Least squares with exponential model

Question

I'm trying to fit values from this model $$y(x)=ae^{−bx}+c$$

where a, b and c are 3 different parameters that I want to find with least squares. So using least squares I want to find the value of a, b and c using this formula

$$ \mathrm{argmin}\sum_i \left( ae^{−bx_i}+c - y_i \right)^2. $$

This leads me to 3 non linear equations after partial derivation of the previous one with respect a, b and c. Surely there are already the general formula for this case but I cannot find them. Can anybody help me?

Answer 1

In this 3 parameter form ($E(y)=ae^{−bx}+c$), your question has a nice simple structure that can be exploited.

In particular, if you condition on $b$, and let $z_i=e^{-bx_i}$, you have

$E(y|b)=az+c$ and need to find the solution to:

$\text{argmin}_{a,c}\sum_i \left( az_i+c - y_i \right)^2\,,$

which is just an ordinary linear regression problem.

With a little additional effort (i.e. substitution of the solution for $\hat{a}$ and $\hat{c}$ as a function of $y$, $x$ and $b$ into the original LS problem), this reduces the problem to a univariate optimization problem (in $b$), which might make life somewhat easier; it could be tackled by routine optimization, for example.

That said, I'd still probably just call a nonlinear least squares function for this problem.

(The same trick could be used on your two parameter version in the other question, but I think any gain there is even smaller and I'd be even less inclined to do it.)