en>Dirac66: /* Commutative operations in mathematics */ Truth functions - added example and clarified obscure notation

2015-01-12T00:37:27Z

Commutative operations in mathematics: Truth functions - added example and clarified obscure notation

← Older revision		Revision as of 02:37, 12 January 2015
Line 1:		Line 1:
	~~[[Image:Homoscedasticity.png\|thumb\|right\|Plot with random data showing homoscedasticity.]]~~		Roberto is what's written concerning his birth certificate fortunately he never really liked that name. South Carolina is your birth place. The favourite hobby for him and as well , his kids is in order to fish and he's previously been doing it for a long time. Auditing is how he supports or perhaps family. Go to his website to obtain out more: http://prometeu.net<br><br>Also visit my web-site [http://prometeu.net clash of clans tricks]
	~~In [[statistics]], a [[sequence]] or a vector of [[random variable]]s~~ is '~~''homoscedastic''' {{IPAc-en\|ˌ\|h\|oʊ\|m\|oʊ\|~~s~~\|k\|ə\|ˈ\|d\|æ\|s\|t\|ɪ\|k}} if all random variables in the sequence or vector have the same [[finite set\|finite]] [[variance]]~~. ~~This~~ is ~~also known as '''homogeneity of variance'''~~. The ~~complementary notion is called [[heteroscedasticity]]. The spellings ''homos'''k'''edasticity''~~ and ~~''heteros'''k'''edasticity'' are also frequently used.<ref>For the Greek etymology of the term~~, see J. Huston McCulloch (1985), [https://docs.google.com/viewer?a=v&q=cache:n9WwLxZX9LsJ:www.ime.usp.br/~abe/lista/pdfZAptC9KazU.pdf+On+Heterosedasticity&hl=en&gl=uk&pid=bl&srcid=ADGEEShKto-i0r96p6JIcxnvPB9w4abkVWuKcpSwR8IUT_hDcvabhUEPS6KW0Hs9lGBESqZOutTYpLCcbe-4vj_ldq4nmkUlYTx6ZiL-c2f4ZFMJpauBi-dMPrJSRMVbQSianWK-FPct&sig=AHIEtbQHgBYoPXlrwLXby_joAnqpVKQ2PQ "On Heterosedasticity"], ''[[Econometrica]]'', 53(2), p. 483.</ref>

	~~The assumption of homoscedasticity simplifies mathematical and computational treatment. <!-- and usually leads to adequate estimation results (e.g. in [[data mining]]) even if the assumption~~ is ~~not true.--> Serious violations~~ in homoscedasticity (assuming a distribution of data is homoscedastic when in actuality it is heteroscedastic {{IPAc-en\|ˌ\|h\|ɛ\|t\|ər\|oʊ\|s\|k\|ə\|ˈ\|d\|æ\|s\|t\|ɪ\|k}}) may result in overestimating the goodness of fit as measured by the [[Pearson product-moment correlation coefficient\|Pearson coefficient]].

	~~==Assumptions of a regression model==~~
	~~As used in describing [[simple linear regression]] analysis, one assumption of the fitted model (~~to ensure that the least-squares estimators are each a [[best linear unbiased estimator]] of the respective population parameters, by the [[Gauss–Markov theorem]]) is that the standard deviations of the error terms are constant and ~~do not depend on the~~ '~~'x''-value. Consequently, each probability distribution~~ for ~~''y'' (response variable) has the same standard deviation regardless of the ''x''-value (predictor)~~. ~~In short, this assumption~~ is ~~homoscedasticity~~. ~~Homoscedasticity is not required for the estimates~~ to ~~be unbiased, consistent, and asymptotically normal {{Citation needed\|date=August 2013}}~~

	~~==Testing==~~
	~~{{expand section\|date=November 2013}}~~
	~~Residuals can be tested for homoscedasticity using the [[Breusch–Pagan test]], which regresses square residuals~~ to independent variables. Since the Breusch–Pagan test is sensitive to normality, the Koenker–Basset or 'generalized Breusch–Pagan' test is used for general purposes{{cn\|date=November 2013}}. Testing for groupwise heteroscedasticity requires the [[Goldfeld–Quandt test]].

	~~==Homoscedastic distributions==~~

	~~Two or~~ more ~~[[normal distribution]]s, <math>N(\mu_i,\Sigma_i)<~~/~~math>, are homoscedastic if they share a common [[covariance matrix\|covariance]] (or [[correlation matrix\|correlation]]) matrix,~~ <~~math~~>~~\Sigma_i = \Sigma_j,\ \forall i,j~~<~~/math~~>~~. Homoscedastic distributions are especially useful to derive statistical [[pattern recognition]] and [[machine learning]] algorithms. One popular example is Fisher's [[linear discriminant analysis]].~~

	~~The concept of homoscedasticity can be applied to distributions on spheres.<ref>Hamsici, Onur C.; Martinez, Aleix M. (2007)~~ [http://~~jmlr.csail.mit.edu/papers/volume8/hamsici07a/hamsici07a~~.~~pdf "Spherical-Homoscedastic Distributions: The Equivalency~~ of ~~Spherical and Normal Distributions in Classification"], ''Journal of Machine Learning Research'', 8, 1583-1623</ref>~~

	~~==See also==~~
	*[[Bartlett's test]]
	*[[Homogeneity (statistics)]]
	*[[Heterogeneous#Statistics\|Heterogeneity]]

	~~{{refimprove\|date=October 2011}}~~

	~~==References==~~
	~~{{reflist}}~~

	~~{{statistics}}~~

	~~[[Category:Statistical deviation and dispersion]]~~

	~~[[de:Homoskedastizität]~~]

en>Jab843: Reverted edits by 108.6.6.60 (talk) to last revision by ClueBot NG (HG)

2014-01-27T23:11:14Z

Reverted edits by 108.6.6.60 (talk) to last revision by ClueBot NG (HG)

Show changes

en>Dirac66: Undid revision 510038308 by 70.95.87.99 (talk) Rv parsing error

2012-08-31T01:56:34Z

Undid revision 510038308 by 70.95.87.99 (talk) Rv parsing error

Show changes

Commutative property - Revision history

en>Dirac66: /* Commutative operations in mathematics */ Truth functions - added example and clarified obscure notation

en>Jab843: Reverted edits by 108.6.6.60 (talk) to last revision by ClueBot NG (HG)

en>Dirac66: Undid revision 510038308 by 70.95.87.99 (talk) Rv parsing error