Results 1 to 3 of 3

Math Help - Need to Derive Inverse Sigmoid Function with Max and Min settings

  1. #1
    Newbie
    Joined
    Aug 2012
    From
    United States
    Posts
    4

    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.
    Need to Derive Inverse Sigmoid Function with Max and Min settings-623_doseflow_1.png

    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?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Sep 2009
    Posts
    242
    Thanks
    1

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

    Quote Originally Posted by jnscollier View Post
    ...
    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.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Aug 2012
    From
    United States
    Posts
    4

    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.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: March 27th 2011, 08:19 AM
  2. Derivative of sigmoid function query
    Posted in the Calculus Forum
    Replies: 2
    Last Post: January 14th 2011, 07:06 AM
  3. Replies: 5
    Last Post: September 3rd 2010, 01:45 AM
  4. Sigmoid function question...
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: April 28th 2009, 10:22 AM
  5. sigmoid function
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: August 7th 2006, 10:18 PM

Search Tags


/mathhelpforum @mathhelpforum