Syllabus

1. Syllabus http://gandalf.psych.umn.edu/~schrater/schrater_lab/courses/MathMod06/ PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2006

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

3. Role of models in science The World The Model Relations between Different World properties Abstraction Experimentation Derivation Data Predictions Relations between World properties And measurement Interpretation Collection of measurements With the procedure for Gathering it PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2006

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

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

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

7. Causal theories specification of causes Some questions can’t 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? PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2006

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

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

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

11. Example: Speech generation Voice Puppetry, M. Brand; Siggraph’99 PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2006

12. Can we fit natural language behavior in this paradigm? Goal of language behavior? Beliefs? Actions? PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2006

13. Can we fit natural language behavior in this paradigm? Goal of language behavior? Convey some meaning Beliefs? Meaning generated by other’s parsing of the sentence Actions? Sentence generation PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2006

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

15. Least action demo Which path will the ball take? Kinetic Energy K = M v2 U = g y http://www.eftaylor.com/software/ActionApplets/LeastAction.html PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2006

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

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

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

19. Modeling Beliefs Roger N. Shepard PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2006

20. Example: Modeling Beliefs Roger N. Shepard PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2006

21. Example: Modeling Beliefs Roger N. Shepard PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2006

22. Example: Modeling Beliefs Roger N. Shepard PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2006

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

24. Homework requires Matlab • “BASIC for people who like linear algebra” • Full programming language • Interpreted language (command) • Scriptable • Define functions (compilable) PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2006

25. 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 = {‘Paul’;’Harry’;’Jane’}; >> A{1} => Paul PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2006

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

27. 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 3 2 4] Vectorwise logical expressions a = [ 1 2 3 1 5 1] a = =1 => [ 1 0 0 1 0 1] size( ), whos, help, lookfor ls, cd, pwd, Indices = find( a = =1 ) => [ 1 4 6 ] PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2006

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