KMS state - Revision history

en>Headbomb: Various citation cleanup + AWB fixes using AWB

2014-08-20T18:02:08Z

Various citation cleanup + AWB fixes using AWB

← Older revision		Revision as of 20:02, 20 August 2014
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	~~{{Regression bar}}~~		The writer is recognized by the name of Numbers Wunder. To collect cash is what her family members and her enjoy. For many years I've been working as a payroll clerk. My family life in Minnesota and my family members enjoys it.<br><br>my web-site: [http://www.neweracinema.com/tube/user/D69E std home test]
	~~{{context\|date=February 2012}}~~
	~~[[File:Isotonic regression.svg\|thumb\|400px\|An example of isotonic regression]]~~

	In [[numerical analysis]], '''isotonic regression''' ('''IR''') involves finding a weighted [[least-squares]] fit <math>x\in \Bbb{R}^n</math> to a [[Euclidean space\|vector]] <math>a\in \Bbb{R}^n</math> with weights vector <math>w\in \Bbb{R}^n</math> subject to a set of non-contradictory constraints of kind <math>x_i \ge x_j</math>.

	~~Such constraints define [[partial order]] or [[total order]] and can be represented as a [[directed graph]] <math>G=(N,E)</math>, where N~~ is the ~~set~~ of ~~variables involved, and E is the set of pairs (i, j) for each constraint <math>x_i \ge x_j</math>~~. ~~Thus, the IR problem corresponds to the following [[quadratic programming\|quadratic program]] (QP):~~

	~~:<math>\min \sum_{i=1}^n w_i (x_i - a_i)^2</math> <math>\text{subject to }x_i\ge x_j~ \text{ for all } (i,j)\in E.</math>~~

	~~In the case when <math>G=(N,E)</math> is a [[total order]], a simple [[iterative algorithm]] for solving this QP~~ is ~~called the [[pool adjacent violators algorithm]] (PAVA). Best~~ and Chakravarti (1990) have studied the problem as an active set identification problem, and have proposed a primal algorithm in O(n), the same [[computational complexity theory\|complexity]] as the PAVA, which can be seen as a dual algorithm.<ref>{{Cite journal \| doi = 10.1007/BF01580873 \| author = Best, M.J.; & Chakravarti N. \| year = 1990 \| title = Active set algorithms for isotonic regression; a unifying framework \| url = \| journal = Mathematical Programming \| volume = 47 \| issue = \| pages = 425–439 }}</ref>

	IR has applications in [[statistical inference]], for example, to fit of an isotonic curve to mean experimental results when an order is expected. A benefit of isotonic regression is that it does not assume any form for the target function, such as linearity assumed by [[linear regression]].

	Another application is nonmetric [[multidimensional scaling]],<ref>{{Cite journal \| doi = 10.1007/BF02289694 \| author = [[Joseph Kruskal\|Kruskal, J. B.]] \| year = 1964 \| title = Nonmetric Multidimensional Scaling: A numerical method \| url = \| journal = Psychometrika \| volume = 29 \| issue = 2\| pages = 115–129 }}</ref> where a low-dimensional [[embedding]] for data points is sought such that order of distances between points in the embedding matches [[order of dissimilarity]] between points. Isotonic regression is used iteratively to fit ideal distances to preserve relative dissimilarity order.

	~~Isotonic regression is also sometimes referred to~~ as ~~''[[monotonic]] regression''. Correctly speaking, ''isotonic'' is used when the direction of the trend is strictly increasing, while ''monotonic'' could imply~~ a ~~trend that is either strictly increasing or strictly decreasing~~.

	~~Isotonic Regression under the <math>L_p</math> for <math>p>0</math> is defined as follows:~~

	~~:<math>\min \sum_{i=1}^n w_i \|x_i - a_i\|^p</math> <math>\mathrm{subject~to~}x_i\ge x_j~ \text{ for all } (i,j)\~~in ~~E.</math>~~

	~~== Simply ordered case ==~~
	~~To illustrate the above, let~~
	~~<math>x_1 \leq x_2 \leq \ldots \leq x_n</math>,~~ and ~~<math>f(x_1) \leq f(x_2) \leq \ldots \leq f(x_n)</math>, and <math>w_i \geq 0 </math>~~.

