I'm trying to fit a function that describes the following data, ideally with as few parameters as possible:
The data has the following properties:
- X and Y go from 0 to 1
- It is an S-Shape growth function.
- The inflection point is not necessarily at X or Y = 0.5
- Growth is initially slow, but then becomes more rapid. The opposite of the Gompertz growth curve to my understanding.
If it helps, here is the data from the graph:
X = [0 0.1818 0.3636 0.5455 0.7273 0.8182 0.9091 1.0000]
Y = [0 0.01 0.025 0.15 0.4 0.8 0.98 1]
I think using a function from the sigmoid family might be a good start. The Gompertz growth curve is part of this family:
$$f(x) = \frac{a}{b + e^{-kx}} + c$$
Where $a$, $b$, $c$, and $k$ are the model parameters.
If this curve is too symmetrical you can try a more generic sigmoid-esque curve:
$$f(x) = \frac{1}{1 + ae^{-k_1 x} + be^{-k_2 x}} + c$$
Hope this helps.