MAT 331, Fall 1997
Project 1: Least-Squares Fitting
Due October 20/21

In this project, you should write a paper explaining and illustrating the very common method of fitting a curve to experimental data known as least-squares. Your paper should outline the basic theory, and give several examples (as suggested below, although you are free to add other ones). It should be complete and self-contained, going over all the necessary background material. In going over the examples, you should include the corresponding work from maple sessions. 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.

Recall that in least squares fitting, you have a collection of data points which are believed to approximate some functional relationship. In this project, you may stick to points in the plane, although the process works the same for points in .

In the most common situation, this functional relation is linear, that is, of the form , where m and b are constants to be determined, and the are assumed to be known exactly while the have some (unknown) amount of error.

In your paper, you should review the theory for this setup, giving an explicit example. Use the function line_pts() from the file /home/mat331/lsq_data to generate your data. You should also cover an example where the are assumed to depend either quadratically or cubically on (that is, or ), but again, the are assumed to be known exactly. You should use either quadratic_pts() or cubic_pts() to generate your data.

The next example you should consider is more unusual: suppose that the points approximate a circle. You should find the ``best fit'' circle to these points, and describe the process.

Recall that the general equation of a circle centered at (a,b) with a radius of r is

Unlike the previous cases, there is no independent variable. (Note that we could try to fit , but not only would the resulting equations be messy, this would bias things very badly; do you see why? You might want to discuss this issue). Nevertheless, we press on. One reasonable measure of the distance between the points and a circle is the ``area difference'', that is

A minor problem with this is that this is not quadratic in a, b, and r (the degree is 4), and so the resulting equations are not linear. They are, however, solvable by maple. But also note that if we let , the resulting functional H(a,b,k) is quadratic, and differentiating with respect to a, b, and k yields three linear equations in three unknowns. Solving these simultaneously gives us the values a, b, and k, and from which we can readily determine the center and radius. (If you discuss this approach, you should probably justify why the minimum for G(a,b,r) is the same as the minimum for H(a,b,k). This is, in fact, true, but it needs to be shown.)

In your paper, you should discuss this derivation in full, and give an explicit example. Use circle_pts() to generate 21 points which approximate a circle. If you like, you could discuss other possible means of measuring the distance between a collection of points and a circle.

A third example you should consider is the following: Suppose the data points are assumed to approximate a line, but both the x-values and the y-values are 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 are data points, you should vary the line to minimize the sum of the quantities . 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. Note that it is not possible to obtain linear equations; there will always be two critical points. 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.

As always, you may collaborate on your efforts, although your write-up must be your own. Also note that each person should have a different collection of data, since it is generated at random. If you work 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 even as a heavily annoted maple worksheet. However, you should include the relevant graphics and maple commands, as needed. These, of course, cannot be handwritten. The entire project should hang together as a single document, rather than as some discussion with a maple session tagged on at the end.

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