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en>Trappist the monk: /* References */replace mr template with mr parameter in CS1 templates; using AWB

2014-09-25T13:26:10Z

References: replace mr template with mr parameter in CS1 templates; using AWB

← Older revision		Revision as of 15:26, 25 September 2014
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	~~'''Error analysis'''~~ is ~~the study of kind and quantity of [[error]] that occurs, particularly in the fields of [[applied mathematics]] (particularly [[numerical analysis]]), [[applied linguistics]] and [[statistics]].~~		Alyson is what my husband enjoys to call me but I don't like when individuals use my complete title. Office supervising is exactly where her primary income arrives from but she's currently utilized for another one. Playing badminton is a thing that he is totally addicted to. I've always cherished residing in Mississippi.<br><br>Review my web page: spirit messages ([http://m-card.co.kr/xe/mcard_2013_promote01/29877 m-card.co.kr])

	~~== Error analysis in numerical modeling ==~~
	~~In numerical simulation or modeling of real systems, error analysis is concerned with the changes in the output of the model as the parameters~~ to ~~the model [[variance\|vary]] about a [[mean]].~~

	For instance, in a system modeled as a function of two variables <math>\scriptstyle z \,=\, f(x,y)</math>. Error analysis deals with the propagation of the [[numerical error]]s in <math>\scriptstyle x</math> and <math>\scriptstyle y</math> (around mean values <math>\scriptstyle\bar{x}</math> and <math>\scriptstyle\bar{y}</math>) to error in <math>\scriptstyle z</math> (around a mean <math>\scriptstyle\bar{z}</math>).<ref>{{cite book\|title~~=Modeling Biological Systems: Principles and Applications\|author=James W~~. ~~Haefner\|pages=186–189\|date=1996\|publisher=Springer\|id=ISBN 0412042010}}</ref>~~

	~~In numerical analysis, error analysis comprises both '''forward error analysis''' and '''[[Numerical stability#Forward, backward, and mixed stability\|backward error analysis]]''~~'. ~~Forward error analysis involves the analysis of a function <math>\scriptstyle z' = f'(a_0,\,a_1,\,\dots,\,a_n)</math> which~~ is ~~an approximation (usually a finite polynomial) to~~ a ~~function <math>\scriptstyle z \,=\, f(a_0,a_1,\dots,a_n)</math> to determine the bounds on the error in the approximation; i.e., to find <math>\scriptstyle\epsilon</math> such~~ that <math>\scriptstyle 0 \,\le\, \|z - z'\| \,\le\, \epsilon</math>. Backward error analysis involves the analysis of the approximation function <math>\scriptstyle z' \,=\, f'(a_0,\,a_1,\,\dots,\,a_n)</math>, to ~~determine the bounds on the parameters <math>\scriptstyle a_i \,=\, \bar{a_i} \,\pm\, \epsilon_i</math> such that the result <math>\scriptstyle z' \,=\, z</math>~~.~~<ref>{{cite book\|title=Schaum~~'~~s Outline of Theory and Problems of Numerical Analysis\|author=Francis J. Scheid\|pages=11\|date=1988\|publisher=McGraw-Hill Professional\|id=ISBN 0070552215}}</ref>~~

	~~=={{anchor\|Error analysis in language teaching}} Error analysis~~ in ~~second language acquisition ==~~
	~~{{Main\|Error analysis (linguistics)}}~~
	~~In [[second language acquisition]], error analysis studies the types and causes of [[speech error\|language errors]]~~. ~~Errors are classified~~<~~ref~~>Cf. Bussmann, Hadumod (1996), ''Routledge Dictionary of Language and Linguistics'', London: Routledge, s.v. error analysis. A comprehensive bibliography was published by Bernd Spillner (1991), ''Error Analysis'', Amsterdam/Philadelphia: Benjamins.<~~/ref~~> ~~according to:~~
	* modality (i.e., level of proficiency in speaking, [[writing]], [[Reading (process)\|reading]], [[listening]])
	* linguistic levels (i.e., [[pronunciation]], [[grammar]], [[vocabulary]], [[style (fiction)\|style]])
	* form (e.g., omission, insertion, substitution)
	* type (systematic errors/errors in competence vs. occasional errors/errors in performance)
	* cause (e.g., [[language transfer\|interference]], [[interlanguage]])
	* norm vs. system

