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I have a large set of {$x_i$}$_{i=1}^n$ and corresponding values of the function {$f(x_i)$}$_{i=1}^n$. My aim is to estimate the function $f(x)$. Therefore, I think that appropriate technique is Lagrange Polynomial Interpolation. The problem is: as I mentioned I have large set of observations (e.g. 100), then using Lagrange Polynomial Interpolation we will find a $n-1$ degree polynomial, which is meaningless to write explicitly. Please guide me in order to pick up the correct technique or use some modification (if any) to estimate the function, which could be written explicitly.

David
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  • Lagrange interpolant is probably a bad idea for a given set of points $(x_i)_i$, see https://en.wikipedia.org/wiki/Runge%27s_phenomenon. Explain what properties the interpolant should have (or what it will be used for). – Joce Dec 18 '18 at 12:40
  • The aim is to estimate a function form the given dataset. In other words, derive from the data the functional form corresponding to $x_i$ and$f(x_i)$. (Kind of regression analysis) – David Dec 18 '18 at 12:50
  • But what will you do with this function? Integrate it, evaluate it outside of the $x_i$'s,...? – Joce Dec 18 '18 at 12:59
  • There are plenty of techniques. Do you have some hints on $f$ properties ? – nicomezi Dec 18 '18 at 13:10
  • @Joce I want to show that the function is increasing, and $f(0)=0$. – David Dec 18 '18 at 13:43
  • @nicomezi Generally, I know about $f$ the following: (i) $f$ is increasing, (ii) $f(0)=0$, (iii) $f(x)<0,$ when $x<0$, $f(x)>0$, when $x>0$. – David Dec 18 '18 at 13:45
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    Have a look at this : https://en.wikipedia.org/wiki/Polynomial_regression – nicomezi Dec 18 '18 at 14:23
  • @nicomezi Should I give the specification of the model for polynomial regerssion? I mean, should i specify the degree of the polynomial or there is any technique which identifies the degree of the polynomial automatically? (Like Lagrange polynomial interpolation $n-1$ degree for n observations) – David Dec 18 '18 at 14:42
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    You can use Cross Validation. Split your data in two sets, determine the polynom over one set and compute the generalization error over the other set. Repeat the process multiple times for every degree. Finally, choose the degree where the generalization error is minimum (in mean). Once you have chosen a degree, compute the final polynom with the whole data set and you are done. Also, since you know that $f(0)=0$, you can get rid of the constant coefficient. – nicomezi Dec 18 '18 at 14:54
  • @nicomezi Thank you very much for you comment. I got it. What do you about spline interpolation? Do you think that it might be useful in the context of this problem? Is there any way to see the estimated polynomial function explicitly (Like it can be done for Lagrange polynomial interpolation)? Thank you. – David Dec 18 '18 at 15:05
  • Nothing prevents you from plotting the graph of the polynoms once the coefficient have been found. I cannot tell you about spline interpolation since I do not know a lot on this subject and your problem. – nicomezi Dec 18 '18 at 16:56
  • @nicomezi Thank you – David Dec 19 '18 at 07:01

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