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Change of Variables(nonlinear function to a linear one)

  1. May 17, 2009 #1
    1. The problem statement, all variables and given/known data
    I'm given a nonlinear model which I am trying to perform a linear regression on. I need to use change of variables in order to convert the nonlinear model into a linear one.

    2. Relevant equations
    I am given [tex]f(x) = \frac{ax}{b+x}[/tex] where a and b are just parameters.
    I need to use change of varibles to change this function into a form of [tex]F(x) = AX + B[/tex]


    3. The attempt at a solution
    I have tried to take the reciprocal of both sides to get
    [tex]\frac{1}{y} = \frac{b+x}{ax} = \frac{b}{ax} + \frac{1}{a}[/tex] , only problem with that is that the x is in the denominator. I have also tried using log transformation to both sides without success. Any help is appreciated thanks!
     
  2. jcsd
  3. May 17, 2009 #2
    see below
     
    Last edited: May 17, 2009
  4. May 17, 2009 #3
    I would rewrite it as b*f(x) - ax = x*f(x)

    or in matrix form

    [tex] \left(\begin{array}{cc}x_{1}&f(x_{1})\\x_{2}&f(x{2})\\.&.\\.&.\\.&.\\x_{n}&f(x_{n})\end{array}\right) \left(\begin{array}{cc}-a\\b\end{array}\right) = \left(\begin{array}{cc}x{1}*f(x_{1})\\x{2}*f(x{2})\\.\\.\\.\\x_{n}*f(x_{n})\end{array}\right)[/tex]
     
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