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May 21st, 2018, 06:10 AM   #1
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Mathematical equations for sigmoidal shape

Hello all,

I am a student working on a project for my college. I want to create sigmoidal shape graphs to show the relationship between wind speed and wind mill suitability scores. The graph should be such that the x axis shows wind speed and Y axis with suitability scores ranging from from 0 to 1.

For example to fit in these values like x1= 5m/s; y1= 0.1, x2= 6m/s; Y2= start of fast growth, x3= 7m/s; Y3= end of fast growth and x4 m/s= 8.8; y4 = 1. Is it possible to create an equation with 4 points defining start point, two intermediate points and end points.

Could someone please help me with this

Thank a lot in advance,
Deepa
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May 21st, 2018, 06:15 AM   #2
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Hi deepamk and welcome to MMF!

Do you have any ideas on where to start? You can help us help you by posting any ideas on where to begin or possible solutions that you may have.
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May 21st, 2018, 04:14 PM   #3
SDK
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There are lots of sigmoidal functions which have been used in a wide variety of applications. Common simple examples include is the logistic function
\[f(x) = \frac{1}{1+e^{-x}} \]
or the inverse tangent function.

A more robust family of these which is heavily used in biology are the Hill functions given by
\[ H(x) = \frac{x^n}{\theta^n + x^n} \]
where $n,\theta$ are parameters. You should plot these in Mathematica or something similar to explore how changing $n,\theta$ vary the shape. You can add additional parameters to satisfy specific constraints.

For example, the function
\[H(x) = H_0 + (H_1 - H_0) \frac{x^n}{\theta^n + x^n} \]
is monotone increasing and takes values from $H_0$ to $H_1$ for $x \in [0, \infty)$. Varying $n,\theta$ allow control of specific features such as the value of $H'(0)$, the steepness of the activation, half life, etc.

Last edited by SDK; May 21st, 2018 at 04:16 PM.
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