主要采用单维度分类策略和规则加例外的分类策略。     

Array models for category learning

A family of models for category learning is developed, all members being based on a common memory array but differing in memory access and decision processes. Within this framework, fully controlled comparisons of exemplar-similarity, feature-frequency, and prototype models reveal isomorphism between models of different types under some conditions but empirically testable differences under others. It is shown that current exemplar-memory models, in which categorization judgments are based on similarities of perceived and remembered category exemplars, can be interpreted as generalized likelihood models but can be modified in a simple way to yield pure similarity models. Distance-based exemplar models are formulated that provide means of investigating issues concerning deterministic versus probabilistic decision rules and links between categorization and properties of perceptual dimensions. Other theoretical issues discussed include aspects of similarity, the role of memory storage versus computation in category judgments, and the limits of applicability of array models.     

On the dangers of averaging across observers when comparing decision bound models and generalized context models of categorization

Averaging across observers is common in psychological research. Often, averaging reduces the measurement error and, thus, does not affect the inference drawn about the behavior of individuals. However, in other situations, averaging alters the structure of the data qualitatively, leading to an incorrect inference about the behavior of individuals. In this research, the influence of averaging across observers on the fits of decision bound models (Ashby, 1992a) and generalized context models (GCM; Nosofsky, 1986) was investigated through Monte Carlo simulation of a variety of categorization conditions, perceptual representations, and individual difference assumptions and in an experiment. The results suggest that (1) averaging has little effect when the GCM is the correct model, (2) averaging often improves the fit of the GCM and worsens the fit of the decision bound model when the decision bound model is the correct model, (3) the GCM is quite flexible and, under many conditions, can mimic the predictions of the decision bound model, whereas the decision bound model is generally unable to mimic the predictions of the GCM, (4) the validity of the decision bound model's perceptual representation assumption can have a large effect on the inference drawn about the form of the decision bound, and (5) the experiment supported the claim that averaging improves the fit of the GCM. These results underscore the importance of performing single-observer analysis if one is interested in understanding the categorization performance of individuals.     

Comparisons between exemplar similarity and mixed prototype models using a linearly separable category structure

Nosofsky and Zaki (2002) found that an exemplar similarity model provided better accounts of individual subject classification and generalization performance than did a mixed prototype model proposed by Smith and Minda (1998; Minda & Smith, 2001). However, these previous tests used a nonlinearly separable category structure. In the present work, the authors extend the previous findings by demonstrating a superiority for the exemplar generalization model over the mixed prototype model in a case involving a linearly separable structure. Because this structure has numerous features that Minda and Smith argued should be conducive to prototype-based processing, the results pose a significant challenge to the mixed prototype view.     

On the nature of implicit categorization

Current categorization models disagree about whether people make a priori assumptions about the structure of unfamiliar categories. Data from two experiments provided strong evidence that people do not make such assumptions. These results rule out prototype models and many decision bound models of categorization. We review previously published neuropsychological results that favor the assumption that category learning relies on a procedural-memory-based system, rather than on an instance- based system (as is assumed by exemplar models). On the basis of these results, a new categorylearning model is proposed that makes no a priori assumptions about category structure and that relies on procedural learning and memory.     

Single versus multiple systems of category learning: Reply to Nosofsky and Kruschke (2002)

Nosofsky and Kruschke (2002) argued that the singlesystem ALCOVE model (Kruschke, 1992) can account for the dual-task category learning data reported by Waldron and Ashby (2001). In our reply, we argue that Nosofsky and Kruschke overstated the ability of ALCOVE to account for the Waldron and Ashby results. In fact, ALCOVE has difficulty with these data, and we show that the only versions of ALCOVE that actually fit the Waldron and Ashby accuracy data make incorrect predictions about other previously unreported features of that experiment. We also show that the dual-system COVIS model (Ashby, Alfonso-Reese, Turken, & Waldron, 1998) naturally predicts these results.     

The effect of category variability in perceptual categorization

Exemplar and distributional accounts of categorization make differing predictions for the classification of a critical exemplar precisely halfway between the nearest exemplars of 2 categories differing in variability. Under standard conditions of sequential presentation, the critical exemplar was classified into the most similar, least variable category, consistent with an exemplar account. However, if the difference in variability is made more salient, then the same exemplar is classified into the more variable, most likely category, consistent with a distributional account. This suggests that participants may be strategic in their use of either strategy. However, when the relative variability of 2 categories was manipulated, participants showed changes in the classification of intermediate exemplars that neither approach could account for.     

