Which of the following examples is the least prototypical for the category furniture

2.1.1 Prototype Models

Prototype models assume that for each category people retain in memory a single specific example (the prototype), and that category members in the world fall in a distribution around the prototype (Fried & Holyoak, 1984; Reed, 1972). Categorization is then a function of how similar the category’s prototype is to the object to be categorized. Some prototype models, for example, assume that the similarity of an item to the prototype of a category is an exponentially decaying function of the psychological distance (or squared distance) between them (Nosofsky, 1987, 1992).

To model the prototype view with the Bayesian network model of Figure 6.1, the C variable would take on states representing different possible categories, and the feature values would depend probabilistically on which category the item is a member of (Figure 6.2). The similarity-based prototype models can be mathematically formalized by specifying a Gaussian distribution over feature values for each category, with the mean and variance of each distribution depending on the category. This method is formally identical to assessing the similarity of an item to the prototype using an exponentially decaying squared-distance function, as described above.

3.08.5.2.2 Prototype model

The prototype model has been suggested as a viable alternative to the current Neo-Kraepelinian approach (Cantor, Smith, French, & Mezzich, 1980; Clarkin, Widiger, Frances, Hurt, & Gilmore, 1983; Horowitz, Post, French, Wallis, & Siegelman, 1981; Livesley, 1985a, 1985b). In this system, patients' symptoms are evaluated in terms of their correlation with a standard or prototypical representation of specific disorders. A prototype consists of the most common features of members of a category, and is the standard against which patients are evaluated (Horowitz et al., 1981).

The prototype model differs from the Neo-Kraepelinian model in several ways. First, the prototype model is based on a philosophy of nominalism, in which diagnostic categories represent concepts used by mental health professionals (Blashfield, 1991). Diagnostic groups are not viewed as discrete, but individuals may warrant membership in a category to a greater or lesser degree. The categories are defined by exemplars, or prototypes, and the presentation of features or symptoms in an individual is neither necessary nor sufficient to determine membership in a category. Rather, the prototype model holds that membership in a category is correlated with the number of representative symptoms the patient has. The prototype model suggests that the degree of membership to a category is correlated with the number of features that a member has, so defining features are neither necessary nor sufficient.

Some authors have described the DSM system as a prototype model, primarily because it uses polythetic, as opposed to monothetic, definitions (Clarkin et al., 1983; Widiger & Frances, 1985). Although the DSM does use polythetic definitions, it does not constitute a prototypical model because specific subsets of symptoms are sufficient for making a diagnosis. Prototype and polythetic models allow variability among features within a category, however, they view category definition differently. Prototype models focus on definition by example, polythetic models focus on category membership as achieved by the presence of certain features that are sufficient. In a prototype model the level of membership is correlated with number of features present, and features are neither necessary nor sufficient since membership is not an absolute. Furthermore, categories in the prototype model have indistinct boundaries, and the membership decision relies largely on clinician judgment. It is likely that the adoption of this model would result in a decrease in reliability compared to the DSM. However, proponents argue that this model is more reflective of real-world categories in psychopathology (Chapman & Chapman, 1969).

12.3.3 The Prototype Model

Calibration produced a tentative prototype model—tentative because validation testing may reveal structural deficiencies that require revision of the model. A series of simulation tests were performed, and several modifications were required. Equations of the current prototype are as follows:

X 2 = 0.0178368 X 3 + 0.0178368 X 4

X 4 = X 1 − X 2 + X 3 − X 5 − X 6

X 5 = 0.000200 X 3 + 0.000200 X 4

X 6 = 0.0892857 X 3 + 0.0892857 X 4

X 7 = 0.0158696 X 9 + 0.0158696 X 10

X 8 = 0.001587 X 9 + 0.001587 X 10

X 10 = 1.75 X 6 + X 7 − X 8 + X 9 − X 11

X 12 = X 13 − X 14 + X 15

X 13 = 1.3698630 X 4 − 0.3478261 X 7 + 0.3478261 X 10

X 15 = 64.0 − 1.3698630 X 4 − 0.3478261 X 10

X3={13.393984initially,X4of previous ΔtthereafterX9={ 91.757840initially,X10 of previous ΔtthereafterX14={13.736306initially,X15of previous Δtthereafter

