PROFESSOR: Hi, and welcome.
Today we're going to do a problem about least squares
approximations.
Here's the question.
Find the quadratic equation through the origin that is a
best fit for these three points (1, 1), (2,
5), and (-1, -2).
I'll give you a minute to work that out on your own.
You can hit the pause button now, and we'll be back in a
minute to work the problem out together.
OK and we're back.
So what's the first step in a problem like this?
The first step is figuring out what our equation looks like,
the one that we're going to find.
So our equation is going to look like ct plus dt
squared equals y.
So that's what we're looking for.
We're looking for a quadratic equation through the origin.
Now if it were just any quadratic equation, then we
would have a constant term, but through the origin just
means that the constant term is 0.
Now what's the next step?
Next we need to set up a matrix equation.
So we need an A, a matrix A. And which matrix is
that going to be?
Well, let me start with the first coordinate of these
three points.
And I'm going to put them in the first column of this
matrix, 1, 2, -1.
And then my second column is going to be the squares of
these coordinates, 1, 4, and 1.
And my x hat--
that's just c and d.
And my b--
this is going to be the second coordinate, 1, 5, and -2.
OK, why did I set this problem up like that?
Well, multiply A times x hat.
The first coordinate of A times x hat is 1 times c
plus 1 times d.
It's just the same as plugging in this first point into the
left hand side of this equation.
And similarly, if I took the second point and plugged it in
here, I would just get 2 times c plus 4 times d, which is
just the same as the second coordinate in the
multiplication A times x hat.
2 times c plus 4 times d.
OK and where did the b come from?
Well, b came from plugging in these points to
the right hand side.
So 1, 5, and -2 are just the y-coordinates
of these three points.
So it would be great if we could solve A x hat equals b.
But we can't solve A x hat equals b, because there isn't
a quadratic equation through the origin that contains these
three points.
But we need to find the best approximation to that, so
that's the same as solving A x hat equals the projection of b
onto the column space of A. Because we only have a chance
of solving A x hat equals something if it's in the
column space of A. And remember from our projections
part of the class that this is the same as solving A
transpose A x hat equals A transpose b.
So that's what we're really going to do.
We're really going to solve A transpose A x hat equals A
transpose b.
And now all we have to do is just the computation.
So let's do it.
So what is A transpose A?
I'm going to do this kind of quickly, because you should be
good at multiplying matrices by now.
Oh this--
I have the negative in the wrong place, thank you.
Thanks for that correction.
I have the negative backwards there.
And what do I get when I multiply these?
I'll let you check.
This is 6, 8, 8, and 10.
And what is A transpose b?
Well, if you multiply this out, I'll let you check that
you get 13 and 19.
So this is just some computation.
And so what we're solving here is 6, 8, 8, 10 times x hat
equals 13, 19.
And we remember how to do this just by using elimination.
We replace the second row by 3 times the second row minus 4
times the first row.
Again, I'm going to do this quickly because you know this
from other parts of the class.
You similarly change the b vector, and we back solve to
get d is -5/2 and c equals--
let's plug that in and see--
think you're going to get 11 over 2.
So what's our final equation?
Our final equation is y equals 11/2 t minus 5/2 t squared.
OK.
So this is our best fit quadratic
equation through the origin.
Now before I end, let me do a couple things.
First, let me go back and review what
the key steps were.
Whenever you're faced with such a best fit equation,
first, you want to see what the general form of the
equation is.
Next you want to write it in terms of matrices.
Write down your matrix A and your vector b.
And lastly, you set up your projection equation.
And then all you have to do is just a computation.
But it might also be worth noting that these three points
certainly aren't on this quadratic equation.
For instance, if I plug in 1 here, I don't get (1, 1), I
get (1, 3).
But that's OK.
This is as close as we can do.
Thanks.