Estimating Spline curve by OLS. Is a good idea to fix the knots at Chebyshev sites?
$begingroup$
I am writing my master's degree thesis on a novel method for fixing knots in an adaptive way and while reading the literature I've found many references to the so-called Chebyshev sites. This sites or points are basically the roots of the Chebyshev polynomials, and there's a proof showing that the Lebesgue constant (a measure that allows us to see how good is a polynomial interpolation) is very close to its lower bound if we put our knots in those sites. A practical guide to splines (De Boor, 1972) for example provides such a proof.
Anyway I am not interested in polynomial interpolation with splines, but in regression using LASSO on a set of B-splines . On Numerical methods in economics (Judd, 1998) it's stated that the results regarding the 'optimality' of the Chebyshev sites hold also for the regression case, but there is no proof showing that the latter is true.
I would like to know if it's a good idea to estimate the regression using B-splines constructed over a knot sequence given by the Chebyshev sites, since in the OLS framework I am not interested on minimising the Lebesgue constant but rather I want to minimise the $||.||_2^2$. In many articles I've found phrases like "we will set the knots at the Chebyshev sites, that are well known to be good..." making allusion to the results presented in De Boor, but ignoring that those refer to the interpolation case.
If you could put some light on the problem I will be very grateful.
polynomials regression interpolation spline chebyshev-polynomials
edited Jan 24 at 12:41
amWhy
1
asked Jan 14 at 14:37
Ramiro ScorolliRamiro Scorolli
637114
$endgroup$
add a comment |
$begingroup$
I am writing my master's degree thesis on a novel method for fixing knots in an adaptive way and while reading the literature I've found many references to the so-called Chebyshev sites. This sites or points are basically the roots of the Chebyshev polynomials, and there's a proof showing that the Lebesgue constant (a measure that allows us to see how good is a polynomial interpolation) is very close to its lower bound if we put our knots in those sites. A practical guide to splines (De Boor, 1972) for example provides such a proof.
Anyway I am not interested in polynomial interpolation with splines, but in regression using LASSO on a set of B-splines . On Numerical methods in economics (Judd, 1998) it's stated that the results regarding the 'optimality' of the Chebyshev sites hold also for the regression case, but there is no proof showing that the latter is true.
I would like to know if it's a good idea to estimate the regression using B-splines constructed over a knot sequence given by the Chebyshev sites, since in the OLS framework I am not interested on minimising the Lebesgue constant but rather I want to minimise the $||.||_2^2$. In many articles I've found phrases like "we will set the knots at the Chebyshev sites, that are well known to be good..." making allusion to the results presented in De Boor, but ignoring that those refer to the interpolation case.
If you could put some light on the problem I will be very grateful.
polynomials regression interpolation spline chebyshev-polynomials
edited Jan 24 at 12:41
amWhy
1
asked Jan 14 at 14:37
Ramiro ScorolliRamiro Scorolli
637114
$endgroup$
$begingroup$
Either let the knots be parameters or choose them according to the density of abscissas of your data.
$endgroup$
– Oppenede
Jan 18 at 13:56
$begingroup$
I have in mind other methods for fitting knots vector, but I am asking particularly about Chebyshev sites. @Oppenede
$endgroup$
– Ramiro Scorolli
Jan 18 at 16:31
add a comment |
$begingroup$
I am writing my master's degree thesis on a novel method for fixing knots in an adaptive way and while reading the literature I've found many references to the so-called Chebyshev sites. This sites or points are basically the roots of the Chebyshev polynomials, and there's a proof showing that the Lebesgue constant (a measure that allows us to see how good is a polynomial interpolation) is very close to its lower bound if we put our knots in those sites. A practical guide to splines (De Boor, 1972) for example provides such a proof.
Anyway I am not interested in polynomial interpolation with splines, but in regression using LASSO on a set of B-splines . On Numerical methods in economics (Judd, 1998) it's stated that the results regarding the 'optimality' of the Chebyshev sites hold also for the regression case, but there is no proof showing that the latter is true.
I would like to know if it's a good idea to estimate the regression using B-splines constructed over a knot sequence given by the Chebyshev sites, since in the OLS framework I am not interested on minimising the Lebesgue constant but rather I want to minimise the $||.||_2^2$. In many articles I've found phrases like "we will set the knots at the Chebyshev sites, that are well known to be good..." making allusion to the results presented in De Boor, but ignoring that those refer to the interpolation case.