Theories of categorisation

One way of considering knowledge systems is as formal mechanisms for classifying and categorising objects. Graphically, a typical ontology resembles a hierarchical taxonomy—though, technically, it is a directed acyclic graph, meaning that concepts can have more than a single ‘parent’ as well as multiple ‘siblings’ and ‘children’. (Ontologies also can support other sorts of conceptual relations, but the relationship of subsumption is axiomatised into the semantics of the OWL directly, as are several other relations.) In such systems, concept application relies on objects meeting necessary and sufficient conditions for class membership. This general model accords well with the broad tradition of category application stretching back to Aristotle. However, ontologies are intended to be machine-oriented representations of conceptualisations, with only an analogical relation to mental cognitive models. What, then, can be gleaned from contemporary theories of categorisation?

Since the 1960s alternative models have been proposed for how mental concepts are organised and applied. Like ontologies, semantic networks, pioneered by Quillian (1967), model cognitive conceptual networks as directed graphs, with concepts connected by one-way associative links. Unlike ontologies these links do not imply any logical (or other) kind of relation between the concepts—only that a general association exists. Semantic networks were adapted for early knowledge representation systems, such as frame systems, which utilise the same graphic structure of conceptual nodes and links: ‘We can think of a frame as a network of nodes and relations’ (Minsky 1974). Minsky also explicitly notes the similarity between frame systems and Kuhnian paradigms—what results from the construction of a frame system as a viewpoint of a slice of the world. By extension, semantic networks can be viewed as proto-paradigms in the Kuhnian sense, though it is not clear what the limits between one network and another might be—this analogy should not, then, be over-strained.

A feature of semantic networks is the lack of underlying logical formalism. While Minskian frame systems and other analogues in the 1970s were ‘updated’ with formal semantic layers, notably through the development of description logics in the 1980s, according to Minsky the lack of formal apparatus is a ‘feature’ rather than a ‘bug’—imposition of checks on consistency, for example, impose an unrealistic constraint on attempts to represent human kinds of knowledge, precisely because humans are rarely consistent in their use of concepts (Minsky 1974). At best they are required to be consistent across a localised portion of their cognitive semantic network, relevant to a given problem at hand, and the associated concepts and reasoning required to handle it. Similarly the authors of semantic network models note the difficulty in assuming neatly structured graphs model mental conceptual organisation: ‘Dictionary definitions are not very orderly and we doubt that human memory, which is far richer, is even as orderly as a dictionary’ (Collins and Quillian 1969). Semantic networks represent an early—and enduring—model of cognition, which continues to be influential in updated models such as neural networks and parallel distributed processing (Rogers and McClelland 2004). Such networks also exhibit two features of relevance to the theory adopted here: first, the emphasis on structural, connectionist models of cognition—that concepts are not merely accumulated quantitatively as entries in a cognitive dictionary, but are also interconnected, so that the addition of new concepts makes a qualitative difference in how existing concepts are applied; and second, the implied coherence of networks, which suggests concepts are not merely arranged haphazardly but form coherent and explanatory schemes or structures.

