en>Yobot: /* See also */WP:CHECKWIKI error fixes using AWB (10093)

2014-05-05T11:28:39Z

See also: WP:CHECKWIKI error fixes using AWB (10093)

en>Addbot: Bot: Migrating 4 interwiki links, now provided by Wikidata on d:q1468336 (Report Errors)

2013-02-28T10:16:13Z

Bot: Migrating 4 interwiki links, now provided by Wikidata on d:q1468336 (Report Errors)

← Older revision		Revision as of 12:16, 28 February 2013
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	The ~~writer~~ is ~~recognized by~~ the ~~name~~ of ~~Numbers Wunder. Doing ceramics~~ is ~~what her family~~ and ~~her appreciate. South Dakota is exactly~~ where I'~~ve usually been living~~. ~~He used to~~ be ~~unemployed but now he is a meter reader~~.<br><br>~~Stop by my web page~~ ..~~. at home std test~~ ([~~http~~:/~~/www~~.~~dangdanghani~~.kr/~~xe/?document_srl=3321100 simply click~~ the ~~next internet site~~])		{{Distinguish2\|the [[Gauss–Markov theorem]] of mathematical statistics}}
			{{merge\|Ornstein–Uhlenbeck process\|discuss=Talk:Gauss–Markov process#Merger proposal\|date=March 2012}}
			{{Unreferenced\|date=December 2009}}

			'''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. The stationary Gauss–Markov process is a very special case because it is unique, except for some trivial exceptions.

			Every Gauss–Markov process ''X''(''t'') possesses the three following properties:

			# If ''h''(''t'') is a non-zero scalar function of ''t'', then ''Z''(''t'') = ''h''(''t'')''X''(''t'') is also a Gauss–Markov process
			# If ''f''(''t'') is a non-decreasing scalar function of ''t'', then ''Z''(''t'') = ''X''(''f''(''t'')) is also a Gauss–Markov process
			# 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).

			==Properties of the Stationary Gauss-Markov Processes==
			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.

			Exponential [[autocorrelation]]:
			:<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]]:

			:<math>\textbf{S}_{x}(j\omega) = \frac{2\sigma^{2}\beta}{\omega^{2} + \beta^{2}}.\,</math>

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

			The above yields the following spectral factorization:

			:<math>\textbf{S}_{x}(s) = \frac{2\sigma^{2}\beta}{-s^{2} + \beta^{2}}
			= \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.

			There are also some trivial exceptions to all of the above.

			{{DEFAULTSORT:Gauss-Markov Process}}
			[[Category:Markov processes]]


			{{probability-stub}}

en>Tamfang: my holy mission is to stamp out "a thing is a term" language

2011-02-05T23:37:46Z

my holy mission is to stamp out "a thing is a term" language

New page

The writer is recognized by the name of Numbers Wunder. Doing ceramics is what her family and her appreciate. South Dakota is exactly where I've usually been living. He used to be unemployed but now he is a meter reader.<br><br>Stop by my web page ... at home std test ([http://www.dangdanghani.kr/xe/?document_srl=3321100 simply click the next internet site])

Overburden pressure - Revision history

en>Yobot: /* See also */WP:CHECKWIKI error fixes using AWB (10093)

en>Addbot: Bot: Migrating 4 interwiki links, now provided by Wikidata on d:q1468336 (Report Errors)

en>Tamfang: my holy mission is to stamp out "a thing is a term" language