Concepts

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Concepts Categorization
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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
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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

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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

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Data Matrix of (dis)similarity
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MDS procedure move points in space to best model
observed similarity relations
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Example 2D solution for bold faces
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2D solution for fruit words
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Whats 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

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Critical Assumptions of Geometric Approach

• Psychological distance should obey three axioms
• Minimality
• Symmetry
• Triangle inequality

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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

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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

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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
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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
• \

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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
• Similarity( Lemon,Orange ) a(3) - b(3) -
  c(3)
• If a10, b6, and c2 Similarity 103-63-236

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Contrast model predicts asymmetries
Suppose weighting parameter b gt c Then,
pomegranate is more similar to apple than vice
versa because pomegranate has fewer distinctive
features
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Contrast model predicts violations of triangle
inequality
If weighting parameters are a gt b gt 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
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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

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Typicality Effects

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

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• turtle precious stone

pants furniture
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
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Typicality Effects

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

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Is this a chair?
Is this a cat?
Is this a dog?
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Categorization Models

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

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Prototypes Representations

• Central Tendency

P
Learning involves abstracting a set of prototypes
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Graded Structure

• Typical items are similar to a prototype
• Typicality effects are naturally predicted

atypical
P
typical
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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

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Problem with Prototype Models

• All information about individual exemplars is
  lost
• category size
• variability of the exemplars
• correlations among attributes

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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
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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

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Theory-based models

• Sometimes similarity does not help to classify.
• Daredevil

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Some Interesting Applications

• 20 Questionshttp//20q.net/
• Google Setshttp//labs.google.com/sets