If you could put some light on the problem I will be very grateful.
polynomials regression interpolation spline chebyshev-polynomials
edited Jan 24 at 12:41
amWhy
1
asked Jan 14 at 14:37
Ramiro ScorolliRamiro Scorolli
637114
$endgroup$
I am writing my master's degree thesis on a novel method for fixing knots in an adaptive way and while reading the literature I've found many references to the so-called Chebyshev sites. This sites or points are basically the roots of the Chebyshev polynomials, and there's a proof showing that the Lebesgue constant (a measure that allows us to see how good is a polynomial interpolation) is very close to its lower bound if we put our knots in those sites. A practical guide to splines (De Boor, 1972) for example provides such a proof.
Anyway I am not interested in polynomial interpolation with splines, but in regression using LASSO on a set of B-splines . On Numerical methods in economics (Judd, 1998) it's stated that the results regarding the 'optimality' of the Chebyshev sites hold also for the regression case, but there is no proof showing that the latter is true.
I would like to know if it's a good idea to estimate the regression using B-splines constructed over a knot sequence given by the Chebyshev sites, since in the OLS framework I am not interested on minimising the Lebesgue constant but rather I want to minimise the $||.||_2^2$. In many articles I've found phrases like "we will set the knots at the Chebyshev sites, that are well known to be good..." making allusion to the results presented in De Boor, but ignoring that those refer to the interpolation case.
If you could put some light on the problem I will be very grateful.
polynomials regression interpolation spline chebyshev-polynomials
polynomials regression interpolation spline chebyshev-polynomials
edited Jan 24 at 12:41
amWhy
1
asked Jan 14 at 14:37
Ramiro ScorolliRamiro Scorolli
637114
edited Jan 24 at 12:41
amWhy
1
asked Jan 14 at 14:37
Ramiro ScorolliRamiro Scorolli
637114
edited Jan 24 at 12:41
amWhy
1
edited Jan 24 at 12:41
amWhy
1
edited Jan 24 at 12:41
amWhy
1
1
asked Jan 14 at 14:37
Ramiro ScorolliRamiro Scorolli
637114
asked Jan 14 at 14:37
Ramiro ScorolliRamiro Scorolli
637114
asked Jan 14 at 14:37
Ramiro ScorolliRamiro Scorolli
637114
637114
$begingroup$
Either let the knots be parameters or choose them according to the density of abscissas of your data.
$endgroup$
– Oppenede
Jan 18 at 13:56
$begingroup$
I have in mind other methods for fitting knots vector, but I am asking particularly about Chebyshev sites. @Oppenede
$endgroup$
– Ramiro Scorolli
Jan 18 at 16:31
add a comment |
$begingroup$
Either let the knots be parameters or choose them according to the density of abscissas of your data.
$endgroup$
– Oppenede
Jan 18 at 13:56
$begingroup$
I have in mind other methods for fitting knots vector, but I am asking particularly about Chebyshev sites. @Oppenede
$endgroup$
– Ramiro Scorolli
Jan 18 at 16:31
$begingroup$
Either let the knots be parameters or choose them according to the density of abscissas of your data.
$endgroup$
– Oppenede
Jan 18 at 13:56
$begingroup$
Either let the knots be parameters or choose them according to the density of abscissas of your data.
$endgroup$
– Oppenede
Jan 18 at 13:56
$begingroup$
I have in mind other methods for fitting knots vector, but I am asking particularly about Chebyshev sites. @Oppenede
$endgroup$
– Ramiro Scorolli
Jan 18 at 16:31
$begingroup$
I have in mind other methods for fitting knots vector, but I am asking particularly about Chebyshev sites. @Oppenede
$endgroup$
– Ramiro Scorolli
Jan 18 at 16:31
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
The key moment is that the polynomial regression requires specific criteria of "optimality". Using statistic approaches, one can obtain the point influence criterion for their selected location.
