Syllabus

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Syllabus
http//gandalf.psych.umn.edu/schrater/schrater_la
b/courses/MathMod06/
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Mathematical Model

• Definition of a Model
• A model is a simplified representation of some
  aspect of the real world.
• Mathematical Models
• representation of relationships between numerical
  or symbolic representations of measurements and
  world properties.
• Relation example
• weight relations between two objects
• x R y if and only if x is heavier than y

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Role of models in science
The World
The Model
Relations between Different World properties
Abstraction
Experimentation
Derivation
Predictions
Data
Relations between World properties And
measurement
Interpretation
Collection of measurements With the procedure for
Gathering it
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Overview

• Mathematical models of Human Behavior
• Types of model
• Descriptive Relations between measurements
• Predictive Relations between world model and
  measurements
• Causal Predictive with directed relations
  between world properties
• Goal of models
• Representing behavior in symbols and relations
• Predicting behavior
• Summarizing large bodies of data
• Making assumptions and theories explicit and
  testable

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Measurements and world properties

• Example Measuring the mind
• What does it mean to measure the mind? How do we
  abstract mind?
• Relations between Measurements
• Why cant we add IQs?
• A silly idea-The IQ of a committee add in
  parallel

For Example
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What kind of model is this?

• Fitts' law is a model of human psychomotor
  behavior for speed/accuracy tradeoffs in rapid,
  aimed movement (not drawing or writing).
  According to Fitts Law, the time to move and
  point to a target of width W at a distance A is a
  logarithmic function of the spatial relative
  error (A/W)
• MT a b log2(2A/W c)
• where
• MT is the movement time
• a and b are empirically determined constants,
  that are device dependent.
• c is a constant of 0, 0.5 or 1
• A is the distance (or amplitude) of movement from
  start to target center
• W is the width of the target, which corresponds
  to accuracy

log2(2A/W c)
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Causal theories specification of causes
Some questions cant be addressed without causal
assumptions. Relations between elements in the
theory are not symmetric.
Natural or man-made? How can we describe what
generated these patterns?
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Abstracting Human Behavior

• World Abstractions
• Events (relations between time, place, and
  objects/agents)
• Outcomes (relations between actions and world)
• Behavior Abstractions
• Goals/Values (Rewards, gains, losses)
• defined over outcomes
• Relations betweens goals/values- (
  utility/preferences)
• Beliefs (Subjective probability)
• Defined over events
• Relations between beliefs (certainty)
• Actions (Moves, choices, decisions,
  communication,etc.)
• Relation between events, actor, and outcomes
• Relations between actions (plans, causes)

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Behavior Modeling

• Behavior theory-
• Define relations between goals, values, and
  beliefs
• Derive actions from goals, values and beliefs
• Behavior measurement
• Methods for quantifying actions
• Only actions are measurable-all other behavior
  properties are theoretical, and require a
  predictive model to connect to measurables.

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Speech Generation

• Goal- Deliver a message
• Events- utterances
• Outcomes - sound fidelity to intention,
  comprehension
• Beliefs- Ideas -gt words words -gt sounds
• Actions- Facial, esophageal, rib cage muscle
  movements

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Example Speech generation
Voice Puppetry, M. Brand Siggraph99
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Can we fit natural language behavior in this
paradigm?
Goal of language behavior? Beliefs? Actions?
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Can we fit natural language behavior in this
paradigm?
Goal of language behavior? Convey some
meaning Beliefs? Meaning generated by
others parsing of the sentence Actions? Sente
nce generation
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Signs of a good theory

• Using a small number of principles, be able to
  derive detailed consequences that can be
  specialized to many different situations.
• Moreover, these consequences can be converted
  into measurable predictions that can be compared
  to experiment.
• Example from physics Classical Mechanics and
  the principle of least action
• The path taken by an object will minimize the
  action (the conversion of potential to kinetic
  energy).

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Least action demo
Which path will the ball take? Kinetic Energy K
M v2 U g y
http//www.eftaylor.com/software/ActionApplets/Lea
stAction.html
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Are there similar principles for human behavior?

• Some of you may operate according to these
  principles
• Sloth principle Minimize effort. Only do what
  you have to?
• Hedonic principle Maximize those good times?
• Power principle Maximize influence? Only I can
  rule the world.
• Evolutionary principle Maximize survival/number
  of progeny?
• Serious proposal
• Maximize value, the expected utility of an action.

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Theoretical framework underlying almost all
models of human behavior

• Decision Theory/Game Theory
• Model human behavior via a Maximization
  Principle Behavior achieves goals by maximizing
  value for the organism.
• People model the world internally and formulate
  beliefs about it.
• People ascribe values to different world states
  and actions

John Nash
Duncan Luce
John von Neumann
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Overview

• Modeling Beliefs
• Belief representation
• Belief formation
• Belief revision
• Modeling Utility for different domains
• Utility for simple cognitive judgments
• Utility for simple perceptual judgments
• Utility for interpersonal interactions
• Utility for simple motor actions (e.g. reaching)
• Utility for mate selection
• Modeling Learning

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Modeling Beliefs
Roger N. Shepard
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Example Modeling Beliefs
Roger N. Shepard
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Example Modeling Beliefs
Roger N. Shepard
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Example Modeling Beliefs
Roger N. Shepard
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Beliefs involve representing certainty about the
presence of abstracted world properties internally
What are the world properties? What is the
abstraction? What is the belief?
Pigment changes Surface changes Material changes
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Homework requires Matlab

• BASIC for people who like linear algebra
• Full programming language
• Interpreted language (command)
• Scriptable
• Define functions (compilable)

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Data

• Basic- Double precision arrays
• A 1 2 3 4 5
• A 1 2 3 4
• B cat(3,A,A) three dimensional array
• Advanced- Cell arrays and structures
• A(1).name Paul
• A(2).name Harry
• A PaulHarryJane
• gtgt A1 gt Paul

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Almost all commands Vectorized

• A 1 2 3 4 5 B 2 3 4 5 6
• C AB
• C A.B
• C AB
• C AB
• sin( C ), exp( C )

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Useful commands

• Colon operator
• Make vectors a 10.910 ind 110
• Grab parts of a vector a(110) a(ind)
• A 1 2 3 4
• A(,2)
• A() 1
• 3
• 2
• 4
• Vectorwise logical expressions
• a 1 2 3 1 5 1
• a 1 gt 1 0 0 1 0 1
• size( ), whos, help, lookfor
• ls, cd, pwd,
• Indices find( a 1 ) gt 1 4 6

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Stats Commands

• Summary statistics, like
• Mean(), Std(), var(), cov(), corrcoef()
• Distributions
• normpdf(),
• Random number generation
• P mod(axb,c)rand(), randn(), binornd()
• Analysis tools
• regress(), etc