Suppose we want to find the best coefficients $a$ and $b$ that fits the data we have according to a model of the form $$ y = a t e^{bt} \text{ or } y = a e^{bt} \text{ or } y = a \left( \frac{x}{b+x} \right) $$ for example. Then I know we can linearize these non-linear least squares problems and find the linear regression of the transformed problem. My question is : will the values of $a$ and $b$ found by solving the linearized problem actually be the optimal parameters for the original non-linear least squares problem ?
I searched a lot on the Internet for information about this and it seems like in some cases we will get the exact values and in other not because the errors according to the linearized problem will behave differently compared to the original non-linear problem. For example computations seem to show that for the first model $y = ate^{bt}$, we get the same $b$ when solving the transformed linear model and numerically calculating the solution of the non-linear model, but $a$ differs a little. But, I do not quite understand. Any help will be very appreciated.
The problem with nonlinear regression is the need of "good" estimates of the model parameters. When linearization is feasible, it is a very good way to get them; when linearization is not feasible, the preliminary task of finding reasonable estimate is quite tedious and may require a lot of effort. Somletime, the model is linear with respect to all parameters except one : in such a case, fix this parameter, perform regression and iterate manually until you find a minimum of the obhjective function. From there, you can start. But, the parameters obtained by linearization can be very far away from those corresponding to the original problem. One of the key reasons is that we do not deal with the same objective function.
Suppose you have a model Y = Exp[a + b / T]; what you want to minimize is the sum of squares of the errors on Y. You can easily linearize this model; but, you will then minimize the sum of squares of the errors on Log(Y) which is totally different. More serious is the fact that regression must be done on what is measured; in my simple case, Y is measured, not Log(Y).