Polynomial Interpolation using large set of data.
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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.
lagrange-interpolation
asked Dec 18 '18 at 12:06
DavidDavid
408
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show 7 more comments
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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.
lagrange-interpolation
asked Dec 18 '18 at 12:06
DavidDavid
408
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Lagrange interpolant is probably a bad idea for a given set of points $(x_i)_i$, see en.wikipedia.org/wiki/Runge%27s_phenomenon. Explain what properties the interpolant should have (or what it will be used for).
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– Joce
Dec 18 '18 at 12:40
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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)
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– David
Dec 18 '18 at 12:50
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But what will you do with this function? Integrate it, evaluate it outside of the $x_i$'s,...?
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– Joce
Dec 18 '18 at 12:59
1
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Have a look at this : en.wikipedia.org/wiki/Polynomial_regression
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– nicomezi
Dec 18 '18 at 14:23
1
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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.
$endgroup$
– nicomezi
Dec 18 '18 at 14:54
|
show 7 more comments
$begingroup$
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.
lagrange-interpolation
asked Dec 18 '18 at 12:06
DavidDavid
408
$endgroup$
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.
lagrange-interpolation
lagrange-interpolation
asked Dec 18 '18 at 12:06
DavidDavid
408
asked Dec 18 '18 at 12:06
DavidDavid
408
edited Dec 18 '18 at 12:17
David
asked Dec 18 '18 at 12:06
DavidDavid
408
asked Dec 18 '18 at 12:06
DavidDavid
408
asked Dec 18 '18 at 12:06
DavidDavid
408
408
$begingroup$
Lagrange interpolant is probably a bad idea for a given set of points $(x_i)_i$, see en.wikipedia.org/wiki/Runge%27s_phenomenon. Explain what properties the interpolant should have (or what it will be used for).
$endgroup$
– Joce
Dec 18 '18 at 12:40
$begingroup$
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)
$endgroup$
– David
Dec 18 '18 at 12:50
$begingroup$
But what will you do with this function? Integrate it, evaluate it outside of the $x_i$'s,...?
$endgroup$
– Joce
Dec 18 '18 at 12:59
1
$begingroup$
Have a look at this : en.wikipedia.org/wiki/Polynomial_regression
$endgroup$
– nicomezi
Dec 18 '18 at 14:23
1
$begingroup$
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.
$endgroup$
– nicomezi
Dec 18 '18 at 14:54
|
show 7 more comments
$begingroup$
Lagrange interpolant is probably a bad idea for a given set of points $(x_i)_i$, see en.wikipedia.org/wiki/Runge%27s_phenomenon. Explain what properties the interpolant should have (or what it will be used for).
$endgroup$
– Joce
Dec 18 '18 at 12:40
$begingroup$
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)
$endgroup$
– David
Dec 18 '18 at 12:50
$begingroup$
But what will you do with this function? Integrate it, evaluate it outside of the $x_i$'s,...?
$endgroup$
– Joce
Dec 18 '18 at 12:59
1
$begingroup$
Have a look at this : en.wikipedia.org/wiki/Polynomial_regression
$endgroup$
– nicomezi
Dec 18 '18 at 14:23
1
$begingroup$
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.
$endgroup$
– nicomezi
Dec 18 '18 at 14:54
$begingroup$
Lagrange interpolant is probably a bad idea for a given set of points $(x_i)_i$, see en.wikipedia.org/wiki/Runge%27s_phenomenon. Explain what properties the interpolant should have (or what it will be used for).
$endgroup$
– Joce
Dec 18 '18 at 12:40
$begingroup$
Lagrange interpolant is probably a bad idea for a given set of points $(x_i)_i$, see en.wikipedia.org/wiki/Runge%27s_phenomenon. Explain what properties the interpolant should have (or what it will be used for).
$endgroup$
– Joce
Dec 18 '18 at 12:40
$begingroup$
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)
$endgroup$
– David
Dec 18 '18 at 12:50
$begingroup$
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)
$endgroup$
– David
Dec 18 '18 at 12:50
$begingroup$
But what will you do with this function? Integrate it, evaluate it outside of the $x_i$'s,...?
$endgroup$
– Joce
Dec 18 '18 at 12:59
$begingroup$
But what will you do with this function? Integrate it, evaluate it outside of the $x_i$'s,...?
$endgroup$
– Joce
Dec 18 '18 at 12:59
1
1
$begingroup$
Have a look at this : en.wikipedia.org/wiki/Polynomial_regression
$endgroup$
– nicomezi
Dec 18 '18 at 14:23
$begingroup$
Have a look at this : en.wikipedia.org/wiki/Polynomial_regression
$endgroup$
– nicomezi
Dec 18 '18 at 14:23
1
1
$begingroup$
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.
$endgroup$
– nicomezi
Dec 18 '18 at 14:54
$begingroup$
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.
$endgroup$
– nicomezi
Dec 18 '18 at 14:54
|
show 7 more comments
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$begingroup$
Lagrange interpolant is probably a bad idea for a given set of points $(x_i)_i$, see en.wikipedia.org/wiki/Runge%27s_phenomenon. Explain what properties the interpolant should have (or what it will be used for).
$endgroup$
– Joce
Dec 18 '18 at 12:40
$begingroup$
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)
$endgroup$
– David
Dec 18 '18 at 12:50
$begingroup$
But what will you do with this function? Integrate it, evaluate it outside of the $x_i$'s,...?
$endgroup$
– Joce
Dec 18 '18 at 12:59
1
$begingroup$
Have a look at this : en.wikipedia.org/wiki/Polynomial_regression
$endgroup$
– nicomezi
Dec 18 '18 at 14:23
1
$begingroup$
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.
$endgroup$
– nicomezi
Dec 18 '18 at 14:54