	Error analysis in SLA was established in the 1960s by [[Stephen Pit Corder]] and colleagues.<ref>{{Cite journal \| last1 = Corder \| first1 = S. P. \| year = 1967 \| title = The significance of learners' errors \| url = \| journal = International Review of Applied Linguistics \| volume = 5 \| issue = \| pages = 160–170 }}</ref> Error analysis was an alternative to [[contrastive analysis]], an approach influenced by [[behaviorism]] through which applied linguists sought to use the formal distinctions between the learners' first and second languages to predict errors. Error analysis showed that contrastive analysis was unable to predict a great majority of errors, although its more valuable aspects have been incorporated into the study of language transfer. A key finding of error analysis has been that many learner errors are produced by learners making faulty inferences about the [[wiktionary:~~rule\|rules]] of the new language.~~

	~~== Error analysis in molecular dynamics simulation ==~~

	~~In [[molecular dynamics]]~~ (~~MD) simulations, there are errors due to inadequate sampling of the phase space or infrequently occurring events, these lead to the statistical error due to random fluctuation in the measurements.~~

	~~For a series of ''M'' measurements of a fluctuating property ''A'', the mean value is:~~

	~~:<math> \langle A \rangle = \frac{1}{M} \sum_{\mu=1}^M A_{\mu}. </math>~~

	~~When these ''M'' measurements are independent, the variance of the mean <''A''> is:~~

	~~:<math> \sigma^{2}( \langle A \rangle ) = \frac{1}{M} \sigma^{2}( A ), </math>~~

	but in most MD simulations, there is correlation between quantity ''A'' at different time, so the variance of the mean <''A''> will be underestimated as the effective number of independent measurements is actually less than ''M''. In such situations we rewrite the variance as :

	~~:<math> \sigma^{2}( \langle A \rangle ) = \frac{1}{M} \sigma^{2}A \left[ 1 + 2 \sum_\mu \left( 1 - \frac{\mu}{M} \right) \phi_{\mu} \right],</math>~~

	~~where <math>\phi_{\mu}</math> is the [[autocorrelation function]] defined by~~

	~~:<math> \phi_{\mu} = \frac{ \langle A_{\mu}A_{0} \rangle - \langle A \rangle^{2} }{ \langle A^{2} \rangle - \langle A \rangle^{2}}.</math>~~

	We can then use the autocorrelation function to estimate the [[error bar]]. Luckily, we have a much simpler method based on [[block averaging]].<ref>D. C. Rapaport, ''The Art of Molecular Dynamics Simulation'', Cambridge University Press.</ref>

	~~== See also ==~~
	* [[Error analysis (linguistics)]]
	* [[Error bar]]
	* [[Errors and residuals in statistics]]
	* [[Propagation of uncertainty]]

	~~== References ==~~
	~~<references />~~

	~~==External links==~~
	* [http://~~teacher.pas~~.~~rochester~~.~~edu~~/~~PHY_LABS~~/~~AppendixB~~/~~AppendixB~~.~~html] All about error analysis~~.

