Concepts & Categorization

1. Concepts & Categorization

2. Geometric (Spatial) Approach • 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

3. 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

4. MDS • Suppose we have N stimuli • 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:

5. Data: Matrix of (dis)similarity

6. MDS procedure: move points in space to best model observed similarity relations

7. Example: 2D solution for bold faces

8. 2D solution for fruit words

9. What’s wrong with spatial representations? • 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

10. Critical Assumptions of Geometric Approach • Psychological distance should obey three axioms • Minimality • Symmetry • Triangle inequality

11. 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

12. 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) • 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

13. 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,b, and c are weighting parameters

14. Example Similarity(X,Y) = a( shared) – b(X but not Y) – c(Y but not X) ` Lemon Orange yellow orange oval round sour sweet trees trees citrus citrus -ade -ade \

15. Example Similarity(X,Y) = a( shared) – b(X but not Y) – c(Y but not X) ` Lemon Orange yellow orange oval round soursweet trees trees citrus citrus -ade -ade 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

16. Contrast model predicts asymmetries Suppose weighting parameter b > c Then, pomegranate is more similar to apple than vice versa because pomegranate has fewer distinctive features

17. Contrast model predicts violations of triangle inequality 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

18. 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

19. Typicality Effects • Typicality Demo • will see X --- Y. • need to judge if X is a member of Y. • finger --- body part • pansy --- animal

20. pants – furniture turtle – precious stone robin – bird dog – mammal turquoise --- precious stone ostrich -- bird poem – reading materials rose – mammal whale – mammal diamond – precious stone book – reading material opal – precious stone

21. Typicality Effects • typical • robin-bird, dog-mammal, book-reading, diamond-precious stone • atypical • ostrich-bird, whale-mammal, poem-reading, turquoise-precious stone

22. Is this a “cat”? Is this a “chair”? Is this a “dog”?

23. Categorization Models • Similarity-based models: A new exemplar is classified based on its similarity to a stored category representation • Types of representation • prototype • exemplar

24. Prototypes Representations • Central Tendency P Learning involves abstracting a set of prototypes

25. Graded Structure • Typical items are similar to a prototype • Typicality effects are naturally predicted atypical P typical

26. 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

27. Problem with Prototype Models • All information about individual exemplars is lost • category size • variability of the exemplars • correlations among attributes

28. 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 dog ?? cat dog dog cat dog cat

29. Exemplars and prototypes • It is hard to distinguish between exemplar models and prototype models • 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

30. Theory-based models • Sometimes similarity does not help to classify. • Daredevil

31. Some Interesting Applications • 20 Questions:http://20q.net/ • Google Sets:http://labs.google.com/sets