Concepts & Categorization

Geometric (Spatial) Approach

• Many prototype and exemplar models assume that similarity is inversely related to distance in some representational space

A

B

C

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

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:

rn

k

r

jkikij xxd1

1

Data: Matrix of (dis)similarity

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

Example: 2D solution for bold faces

2D solution for fruit words

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

Critical Assumptions of Geometric Approach

• Psychological distance should obey three axioms

– Minimality

– Symmetry

– Triangle inequality

0),(),(),( bbdaadbad

),(),( abdbad

),(),(),( cadcbdbad

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 ),(),( abdbad

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

),(),(),( cadcbdbad

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

Example

Similarity(X,Y) = a( shared) – b(X but not Y) – c(Y but not X)

` Lemon Orangeyellow orangeoval roundsour sweettrees treescitrus citrus-ade -ade

\

Example

Similarity(X,Y) = a( shared) – b(X but not Y) – c(Y but not X)

` Lemon Orangeyellow orangeoval roundsour sweettrees treescitrus 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

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

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

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

Typicality Effects

• Typicality Demo– will see X --- Y.

– need to judge if X is a member of Y. • finger --- body part• pansy --- animal

turtle – precious stone

pants – furniture

robin – birddog – mammal

turquoise --- precious stone

ostrich -- birdpoem – reading materials

rose – mammal

whale – mammaldiamond – precious stone

book – reading materialopal – precious stone

Typicality Effects

• typical– robin-bird, dog-mammal, book-reading,

diamond-precious stone

• atypical– ostrich-bird, whale-mammal, poem-reading,

turquoise-precious stone

Is this a “chair”? Is this a “cat”?

Is this a “dog”?

Categorization Models

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

• Types of representation– prototype– exemplar

Prototypes Representations• Central Tendency

Learning involves abstracting a set of prototypes

Graded Structure

• Typical items are similar to a prototype

• Typicality effects are naturally predicted

atypical

typical

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

Prototype Small Distortion

Medium Distortion

Large Distortion

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

dog

dog

dog

cat

cat

cat

dog

??

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

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

– Daredevil

Some Interesting Applications

• 20 Questions:http://20q.net/

• Google Sets:http://labs.google.com/sets