M314 Fall 2005 Project
Due Dates: Completed Project due Monday, November 21 (or earlier)
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.
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
ATAx = ATb.
It turns out this new system always has a solution,
and any solution
x = (x1, ... , xn)
of this new system has the property that
[a1-b1]2 + ... +
is as small as possible, where
a = (a1, ... , an)
is just Ax and
(b1, ... , bn) is just b.
Thus this method is called
the ``method of least squares''.
Here's the project. It has two parts.
Make sure your project write up includes the following:
- 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
- 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 pA and team B
has a power rating of pB, then when team A plays team B,
the difference in the scores should be the difference, pA - pB.
- 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 pA - pB = dAB,
where dAB 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.
Here is a link
to the Big Twleve football schedule, with scores of games played.
- 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.
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.