Thread: Need to Derive Inverse Sigmoid Function with Max and Min settings

1. Need to Derive Inverse Sigmoid Function with Max and Min settings

Q: What am I trying to accomplish?
A: I am trying to derive a mathematical function for an inverse signoid line that starts out at the max value and over time declines to a min value that I define.

Example
$CPC is plotted on the Y axis, Total cost is plotted on the X axis. I am trying to determine what the$ CPC (between $0.00 &$10 for example) should be given a certain level of total cost (between $0.00 &$100 for example)

Here's a picture the function I am trying to derive.

I basically want the formula to start out at the Max CPC (Let's say $10) and as cost increases, to decrease logarithmically but not below a MIN CPC value (Let's say$0.50). At the MIN CPC value, I want it to be a horizontal asymptote.

Is there a function that will let me input the Max CPC and MIN CPC to be the Ceiling and Floor and then essentially decay based on the level of Total Cost?

2. Re: Need to Derive Inverse Sigmoid Function with Max and Min settings

Originally Posted by jnscollier
...
A: I am trying to derive a mathematical function for an inverse signoid line that starts out at the max value and over time declines to a min value that I define.
...
I basically want the formula to start out at the Max CPC (Let's say $10) and as cost increases, to decrease logarithmically but not below a MIN CPC value (Let's say$0.50). At the MIN CPC value, I want it to be a horizontal asymptote.

Is there a function that will let me input the Max CPC and MIN CPC to be the Ceiling and Floor and then essentially decay based on the level of Total Cost?
You have a few choices. If you haven't already, look up sigmoid function on wikipedia: Sigmoid function - Wikipedia, the free encyclopedia

In general, to get the curve to take only positive x axis values you will have to take the natural log of the argument x. And to get it to face the direction you want (like in your graph) you will have to multiply the function by -1 and add 1. Here are three functions I just tried out on my grapher which seem to match what you're asking for:

Below, the constant "a" is just "Max CPC" minus "Min CPC". You will have to interpret 0 as the Min CPC.

Using the error function:

$\frac{a}{2}(1 - erf(\ln(x)))$

Manipulating the log-logistic function:

$a \left( 1-\frac{1}{(1 + 1/x^2)} \right)$

(you can also try removing the square from the x)

And my personal favorite:

$a\left( 1-e^{-1/x} \right)$

They each "decay" according to different principles. I like the last one the best because 1/x is the elasticity of the function with respect to x, which to my mind models the rate of decay to total cost relation in a clear way.

3. Re: Need to Derive Inverse Sigmoid Function with Max and Min settings

This information is so valuable to me. Thank you so much rainer. I've just begun modifying your equations and I've already come up with the solution I was after.