	~~The isotonic estimator,~~ <~~math~~>g^*<~~/math~~>~~, minimizes the weighted least squares~~-~~like condition~~:

	:~~<math>\min_g \sum_{i=1}^n w_i (g(x_i) - f(x_i))^2<~~/~~math>~~

	~~Where <math>g</math> is the unknown function we are estimating, and <math>f</math> is a known function.~~

	Software has been developed in the R statistical package for computing isotone (monotonic) regression.<ref>{{cite journal\|last=De Leeuw\|first=Jan\|coauthors=K. Hornik, P. Mair\|title=Isotone Optimization in R: Pool-Adjacent-Violators Algorithm (PAVA) and Active Set Methods\|journal=Journal of statistical software\|year=2009\|volume=32\|issue=5\|pages=1}}</~~ref>~~

	~~== References ==~~
	~~{{Reflist}}~~

	~~== Further reading ==~~
	* {{cite book \|last=Robertson \|first=T. \|last2=Wright \|first2=F. T. \|last3=Dykstra \|first3=R. L. \|year=1988 \|title=Order restricted statistical inference \|location=New York \|publisher=Wiley \|isbn=0-471-91787-7 }}
	* {{cite book \|last=Barlow \|first=R. E. \|last2=Bartholomew \|first2=D. J. \|last3=Bremner \|first3=J. M. \|last4=Brunk \|first4=H. D. ~~\|year=1972 \|title=Statistical inference under order restrictions; the theory and application of isotonic regression \|location=New York \|publisher=Wiley \|year=1972 \|isbn=0-471-04970-0 }}~~
	* {{Cite journal \| doi = 10.1111/j.1467-9868.2008.00677.x \| author = Shively, T.S., Sager, T.W., Walker, S.G. \| year = 2009 \| title = A Bayesian approach to non-parametric monotone function estimation \| url = \| journal = Journal of the Royal Statistical Society, Series B \| volume = 71 \| issue = 1\| pages = 159–175 }}
	* {{Cite journal \| doi = 10.1093/~~biomet~~/88.3.793 \| author = Wu, W. B.; Woodroofe, M.; & Mentz, G. \| year = 2001 \| title = Isotonic regression: Another look at the changepoint problem \| url = \| journal = Biometrika \| volume = 88 \| issue = 3\| pages = 793–804 }}

	~~{{Statistics\|correlation}}~~

	~~{{DEFAULTSORT:Isotonic Regression}}~~
	~~[[Category:Regression analysis]]~~
	~~[[Category:Nonparametric regression]]~~
	~~[[Category:Non-parametric Bayesian methods]]~~
	~~[[Category:Numerical analysis]~~]

en>Addbot: Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q3591812

2013-03-14T17:44:56Z

Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q3591812

← Older revision		Revision as of 19:44, 14 March 2013
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	~~The title~~ of the author is ~~Figures~~. ~~To do aerobics~~ is a ~~factor~~ that ~~I'm completely addicted~~ to. ~~North Dakota~~ is ~~where me and my spouse live~~. ~~I am~~ a ~~meter reader~~.<br><br>~~Feel free~~ to ~~visit my weblog~~ - ~~[http~~://~~gcjcteam~~.~~org/index~~.~~php?mid~~=~~etc_video&document_srl~~=~~655020&sort_index~~=~~regdate&order_type~~=~~desc http~~://~~gcjcteam~~.~~org~~/~~index~~.~~php?mid~~=~~etc_video~~&~~document_srl~~=~~655020&sort_index~~=~~regdate&order_type~~=~~desc~~]		{{Regression bar}}
			{{context\|date=February 2012}}
			[[File:Isotonic regression.svg\|thumb\|400px\|An example of isotonic regression]]

			In [[numerical analysis]], '''isotonic regression''' ('''IR''') involves finding a weighted [[least-squares]] fit <math>x\in \Bbb{R}^n</math> to a [[Euclidean space\|vector]] <math>a\in \Bbb{R}^n</math> with weights vector <math>w\in \Bbb{R}^n</math> subject to a set of non-contradictory constraints of kind <math>x_i \ge x_j</math>.

			Such constraints define [[partial order]] or [[total order]] and can be represented as a [[directed graph]] <math>G=(N,E)</math>, where N is the set of variables involved, and E is the set of pairs (i, j) for each constraint <math>x_i \ge x_j</math>. Thus, the IR problem corresponds to the following [[quadratic programming\|quadratic program]] (QP):

			:<math>\min \sum_{i=1}^n w_i (x_i - a_i)^2</math> <math>\text{subject to }x_i\ge x_j~ \text{ for all } (i,j)\in E.</math>

			In the case when <math>G=(N,E)</math> is a [[total order]], a simple [[iterative algorithm]] for solving this QP is called the [[pool adjacent violators algorithm]] (PAVA). Best and Chakravarti (1990) have studied the problem as an active set identification problem, and have proposed a primal algorithm in O(n), the same [[computational complexity theory\|complexity]] as the PAVA, which can be seen as a dual algorithm.<ref>{{Cite journal \| doi = 10.1007/BF01580873 \| author = Best, M.J.; & Chakravarti N. \| year = 1990 \| title = Active set algorithms for isotonic regression; a unifying framework \| url = \| journal = Mathematical Programming \| volume = 47 \| issue = \| pages = 425–439 }}</ref>

			IR has applications in [[statistical inference]], for example, to fit of an isotonic curve to mean experimental results when an order is expected. A benefit of isotonic regression is that it does not assume any form for the target function, such as linearity assumed by [[linear regression]].