In the mid-1970s prototype theory, another cognitive model, was proposed for describing concept use. Building on Wittgenstein’s development of ‘language games’ (Wittgenstein 1967), Rosch (1975) demonstrated through a series of empirical experiments that the process of classifying objects under conceptual labels was generally not undertaken by looking for necessary and sufficient conditions for concept-hood. Rather, concepts are applied based on similarities between a perceived object and a conceptual ‘prototype’—a typical or exemplary instance of a concept. Possession of necessary and sufficient attributes is a weaker indicator for object inclusion within a category than the proximation of the values of particularly salient attributes—markers of family resemblance—to those of the ideal category member. For example, a candidate dog might be classified so by virtue of the proximity of key perceptual attributes to those of an ideal ‘dog’ in the mind of the perceiver—fur, number of legs, size, shape of head, and so on. Applying categories on the basis of family resemblances rather than criterial attributes suggests that, at least in everyday circumstances, concept application is a vague and error-prone affair, guided by fuzzy heuristics rather than strict adherence to definitional conditions. Also, by implication, concept application is part of learning—repeated use of concepts results in prototypes which are more consistent with those used by other concept users. This would suggest a strong normative and consensual dimension to concept use. Finally, Rosch (1975) postulated that there exist ‘basic level semantic categories’, containing concepts most proximate to human experience and cognition. Superordinate categories have less contrastive features, while subordinate categories have less common features—hence basic categories tend to be those with more clearly identifiable prototypical instances, and so tend to be privileged in concept learning and use.

While semantic network and prototype models provide evocative descriptive theories that seem to capture more intuitive features of categorisation, they provide relatively little causal explanation of how particular clusters of concepts come to be organised cognitively. Several new theories were developed in the 1980s with a stronger explanatory emphasis (Komatsu 1992). Medin and Schaffer (1978), for example, propose an exemplar-based ‘context’ theory rival to prototype theory, which eschews the inherent naturalism of ‘basic level’ categorial identification for a more active role of cognition in devising ‘strategies and hypotheses’ when retrieving memorised category exemplar candidates. Concept use, then, involves agents not merely navigating a conceptual hierarchy or observing perceptual family resemblances when they apply concepts; they are also actively formulating theories derived from the present context, and drawing on associative connections between concept candidates and other associated concepts. In this model, concept use involves scientific theorising; in later variants, the model becomes ‘theory theory’ (Medin 1989). As one proponent puts it:

In particular, children develop abstract, coherent systems of entities and rules, particularly causal entities and rules. That is, they develop theories. These theories enable children to make predictions about new evidence, to interpret evidence, and to explain evidence. Children actively experiment with and explore the world, testing the predictions of the theory and gathering relevant evidence. Some counter-evidence to the theory is simply reinterpreted in terms of the theory. Eventually, however, when many predictions of the theory are falsified, the child begins to seek alternative theories. If the alternative does a better job of predicting and explaining the evidence it replaces the existing theory (Gopnik 2003, p. 240).

Empirical research on cognitive development in children (Gopnik 2003) and cross-cultural comparisons of conceptual organisation and preference (Atran et al. 1999; Medin et al. 2006; Ross and Medin 2005) has shown strong support for ‘theory theory’ accounts. Quine’s view of science as ‘self-conscious common sense’ provides a further form of philosophical endorsement to this view.

For the purposes of this study, a strength of the ‘theory theory’ account is its orientation towards conceptual holism and schematism—concepts do not merely relate to objects in the world, according to this view (although assuredly they do this too); they also stand within a dynamic, explanatory apparatus, with other concepts, relations and rules. Moreover theories are used by agents not to explain phenomena to themselves, but also to others; concept use has then a role both in one’s own sense making of the world, and also in how one describes, explains, justifies and communicates with others. In short, concepts are understood as standing not only in relation to objects in the world, as a correspondence theory would have it; they stand in relation to one another, to form at least locally coherent mental explanations; and they also bind together participating users into communities and cultures. The account presented here similarly draws on supplemental coherentist and consensual notions of truth to explain commensurability.

