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'''Least trimmed squares''' ('''LTS'''), or '''least trimmed sum of squares''', is a [[robust statistics|robust statistical method]] that fits a function to a set of data whilst not being unduly affected by the presence of [[outlier]]s. It is one of a number of methods for [[robust regression]].
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.


== Description of method ==
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]
Instead of the standard [[least squares]] method, which minimises the [[Residual sum of squares|sum of squared residuals]] over ''n'' points, the LTS method attempts to minimise the sum of squared residuals over a subset, ''k'', of those points. The ''n-k'' points which are not used do not influence the fit.
* Only registered users will be able to execute this rendering mode.
* Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.


In a standard least squares problem, the estimated parameter values, β, are defined to be those values that minimise the objective function,  ''S''(β), of squared residuals
Registered users will be able to choose between the following three rendering modes:  
:<math>S=\sum_{i=1}^{n}{r_i(\beta)}^2</math>,
where the [[errors and residuals in statistics|residuals]] are defined as the differences between the values of the [[Dependent and independent variables|dependent variables]] (observations) and the model values


:<math>r_i(\beta)= y_i - f(x_i, \beta),</math>
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


and where ''n'' is the overall number of data points. For a least trimmed squares analysis, this objective function is replaced by one constructed in the following way. For a fixed value of &beta;, let <math> r_{(j)}(\beta) </math> denote the set of ordered absolute values of the residuals (in increasing order of absolute value). In this notation, the standard sum of squares function is
<!--'''PNG''' (currently default in production)
:<math>S(\beta)=\sum_{j=1}^n (r_{(j)}(\beta))^2,</math>
:<math forcemathmode="png">E=mc^2</math>
while the objective function for LTS is
:<math>S_k(\beta)=\sum_{j=1}^k (r_{(j)}(\beta))^2.</math>


== Computational considerations ==
'''source'''
Because this method is binary, in that points are either included or excluded, no closed form solution exists. As a result, methods which try to find a LTS solution through a problem sift through combinations of the data, attempting to find the ''k'' subset which yields the lowest sum of squared residuals. Methods exist for low ''n'' which will find the exact solution, however as ''n'' rises, the number of combinations grows rapidly, thus yielding methods which attempt to find approximate (but generally sufficient) solutions.
:<math forcemathmode="source">E=mc^2</math> -->


== References ==
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].
* [[Peter Rousseeuw|Rousseeuw, P. J.]] (1984) "Least Median of Squares Regression" ''Journal of the American Statistical Association'', 79, 871&ndash;880. {{JSTOR|2288718}}
*Rousseeuw, P. J., Leroy A.M.  (1987) ''Robust Regression and Outlier Detection'', Wiley. ISBN 978-0-471-85233-9 (Published online 2005 {{DOI| 10.1002/0471725382}} )
*Li, L.M. (2005) "An algorithm for computing exact least-trimmed squares estimate of simple linear regression with constraints", ''Computational Statistics & Data Analysis'', 48 (4), 717&ndash;734. {{DOI| 10.1016/j.csda.2004.04.003.}}
*Atkinson, A.C., Cheng, T.-C. (1999) "Computing least trimmed squares regression with the forward search", ''Statistics and Computing'', 9 (4), 251&ndash;263. {{DOI| 10.1023/A:1008942604045}}
*Jung, Kang-Mo (2007) "Least Trimmed Squares Estimator in the Errors-in-Variables Model", ''Journal of Applied Statistics'', 34 (3), 331&ndash;338. {{DOI| 10.1080/02664760601004973}}


[[Category:Robust statistics]]
==Demos==
[[Category:Robust regression]]
 
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:
 
 
* accessibility:
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.
 
==Test pages ==
 
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
*[[Displaystyle]]
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]
 
*[[Inputtypes|Inputtypes (private Wikis only)]]
*[[Url2Image|Url2Image (private Wikis only)]]
==Bug reporting==
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .

Latest revision as of 22:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML

E=mc2


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

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