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Sigmoid curve

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Weaver:
I am lusting after a sigmoid curve function just now. I tried the well-known one
    1 / ( 1 + e-x ).

I stuck it into a spreadsheet, translated and rescaled it a bit and got a pleasing result.

My problem: how do decide how to parametrise. I want the bulk of the thing to fit into a box of certain dimensions. I can rescale and / or translate it vertically to help with that. How do I decide where to position the transition and how fast to make the transition?

That clearly involves x-dimension rescaling translation. I have got one known point at the top of the slope, so that gives me some idea of where the right hand end should be horizontal line (as well as vertically). If I put more scaling factors in then I get a number of additional unknowns. With two known points, if I can get a second point somewhere on it, then I have four values so I can solve for four unknowns - y-translation and scaling could eat up two, that might leave me two more to deal with which would be x-translation and scaling.

So, I need to decide what that other ‘known’ given point could be. And then use that as a basis for back-solving for the scaling parameters that I need to have correct in the midst of the equation. Some k (x-x0) instead of just plain k (x-x0)  and then two for y.

burakkucat:
I can't really help you make a decision.

But I will say that curve stirs my memory and brings thoughts of the A-law or μ-law algorithm, for quantisation, in Pulse Code Modulation to the surface!  :D

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