Nonlinear least squares with analytical solution











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I want to find a "true" nonlinear least squares problem which does have an analytical solution.



I tried to construct something with a Dirac-Delta function and ended up with $y_n = c^2delta(x_n-x_1)x_n+varepsilon_n$, in which I assumed a dataset $mathcal{D}=left{(x_1,y_1),ldots,(x_N,y_N) right}$. This equation is not really nonlinear in the coefficients as we can reformulate the regression equation to $y_n=tilde{c}delta(x_n-x_1)x_n+varepsilon_n$ with $tilde{c}geq 0$, hence this would still count as a linear regression with nonlinear basis functions.



Is there a well-known example of a nonlinear least squares problem that does have an analytical closed-form solution? References are appreciated.






closed-form least-squares







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    up vote
    1
    down vote

    favorite












    I want to find a "true" nonlinear least squares problem which does have an analytical solution.



    I tried to construct something with a Dirac-Delta function and ended up with $y_n = c^2delta(x_n-x_1)x_n+varepsilon_n$, in which I assumed a dataset $mathcal{D}=left{(x_1,y_1),ldots,(x_N,y_N) right}$. This equation is not really nonlinear in the coefficients as we can reformulate the regression equation to $y_n=tilde{c}delta(x_n-x_1)x_n+varepsilon_n$ with $tilde{c}geq 0$, hence this would still count as a linear regression with nonlinear basis functions.



    Is there a well-known example of a nonlinear least squares problem that does have an analytical closed-form solution? References are appreciated.






    closed-form least-squares







    share|cite|improve this question
























      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite











      I want to find a "true" nonlinear least squares problem which does have an analytical solution.



      I tried to construct something with a Dirac-Delta function and ended up with $y_n = c^2delta(x_n-x_1)x_n+varepsilon_n$, in which I assumed a dataset $mathcal{D}=left{(x_1,y_1),ldots,(x_N,y_N) right}$. This equation is not really nonlinear in the coefficients as we can reformulate the regression equation to $y_n=tilde{c}delta(x_n-x_1)x_n+varepsilon_n$ with $tilde{c}geq 0$, hence this would still count as a linear regression with nonlinear basis functions.



      Is there a well-known example of a nonlinear least squares problem that does have an analytical closed-form solution? References are appreciated.






      closed-form least-squares







      share|cite|improve this question













      I want to find a "true" nonlinear least squares problem which does have an analytical solution.



      I tried to construct something with a Dirac-Delta function and ended up with $y_n = c^2delta(x_n-x_1)x_n+varepsilon_n$, in which I assumed a dataset $mathcal{D}=left{(x_1,y_1),ldots,(x_N,y_N) right}$. This equation is not really nonlinear in the coefficients as we can reformulate the regression equation to $y_n=tilde{c}delta(x_n-x_1)x_n+varepsilon_n$ with $tilde{c}geq 0$, hence this would still count as a linear regression with nonlinear basis functions.



      Is there a well-known example of a nonlinear least squares problem that does have an analytical closed-form solution? References are appreciated.






      closed-form least-squares




      closed-form least-squares






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked 15 hours ago









      MrYouMath

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