Mathematics 191Research Seminar in Mathematical Modeling

Lecture 3February 1st, 2005

Overview● Syllabus and Deadlines● Project Requirements● Mathematical Paper Writing● Functional Dependencies● Assumptions: Types and Limitations● Introducing Mathematical Structure● Case Study: Homing Pigeons● Case Study: Elevator Control Theory● Wide World of Models (time permitting)

Sneak Preview● Feb 1, 3: Assumptions, basic model-building, parameters, naïve models,

introducing mathematical structure, elementary mathematical models, model-construction practice.

● Feb 8, 11: Project talks.

● Feb 15, 17: The need for model validation. Mathematical programming. Simulation. Methods of validation and approximation. Modeling data-sets. Statistical and aggregate methods.

● Feb 22, 24: Model validation, continued.

Deadlines to note:

● Feb 28th: Draft project prospectus due.

● March 1: Project 2 due (tentative).

● March 3: Final project prospectus due. Student lessons signup.

● Project Requirements (give out on handout)Work in teams of 1, 2 or 3.Prepare a 30 minute talk on your model. Present your model in a mathematical paper similar to the MCM papers, in LaTeX.Consult an expert in the field.Participation in the MCM will serve as your paper topic.Short class on Tuesday due to MCM.

● Email us your topic choice for approval ASAP – preferably by tomorrow night.Projects are due Feb 11th. This is not an easy assignment! Don't put it off!

● Use of existing papers is recommended, but only to a point.

Structure of a Mathematical/MCM Paper

● Title ● Abstract ● Introduction● Background information● Definition of variables● Assumptions● Development of model● Construction of basic model● Construction of advanced model● Balidation, analysis, strengths &

weaknesses, conclusions/future work, bibliography, acknowledgements.

● There are special rules for MCM teams, which are posted on the COMAP website. A TeX template is available on the class website for papers. Find sample papers.

Taking Modeling to the Streets

● Real-world modeling on the streets of San Francisco.● You’ll get a flavor for data collection, analysis, and the

wide range of possible modeling topics.● Our task in the data analysis and model validation is to

either: 1) Develop a model beforehand and analyze its effectiveness onsite 2) pick a system beforehand and collect data in order to predict behavior

● Date selection, if there are too many conflicts. What if you can’t make it?

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 (the inverse of the abstraction process)

● Sensitivity Analysis

● Relax Assumptions

● Iterate

● Assess Model Limitations

Today’s Goals

● Once we know the variables involved in our system, we can map out the relationships between these variables as functions

● Proceeding carefully, we’ll discuss how to reduce the massive number of relationships and quantities into something manageable

● From this, we’ll see how to pick out a basic mathematical structure.

Reducing Complex Systems● Consider a complex problem – for example,

tsunami modeling from last time● How do we deal with the unwieldy number of

variables?● What if we don't know the value of a given

quantity?● What about processes that seem too complicated

to represent in a paper whose primary focus lies elsewhere? What if we know the process requires advanced mathematics we don't have time to learn?

Assumptions in Modeling● We assume that...tsunamis radiate outward from the epicenter of

an earthquake in all directions with equal speed● Masters and Johnson [39] demonstratde using empirical data that

the wave height was equal at all points equidistant from the epicenter in each of 200 tsunamis in 1958.

● Assumptions are good in that they render an otherwise intractable problem tractable.

● Assumptions should be used sparingly! Assumptions about our system hinder the generality of our model.

● In a paper, declare your assumptions up front, prior to discussing construction of your model.

When MCM teams should use assumptions

● “Modeling assumptions fall into two broad categories: physical assumptions requiring justification with discussion, and numerical parameter assumptions that may result from citations noted. The plausibility and applicability of either type directly depended on how well teams linked a particular assumption to the problem as stated in the MCM, rather than to some problem stated in the reference source document.” - Judges' Comments, Asteroid Impact

● “For the MCM, useful assumptions typically arise in one of two settings: either a team needs specifc information concerning the problem that they do not have (and cannot get in the time allotted) and hence must make an assumption in order to carry on; or a team decides to make an assumption that simplifes some detail(s) of the problem in order to use the mathematics they are familiar with or risk not being able to complete their modeling effort in the time allotted.” - Judges' Comments, Wind and Waterspray

● Unnecessary or unrealistic assumptions are specifically cited by judges as reasons papers don't win.

● Teams *MUST* discuss the effect of any limiting assumptions in the Strengths and Weaknesses section of their paper!

Mathematical Structure● The mathematical structure we apply should simply

emerge naturally from the problem.● The most important point: think about how the system

should behave.● Consider endpoints of your model, and any fixed values.

How should things change over time?● Of course, there are usually multiple ways to proceed

with our mathematical representation. Reach into the toolbox of mathematical structures and go for it.

● This is where exposure to a wide range of mathematical methods becomes most useful.

Homing Pigeon Models● We'll do just one example before we start doing practice.● WWI Modeling: How far can a pigeon fly without

stopping?● Variables: Bird speed, bird endurance, bird weight, bird

size, wingspan, time. Let’s just call the ones we choose not to model “biological parameters”, and assume that they are given.

● All we’ll try to model is bird speed as a function of time.

How to proceed

● Consider endpoints of the system● How should the system behave? ● Now you try it. How far can a homing pigeon fly

if we allow it to stop and rest?

So, to sum it up● Assumptions permit us to simplify our system in

order to use mathematics we are familiar with or use parameters that we are estimating

● Assumptions should be used sparingly, declared up-front, and relaxed or assessed afterwards

● The choice of mathematics will often emerge naturally from our knowledge of how the system should behave

● Using known or extreme values of a system will assist us in picking our mathematical structure

A few words on time management...

● Because this class will prepare us for the MCM, we will need to be able to derive models very rapidly (~20 minutes).

● As such, every day we'll work on a little bit of modeling and see how fast we can move from open-ended question to mathematical solution.

Evans Hall Elevators● Evans Hall has twelve floors and three main elevators centrally

located on each floor. Everyone complains that the elevators often behave unintelligently, sending multiple elevators to a given floor where only one is required, and forcing individuals to wait when there are free elevators.

● Develop a model to represent elevator operation and design an optimal algorithm for control of these elevators to improve performance. Use your model to assess the effectiveness of your algorithm. Keep in mind the limited amount of information available to the elevator.

Problems within the Purview of Modeling

● Most disciplines are well-established, so the models used have been in existence for hundreds of years. As undergraduates we often consider only existing models in class.

● This process is so established that we encounter models on a daily basis without always recognizing them as models.

● However, glory lies in development of our own models. Let's try some samples.

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?

How can we detect moving objects in an ambient noise field?

● The world's oceans contain an ambient noise field. Seismic disturbances, surface shipping, and marine mammals are sources that, in different frequency ranges, contribute to this field. We wish to consider how this ambient noise might be used to detect large maving objects, e.g., submarines located below the ocean surface. Assuming that a submarine makes no intrinsic noise, develop a method for detecting the presence of a moving submarine, its speed, its size, and its direction of travel, using only information obtained by measuring changes to the ambient noise field. Begin with noise at one fixed frequency and amplitude.

How Did Velociraptors Hunt?

● 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 many people should you date in your life?

● Consider the sequence of individuals we choose to date over the course of a lifetime (100 or so).

● Assume for a simple model that once we dump someone, we never see them again.

● What is the optimal number of people to date before “settling down” to maximize the probability that we'll find Mr./Ms. Right?

● In the advanced model, we can “play the field”...

Fin