Quasinormal operator - Revision history

en>Yobot: WP:CHECKWIKI error fixes using AWB (10093)

2014-05-05T12:17:34Z

WP:CHECKWIKI error fixes using AWB (10093)

← Older revision		Revision as of 14:17, 5 May 2014
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	In [[statistics]], the '''generalized linear array model'''('''GLAM''') is used for analyzing data sets with array structures. It based on the [[generalized linear model]] with the [[design matrix]] written as a [[Kronecker product]].		I am Oscar and I totally dig that title. Hiring is her working day occupation now but std [https://hcbsales.com/node/3739 at home std testing] test she's usually wanted her own company. One of [http://www.revleft.com/vb/member.php?u=160656 over the counter std test] extremely best issues in the world for me is to do aerobics and I've been doing it for fairly a while. South Dakota is her std home test beginning location but she requirements to transfer because of her family.<br><br>Here is my homepage ... home [http://Samedaystdtesting.com/testing-clinics/florida-fl/hialeah-std-testing/2750-west-68th-street-suite-225-226-33016 std test] kit - [http://www.onbizin.co.kr/xe/?document_srl=354357 visit the up coming internet page],

	~~== Overview ==~~
	~~The generalized linear array model or GLAM was introduced in 2006.<ref>Currie,~~ I.~~D.;Durban, M.;Eilers, P. H. C.~~ ~~(2006) "Generalized linear array models with applications to multidimensional smoothing",''~~[~~[Journal of the Royal Statistical Society]]'', 68(2), 259-280~~.</~~ref> Such models provide a structure and a computational procedure for fitting [[generalized linear model]~~]s ~~or GLMs whose model matrix can be written as a Kronecker product and whose data can be written as an array~~. ~~In a large GLM, the GLAM approach gives very substantial savings in both storage and computational time over the usual GLM algorithm.~~

	~~Suppose that the data <math>\mathbf Y<~~/~~math> is arranged in a <math>d<~~/~~math>-dimensional array with size <math>n_1\times n_2\times\ldots\times n_d<~~/~~math>; thus,the corresponding data vector <math>\mathbf y = \textbf{vec}(\mathbf Y)<~~/~~math> has size <math>n_1n_2n_3\cdots n_d</math>~~. ~~Suppose also that~~ the ~~[[design matrix]~~] is ~~of the form~~
	~~:<math>\mathbf X = \mathbf X_d\otimes\mathbf X_{d-1}\otimes\ldots\otimes\mathbf X_1.</math>~~

	~~The standard analysis of a GLM with data vector <math>\mathbf y</math>~~ and ~~design matrix <math>\mathbf X</math> proceeds by repeated evaluation of the scoring algorithm~~

	~~:<math> \mathbf X~~'~~\tilde{\mathbf W}_\delta\mathbf X\hat{\boldsymbol\theta} = \mathbf X'\tilde{\mathbf W}_\delta\tilde{\mathbf z} ,</math>~~

	~~where <math>\tilde{\boldsymbol\theta}</math> represents the approximate solution of <math>\boldsymbol\theta</math>, and <math>\hat{\boldsymbol\theta}</math>~~ is ~~the improved~~ ~~value~~ of ~~it;~~ <~~math~~>~~\mathbf W_\delta~~<~~/math~~> is ~~the diagonal weight matrix with elements~~

	~~:<math> w_{ii}^{-1} = \left(\frac{\partial\eta_i}{\partial\mu_i}\right)^2\text{var}(y_i),</math>~~

	~~and~~
	~~:<math>\mathbf z = \boldsymbol\eta + \mathbf W_\delta^{-1}(\mathbf y - \boldsymbol\mu)</math>~~
	~~is the working variable~~.

	~~Computationally, GLAM provides array algorithms to calculate the linear predictor,~~
	:~~<math> \boldsymbol\eta = \mathbf X \boldsymbol\theta <~~/~~math>~~
	~~and the weighted inner product~~
	~~:<math> \mathbf X'\tilde{\mathbf W}_\delta\mathbf X <~~/~~math>~~
	~~without evaluation of the model matrix <math> \mathbf X~~ .</~~math>~~

	~~===Example===~~

	~~In 2 dimensions, let <math>\mathbf X = \mathbf X_2\otimes\mathbf X_1,<~~/~~math> then the linear predictor is written <math>\mathbf X_1 \boldsymbol\Theta \mathbf X_2' <~~/~~math> where <math>\boldsymbol\Theta <~~/math> is the matrix of coefficients; the weighted inner product is obtained from <math>G(\mathbf X_1)' \mathbf W G(\mathbf X_2)</math> and <math> \mathbf W </math> is the matrix of weights; here <math>G(\mathbf M) </math> is the row tensor function of the <math> r \times c</math> matrix <math> \mathbf M </math> given by

	:~~<math>G(\mathbf M) = (\mathbf M \otimes \mathbf 1') * (\mathbf 1' \otimes \mathbf M)<~~/~~math>~~
	~~where <math>*<~~/~~math> means element by element multiplication and <math>\mathbf 1</math> is a vector of 1's of length <math> c</math>~~.

	~~These low storage high speed formulae extend to <math>d</math>-dimensions~~.