			Another application is nonmetric [[multidimensional scaling]],<ref>{{Cite journal \| doi = 10.1007/BF02289694 \| author = [[Joseph Kruskal\|Kruskal, J. B.]] \| year = 1964 \| title = Nonmetric Multidimensional Scaling: A numerical method \| url = \| journal = Psychometrika \| volume = 29 \| issue = 2\| pages = 115–129 }}</ref> where a low-dimensional [[embedding]] for data points is sought such that order of distances between points in the embedding matches [[order of dissimilarity]] between points. Isotonic regression is used iteratively to fit ideal distances to preserve relative dissimilarity order.

			Isotonic regression is also sometimes referred to as ''[[monotonic]] regression''. Correctly speaking, ''isotonic'' is used when the direction of the trend is strictly increasing, while ''monotonic'' could imply a trend that is either strictly increasing or strictly decreasing.

			Isotonic Regression under the <math>L_p</math> for <math>p>0</math> is defined as follows:

			:<math>\min \sum_{i=1}^n w_i \|x_i - a_i\|^p</math> <math>\mathrm{subject~to~}x_i\ge x_j~ \text{ for all } (i,j)\in E.</math>

			== Simply ordered case ==
			To illustrate the above, let
			<math>x_1 \leq x_2 \leq \ldots \leq x_n</math>, and <math>f(x_1) \leq f(x_2) \leq \ldots \leq f(x_n)</math>, and <math>w_i \geq 0 </math>.

			The isotonic estimator, <math>g^*</math>, minimizes the weighted least squares-like condition:

			:<math>\min_g \sum_{i=1}^n w_i (g(x_i) - f(x_i))^2</math>

			Where <math>g</math> is the unknown function we are estimating, and <math>f</math> is a known function.

			Software has been developed in the R statistical package for computing isotone (monotonic) regression.<ref>{{cite journal\|last=De Leeuw\|first=Jan\|coauthors=K. Hornik, P. Mair\|title=Isotone Optimization in R: Pool-Adjacent-Violators Algorithm (PAVA) and Active Set Methods\|journal=Journal of statistical software\|year=2009\|volume=32\|issue=5\|pages=1}}</ref>

			== References ==
			{{Reflist}}

			== Further reading ==
			* {{cite book \|last=Robertson \|first=T. \|last2=Wright \|first2=F. T. \|last3=Dykstra \|first3=R. L. \|year=1988 \|title=Order restricted statistical inference \|location=New York \|publisher=Wiley \|isbn=0-471-91787-7 }}
			* {{cite book \|last=Barlow \|first=R. E. \|last2=Bartholomew \|first2=D. J. \|last3=Bremner \|first3=J. M. \|last4=Brunk \|first4=H. D. \|year=1972 \|title=Statistical inference under order restrictions; the theory and application of isotonic regression \|location=New York \|publisher=Wiley \|year=1972 \|isbn=0-471-04970-0 }}
			* {{Cite journal \| doi = 10.1111/j.1467-9868.2008.00677.x \| author = Shively, T.S., Sager, T.W., Walker, S.G. \| year = 2009 \| title = A Bayesian approach to non-parametric monotone function estimation \| url = \| journal = Journal of the Royal Statistical Society, Series B \| volume = 71 \| issue = 1\| pages = 159–175 }}
			* {{Cite journal \| doi = 10.1093/biomet/88.3.793 \| author = Wu, W. B.; Woodroofe, M.; & Mentz, G. \| year = 2001 \| title = Isotonic regression: Another look at the changepoint problem \| url = \| journal = Biometrika \| volume = 88 \| issue = 3\| pages = 793–804 }}

			{{Statistics\|correlation}}

			{{DEFAULTSORT:Isotonic Regression}}
			[[Category:Regression analysis]]
			[[Category:Nonparametric regression]]
			[[Category:Non-parametric Bayesian methods]]
			[[Category:Numerical analysis]]

en>Bibcode Bot: Adding 0 arxiv eprint(s), 1 bibcode(s) and 0 doi(s). Did it miss something? Report bugs, errors, and suggestions at User talk:Bibcode Bot

2012-07-30T19:48:02Z

Adding 0 arxiv eprint(s), 1 bibcode(s) and 0 doi(s). Did it miss something? Report bugs, errors, and suggestions at User talk:Bibcode Bot

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The title of the author is Figures. To do aerobics is a factor that I'm completely addicted to. North Dakota is where me and my spouse live. I am a meter reader.<br><br>Feel free to visit my weblog - [http://gcjcteam.org/index.php?mid=etc_video&document_srl=655020&sort_index=regdate&order_type=desc http://gcjcteam.org/index.php?mid=etc_video&document_srl=655020&sort_index=regdate&order_type=desc]