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Quick formulation of the problem:
Given two points: $(x_l, y_l)$ and $(x_u, y_u)$
with: $x_l < x_u$ and $y_l < y_u$,
and given lower asymptote=0 and higher asymptote=1, what's the logistic function that passes through the two points?

Explanatory image:

Other details:
I'm given two points in the form of Pareto 90/10 (green in the example above) or 80/20 (blue in the example above), and I know that the upper bound is one and the lower bound is zero.
How do I get the formula of a sigmoid function (such as the logistic function) that has a lower asymptote on the left and higher asymptote on the right and passes via the two points?


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up vote 7 down vote accepted

I believe you're looking for constants $a$ and $b$ so that $f(x_\ell) = y_\ell$ and $f(x_u) = y_u$ where $f(x) = \exp(a + bx) / (1 + \exp(a + bx))$.

This is equivalent to the linear system $a + b x_\ell = g(y_\ell)$ and $a + b x_u = g(y_u)$ where $g(y) = f^{-1}(y) = \log(y/(1-y))$.

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Correct, thanks! – Mino Oct 28 '10 at 23:19

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