Spatial Models: Multidimensional Scaling

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Spatial Models Multidimensional Scaling

• ISYE/CS/PSYC 7790
• Cognitive Modeling
• Fall 2003

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Topics today

• Administrivia
• Five aspects of mediating states
• Spatial models of representation

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Administrivia

• Will meet in Room 16 for lab today
• Down in basement Chemistry area near the coffee
  cart
• Part 1 Discussion of feature-based models
• Part 2 Forming project groups

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Forming project groups

• Looking for projects in the main areas of the
  class
• Problem-solving, memory, analogy, visual
  attention and processing, and (possibly)
  categorization
• Open to other topics
• Think about a project that youd like to propose

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Five aspects of mediating states

• Enduring
• Discrete
• Composable
• Abstract
• Rule-driven

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Lecture Outline

• Space representing space (5)
• Mechanisms for spatial encoding (5)
• MDS models (5)
• Using MDS models for conceptual information (5)
• MDS models and the color circle (5)
• Kosslyns computational model (5)
• Tverskys criteria for MDS models (5)

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Space representing space

• Space in the representing world is equivalent to
  space (or some linear dimension) in the
  represented world
• Real-life representing mechanisms
• Thermostat (space represents temperature)
• Piano roll (length of hole represents duration of
  note)
• Pie or bar charts (many other charts, of course)

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Cognitive spatial representations

• Imaginary movement
• Dead reckoning
• Color spaces

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Examples of spatial representation mechanisms

• Example 1 Kosslyns Island

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Examples of spatial representation mechanisms

• Example 2 Dead reckoning in animals
• Gerbils (Mittelstaedt Mittelstaedt, 1980
  Gallistel, 1990)
• Take pup from nest
• Mother will dead reckon old location of nest, and
  return after retrieving pup.

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Examples of spatial representation mechanisms

• Example 3 Color similarity

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What mechanisms are used?

• Not always clear
• Direct encoding
• Analog encoding
• Others?

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Multidimensional scaling (MDS) models

• Input A proximity matrix
• Output A multidimensional space with a location
  for each item

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Proximity matrix for major US cities
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Results of MDS algorithm on city proximity data
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Proximity matrix for color similarity
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Results of MDS algorithm on color similarity data
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Results of MDS algorithm in numeral similarity
data
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Conceptual MDS models

• Concepts fit into a similarity space
• Closer items are more similar
• Distant items are more dissimilar

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Rips, Fitts Shoben (1973)
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Rumelhart Abrahamson (1973)

• ABC? problems
• Draw line from A-gtB, then from B-gtC, then make a
  parallelogram

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RobinSparrow as Duck ??
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PigeonParrot as Goose ??
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Sadler Shoben (1993)

• Counterargument Subjects are using the closest
  item in the space to C.

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Tverskys Axioms

• Minimality
• d(x,x) d(y,y) 0.
• Symmetry
• d(x,y) d(y,x).
• Triangle Inequality
• d(x,y)lt d(x,z)d(y,z)

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Minimality

• d(x,x) d(y,y) 0.
• Everything is most similar (or proximate) to
  itself
• Each thing is as similar to itself as another
  item is similar to itself.
• Dog, Dog
• Freedom, Freedom
• George Washington, George Washington
• 1.23 , 1.23

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Symmetry

• d(x,y) d(y,x).
• A is as similar to B as B is to A.
• d(Cuba, China) d(China, Cuba)
• d(butcher, surgeon) d(surgeon, butcher)
• d(FDR, W) d(W, FDR)

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Triangle Inequality

• d(x,y)lt d(x,z)d(y,z)
• d(atlanta,chicago) lt d(atlanta,indianapolis)
  d(indianapolis, chicago)
• d(goat,sheep) lt d(goat, pig) d(pig,
  sheep).

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Questions

• Where does MDS fit in MDs aspects of mediating
  states?
• Where does it fall short?
• What sorts of cognitive predictions does it make?

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Coming Up

• Later today Feature-based models
• Choosing projects