Math 411 - Spring 1997 - Assignment 2
Non-linear scaling and linear regression

Consider a collection of data which appears to reasonably be
described by an exponential 
           y(t) = c exp(kt)                  (1) 
where c and k are constants. Thus, you have a collection of data
points (ti,yi) which might be reasonably described by this
function.

1. Consider two alternative methods to estimate the parameters c
and k. First, do a non-linear scaling, writing the equation (1) as
ln(y(t))=ln(c) + kt
and then use linear least squares (such as the LR key on a
calculator) to estimate the parameters. Second, consider choosing
the parameters which best fit (1) directly, by minimizing the least
squares estimate given by the sum over i of (y(ti)-yi)^2
Will these two methods give the same estimates of the parameters?
Why or why not?

2. Under what circumstances would you expect the two methods
described above to give close to the same estimates, and when do
you expect that they will be quit different? Illustrate your
comments with appropriate data examples.