Backfitting algorithm - Revision history

198.102.151.244 at 17:36, 20 September 2024

2024-09-20T17:36:18Z

← Previous revision		Revision as of 17:36, 20 September 2024
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	Additive models are a class of [[Nonparametric regression\|non-parametric regression]] models of the form:		Additive models are a class of [[Nonparametric regression\|non-parametric regression]] models of the form:

	: <math> Y_i = \alpha + \sum_{j=1}^p f_j(X_{j}) + \epsilon_i </math>		: <math> Y_i = \alpha + \sum_{j=1}^p f_j(X_{ij}) + \epsilon_i </math>

	where each <math>X_1, X_2, \ldots, X_p </math> is a variable in our <math>p</math>-dimensional predictor <math>X</math>, and <math>Y</math> is our outcome variable. <math>\epsilon</math> represents our inherent error, which is assumed to have mean zero. The <math>f_j</math> represent unspecified smooth functions of a single <math>X_j</math>. Given the flexibility in the <math>f_j</math>, we typically do not have a unique solution: <math>\alpha</math> is left unidentifiable as one can add any constants to any of the <math>f_j</math> and subtract this value from <math>\alpha</math>. It is common to rectify this by constraining		where each <math>X_1, X_2, \ldots, X_p </math> is a variable in our <math>p</math>-dimensional predictor <math>X</math>, and <math>Y</math> is our outcome variable. <math>\epsilon</math> represents our inherent error, which is assumed to have mean zero. The <math>f_j</math> represent unspecified smooth functions of a single <math>X_j</math>. Given the flexibility in the <math>f_j</math>, we typically do not have a unique solution: <math>\alpha</math> is left unidentifiable as one can add any constants to any of the <math>f_j</math> and subtract this value from <math>\alpha</math>. It is common to rectify this by constraining

141.223.38.39 at 11:44, 18 March 2024

2024-03-18T11:44:17Z

← Previous revision		Revision as of 11:44, 18 March 2024
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	Additive models are a class of [[Nonparametric regression\|non-parametric regression]] models of the form:		Additive models are a class of [[Nonparametric regression\|non-parametric regression]] models of the form:

	: <math> Y_i = \alpha + \sum_{j=1}^p f_j(X_{ij}) + \epsilon_i </math>		: <math> Y_i = \alpha + \sum_{j=1}^p f_j(X_{j}) + \epsilon_i </math>

	where each <math>X_1, X_2, \ldots, X_p </math> is a variable in our <math>p</math>-dimensional predictor <math>X</math>, and <math>Y</math> is our outcome variable. <math>\epsilon</math> represents our inherent error, which is assumed to have mean zero. The <math>f_j</math> represent unspecified smooth functions of a single <math>X_j</math>. Given the flexibility in the <math>f_j</math>, we typically do not have a unique solution: <math>\alpha</math> is left unidentifiable as one can add any constants to any of the <math>f_j</math> and subtract this value from <math>\alpha</math>. It is common to rectify this by constraining		where each <math>X_1, X_2, \ldots, X_p </math> is a variable in our <math>p</math>-dimensional predictor <math>X</math>, and <math>Y</math> is our outcome variable. <math>\epsilon</math> represents our inherent error, which is assumed to have mean zero. The <math>f_j</math> represent unspecified smooth functions of a single <math>X_j</math>. Given the flexibility in the <math>f_j</math>, we typically do not have a unique solution: <math>\alpha</math> is left unidentifiable as one can add any constants to any of the <math>f_j</math> and subtract this value from <math>\alpha</math>. It is common to rectify this by constraining

	: <math>\sum_{i = 1}^N f_j(X_{j}) = 0</math> for all <math>j</math>		: <math>\sum_{i = 1}^N f_j(X_{ij}) = 0</math> for all <math>j</math>

	leaving		leaving

141.223.38.39 at 11:43, 18 March 2024

2024-03-18T11:43:49Z

← Previous revision		Revision as of 11:43, 18 March 2024
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	where each <math>X_1, X_2, \ldots, X_p </math> is a variable in our <math>p</math>-dimensional predictor <math>X</math>, and <math>Y</math> is our outcome variable. <math>\epsilon</math> represents our inherent error, which is assumed to have mean zero. The <math>f_j</math> represent unspecified smooth functions of a single <math>X_j</math>. Given the flexibility in the <math>f_j</math>, we typically do not have a unique solution: <math>\alpha</math> is left unidentifiable as one can add any constants to any of the <math>f_j</math> and subtract this value from <math>\alpha</math>. It is common to rectify this by constraining		where each <math>X_1, X_2, \ldots, X_p </math> is a variable in our <math>p</math>-dimensional predictor <math>X</math>, and <math>Y</math> is our outcome variable. <math>\epsilon</math> represents our inherent error, which is assumed to have mean zero. The <math>f_j</math> represent unspecified smooth functions of a single <math>X_j</math>. Given the flexibility in the <math>f_j</math>, we typically do not have a unique solution: <math>\alpha</math> is left unidentifiable as one can add any constants to any of the <math>f_j</math> and subtract this value from <math>\alpha</math>. It is common to rectify this by constraining

