|
|
| Line 1: |
Line 1: |
| '''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]]
| |
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.
Review my web page: spirit messages (m-card.co.kr)