At the moment I try to find the best (easiest) way to fit the bounded growth model
$$ y_i = a-ae^{bx_i},$$
with parameter $a,b∈ℝ$, and data points $(x_i,y_i)$. (This form is already simplified, such that the growth starts with $y(0)=0$.)
That model is non-linear, so the first way is to use Non-Linear Least Squares. Sometimes, like in this question it is possible to linearise the model function by using the logarithm. Then you can use Linear Least-Squares to fit the model.
I have tried that, but failed, because of the first summand $a-[...]$.
Is there a way to linearise this model?
Context: The data looks like this
The value $y_i$ will rise and follow the bounded growth model, see red box. The goal is to estimate the bound of that growth, i.e. the purple horizontal line. That is the parameter $a$ in the model. If that is lower/higher than the desired bound, you change system-parameters to achieve the desired bound. In the picture you can see two corrections done to the system.
Any help is appreciated. Also hints to literature are very welcome, as I couldn't find much on that topic. Which surprised me, as I thought that this is some kind of standard growth model.
edit: One way I found is using differentiation on the problem and then use the logarithm resulting in $$\ln(y_i') = \ln(a)+\ln(b)+bx_i,$$ but differentiating measured data is usually not a good idea.
The model being $$y = a(1-e^{bx})$$ in a first step fix $b$ at a given value and define $z_i=1-e^{bx_i}$. So, a very simple linear regression gives $$a=\frac{\sum_{i=1}^n y_iz_i}{\sum_{i=1}^n z_i^2}$$ Now, compute the sum of squares for this given value of $b$. Run a few cases and plot the results. When you see more or less a minimum, you have an estimate of $a,b$ and you can start the nonlinear regression.
But, there is no way to make the model linear (except if you know one of the parameters).
Edit
If you have a good estimate of $a$, then write $$\log \left(\frac{a-y}{a}\right)=bx$$ Define $z_i=\log \left(\frac{a-y_i}{a}\right)$ so the model is $z=b x$ and use $$b=\frac{\sum_{i=1}^n x_iz_i}{\sum_{i=1}^n x_i^2}$$