	: <math>\sum_{i = 1}^N f_j(X_{ij}) = 0</math> for all <math>j</math>		: <math>\sum_{i = 1}^N f_j(X_{j}) = 0</math> for all <math>j</math>

	leaving		leaving

Qninja at 12:30, 29 May 2022

2022-05-29T12:30:03Z

← Previous revision		Revision as of 12:30, 29 May 2022
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	{{Short description\|Iterative procedure}}		{{Short description\|Iterative procedure}}
	In [[statistics]], the '''backfitting algorithm''' is a simple iterative procedure used to fit a [[generalized additive model]]. It was introduced in 1985 by [[Leo Breiman]] and [[Jerome H. Friedman\|Jerome Friedman]] along with generalized additive models. In most cases, the backfitting algorithm is equivalent to the [[Gauss–Seidel\|Gauss–Seidel method]] algorithm for solving a certain [[linear system of equations]].		In [[statistics]], the '''backfitting algorithm''' is a simple iterative procedure used to fit a [[generalized additive model]]. It was introduced in 1985 by [[Leo Breiman]] and [[Jerome H. Friedman\|Jerome Friedman]] along with generalized additive models. In most cases, the backfitting algorithm is equivalent to the [[Gauss–Seidel\|Gauss–Seidel method]], an algorithm used for solving a certain [[linear system of equations]].

	==Algorithm==		==Algorithm==

VaudevillianScientist at 14:23, 5 May 2022

2022-05-05T14:23:04Z

← Previous revision		Revision as of 14:23, 5 May 2022
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	{{Short description\|Iterative procedure}}		{{Short description\|Iterative procedure}}
	In [[statistics]], the '''backfitting algorithm''' is a simple iterative procedure used to fit a [[generalized additive model]]. It was introduced in 1985 by Leo Breiman and Jerome Friedman along with generalized additive models. In most cases, the backfitting algorithm is equivalent to the [[Gauss–Seidel\|Gauss–Seidel method]] algorithm for solving a certain linear system of equations.		In [[statistics]], the '''backfitting algorithm''' is a simple iterative procedure used to fit a [[generalized additive model]]. It was introduced in 1985 by [[Leo Breiman]] and [[Jerome H. Friedman\|Jerome Friedman]] along with generalized additive models. In most cases, the backfitting algorithm is equivalent to the [[Gauss–Seidel\|Gauss–Seidel method]] algorithm for solving a certain [[linear system of equations]].

	==Algorithm==		==Algorithm==
	Additive models are a class of non-parametric regression models of the form:		Additive models are a class of [[Nonparametric regression\|non-parametric regression]] models of the form:

	: <math> Y_i = \alpha + \sum_{j=1}^p f_j(X_{ij}) + \epsilon_i </math>		: <math> Y_i = \alpha + \sum_{j=1}^p f_j(X_{ij}) + \epsilon_i </math>

Qwerfjkl (bot): Capitalising short description "iterative procedure" per WP:SDFORMAT (via Bandersnatch)

2022-02-06T21:01:36Z

Capitalising short description "iterative procedure" per WP:SDFORMAT (via Bandersnatch)

← Previous revision		Revision as of 21:01, 6 February 2022
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	{{Short description\|~~iterative~~ procedure}}		{{Short description\|Iterative procedure}}
	In [[statistics]], the '''backfitting algorithm''' is a simple iterative procedure used to fit a [[generalized additive model]]. It was introduced in 1985 by Leo Breiman and Jerome Friedman along with generalized additive models. In most cases, the backfitting algorithm is equivalent to the [[Gauss–Seidel\|Gauss–Seidel method]] algorithm for solving a certain linear system of equations.		In [[statistics]], the '''backfitting algorithm''' is a simple iterative procedure used to fit a [[generalized additive model]]. It was introduced in 1985 by Leo Breiman and Jerome Friedman along with generalized additive models. In most cases, the backfitting algorithm is equivalent to the [[Gauss–Seidel\|Gauss–Seidel method]] algorithm for solving a certain linear system of equations.