	~~[[Category:Numerical analysis]]~~
	~~[[Category:Error]~~]

en>Addbot: Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q3039873

2013-03-16T16:31:43Z

Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q3039873

← Older revision		Revision as of 18:31, 16 March 2013
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	[~~http:~~/~~/www~~.~~dreammakersdancewear.com~~/~~UserProfile~~/~~tabid~~/78/~~userId~~/~~16490~~/~~Default~~.~~aspx extended auto warranty~~] ~~car warranty I~~'~~m Luther but I never truly liked that title~~. ~~Meter studying is~~ ~~extended car warranty where my primary income comes from but quickly [http:~~//~~www~~.~~spa-it~~.al/~~ActivityFeed~~/~~MyProfile~~/~~tabid~~/60/~~userId/11049/Default~~.~~aspx extended auto warranty] car warranty I~~'~~ll be on my personal~~. ~~He currently lives~~ in ~~Idaho~~ and ~~his parents live close by~~. ~~The favorite hobby for my kids~~ and ~~me [https~~://~~www~~.~~hyundaiusa~~.~~com~~/~~assurance~~/~~america-best-warranty~~.~~aspx extended auto~~] [~~http://www~~.~~popularmechanics~~.~~com~~/~~cars/how-~~to~~/repair/how-~~to~~-get-~~a~~-used-car-warranty-and-not-get-screwed-6654348 warranty~~] ~~is taking part~~ in ~~crochet and now I~~'~~m attempting to earn cash with it~~.<br><br>~~Look~~ at ~~my blog post~~: [~~http:~~//~~pupo~~.ch/~~mypupo~~/~~panda~~/~~panda2/index.php?mod~~=~~users&action~~=~~view&id~~=~~161~~ http://~~pupo~~.ch/~~mypupo~~/~~panda~~/~~panda2/index~~.~~php?mod=users&action=view&id=161~~]		'''Error analysis''' is the study of kind and quantity of [[error]] that occurs, particularly in the fields of [[applied mathematics]] (particularly [[numerical analysis]]), [[applied linguistics]] and [[statistics]].

			== Error analysis in numerical modeling ==
			In numerical simulation or modeling of real systems, error analysis is concerned with the changes in the output of the model as the parameters to the model [[variance\|vary]] about a [[mean]].

			For instance, in a system modeled as a function of two variables <math>\scriptstyle z \,=\, f(x,y)</math>. Error analysis deals with the propagation of the [[numerical error]]s in <math>\scriptstyle x</math> and <math>\scriptstyle y</math> (around mean values <math>\scriptstyle\bar{x}</math> and <math>\scriptstyle\bar{y}</math>) to error in <math>\scriptstyle z</math> (around a mean <math>\scriptstyle\bar{z}</math>).<ref>{{cite book\|title=Modeling Biological Systems: Principles and Applications\|author=James W. Haefner\|pages=186–189\|date=1996\|publisher=Springer\|id=ISBN 0412042010}}</ref>

			In numerical analysis, error analysis comprises both '''forward error analysis''' and '''[[Numerical stability#Forward, backward, and mixed stability\|backward error analysis]]'''. Forward error analysis involves the analysis of a function <math>\scriptstyle z' = f'(a_0,\,a_1,\,\dots,\,a_n)</math> which is an approximation (usually a finite polynomial) to a function <math>\scriptstyle z \,=\, f(a_0,a_1,\dots,a_n)</math> to determine the bounds on the error in the approximation; i.e., to find <math>\scriptstyle\epsilon</math> such that <math>\scriptstyle 0 \,\le\, \|z - z'\| \,\le\, \epsilon</math>. Backward error analysis involves the analysis of the approximation function <math>\scriptstyle z' \,=\, f'(a_0,\,a_1,\,\dots,\,a_n)</math>, to determine the bounds on the parameters <math>\scriptstyle a_i \,=\, \bar{a_i} \,\pm\, \epsilon_i</math> such that the result <math>\scriptstyle z' \,=\, z</math>.<ref>{{cite book\|title=Schaum's Outline of Theory and Problems of Numerical Analysis\|author=Francis J. Scheid\|pages=11\|date=1988\|publisher=McGraw-Hill Professional\|id=ISBN 0070552215}}</ref>