Comparing categorization models

Four experiments are presented that competitively test rule- and exemplar-based models of human categorization behavior. Participants classified stimuli that varied on a unidimensional axis into 2 categories. The stimuli did not consistently belong to a category; instead, they were probabilistically assigned. By manipulating these assignment probabilities, it was possible to produce stimuli for which exemplar- and rule-based explanations made qualitatively different predictions. F. G. Ashby and J. T. Townsend's (1986) rule-based general recognition theory provided a better account of the data than R. M. Nosofsky's (1986) exemplar-based generalized context model in conditions in which the to-be-classified stimuli were relatively confusable. However, generalized context model provided a better account when the stimuli were relatively few and distinct. These findings are consistent with multiple process accounts of categorization and demonstrate that stimulus confusion is a determining factor as 10 which process mediates categorization.     

Categorization of multidimensional geometrical figures by chickens (Gallus gallus f. domestica): fit of basic assumptions from exemplar, feature and prototype theory

Five hens, experienced in discrimination of two categories of multidimensional geometrical figures presented in fixed pairs in a simultaneous discrimination, were tested with familiar figures arranged as new pairs to assess the dependence of categorization performance on learned relational or configural cues. Test performance did not differ from training: relational or configural cues still influenced discrimination performance. It was suggested that – in accordance with exemplar theories – this influence depended on differences between pairs of probe exemplars that facilitate retrieval of learned category members. To test whether exemplar, feature or prototype theory was most suitable to explain categorization by chickens, the rates of pecking at exemplars were analysed using principal components analysis (PCA). The distribution of the exemplars' component loads on the single component obtained was examined in the light of the conditions dictated by the three types of theories on how representative category exemplars should be. The least constraining theory, i.e. the exemplar theory, was most suitable. Defining factors of classificatory behaviour are discussed with a special emphasis on the characteristics of category-defining stimulus attributes. Accepted after revision: 29 May 2001 Electronic Publication     

Generalization and similarity in exemplar models of categorization: Insights from machine learning

Exemplar theories of categorization depend on similarity for explaining subjects’ ability to generalize to new stimuli. A major criticism of exemplar theories concerns their lack of abstraction mechanisms and thus, seemingly, of generalization ability. Here, we use insights from machine learning to demonstrate that exemplar models can actually generalize very well. Kernel methods in machine learning are akin to exemplar models and are very successful in real-world applications. Their generalization performance depends crucially on the chosen similarity measure. Although similarity plays an important role in describing generalization behavior, it is not the only factor that controls generalization performance. In machine learning, kernel methods are often combined with regularization techniques in order to ensure good generalization. These same techniques are easily incorporated in exemplar models. We show that the generalized context model (Nosofsky, 1986) and ALCOVE (Kruschke, 1992) are closely related to a statistical model called kernel logistic regression. We argue that generalization is central to the enterprise of understanding categorization behavior, and we suggest some ways in which insights from machine learning can offer guidance.     

Measures of similarity in models of categorization

This paper concerns the use of similarities based on geometric distance in models of categorization. Two problematic implications of such similarities are outlined. First, in a comparison between two stimuli, geometric distance implies that matching features are not taken into account. Second, missing features are assumed not to exist. Only nonmatching features enter into calculations of similarity. A new model is constructed that is based on the ALCOVE model (Kruschke, 1992), but it uses a feature-matching similarity measure (see, e.g., Tversky, 1977) rather than a geometric one. It is an on-line model in the sense that both dimensions and exemplars are constructed during the categorization process. The model accounts better than ALCOVE does for data with missing features (Experiments 1 and 2) and at least as well as ALCOVE for a data set without missing features (Nosofsky, Kruschke, & McKinley, 1992). This suggests that, at least for some stimulus materials, similarity in categorization is more akin to a feature-matching procedure than to geometric distance calculation.     

Perceptual fluency can be used as a cue for categorization decisions

Learning in the prototype distortion task is thought to involve perceptual learning in which category members experience an enhanced visual response (Ashby & Maddox. Annual Review of Psychology, 56, 149-178, 2005). This response likely leads to more-efficient processing, which in turn may result in a feeling of perceptual fluency for category members. We examined the perceptual-fluency hypothesis by manipulating fluency independently from category membership. We predicted that when perceptual fluency was induced using subliminal priming, this fluency would be misattributed to category membership and would affect categorization decisions. In a prototype distortion task, the participants were more likely to judge stimuli that were not members of the category as category members when the nonmembers were made perceptually fluent with a matching subliminal prime. This result suggests that perceptual fluency can be used as a cue during some categorization decisions. In addition, the results provided converging evidence that some types of categorization are based on perceptual learning.