(b)
MATH 261, Spring 2001 Due Date: Name(s):
Honors Project 20: Least Squares
In many science and engineering applications, one is often given a set of data points {(xi, yi), i = 1, ..., N}
in R2
, and interested in finding the line which "best fits this data" (see Fig. (a) below). One solution to
this problem is provided by classical least squares: finding the line for which the sum of the squares of the
distances di from the data points to the line in the direction of the dependent axis is a minimum (see Fig. (b)
below).
As an application of optimization of functions of two variables -- specifically, finding the m and b that
minimize the error term
)i = 1N
(yi - (mxi + b))2
,
-- it follows that the line y = mx+b which best approximates the data in the sense of classical least squares
is given by
m =
N xiyi - ( xi) ( yi)
N x2
i - ( xi)
2 and b =
( yi) x2
i - ( xi) ( xiyi)
N x2
i - ( xi)
2 .
Tasks
1. Verify the above formulas.
2. Devise a procedure for finding the plane  whose equation is of the form z = ax+by+c which "best fits"
the data {(xi, yi, zi), i = 1, ..., N} in R3
. In the end you should arrive at a system of linear equations
that need to be solved. You do not have to solve this system here, however. Indeed, to do Task 3 below
you can use Maples solve command.
3. Apply the procedure you developed above to the following data set:
xi 0.9 1.1 1.2 1.4 2.3 2.9 3.5 3.6 4.1 4.8
yi 1.2 3.6 3.5 2.4 2.2 0.6 1.6 4.3 1.5 2.7
zi 10.8 19.2 18.9 15.4 15.8 11.1 15.2 24.5 15.6 20.5
4. Without going through any analysis, what system of linear equations would you guess you would have
to solve to find the hyperplane whose equations is w = ax + by + cz + d which "best fits" the data
{(xi, yi, zi, wi), i = 1, ..., N} in R4
.