	~~==Applications==~~
	~~GLAM is designed to be used in <math>d<~~/~~math>-dimensional smoothing problems where the data are arranged in an array and the smoothing matrix is constructed as a Kronecker product of <math>d<~~/~~math> one-dimensional smoothing matrices.~~

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

	~~[[Category:Multivariate statistics]]~~
	~~[[Category:Generalized linear models]~~]

en>Mark viking: Added wl

2014-01-17T05:56:36Z

Added wl

← Older revision		Revision as of 07:56, 17 January 2014
Line 1:		Line 1:
	~~Hello and welcome~~. ~~My title is Figures Wunder~~. ~~My family members lives~~ in ~~Minnesota and my family members loves it~~. ~~He utilized~~ to be ~~unemployed but now he~~ is a ~~meter reader~~. ~~Playing baseball~~ is the ~~hobby he will by no means stop performing~~.<br><br>~~Also visit my web page [http~~://~~drupal~~.~~12thirty4~~.~~com~~/~~gmaps~~/~~node~~/~~4991 http~~://~~drupal~~.~~12thirty4~~.~~com~~/]		In [[statistics]], the '''generalized linear array model'''('''GLAM''') is used for analyzing data sets with array structures. It based on the [[generalized linear model]] with the [[design matrix]] written as a [[Kronecker product]].

			== Overview ==
			The generalized linear array model or GLAM was introduced in 2006.<ref>Currie, I.D.;Durban, M.;Eilers, P. H. C. (2006) "Generalized linear array models with applications to multidimensional smoothing",''[[Journal of the Royal Statistical Society]]'', 68(2), 259-280.</ref> Such models provide a structure and a computational procedure for fitting [[generalized linear model]]s or GLMs whose model matrix can be written as a Kronecker product and whose data can be written as an array. In a large GLM, the GLAM approach gives very substantial savings in both storage and computational time over the usual GLM algorithm.

			Suppose that the data <math>\mathbf Y</math> is arranged in a <math>d</math>-dimensional array with size <math>n_1\times n_2\times\ldots\times n_d</math>; thus,the corresponding data vector <math>\mathbf y = \textbf{vec}(\mathbf Y)</math> has size <math>n_1n_2n_3\cdots n_d</math>. Suppose also that the [[design matrix]] is of the form
			:<math>\mathbf X = \mathbf X_d\otimes\mathbf X_{d-1}\otimes\ldots\otimes\mathbf X_1.</math>

			The standard analysis of a GLM with data vector <math>\mathbf y</math> and design matrix <math>\mathbf X</math> proceeds by repeated evaluation of the scoring algorithm

			:<math> \mathbf X'\tilde{\mathbf W}_\delta\mathbf X\hat{\boldsymbol\theta} = \mathbf X'\tilde{\mathbf W}_\delta\tilde{\mathbf z} ,</math>

			where <math>\tilde{\boldsymbol\theta}</math> represents the approximate solution of <math>\boldsymbol\theta</math>, and <math>\hat{\boldsymbol\theta}</math> is the improved value of it; <math>\mathbf W_\delta</math> is the diagonal weight matrix with elements

			:<math> w_{ii}^{-1} = \left(\frac{\partial\eta_i}{\partial\mu_i}\right)^2\text{var}(y_i),</math>

			and
			:<math>\mathbf z = \boldsymbol\eta + \mathbf W_\delta^{-1}(\mathbf y - \boldsymbol\mu)</math>
			is the working variable.

			Computationally, GLAM provides array algorithms to calculate the linear predictor,
			:<math> \boldsymbol\eta = \mathbf X \boldsymbol\theta </math>
			and the weighted inner product
			:<math> \mathbf X'\tilde{\mathbf W}_\delta\mathbf X </math>
			without evaluation of the model matrix <math> \mathbf X .</math>

			===Example===

			In 2 dimensions, let <math>\mathbf X = \mathbf X_2\otimes\mathbf X_1,</math> then the linear predictor is written <math>\mathbf X_1 \boldsymbol\Theta \mathbf X_2' </math> where <math>\boldsymbol\Theta </math> is the matrix of coefficients; the weighted inner product is obtained from <math>G(\mathbf X_1)' \mathbf W G(\mathbf X_2)</math> and <math> \mathbf W </math> is the matrix of weights; here <math>G(\mathbf M) </math> is the row tensor function of the <math> r \times c</math> matrix <math> \mathbf M </math> given by

			:<math>G(\mathbf M) = (\mathbf M \otimes \mathbf 1') * (\mathbf 1' \otimes \mathbf M)</math>
			where <math>*</math> means element by element multiplication and <math>\mathbf 1</math> is a vector of 1's of length <math> c</math>.

			These low storage high speed formulae extend to <math>d</math>-dimensions.

			==Applications==
			GLAM is designed to be used in <math>d</math>-dimensional smoothing problems where the data are arranged in an array and the smoothing matrix is constructed as a Kronecker product of <math>d</math> one-dimensional smoothing matrices.

			==References==
			{{reflist}}

			[[Category:Multivariate statistics]]
			[[Category:Generalized linear models]]

en>Kiefer.Wolfowitz: Adding category :Category:Invariant subspaces (using HotCat)

2010-05-23T16:04:04Z

Adding category Category:Invariant subspaces (using HotCat)

New page

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