M314 Fall 2005 Project

Due Dates: Completed Project due Monday, November 21 (or earlier)

Introduction

Sometimes, linear systems of equations involve real-life data. Such data often involve measurements which can have errors in them, due, if for no other reason, to unavoidable statistical variability. Thus systems that you might expect to have a solution still might be inconsistent. Although exact solutions might not exist, you nonetheless might be interested in approximate solutions, particularly best possible approximate solutions. In this project you will study a method for getting best possible approximate solutions, in the context of predicting, based on early season results, how the Husker football team will do in the North Division of the Big Twelve.

Some Background

Here's some background. Consider a system Ax = b of equations. If it is consistent, then you can go ahead and solve it using the methods you've learned in this class. But suppose it's not consistent. Then look instead at A^TAx = A^Tb. It turns out this new system always has a solution, and any solution x = (x₁, ... , x_n) of this new system has the property that [a₁-b₁]² + ... + [a_n-b_n]² is as small as possible, where a = (a₁, ... , a_n) is just Ax and (b₁, ... , b_n) is just b. Thus this method is called the ``method of least squares''.

The Project

Here's the project. It has two parts.

First Part:
- Demonstrate how the method works by using it to find the best fit line (called a linear regression line) through the following three points: A=(0,0), B=(1,1), C=(2,1). (I.e., write a system of equations that m and b would have to satisfy if y = mx + b were to be the equation of a line through these three points. Since the points do not lie on a line, your system of equations will be inconsistent, but you can still get a best fit line using the method described above.)
- Go to the library and get the formula for the linear regression line from a stat book (cite your reference), and get the equation of the line using the formula. Put the formula in your project write up, and compare what it gives you with what you got by the matrix method. (Note: most calculators can do linear regression.)
Second Part:
- For the second part, we want to predict the difference in scores when any two teams of the North Division of the Big Twelve play each other. It seems reasonable to assume that stronger teams will beat weaker teams, and that the stronger a team is the more it will win by. So we want to assign a number, call it a power rating, to each of the teams in the Big Twelve North such that if team A has a power rating of p_A and team B has a power rating of p_B, then when team A plays team B, the difference in the scores should be the difference, p_A - p_B.
- So, for each game played before November 13 between any two teams A and B of the Big Twelve North, write down an equation of the form p_A - p_B = d_AB, where d_AB is the actual difference in scores of that game. The power ratings are the unknowns which you will try to solve for.
- Since by November 13 there will have been 12 games played between the 6 North Division teams, you will have a system of 12 equations in 6 unknowns.
  - Between November 13 and November 18, find the best possible solution to the system of equations.
  - Use your solution to predict the score differential, and hence the winner, of the Colorado/Nebraska game.
  - Also, determine what prediction this method of least squares gives for the outcome of the Kansas State/Nebraska game November 12, based on North Division games played before November 12, and discuss how well you feel the method worked.

Make sure your project write up includes the following:

An introduction where you briefly explain that you are demonstrating how to apply least squares to make predictions. (Aside: the first use of least squares was by C. F. Gauss, regarding astronomical orbital predictions.)
Use part I as an example of the method.
For part II, separately for the Colorado/Nebraska and the Kansas State/Nebraska games, state whether the system is consistent or inconsistent. Discuss why you might or might not expect this system to be consistent.
Include enough of the main steps so that I can check how you found your least squares solutions for each of the two games.
Be sure to include an explicit prediction of which team will win the Colorado/Nebraska game, and by how much.
Discuss why you think the Kansas State/Nebraska game prediction did or did not work well, taking into account the actual outcome of the game.

Here is a link to the Big Twleve football schedule, with scores of games played.

Grading

You may work on this project in groups. All members of a given group will get the same grade. Groups of up to 4 are fine. Please check with me if you wish to form a group of more than 4. Your project grade will depend partly on spelling and grammar, in addition to the correctness of the mathematical results and the clarity of your exposition, so I recommend carefully proofreading your write up and using a spell checker.