I have the following data:
y x
1275 230
1350 235
1650 250
2000 277
3750 522
4222 545
5018 625
6125 713
6200 735
8150 820
9975 992
12200 1322
12750 1900
13014 2022
13275 2155
I would like to find reasonable initial values for the model
$$y=\alpha+\beta_1\text{exp}(-\beta_2 e^{-\beta_3 x})+\epsilon$$
What I know:
For the Gompertz model, the inflection point satisfies
$$x=\text{log}\left(\frac{\beta_2}{\beta_3}\right)$$$$x=\frac{\text{log}(\beta_2)}{\beta_3}$$
For the Gompertz model we have
$$\lim_{x\rightarrow\infty} \beta_1\text{exp}(-\beta_2 e^{-\beta_3 x})=\beta_1$$
so presumably with an intercept we have
$$\lim_{x\rightarrow\infty} \alpha+\beta_1\text{exp}(-\beta_2 e^{-\beta_3 x})=\alpha+\beta_1$$
so we can set $\alpha+\beta_1=13275$, the maximum value of $y$ in the dataset.
However, I can't seem to combine what I know to find initial values.
I would like to find reasonable initial values and not rely on specifying an exhaustive grid of values.
Any ideas or suggestions would be appreciated.
Update:
I read on wikipedia that the halfway point is found to be
$${\displaystyle x_{\text{hwp}}=-{\frac {\ln\left(\frac{ln(2)}{\beta_2}\right)}{\beta_3}}}$$
I let $x_{\text{hwp}}=713$, the median of the $x$'s.
As previously stated, I have $$x=\frac{\text{log}(\beta_2)}{\beta_3}$$
I let $x=1000$, since a plot of the data shows that a possible inflection point is around there.
By software, this system of equation results in $\beta_2=0.279$ and $\beta_3=-0.0013$.
I (randomly) decided to let $\alpha=13275$ and $\beta_1=-13275$, where $13275$ is the maximum value of $y$.
The model then converges after 15 iterations to
$$\hat{y}=12934.4-14349.2\cdot\text{exp}(-0.1214e^{0.00257x})$$
which are reasonably close to my initial estimates. I'm not sure why it would make sense to have $\hat{\alpha}=13275$ and $\hat{\beta_1}=-13275$ though.