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Minimizing distance between two function

  1. Apr 13, 2013 #1
    Consider the functions
    [tex] g(x)=e^x/(1+e^x)^2 [/tex]
    and
    [tex]f(x) = vf_1(x) + (1-v)f_2(x) [/tex].
    where
    [tex] f_1(x)=\dfrac{1}{\sqrt{2\pi\sigma^2_1}}e^{-\dfrac{x^2}{2\sigma^2_1}},\,\,\, f_2(x)=\dfrac{1}{\sqrt{2\pi\sigma^2_2}}e^{-\dfrac{x^2}{2\sigma^2_2}}[/tex]
    and
    [tex] -\infty < x < \infty, \,\,\, 0 < v < 1, \,\,\, \sigma^2_1 > 0, \,\,\, \sigma^2_2 > 0. [/tex]

    What I want to do basically is to approximate the function g(x) with f(x) by selecting proper values of [tex] v, \sigma^2_1, \sigma^2_2. [/tex] Just by inspection and manual iteration I concluded that something like [tex] v=0.3, \sigma^2_1=6.2, \sigma^2_2=1.95 [/tex] provides a fairly good approximation, but of course, I want to carry out the analysis more formally.

    First I considered minimizing
    [tex] \int_{-\infty}^{\infty} ( g(x) - (vf_1(x) + (1-v)f_2(x)) )^2 dx. [/tex]
    The problems is that this integral will not yield an analytical result.

    Second, what could also be of interest is to minimize abs(g(x)-f(x)) for all x so that we will at least know the maximum error that could occur.

    My question is therefore twofold:
    1) Do you have any suggestions about solving this problem analytically?
    2) If resorting to numerical optimization when minimizing the integral above, how do I implement it in Mathematica? I tried the following code but it failed.

    fz1 := (1/Sqrt[2*Pi*vz1])*Exp[-z^2/(2*vz1)]
    fz2 := (1/Sqrt[2*Pi*vz2])*Exp[-z^2/(2*vz2)]
    gz := Exp[z]/(1 + Exp[z])^2
    uz := gz - (v*fz1 + (1 - v)*fz2)
    Uz[vz1_?NumericQ, vz2_?NumericQ, v_?NumericQ] :=
    NIntegrate[uz^2, {z, -Infinity, Infinity}]
    NMinimize[{Uz, vz1 > 0 && vz2 > 0 && v > 0 && v < 1}, {vz1, vz2, v}]

    All suggestions are most welcome!
     
  2. jcsd
  3. Apr 13, 2013 #2

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    Even if an analytic approach is possible (I don't know), a numeric solution is way easier.

    How does Mathematica fail? Can you give some starting values?
    You could try to remove the constraints on the variables. If the restrictions for v are really necessary, but problematic, you can use something like v=1/(1+e^w) with unbounded, free w.
     
  4. Apr 13, 2013 #3
    Mathematica reports:
    NMinimize::nnum: The function value Uz is not a number at {v,vz1,vz2} = {0.652468,1.26614,1.36563}
    but I don't see the error.

    The restrictions are necessary, but I will try you suggestion!


    Edit:
    Now I tried the following code below and although I got several error messages the result
    {2.9583*10^-6, {v -> 0.107599, vz1 -> 1.53645, vz2 -> 4.66904}}
    seems good when plotting the functions.

    fz1 := (1/Sqrt[2*Pi*vz1])*Exp[-z^2/(2*vz1)]
    fz2 := (1/Sqrt[2*Pi*vz2])*Exp[-z^2/(2*vz2)]
    gz := Exp[z]/(1 + Exp[z])^2
    uz[vz1_?NumericQ, vz2_?NumericQ, v_?NumericQ] :=
    gz - ((1/(1 + Exp[v]))*fz1 + (1 - (1/(1 + Exp[v])))*fz2)
    NMinimize[{NIntegrate[uz^2, {z, -Infinity, Infinity}], vz1 > 0 && vz2 > 0}, {v, vz1, vz2}]
     
    Last edited: Apr 13, 2013
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