en>Mark viking: moved wl to earlier instance, also disambiguated and removed tag

2013-09-13T20:01:39Z

moved wl to earlier instance, also disambiguated and removed tag

← Older revision		Revision as of 22:01, 13 September 2013
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	~~Friends call him Royal Cummins~~. ~~To play croquet~~ is ~~something~~ that ~~I've done for many years~~. ~~Bookkeeping~~ is ~~how he supports his family members~~ and ~~his salary~~ has ~~been truly satisfying~~. ~~My home~~ is ~~now in Kansas~~.<br><br>~~my blog~~ - [http://~~mcdonnell~~.~~exiondev~~.~~com~~/~~node~~/~~325 auto warranty~~]		A '''whitening transformation''' is a [[decorrelation]] transformation that transforms a set of random variables having a known [[covariance matrix]] <math>M</math> into a set of new random variables whose covariance is the [[identity matrix]] (meaning that they are [[uncorrelated]] and all have [[variance]] 1).

			The transformation is called "whitening" because it changes the input vector into a [[white noise\|white noise vector]]. It differs from a general decorrelation transformation in that the latter only makes the covariances equal to zero, so that the correlation matrix may be any diagonal matrix.

			The inverse [[coloring transformation]] transforms a vector <math>Y</math> of uncorrelated variables (a white random vector) into a vector <math>X</math> with a specified covariance matrix.

			== Definition ==
			Suppose <math>X</math> is a [[random vector\|random (column) vector]] with covariance matrix <math>M</math> and mean 0. The matrix <math>M</math> can be written as the expected value of the [[outer product]] of <math>X</math> with itself, namely:
			:<math> M = \operatorname{E}[X X^\top]</math>
			Since <math>M</math> is [[symmetric matrix\|symmetric]] and [[Positive-semidefinite matrix\|positive semidefinite]], it has a [[square root of a matrix\|square root]] <math>M^{1/2}</math>, such that <math> M^{1/2}( M^{1/2})^\top = M</math>. If <math>M</math> is positive definite, <math>M^{1/2}</math> is invertible. Then the vector <math>Y = M^{-1/2}X</math> has covariance matrix:

			<math> \operatorname{Cov}(Y) = \operatorname{E}[Y Y^\top] = M^{-1/2} \operatorname{E}[X X^\top] (M^{-1/2})^\top </math>

			<math> \operatorname{Cov}(Y)= M^{-1/2} M (M^{-1/2})^\top = (M^{-1/2} M^{1/2}) ((M^{1/2})^\top (M^{-1/2})^\top ) = I</math>

			and is therefore a white random vector.

			Since the square root of a matrix is not unique, the whitening transformation is not unique either.

			If <math>M</math> is not positive definite, then <math>M^{1/2}</math> is not invertible, and it is impossible to map <math>X</math> to a white vector of the same length. In that case the vector <math>X</math> can still be mapped to a smaller white vector <math>Y</math> with <math>m</math> elements, where <math>m</math> is the number of non-zero [[eigenvalue]]s of <math>M</math>.

			==See also==
			* [[Decorrelation]]
			* [[Randomness extractor]]
			* [[Hardware random number generator]]
			* [[Principal component analysis]]

			==References==
			{{reflist}}

			== External links ==
			* http://courses.media.mit.edu/2010fall/mas622j/whiten.pdf

			[[Category:Classification algorithms]]

en>Maxal: /* Definition */

2012-02-09T03:03:47Z

Definition

New page

Friends call him Royal Cummins. To play croquet is something that I've done for many years. Bookkeeping is how he supports his family members and his salary has been truly satisfying. My home is now in Kansas.<br><br>my blog - [http://mcdonnell.exiondev.com/node/325 auto warranty]

Hilbert basis (linear programming) - Revision history

en>Mark viking: moved wl to earlier instance, also disambiguated and removed tag

en>Maxal: /* Definition */