5.4 Observed usage

Usecues and user actions can best be investigated by means of explorative research into user–model/–prototype/–product interaction. The actions that users carry out in operating a new design can be observed (unobtrusively, if desirable and possible). In order to access the situatedness of action, explorative research into product usage should be carried out in as natural a context as possible. For the purposes of design, however, it is often preferable to test a product early in a design process. Simulation of usage situations offers ways to test early. There are many ways and degrees that everyday usage situations can be observed in simulation before a design is finalized. For reasons of observability and confidentiality, users can be asked to try out a product in an environment that can easily be recorded on video. Users can be asked to carry out simulations of real product usage, and they can be asked to try out products at various stages of design. A design can be tested at the stage of a drawing, an early model, an appearance model or a functional prototype (Rooden, 2001). All of these simulations of real usage can be evaluated at appropriate stages of a design process.

2.29.2.2 Evaluations of Prototype and Exemplar Models

There have been many (many) comparisons of prototype and exemplar models (for a review, see Murphy, 2002). The general result is that exemplar models do as well or better than prototype models in most cases (though see Smith and Minda, 1998, 2001). Our view, summarizing over many results, is that the advantage is largely a result of two factors: similarity calculation and selectivity (Ross and Makin, 1999). First, the exemplar models’ multiplicative similarity (compared to the prototype models’ usual assumption of additive similarity) means that the model does not combine features independently but, rather, is sensitive to the relational information among the features (e.g., Medin and Schaffer, 1978). The combination of features is being used beyond their separate contribution to determining classification. Thus, if one encountered small birds that sang and large birds that squawked, a prototype representation would not be sensitive to that particular relational (cooccurrence) information, whereas an exemplar model would. Although it is not a usual assumption, prototype models might also incorporate multiplicative similarity (e.g., Smith and Minda, 1998); this helps the fit, but it does not mimic the predictions of the exemplar model. (Multiplicative similarity is a nonlinear function, so calculating multiplicative similarity on the mean (prototype) is not the same as the mean of multiplicative similarities on the individual instances.) In fact, exemplar models implicitly keep all the statistical information (e.g., frequency, variability, cooccurrence) by keeping all the exemplars. One might argue that a prototype model could also keep various statistical information around to make it equivalent to such exemplar models (e.g., Barsalou, 1990), but no one has proposed such a formal model. Second, because exemplar models have no summary representation, the same knowledge is not used for all the different decisions. Thus, even unusual items can be classified by similarity to earlier unusual similar members. The ability to use different knowledge for different decisions means that the exemplar model can classify unusual items without compromising its ability to easily classify more typical items. This flexibility is important in allowing the exemplar model to fit a variety of classification data.

The exemplar model fits the data well for many classification experiments but has difficulties with other aspects of category-related judgments. One major issue is that it has no place for these summary representations that we all find attractive in thinking about concepts. In particular, to answer the question as to why these items are all members of the same category, the exemplar view is left with the unsatisfying answer that “they all have the same category label.” That may be fine for arbitrary experimenter-defined categories in the laboratory but seems woefully inadequate for permutation problems, extroverts, and love triangles. (Note that the classical view can point to a definition and the prototype view to similarity to some common summary representation.)

2 Selection by Analogy

The preceding view is abstractionist in the sense that it assumes that people have abstracted from their experience the relative likelihood that a particular concept will combine in a particular way. In this sense, it is a kind of prototype model that discards particular information and retains only the summary.

The analogy view is just the opposite, in the sense that it retains no summary information. Although one can have an analogy to a prototype or other summary representation, analogy is used here as a kind of single exemplar model that bases its best guess on a known exemplar that is presumably retrieved by similarity. For example, if one were asked what a chocolate dog was, one might check first to see that there was no entry for this combination and then retrieve “chocolate bunny.” Assuming one knew that a chocolate bunny was a bunny made out of chocolate, one could then make the analogy that “chocolate dog” had the same kind of interpretation as “chocolate bunny.”

15.3.2. Regression models and transfer

Given the previous provisions, in this section we show an example of benefit transfer in the case of informal beach recreation. A benefit transfer function is usually linear, at least in the sense of first degree approximation. To formalise the model, start with the prototype model from Brouwer (2000):

(15.1)Vi=α+βXi+γYi+δZi+ɛi

where α β γ δ are parameters, V is the value per site per visit for a given policy, X, Y and Z have been defined above and i indexes the studies. Because we have no data on several variables that could explain the value, such as beach width and length or respondents’ income, Ordinary Least Squares (OLS) estimation of the Brouwer's linear model will be biased. This is a standard result with OLS: missing regressors lead to bias.

