PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2006 Syllabus schrater/schrater_lab/courses/MathMod06

PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2006

Syllabus

http://gandalf.psych.umn.edu/~schrater/schrater_lab/courses/MathMod06/

http://gandalf.psych.umn.edu/~schrater/schrater_lab/courses/MathMod05/


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


Role of models in science

The World The Model

PredictionsData

Derivation

Interpretation

Experimentation

Abstraction Relations betweenDifferent World properties

Relations betweenWorld properties And measurementCollection of measurements

With the procedure for Gathering it


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


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 can’t we add IQs? A silly idea-The IQ of a committee add in parallel:

€

IQgroup =1

1

IQ1

+1

IQ2

+ ...+1

IQN

€

IQgroup =1

1

120+

1

140+

1

150

= 45For Example:


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)


Causal theories specification of causes

Natural or man-made?

How can we describe what generated these patterns?

Some questions can’t be addressed without causal assumptions.Relations between elements in the theory are not symmetric.


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)


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.


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

movements


Example: Speech generationVoice Puppetry, M. Brand;

Siggraph’99


Can we fit natural language behavior in this paradigm?

Goal of language behavior?

Beliefs?

Actions?


Can we fit natural language behavior in this paradigm?

Goal of language behavior?

Convey some meaning

Beliefs?

Meaning generated by other’s parsing ofthe sentence

Actions?

Sentence generation


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).


Least action demo

http://www.eftaylor.com/software/ActionApplets/LeastAction.html

Which path will the ball take?

Kinetic Energy

K = M v2

U = g y

http://www.eftaylor.com/software/ActionApplets/LeastAction.html


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.


Theoretical framework underlying almost all models of human behavior

• Decision Theory/Game TheoryModel human behavior via a Maximization Principle: Behavior

achieves goals by maximizing value for the organism. 1. People model the world internally and formulate beliefs about it.2. People ascribe values to different world states and actions

John von Neumann John Nash Duncan Luce


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


Modeling BeliefsRoger N. Shepard


Roger N. Shepard

Example: Modeling Beliefs


Roger N. Shepard

Example: Modeling Beliefs


Example: Modeling BeliefsRoger N. Shepard


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 changesSurface changesMaterial changes


Homework requires Matlab• “BASIC for people who like linear algebra”• Full programming language

– Interpreted language (command)– Scriptable– Define functions (compilable)


Data

• Basic- Double precision arraysA = [ 1 2 3 4 5]A = [ 1 2; 3 4]B = cat(3,A,A) %three dimensional array

Advanced- Cell arrays and structuresA(1).name = ‘Paul’A(2).name = ‘Harry’

A = {‘Paul’;’Harry’;’Jane’}; >> A{1} => Paul


Almost all commands Vectorized• A = [ 1 2 3 4 5 ] ; B = [ 2 3 4 5 6]

– C = A+B – C = A.*B – C = A*B’– C = [A;B]– sin( C ), exp( C )


Useful commands• Colon operator

– Make vectors: a = 1:0.9:10; ind = 1:10– Grab parts of a vector: a(1:10) = a(ind)– A = [ 1 2; 3 4]– A(:,2)– A(:) = [ 1

324]

Vectorwise logical expressionsa = [ 1 2 3 1 5 1]a = =1 => [ 1 0 0 1 0 1]

size( ), whos, help, lookforls, cd, pwd, Indices = find( a = =1 ) => [ 1 4 6 ]


Stats Commands• Summary statistics, like

– Mean(), Std(), var(), cov(), corrcoef()• Distributions:

– normpdf(),• Random number generation

– P = mod(a*x+b,c)rand(), randn(), binornd()

• Analysis tools– regress(), etc

Documents

PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2006 Syllabus schrater/schrater_lab/courses/MathMod06