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I'm working with the four-parameter logistic function.

$y = a + \frac{b-a}{1+e^{c(d-x)}}$

There are two points on the curve at which the oblique portion of the curve meets the lower and upper plateaus of the function.

What methods would be recommended for characterizing these two points? That is, how to characterize the points on the 4-parameter logistic function at which there is maximum 'bend'?

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1 Answer 1

In my experience, we call this function a sigmoid. And what I think you are after, if I am interpreting you correctly, is where the third derivative vanishes, i.e, where the second derivative is maximized. To illustrate: let $f(x) = \frac{1}{1+e^{-x}}$; then

$$f'''(x) = 0 \implies \frac{e^x}{(1+e^x)^4} (1-4 e^x+e^{2 x}) = 0$$

which is satisfied when $x=\log{(2 \pm \sqrt{3})} \approx \pm 1.317$.

Please let me know if that helps. Of course, you have a lot more parameters, but the idea is the same.

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This method seems to overestimate what I'm looking for slightly. I've the vertical red line, here shows the value given at f''(x). I'm looking to characterise the portion of the curve that might be described as the 'knee'. Or the point in the middle of the region that appears to have maximum curvature. Cheers. –  skleene Jan 30 '13 at 9:10
    
Not sure what you mean by $f'(x)$. The plot on the right looks like $f''(x)$ to me, and you want the extrema of this plot, for which you would solve $f'''(x)=0$ to obtain as I outlined in my answer. My number was simply an example; you would need to carry out the analysis for your particular parameters. –  Ron Gordon Jan 30 '13 at 10:41
    
My apologies, my comment was a little sloppy. The plot on the right is indeed $f''(x)$. The vertical red line indicates the derived max value for $f''(x)$ and I have plotted this location on $f(x)$ on the lefthand plot. This method seems to give me a point at the end of the bended region, whereas I'm interested in characterizing the middle of the bended region (it's 'knee'). –  skleene Jan 31 '13 at 12:32

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