NAddleman: /* Motivation */ add inline math formatting for variables (minor)

2021-12-29T16:22:10Z

Motivation: add inline math formatting for variables (minor)

← Previous revision		Revision as of 16:22, 29 December 2021
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	: <math> \hat{S}f = QY \, </math>		: <math> \hat{S}f = QY \, </math>

	An exact solution of this is infeasible to calculate for large np, so the iterative technique of backfitting is used. We take initial guesses <math>f_j^{(0)}</math> and update each <math>f_j^{(\ell)}</math> in turn to be the smoothed fit for the residuals of all the others:		An exact solution of this is infeasible to calculate for large ''np'', so the iterative technique of backfitting is used. We take initial guesses <math>f_j^{(0)}</math> and update each <math>f_j^{(\ell)}</math> in turn to be the smoothed fit for the residuals of all the others:

	: <math> \hat{f_j}^{(\ell)} \leftarrow \text{Smooth}[\lbrace y_i - \hat{\alpha} - \sum_{k \neq j} \hat{f_k}(x_{ik}) \rbrace_1^N ]</math>		: <math> \hat{f_j}^{(\ell)} \leftarrow \text{Smooth}[\lbrace y_i - \hat{\alpha} - \sum_{k \neq j} \hat{f_k}(x_{ik}) \rbrace_1^N ]</math>

Rlink2: /* External links /archive link repair, may include: archive. -> archive.today, and http->https for ghostarchive.org and archive.org (wp:el#Specifying_protocols)

2021-12-05T14:01:21Z

External links: archive link repair, may include: archive.* -> archive.today, and http->https for ghostarchive.org and archive.org (wp:el#Specifying_protocols)

← Previous revision		Revision as of 14:01, 5 December 2021
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	==External links==		==External links==
	*[https://archive.is/20121211125906/http://rss.acs.unt.edu/Rdoc/library/gam/html/gam.html R Package for GAM backfitting]		*[https://archive.today/20121211125906/http://rss.acs.unt.edu/Rdoc/library/gam/html/gam.html R Package for GAM backfitting]
	*[https://web.archive.org/web/20061121130651/http://pbil.univ-lyon1.fr/library/mda/html/bruto.html R Package for BRUTO backfitting]		*[https://web.archive.org/web/20061121130651/http://pbil.univ-lyon1.fr/library/mda/html/bruto.html R Package for BRUTO backfitting]

Hthrhthr12: /* Motivation */ Fixed typo

2021-06-26T21:40:31Z

Motivation: Fixed typo

← Previous revision		Revision as of 21:40, 26 June 2021
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	: <math> \hat{S}f = QY \, </math>		: <math> \hat{S}f = QY \, </math>

	An exact solution of this is infeasible to calculate for large np, so the iterative technique of backfitting is used. We take initial guesses <math>~~f_i~~^{(0)}</math> and update each <math>~~f_i~~^{(j)}</math> in turn to be the smoothed fit for the residuals of all the others:		An exact solution of this is infeasible to calculate for large np, so the iterative technique of backfitting is used. We take initial guesses <math>f_j^{(0)}</math> and update each <math>f_j^{(\ell)}</math> in turn to be the smoothed fit for the residuals of all the others:

	: <math> \hat{~~f_i~~}^{(j)} \leftarrow \text{Smooth}[\lbrace y_i - \hat{\alpha} - \sum_{k \neq j} \hat{f_k}(x_{ik}) \rbrace_1^N ]</math>		: <math> \hat{f_j}^{(\ell)} \leftarrow \text{Smooth}[\lbrace y_i - \hat{\alpha} - \sum_{k \neq j} \hat{f_k}(x_{ik}) \rbrace_1^N ]</math>

	Looking at the abbreviated form it is easy to see the backfitting algorithm as equivalent to the [[Gauss–Seidel\|Gauss–Seidel method]] for linear smoothing operators ''S''.		Looking at the abbreviated form it is easy to see the backfitting algorithm as equivalent to the [[Gauss–Seidel\|Gauss–Seidel method]] for linear smoothing operators ''S''.

Proteusiscrao: #suggestededit-add 1.0

2021-04-05T09:06:01Z

#suggestededit-add 1.0

← Previous revision		Revision as of 09:06, 5 April 2021
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			{{Short description\|iterative procedure}}
	In [[statistics]], the '''backfitting algorithm''' is a simple iterative procedure used to fit a [[generalized additive model]]. It was introduced in 1985 by Leo Breiman and Jerome Friedman along with generalized additive models. In most cases, the backfitting algorithm is equivalent to the [[Gauss–Seidel\|Gauss–Seidel method]] algorithm for solving a certain linear system of equations.		In [[statistics]], the '''backfitting algorithm''' is a simple iterative procedure used to fit a [[generalized additive model]]. It was introduced in 1985 by Leo Breiman and Jerome Friedman along with generalized additive models. In most cases, the backfitting algorithm is equivalent to the [[Gauss–Seidel\|Gauss–Seidel method]] algorithm for solving a certain linear system of equations.