To estimate the point influence for their given location, can be recommended the theory of the orthogonal polynomial regression, based on the article Orthogonal Polynomial Curve Fitting, Jeff Reid. The first step of this approach is the synthesis of the family of orthogonal polynomials by the given abscissa array, which does not use the ordinates (see the plot of polynomials for the common case of abscissa arrangement).
The sample variances, using absolute values of the normalized high-degree polynomials in the nodes, can be considered as the estimation of this influence.
answered Jan 24 at 12:39
Yuri NegometyanovYuri Negometyanov
11.4k1728
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3073300%2festimating-spline-curve-by-ols-is-a-good-idea-to-fix-the-knots-at-chebyshev-sit%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The key moment is that the polynomial regression requires specific criteria of "optimality". Using statistic approaches, one can obtain the point influence criterion for their selected location.
To estimate the point influence for their given location, can be recommended the theory of the orthogonal polynomial regression, based on the article Orthogonal Polynomial Curve Fitting, Jeff Reid. The first step of this approach is the synthesis of the family of orthogonal polynomials by the given abscissa array, which does not use the ordinates (see the plot of polynomials for the common case of abscissa arrangement).
The sample variances, using absolute values of the normalized high-degree polynomials in the nodes, can be considered as the estimation of this influence.
answered Jan 24 at 12:39
Yuri NegometyanovYuri Negometyanov
11.4k1728
$endgroup$
add a comment |
$begingroup$
The key moment is that the polynomial regression requires specific criteria of "optimality". Using statistic approaches, one can obtain the point influence criterion for their selected location.
To estimate the point influence for their given location, can be recommended the theory of the orthogonal polynomial regression, based on the article Orthogonal Polynomial Curve Fitting, Jeff Reid. The first step of this approach is the synthesis of the family of orthogonal polynomials by the given abscissa array, which does not use the ordinates (see the plot of polynomials for the common case of abscissa arrangement).
The sample variances, using absolute values of the normalized high-degree polynomials in the nodes, can be considered as the estimation of this influence.
answered Jan 24 at 12:39
Yuri NegometyanovYuri Negometyanov
11.4k1728
$endgroup$
add a comment |
$begingroup$
The key moment is that the polynomial regression requires specific criteria of "optimality". Using statistic approaches, one can obtain the point influence criterion for their selected location.
To estimate the point influence for their given location, can be recommended the theory of the orthogonal polynomial regression, based on the article Orthogonal Polynomial Curve Fitting, Jeff Reid. The first step of this approach is the synthesis of the family of orthogonal polynomials by the given abscissa array, which does not use the ordinates (see the plot of polynomials for the common case of abscissa arrangement).
The sample variances, using absolute values of the normalized high-degree polynomials in the nodes, can be considered as the estimation of this influence.
answered Jan 24 at 12:39
Yuri NegometyanovYuri Negometyanov
11.4k1728
$endgroup$
The key moment is that the polynomial regression requires specific criteria of "optimality". Using statistic approaches, one can obtain the point influence criterion for their selected location.
To estimate the point influence for their given location, can be recommended the theory of the orthogonal polynomial regression, based on the article Orthogonal Polynomial Curve Fitting, Jeff Reid. The first step of this approach is the synthesis of the family of orthogonal polynomials by the given abscissa array, which does not use the ordinates (see the plot of polynomials for the common case of abscissa arrangement).
The sample variances, using absolute values of the normalized high-degree polynomials in the nodes, can be considered as the estimation of this influence.
answered Jan 24 at 12:39
Yuri NegometyanovYuri Negometyanov
11.4k1728
answered Jan 24 at 12:39
Yuri NegometyanovYuri Negometyanov
11.4k1728
answered Jan 24 at 12:39
Yuri NegometyanovYuri Negometyanov
11.4k1728
answered Jan 24 at 12:39
Yuri NegometyanovYuri Negometyanov
11.4k1728
11.4k1728
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3073300%2festimating-spline-curve-by-ols-is-a-good-idea-to-fix-the-knots-at-chebyshev-sit%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
Either let the knots be parameters or choose them according to the density of abscissas of your data.
$endgroup$
– Oppenede
Jan 18 at 13:56
$begingroup$
I have in mind other methods for fitting knots vector, but I am asking particularly about Chebyshev sites. @Oppenede
$endgroup$
– Ramiro Scorolli
Jan 18 at 16:31