Mathematics 191
Research Seminar in Mathematical Modeling
Validation and Simulation I
February 10th, 2005
Project Comments
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Project comments
Project two should form groups and start now
Formal project requirements will be up online
Make sure you send a final prospectus to us by the
due date – 2 weeks!
• Field trip for validation
Overview
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Syllabus and Projects
Introduction to Validation
Validation versus Simulation
Methods of Validation
Vote-Counting and Velociraptor Validation
Giving Mathematical Talks
Content and Style
Presentations
Section Two: Validation and
Simulation
Over the next two weeks, we’ll go over:
• Methods of validation
• Statistical methods
• Simulation design and analysis
• Data generation and distributions
• Mathematical Programming?
The Modeling Process
• Statement of Problem (abstraction)
• Define Model Objective / Objective Function
• Definitions and Identification of Variables (background research and common
sense)
• Assumptions (for tractability)
• Establish Informal Relationships Based on System
• Construct Mathematical Statements
• Construct Base Model
• Estimate Parameters
• Apply Mathematical Methods
• Pure Mathematical Solution
• Simulation and Validation
• Sensitivity Analysis
• Relax Assumptions
• Iterate
• Assess Model Limitations
The Need for Justification
• Validation is the assessment of how closely our
model meets a given set of criteria, typically,
how well the model represents real-world data.
• If we’re using models to make real-world
predictions, our model is only effective if it can
predict real-world data correctly
Three Options for Validation /
Justification
1) Evaluate model using objective function
- In the tollbooth model, throughput
2) “Evaluate” (compare) model using another
model, with orthogonal assumptions
- In the tollbooth model, discrete simulations
3) Evaluate model using real-world data (might be
from a similar system, not necessarily
equivalent)
- What’s the potential problem here?
Another option - Goodness of Fit
• We are concerned in many contexts with
“goodness of fit” and error - how close we come
to predicting actual performance in terms of
deviation from sets of data points
• Frequently occurs in natural-science contexts
• In other contexts, we don't – what about
psychological models, for instance?
• Statistical tools are coming up later
Simulations and Validation
• Simulations play two roles:
• As modeling tools, they can numerically evaluate a
system over time that is too complicated to solve
analytically – usually this means there are a very large
number of variables, or that there are coupled equations
we can only solve numerically.
• In addition to being tools, they can be used to generate
independent models and data for validation purposes,
although we don't necessarily need a simulation to
validate our data.
A Very Crude Summary of Simulation
Construction
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Define relevant variables
Come up with equations and rules for behavior
Implement the rules as pseudocode
Implement the rules as real code
Evolve the system over time
Evans Hall Elevators, Revisited
• Recall that a few weeks
ago we were interested
in designing optimal
algorithms for
performance.
• In teams of two, write a
pseudocode simulation
to represent the
behavior of the
elevators.
How Did Velociraptors Hunt?
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Part 1. Assuming the velociraptor is a solitary hunter, design
a mathematical model that describes a hunting strategy for a
single velociraptor stalking and chasing a single
thescelosaurus as well as the evasive strategy of the prey.
Assume that the thescelosaurus can always detect the
velociraptor when it comes within 15 meters, but may
detect the predator at even greater ranges (up to 50 meters)
depending upon the habitat and weather conditions.
Additionally, due to its physical structure and strength, the
velociraptor has a limited turning radius when running at
full speed. This radius is estimated to be three times the
animal's hip height. On the other hand, the thescelosaurus is
extremely agile and has a turning radius of 0.5 meters.
Part 2. Assuming more realistically that the velociraptor
hunted in pairs, design a new model that describes a hunting
strategy for two velociraptors stalking and chasing a single
thescelosaurus as well as the evasive strategy of the prey.
Use the other assumptions given in Part 1.
Here, some constraints and assumptions are given to us by
the problem.
What sort of mathematical approaches might we use to
solve this problem?
How can we hold fair elections in
a dangerous or uncertain
environment?
• Iraq's electoral system is based loosely on our
own. Iraq is divided into a certain set of
precincts, but turnout is expected to be fairly
low, and perhaps unfairly biased towards certain
groups.
• As an added complication, nobody knows who's
really running.
• In the US election, random votes were lost on
certain machines in certain areas. User and
machine error, and rarely, fraud, also
contributed to mis-votes. Rerunning an election
is costly and highly undesirable.
• How can we still determine a “fair” winner in
these situations?
Why Give Talks in a Math Class?
• It's what mathematicians and nonmathematicians do.
• You'll get better at it, one way or another.
A few tips on effective
presentation
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Eye contact, posture, body language, volume
Tone and voice inflection
Face the audience, not your screen or notes
Engage your audience through examples and
interaction
• PowerPoint content
• Be prepared!
Suggested Outline
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“Hook”
Overview
Motivate your problem.
Explain prior approaches.
Explain your approach.
Explain your results.
Suggest future work.
Applications.
Conclude
Fin