Cognitive Models of Subitization

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Cognitive Models of Subitization

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

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Administrivia

• Another lecture at 4 pm
• Will be discussing subitization
• Will use computers, but wont have a lab
  assignment
• Ways to take advantage of tickets?
• Collect readings for projects
• 1-2 readings each
• Figure out scheduling later?

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Outline

• Cognitive models of perception
• The problem of subitization
• What it is
• Why it is a useful model for us to start with
• Building a subitization model using a
  connectionist network
• Building a subitization model with ACT-R (in Lab)

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Basic constraints of perception

• Extremely short processing time
• Parallel
• Attention mechanism
• Fovea
• Integrating parts into whole
• Color/place/movement detectors
• Spatial relations
• Top-down influences

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Basic issues Attention

• Cant process everything at once how do we
  choose?
• Pre-attentive processing
• Color, line-thickness, other
• Spotlight metaphor
• FINSTs
• Other models of attention?

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What well (probably) cover in this class

• Subitization
• Pre-attentive processing
• ACT-R/PM
• APEX/GOMS
• Visual routines
• Qualitative spatial reasoning
• (with Analogy) symmetry detection

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Outline

• Cognitive models of perception
• The problem of subitization
• What is it?
• Why it is a useful model for us to start with
• Building a subitization model using a
  connectionist network
• Building a subitization model with ACT-R (in Lab)

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Subitizing
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Subitizing
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Subitizing
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Subitizing
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Subitizing
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Subitizing
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General Characteristics of Subitized Figures

• Limit is somewhere between 4 and 6 objects.
• From 2-4 objects Counting takes about 50 ms per
  item
• From 6 up Counting takes 250-300 ms per item

Peterson Simon (2000). Computational evidence
for the subitizing phenomenon. Cognitive Science.
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Explanation 1

• Humans have a fixed set of attentional tags
  (FINSTs) that are used for tracking
• Trick and Pulyshyn (1993, 1994)
• Tags serve additional purpose of tracking moving
  objects
• Problems with this explanation
• Why 4? Why not some much larger number?

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Explanation 2

• ACT-R Model
• Anderson, Matessa Lebiere, 1997 Klahr
  Wallace, 1976
• Pre-code set of instant recognition operators
  for recognizing up to four instances
• Have counting rules (just like the ones we saw)
  for instances after that
• Problems
• Again, why 4?

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Peterson Simons Goals

• Why this particular low number?
• Why does this number vary from person to person?
• Non-nativist explanation?
• Theory
• Subitizing is the result of learned patterns in
  long-term memory
• Smaller, less variable patterns provide better
  and more reliable priming
• A set of elements greater than 5 has too many
  variations to be effectively primed

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Outline

• Cognitive models of perception
• The problem of subitization
• What is it?
• Why it is a useful model for us to start with
• Building a subitization model using a
  connectionist network
• Building a subitization model with ACT-R (in Lab)

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PDP (Neural Net) Model

• Model subitization as NN classifier
• 3-layer, fully connected
• Simple backprop algorithm
• Limitations
• NN cant model iterative processes like counting

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Domain model

• 4x4 grid
• Between 1-6 visible objects distributed over grid
• For modeling purposes,we can treat as16
  locations (L1 L16).

N4.
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Setup

• Pattern 22 item vector
• 16 items indicating 16 locations in grid
• 6 items, one for each count value (1..6)

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Results
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Discussion

• Results for PDP generalized for other factors
• Different number of hidden units
• Larger grid
• Larger numerosities
• Odd results for 6?
• End effect
• Results still seem somewhat muddy
• Weakened by lack of counting model to provide
  contrast

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Outline

• Cognitive models of perception
• The problem of subitization
• What is it?
• Why it is a useful model for us to start with
• Building a subitization model using a
  connectionist network
• Building a subitization model with ACT-R (in Lab)

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Setup for ACT-R Model

• Domain model and problem instances
• Chunks and chunk types
• Productions
• Memory and production strengthening

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Domain model

• 4x4 grid
• Between 1-6 visible objects distributed over grid
• For modeling purposes,we can treat as16
  locations (L1 L16).

N4.
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Training and testing sets

• Training sets
• Between 1,000 and 25,000 example patterns
• Divided evenly by numerosity
• Examine training at 1K, 3K, 5K, 10K, etc.
• Test set
• 25 patterns for each of six numerosities
• 150 patterns total

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Variability increases with numerosity
Peterson Simon (2000).
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Chunks and Chunk Types

• Items to represent
• Objects
• Locations in the grid
• Counting information
• Taken from previous counting model

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Chunk Types for Objects and Locations

• (Chunk-Type object
• attended Have I looked at it yet?
• visible Is it visible?
• location Which grid location is it in?
• )
• (Chunk-Type location
• attended Have I looked at it yet?
• occupied Is there anything in it?
• )

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Chunk Types for Goals

• (Chunk-Type method Counting strategy
• pattern Trial pattern under consideration
• strategy Always COUNT
• total Tally of items.
• )
• (Chunk-Type identify Recognition strategy
• pattern Trial pattern under consideration
• strategy ??
• total Total of items.
• )

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Chunk Types for Patterns

• (chunk-type test-pattern
• item1 item2 item3 item4 item5 item6
• amount)
• (chunk-type store-pattern Actually another
  goal
• pattern strategy total)
• (chunk-type stored-pattern
• item1 item2 item3 item4 item5 item6
• amount)

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Productions

• Counting
• Recognition
• Control knowledge
• Done counting?
• Storing new pattern instances in memory
• Strengthening old pattern instances

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Counting Productions

• SET-GOAL-COUNT Change the current goal focus
  from the IDENTIFY goal (which uses recognition)
  to the METHOD goal (which uses counting).
• GET-NEXT-OBJECT Find an unattended object in the
  grid, add one to the current tally, and mark
  the object as attended.

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Recognition (Match-N) Productions

• When
• System has IDENTIFY goal, and
• There is a stored pattern, and
• The locations in the stored pattern are
  unattended and occupied
• Then
• Attend to all locations simultaneously, and
  announce the number from the stored pattern

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Problem Which productions fire when?

• Wouldnt MATCH-ONE interfere with MATCH-TWO,
  MATCH-THREE, etc?
• No attempt to determine that the given set of
  objects is the only set available
• Solution
• Set the R values of MATCH productions
• Match-six (0.9), match-five (0.85), match-four
  (0.8), match-three (0.75), match-two (0.7),
  match-one (0.65)
• Set-Goal-Count is set very low, at 0.1. Why?

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Control Knowledge Productions

• DONE-COUNTING Determines when the counting is
  done.
• STRENGTHEN-INSTANCE Strengthen a stored pattern.
• STORE-INSTANCE Store a given pattern.

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Prediction

• Run on a large number of trials
• 1K to 50K
• Decay rate set very low
• Rule strengthening should for subitization
• Learned patterns contribute to rule strengthening
• Patterns added or strengthened at end of counting
  procedure

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Peterson Simon (2000).
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Peterson Simon (2000).
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Other Items for Discussion

• What does this model say about the individuation
  process? Is this important?
• These models are quite different what do they
  hold in common?
• Is memory priming a good mechanism for explaining
  subitization?
• Is this work cross-modal (audition)?