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{{Distinguish2|the [[Gauss–Markov theorem]] of mathematical statistics}}
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.
{{merge|Ornstein–Uhlenbeck process|discuss=Talk:Gauss–Markov process#Merger proposal|date=March 2012}}


'''Gauss–Markov stochastic processes''' (named after [[Carl Friedrich Gauss]] and [[Andrey Markov]]) are [[stochastic process]]es that satisfy the requirements for both [[Gaussian process]]es and [[Markov process]]es.<ref name=Rasmussen2006>{{cite book|last=C. E. Rasmussen & C. K. I. Williams,|title=Gaussian Processes for Machine Learning|date=2006|publisher=MIT Press|isbn=026218253X|page=Appendix B|url=http://www.gaussianprocess.org/gpml/chapters/RWB.pdf}}</ref><ref name=Lamon2008>{{cite book|last=Lamon|first=Pierre|title=3D-Position Tracking and Control for All-Terrain Robots|date=2008|publisher=Springer|isbn=978-3-540-78286-5|pages=93-95}}</ref> The stationary Gauss–Markov process is a very special case because it is unique, except for some trivial exceptions.  
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Every Gauss–Markov process ''X''(''t'') possesses the three following properties:
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# If ''h''(''t'') is a non-zero scalar function of ''t'', then ''Z''(''t'') = ''h''(''t'')''X''(''t'') is also a Gauss–Markov process
'''MathML'''
# If ''f''(''t'') is a non-decreasing scalar function of ''t'', then ''Z''(''t'') = ''X''(''f''(''t'')) is also a Gauss–Markov process
:<math forcemathmode="mathml">E=mc^2</math>
# There exists a non-zero scalar function ''h''(''t'') and a non-decreasing scalar function ''f''(''t'') such that ''X''(''t'') = ''h''(''t'')''W''(''f''(''t'')), where ''W''(''t'') is the standard [[Wiener process]].


Property (3) means that every Gauss–Markov process can be synthesized from the standard Wiener process (SWP).
<!--'''PNG'''  (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


==Properties of the Stationary Gauss-Markov Processes==
'''source'''
A stationary Gauss–Markov process with [[variance]] <math>\textbf{E}(X^{2}(t)) = \sigma^{2}</math> and [[time constant]] <math>\beta^{-1}</math> has the following properties.
:<math forcemathmode="source">E=mc^2</math> -->


Exponential [[autocorrelation]]:
<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].
:<math>\textbf{R}_{x}(\tau) = \sigma^{2}e^{-\beta |\tau|}.\,</math>


A power [[spectral density]] (PSD) function that has the same '''shape''' as the [[Cauchy distribution]]:
==Demos==


:<math>\textbf{S}_{x}(j\omega) = \frac{2\sigma^{2}\beta}{\omega^{2} + \beta^{2}}.\,</math>
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


(Note that the Cauchy distribution and this spectrum differ by  scale factors.)


The above yields the following spectral factorization:
* 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.


:<math>\textbf{S}_{x}(s) = \frac{2\sigma^{2}\beta}{-s^{2} + \beta^{2}}
==Test pages ==
                        = \frac{\sqrt{2\beta}\,\sigma}{(s + \beta)}
                          \cdot\frac{\sqrt{2\beta}\,\sigma}{(-s + \beta)}.
</math>


which is important in Wiener filtering and other areas.
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
*[[Displaystyle]]
*[[MathAxisAlignment]]
*[[Styling]]
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*[[Help:Formula]]


There are also some trivial exceptions to all of the above.
*[[Inputtypes|Inputtypes (private Wikis only)]]
 
*[[Url2Image|Url2Image (private Wikis only)]]
==References==
==Bug reporting==
{{reflist}}
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 .
 
{{DEFAULTSORT:Gauss-Markov Process}}
[[Category:Markov processes]]
 
 
{{probability-stub}}

Latest revision as of 23:52, 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


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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