			=={{anchor\|Error analysis in language teaching}} Error analysis in second language acquisition ==
			{{Main\|Error analysis (linguistics)}}
			In [[second language acquisition]], error analysis studies the types and causes of [[speech error\|language errors]]. Errors are classified<ref>Cf. Bussmann, Hadumod (1996), ''Routledge Dictionary of Language and Linguistics'', London: Routledge, s.v. error analysis. A comprehensive bibliography was published by Bernd Spillner (1991), ''Error Analysis'', Amsterdam/Philadelphia: Benjamins.</ref> according to:
			* modality (i.e., level of proficiency in speaking, [[writing]], [[Reading (process)\|reading]], [[listening]])
			* linguistic levels (i.e., [[pronunciation]], [[grammar]], [[vocabulary]], [[style (fiction)\|style]])
			* form (e.g., omission, insertion, substitution)
			* type (systematic errors/errors in competence vs. occasional errors/errors in performance)
			* cause (e.g., [[language transfer\|interference]], [[interlanguage]])
			* norm vs. system

			Error analysis in SLA was established in the 1960s by [[Stephen Pit Corder]] and colleagues.<ref>{{Cite journal \| last1 = Corder \| first1 = S. P. \| year = 1967 \| title = The significance of learners' errors \| url = \| journal = International Review of Applied Linguistics \| volume = 5 \| issue = \| pages = 160–170 }}</ref> Error analysis was an alternative to [[contrastive analysis]], an approach influenced by [[behaviorism]] through which applied linguists sought to use the formal distinctions between the learners' first and second languages to predict errors. Error analysis showed that contrastive analysis was unable to predict a great majority of errors, although its more valuable aspects have been incorporated into the study of language transfer. A key finding of error analysis has been that many learner errors are produced by learners making faulty inferences about the [[wiktionary:rule\|rules]] of the new language.

			== Error analysis in molecular dynamics simulation ==

			In [[molecular dynamics]] (MD) simulations, there are errors due to inadequate sampling of the phase space or infrequently occurring events, these lead to the statistical error due to random fluctuation in the measurements.

			For a series of ''M'' measurements of a fluctuating property ''A'', the mean value is:

			:<math> \langle A \rangle = \frac{1}{M} \sum_{\mu=1}^M A_{\mu}. </math>

			When these ''M'' measurements are independent, the variance of the mean <''A''> is:

			:<math> \sigma^{2}( \langle A \rangle ) = \frac{1}{M} \sigma^{2}( A ), </math>

			but in most MD simulations, there is correlation between quantity ''A'' at different time, so the variance of the mean <''A''> will be underestimated as the effective number of independent measurements is actually less than ''M''. In such situations we rewrite the variance as :

			:<math> \sigma^{2}( \langle A \rangle ) = \frac{1}{M} \sigma^{2}A \left[ 1 + 2 \sum_\mu \left( 1 - \frac{\mu}{M} \right) \phi_{\mu} \right],</math>

			where <math>\phi_{\mu}</math> is the [[autocorrelation function]] defined by

			:<math> \phi_{\mu} = \frac{ \langle A_{\mu}A_{0} \rangle - \langle A \rangle^{2} }{ \langle A^{2} \rangle - \langle A \rangle^{2}}.</math>

			We can then use the autocorrelation function to estimate the [[error bar]]. Luckily, we have a much simpler method based on [[block averaging]].<ref>D. C. Rapaport, ''The Art of Molecular Dynamics Simulation'', Cambridge University Press.</ref>

			== See also ==
			* [[Error analysis (linguistics)]]
			* [[Error bar]]
			* [[Errors and residuals in statistics]]
			* [[Propagation of uncertainty]]

			== References ==
			<references />

			==External links==
			* [http://teacher.pas.rochester.edu/PHY_LABS/AppendixB/AppendixB.html] All about error analysis.

			[[Category:Numerical analysis]]
			[[Category:Error]]

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2012-03-21T03:27:07Z

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