MAT 331, Spring 98

**Project 1: Least-Squares Fitting**

*Due 3 March 1998*

This project, like all in this class, have a significant component which is expository in nature. Please pay attention to organization, sentence structure, and so on. You will be graded on both the quality of your mathematical exposition and on the correctness of your computer work. You should treat this like a paper, rather than a typical homework assignment. A good paper should be complete and self-contained, going over all the necessary background material.

**Part 1. **
In this part, you are given a collection of points which you
believe approximate a circle. You should find the ``best fit''
circle to these points using least squares. In your write-up, you should
not only find the circle, but explain clearly what you are doing at each
step and why.

As we saw in class, the desired circle corresponds to the minimum of the
function
While `maple` can find the critical points of this equation directly, you
should do the following:

Let *k*=*a ^{2}*+

In your project, you should give a proof that the circle which gives a
minimum for *G* always corresponds to the minimum for *H* . You could do
this by showing that

In your paper, you should discuss this case in full detail, and give an
example. Use the routine `circle_pts()`

to generate some data points
which approximate a circle.

Optionally, you could also discuss other possible means of measuring the distance between a collection of points and a circle. For example, what would be the disadvantage to using the ``usual'' method of least squares to fit the relationship ?

**Part 2. **
Suppose the data points (*x*_{i},*y*_{i}) are assumed to approximate a line, but
both the *x* -values and the *y* -values are viewed as approximate. In this
case, it makes sense to minimize the sum of the squares of the shortest
distance to the line, rather than the vertical distance. That is, if
(*x*,*y*) are points on the line and (*x*_{i},*y*_{i}) are data points, you should
vary the line to minimize the sum of the quantities (*x*-*x*_{i})^{2} + (*y*-*y*_{i})^{2} .
Note that there are several (equivalent) ways to represent the line and/or
compute the distance. You should, of course, choose whichever seems best to
you.

You are to derive a least-squares fitting method for this case, and illustrate it by an example. Compare the line you find in this example to the one given by ``usual'' least squares. Do they give essentially the same line? When do they differ? Explain.

Note that it is *not possible* to obtain linear equations; there will
always be more than one critical point. Only one of them corresponds to the
minimum- what is the geometric interpretation of the other one? Be sure to
describe the mathematics of what you are doing in sufficient detail. In
particular, you will need to derive a formula for the smallest distance
between a point (*x*_{i}, *y*_{i}) and a line *y*=*mx*+*b* . *Do not* merely
present the formula; derive it and explain your derivation.

**Note:**
While it *is* acceptable to discuss the problems with each other and how
to solve them, the write-up you present *must* be your own. Also note
that each person should have a different collection of data, since it is
generated at random. If you have significant discussions with others, it
would be nice to acknowledge who they are. You can produce your
paper in any word-processor you like (or even write it by hand, if your
writing is quite legible), or as a heavily annotated `maple` worksheet.
In all cases, you should include the relevant graphics and `maple` commands, as
needed. These, of course, cannot be handwritten, but may be inserted via
``cut and paste'' if you like. Each part of the project should hang
together as a single document, rather than as some discussion with a `maple` session tagged on at the end.