Least Squares
Part 3: The normal equations
If p = b1
+ mx is the projection of y onto W = Span(1, x),
then we can write
p = Xv,
where X is the matrix with
columns 1 and x, and v is the solution vector (b,
m)^{T}. Also, the vector y - p is orthogonal to the
entire subspace W, in particular, to both 1 and x. This orthogonality
can be expressed as
X^{T}(y
- p) = 0,
or
X^{T}y
= X^{T}p = X^{T}Xv.
The system of equations
X^{T}Xv
= X^{T}y,
in which everything is known
except v, constitutes the normal equations. Thus, the coefficients
for the best fitting line are the solutions of the normal equations.
- Construct the matrix X
for the cancer death data, and solve the normal equations. Make sure your
numbers m and b agree with what you see in the figure showing the least
squares line.
- Confirm that your coefficients
m and b give the same projection p as the one you computed in Part
2.
- If there were a location
in the Columbia River area with exposure index 5.5, what would you predict
about its cancer death rate? What portion of this death rate would you
attribute to radioactive wastes from the Hanford plant?
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