Concepts & Categorization

Transcription

1 Geometric (Spatial) Approach Concepts & Categorization Many prototype and exemplar models assume that similarity is inversely related to distance in some representational space B C A distance A,B small psychologically similar distance B,C large psychologically dissimilar Multidimensional Scaling Represent observed similarities by a multidimensional space close neighbors should have high similarity Multidimensional Scaling (MDS): iterative procedure to place points in a (low) dimensional space to model observed similarities Suppose we have N stimuli MDS Measure the (dis)similarity between every pair of stimuli (N x (N-1) / 2 pairs). Represent each stimulus as a point in a multidimensional space. Similarity is measured by geometric distance, e.g., Minkowski distance metric: d ij n = x k= 1 ik x 1 r r jk Data: Matrix of (dis)similarity MDS procedure: move points in space to best model observed similarity relations 1

2 Example: 2D solution for bold faces 2D solution for fruit words What s wrong with spatial representations? Critical Assumptions of Geometric Approach Tversky argued that similarity is more flexible than can be predicted by distance in some psychological space Distances should obey metric axioms Metric axioms are sometimes violated in the case of conceptual stimuli Psychological distance should obey three axioms Minimality d( a, d( a, a) = d( = 0 Symmetry d ( a, = d( a) Triangle inequality d( a, + d( d( a, Similarities can be asymmetric North-Korea is more similar to China than vice versa Pomegranate is more similar to Apple than vice versa Violates symmetry d ( a, = d( a) Violations of triangle inequality Spatial representations predict that if A and B are similar, and B and C are similar, then A and C have to be somewhat similar as well (triangle inequality) d( a, + d( d( a, However, you can find examples where A is similar to B, B is similar to C, but A is not similar to C at all violation of the triangle inequality Example: RIVER is similar to BANK MONEY is similar to BANK RIVER is not similar to MONEY 2

3 Feature Contrast Model (Tversky, 1977) Model addresses problems of geometric models of similarity Represent stimuli with sets of discrete features Similarity is a flexible function of the number of common and distinctive features # shared features # features unique to X #features unique to Y Similarity(X,Y) = a( shared) b(x but not Y) c(y but not X) a, and c are weighting parameters Example Similarity(X,Y) = a( shared) b(x but not Y) c(y but not X) ` Lemon Orange yellow orange oval round sour sweet \ Example Similarity(X,Y) = a( shared) b(x but not Y) c(y but not X) Contrast model predicts asymmetries ` Lemon Orange yellow orange oval round sour sweet Similarity( Lemon, Orange ) = a(3) - b(3) - c(3) If a=10, b=6, and c=2 Similarity = 10*3-6*3-2*3=6 Suppose weighting parameter b > c Then, pomegranate is more similar to apple than vice versa because pomegranate has fewer distinctive features Contrast model predicts violations of triangle inequality Nearest neighbor problem (Tversky & Hutchinson (1986) In similarity data, Fruit is nearest neighbor in 18 out of 20 items In 2D solution, Fruit can be nearest neighbor of at most 5 items High-dimensional solutions might solve this but these are less appealing If weighting parameters are: a > b > c (common feature weighted more) Then, model can predict that while Lemon is similar to Orange and Orange is similar to Apricot, the similarity between Lemon and Apricot is still low 3

4 Typicality Demo will see X --- Y. Typicality Effects need to judge if X is a member of Y. finger --- body part pansy --- animal turtle precious stone pants furniture robin bird mammal turquoise --- precious stone ostrich -- bird poem reading materials rose mammal whale mammal diamond precious stone book reading material opal precious stone Typicality Effects typical robin-bird, -mammal, book-reading, diamond-precious stone Is this a chair? Is this a cat? atypical ostrich-bird, whale-mammal, poem-reading, turquoise-precious stone Is this a? Categorization Models Similarity-based models: A new exemplar is classified based on its similarity to a stored category representation Prototypes Representations Central Tendency Types of representation prototype exemplar Learning involves abstracting a set of prototypes 4

5 Graded Structure Typical items are similar to a prototype Typicality effects are naturally predicted atypical Classification of Prototype If there is a prototype representation Prototype should be easy to classify Even if the prototype is never seen during learning Posner & Keele typical Prototype Small Distortion Medium Distortion Large Distortion Problem with Prototype Models All information about individual exemplars is lost category size variability of the exemplars correlations among attributes Exemplar model category representation consists of storage of a number of category members New exemplars are compared to known exemplars most similar item will influence classification the most?? cat cat cat Exemplars and prototypes It is hard to distinguish between exemplar models and prototype models Theory-based models Sometimes similarity does not help to classify. Daredevil Both can predict many of the same patterns of data Graded typicality How many exemplars is new item similar to? Prototype classification effects Prototype is similar to most category members 5

6 Some Interesting Applications 20 Questions: Google Sets: 6