Clustering Using the Contrast Model - eScholarship
Clustering Using the Contrast Model Daniel J. Navarro and Michael D. Lee daniel.navarro,michael.lee @psychology.adelaide.edu.au Department of Psychology, University of Adelaide South Australia, 5005, AUSTRALIA Abstract An algorithm is developed for generating featural rep- resentations from similarity data using Tversky’s (1977) Contrast Model. Unlike previous additive clustering ap- proaches, the algorithm fits a representational model that allows for stimulus similarity to be measured in terms of both common and distinctive features. The important is- sue of striking an appropriate balance between data fit and representational complexity is addressed through the use of the Geometric Complexity Criterion to guide model selection. The ability of the algorithm to recover known featural representations from noisy data is tested, and it is also applied to real data measuring the similarity of kin- ship terms. Introduction Understanding human mental representation is necessary for understanding human perception, cognition, decision making, and action. Mental representations play an im- portant role in mediating adaptive behavior, and form the basis for the cognitive processes of generalization, infer- ence and learning. Different assumptions regarding the nature and form of mental representation lead to different constraints on formal models of these processes. For this reason, Pinker (1998) argues that “pinning down men- tal representation is the route to rigor in psychology” (p. 85). Certainly, it is important that cognitive models use principled mental representations, since the ad hoc defi- nition of stimuli on the basis of intuitive reasonableness is a highly questionable practice (Brooks 1991, Komatsu 1992, Lee 1998). One appealing and widely used approach for deriving stimulus representations is to base them on measures of stimulus similarity. Following Shepard (1987), similarity may be understood as a measure of the degree to which the consequences of one stimulus generalize to another, and so it makes adaptive sense to give more similar stim- uli mental representations that are themselves more sim- ilar. For a domain with n stimuli, similarity data take the form of an n × n similarity matrix, S = s i j , where s i j is the similarity of the ith and jth stimuli. The goal of similarity-based representation is then to define stimulus representations that, under a given similarity model, cap- ture the constraints implicit in the similarity matrix by approximating the data. Goldstone’s (in press) recent review identifies four broad model classes for stimulus similarity: geomet- ric, featural, alignment-based, and transformational. Of these, the two most widely used approaches are the ge- ometric, where stimuli are represented in terms of their values on different dimensions, and the featural, where stimuli are represented in terms of the presence or ab- sence of weighted features. The geometric approach is most often used in formal models of cognitive pro- cesses, partly because of the ready availability of tech- niques such as multidimensional scaling (e.g., Kruskal 1964; see Cox & Cox 1994 for an overview), which gen- erate geometric representations from similarity data. The featural approach to stimulus representation, however, is at least as important as the geometric approach, and war- rants the development of techniques analogous to multi- dimensional scaling. Accordingly, this paper describes an algorithm that generates featural representations from similarity data. The optimization processes used in the algorithm are standard ones, and could almost certainly be improved. In this regard, we draw on Shepard and Arabie’s (1979) distinction between the psychological model that is be- ing fit, and the algorithm that does the fitting. We make no claims regarding the significance of the algorithm it- self (and certainly do not claim it is a model of the way humans learn mental representations), but believe that the psychological representational model that it fits has three important properties. First, it allows for the arbi- trary definition of features, avoiding the limitations of partitioning or hierarchical clustering. Second, it uses a more general model of featural stimulus similarity than has previously been considered. Third, it generates feat- ural representations in a way that balances the competing demands of data-fit and representational complexity. Featural Representation Within a featural representation, stimuli are defined by the presence or absence of a set of saliency weighted fea- tures or properties. Formally, if a stimulus domain con- tains n stimuli and m features, a featural representation is given by the n × m matrix F = f ik , where f ik = if stimulus i has feature k otherwise, together with a vector w = (w 1 , . . . , w m ) giving the (pos- itive) weights of each of the features.