Since in the current dataset, there is often more than one observation for a single site, the model can be written as:

(15.2)Vit=αιi+β Xit+γYit+δZit+ɛit

where Vit. the value for site i under the circumstance t. The circumstance can refer to a different point in time (a different year), or to some hypothetical situation (for example, the site is eroded). This is a panel data model, the main difference with Brouwer's linear model is that the intercept term a is now specific to each site because it is indexed by i. This is critical because the site-specific intercept term will account for all the differences in values across sites not accounted for in the regressors, and thus avoid the bias problem referred to above.

When the goal of the study is to predict the value of one site given some characteristics, bias in the estimated coefficients is not important. Therefore Brouwer's linear model can be estimated using OLS. When the goal of the study is to estimate the marginal effect of some characteristic of the beach, it is critical to estimate the coefficients without bias and then the panel data model is best. This is illustrated below.

The date (T) of the study is inserted in the regressions as a natural trend starting in 1975 (normalised to 1). The 4 categories of visitors (local residents, day visitors, stay visitors and unspecified type) are represented using three dichotomous variables (Local, Day, Stay), with the omitted category being the unspecified type. The 3 remaining categories of quality of the site (eroded, current quality, defended) are represented using two dichotomous variables (Eroded, Defended), the omitted category is the current quality.

The concept of value has three categories (VOE, WTP for use, Consumer Surplus). The 3 categories have been represented by 2 dichotomous variables (WTP, CS), the omitted category being VOE. In the panel data model, it turns out that the sum of these 2 variables is a vector of zeros and ones identical to the sum of certain site-specific constants. Therefore, one of these 2 variables had to be removed to enable estimation. Since the decision to remove is arbitrary, we present the 2 sets of results: in the first one (Table 15.8.a) the variable removed is the dummy indicating the Consumer Surplus, in the second one (Table 15.8.b) it is the dummy indicating the WTP for use.

Table 15.8. Panel data estimates.

The tables are quite similar with the exception of the intercept term, this is reasonable because of the two different dummies (WTP or CS). Neither the effect of time (T) nor of the type of respondents (Local residents, Day visitors, Stay visitors or Unspecified) are statistically significant.

The quality of the site (Current, Defended, Eroded) is very significant. «Current» refers to the beach as it is at the moment of the study; it denotes a coastal site that is enjoyable under normal conditions. «Eroded» indicates a state, usually hypothetical, in which only a narrow range of the beach remains in place, if any. «Defended» indicates that a coastal defence scheme, also usually hypothetical, is implemented that partially modifies the aspect of the beach and may enlarge it.

Finally, the high significance of the concept of value used (VOE, WTP for use, Consumer surplus) is worrisome. It is acceptable that different concepts of value yield different values, but the problem is that different survey design (Open-ended CV or Travel cost model) have been used for the different concepts. Therefore, we cannot tell whether the differences in value are genuine or are led by the method used. If it is the former, we would still have to decide which concept of value is more appropriate. If it is the latter, then benefit transfer of informal beach recreation is flawed since a different method leads to a different value for the same beach. These are the conclusions of the panel data models regarding the effect of invidual characteristics on the site value.

The results of estimating Brouwer's model directly by OLS are shown in Table 15.9. Since the OLS estimates are biased, they are not interpreted.

Table 15.9. OLS (biased) estimates.

To run a transfer exercise on the basis of the regressions above, for each site run the above regressions (the 2 panel data regressions and the OLS) without this site's observation(s) and predict its value using the level of the regressors specific to this site. Then, to measure the gain of precision obtained by carrying a new study, compare the predicted value with the one obtained from the original study. The measure of prediction error is the proportion of deviation from the value(s) reported for the site in absolute term. We also present the simple value transfer prediction which consists in predicting for one site the average value of the other sites.

