I have some "pre-digested" data, that has been described as a sum of sigmoid functions.
Ascending sigmoids range from 0 to 1. Descending sigmoids range from 0 to -1. Each sigmoid is characterised by 3 parameters (p,q,r), where r is the gradient at the inflection point (p,q): $-\infty \lt x \lt \infty$. Note, this is non-symetrical in that the inflection does not occur at y = 0.5, but at y = q.
I can see why they have done this as it allows an easy first order representation of the data. Trouble is they have not specified the function used for the sigmoid.
So I am looking for a differentiable function for a sigmoid expressed in terms of p,q,r.
I suspect this will be a relatively simple equation, but all the sigmoids I've tried to refactor get really complicated, or can only be done peicewise.
I think there will be multiple solutions, so the cigar goes to the simplest one that satisfies the requirements.
Is anyone familiar with the function I'm looking for?