- #1
bers
- 4
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Hi,
I want to solve an overdetermined non-linear equation with the method of least squares. Assume it's f(x) = 1 + ax + a^2 + b, and I want to estimate a and b. This is non-linear, as I said, so the derivatives of the squared residuals involve a^3 terms and are difficult to solve.
Now I thought about substituting c = a^2 + b, which would turn this into a linear system that LOOKS pretty easy to solve. But when calculating the derivatives of the residuals, do I need to take into account that dc/da != 0 (which would make the terms quite complicated again), or can I simply proceed as if c had been the original variable?
If I am allowed to go this way (I guess I am), what are the conditions - when am I allowed to substitute to turn a non-linear least-squares system into a linear one? Can you point me at any fundamental rule for this? I searched Google quite a bit, but have not found any adequate.
Thanks
bers
I want to solve an overdetermined non-linear equation with the method of least squares. Assume it's f(x) = 1 + ax + a^2 + b, and I want to estimate a and b. This is non-linear, as I said, so the derivatives of the squared residuals involve a^3 terms and are difficult to solve.
Now I thought about substituting c = a^2 + b, which would turn this into a linear system that LOOKS pretty easy to solve. But when calculating the derivatives of the residuals, do I need to take into account that dc/da != 0 (which would make the terms quite complicated again), or can I simply proceed as if c had been the original variable?
If I am allowed to go this way (I guess I am), what are the conditions - when am I allowed to substitute to turn a non-linear least-squares system into a linear one? Can you point me at any fundamental rule for this? I searched Google quite a bit, but have not found any adequate.
Thanks
bers