Figure 15.1 reports the proportion (vertical axis) of predictions that falls below the error level indicated on the horizontal axis. We call that the cumulative distribution of prediction errors. For example, the proportion of predictions of less than a 40% error is about 70% for OLS and 55% when the prediction is the average of the values of the other sites. We say that model A predicts better than model B when the cumulative distribution of prediction errors of model A is above that of model B. In that sense, the panel data models are worse than a simple average of values (but that does not undermine their qualities for an unbiased estimation of regression coefficients). For prediction purposes, our best model is the OLS.

In summary, we have shown that to transfer benefit Brouwer's equation could be estimated by OLS. Figure 15.1 reports the risk of error in doing so. To find out about the marginal effect of some characteristic, panel data models could be used.

4 Summary

The initial flush of success of prototype theory proved to be a poor predictor of its future. One of the main functions of classification is that it allows us to make inferences and predictions on the basis of partial information. In general, the pairs of storage and retrieval assumptions associated with exemplar models preserve much more information than prototype models, information that people show sensitivity to. The context sensitivity of exemplar models is also consistent with much of the memory literature (e.g., Tulving, 1983).

The exemplar view of conceptual structure has a number of characteristics than distinguish it from other probabilistic view models. The prototype view claimed that categories were represented in terms of characteristic properties that worked together to create (linearly separable) categories where examples could be successfully classified on the basis of their similarity to prototypes. The exemplar view has no such requirement. The features used to categorize are the features of the category examples, and these need not be characteristic of the category overall. Some models with the Exemplar framework allow feature weighting to vary from example to example (e.g., Medin & Edelson, 1988). In short, the exemplar view appears to imply virtually no constraints on category membership. This is an issue we’ll return to, but first we add a few more complications.

Prototypicality

In order to recognize things, we tend to classify all things into groups of objects which share some properties. For those object categories for which there are many exemplars, such as human faces, cars, toasters, or cubist paintings, it seems that through experience we build so-called prototypes. These are typical representations which allow us to trigger appropriate responses and which summarize information that all objects of that class have in common. This is not to say that the prototype is represented by a certain category member; a prototype is ‘simply a convenient grammatical fiction; what is really referred to are judgments of degree of prototypicality’ (Rosch, 1978, p. 40). Whitfield and his colleagues (1983; Whitfield and Slatter, 1979) carried out pioneering work concerning the effect of prototypicality on preference. They directly tested a preference-for-prototypes model (see also Martindale, 1984) against Berlyne's ‘collative-motivation’ model predicting an inverted U-shaped relationship between preference and novelty/complexity. They measured appreciation for different kinds of chairs that varied in prototypicality as a result of belonging to different styles, assuming that ‘Georgian chairs’ are more prototypical than ‘Modern style’ chairs, and these more prototypical than ‘Art Nouveau’ chairs. Moreover, the authors directly measured subjective impressions of typicality (as well as complexity and novelty) for all chair models investigated. As expected, more prototypical chairs were liked better, and typicality was negatively correlated with novelty, indicating that prototypicality is opposite to novelty. Contrary to what Berlyne's model would have predicted, ‘complexity’ did not account for differences in aesthetic appreciation. Subsequent studies in which both models were empirically tested against each other were performed for diverse categories such as houses (Purcell, 1984), cubist paintings (Hekkert and van Wieringen, 1990), and musical performances (Repp, 1997), all confirming a linear relationship between preference and prototypicality.

Although familiarity is not the only defining variable of (proto)typicality (Barsalou, 1985), the two concepts are clearly related. They both find their aesthetic attractiveness in ease of classification or processing (Reber et al., 2004). But ease of processing is not what people are always after. At various occasions people look for novel or original instances and especially children have a bias towards novelty in their early ages (e.g. Uehara, 2000).