Expander walk sampling - Revision history

178.38.179.63: /* Statement */

2014-12-11T11:00:31Z

Statement

← Older revision		Revision as of 13:00, 11 December 2014
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	~~In [[statistics]]~~, ~~a '''central composite design'''~~ is an experimental design, useful in [[response surface methodology]], for building a second order (quadratic) model for the [[response variable]] without needing to use a complete three-level [[factorial experiment]].		Nice to meet you, my name is Ling and I totally dig that title. Arizona has usually been my living place but my spouse wants us to transfer. To play croquet is some thing that I've carried out for years. Interviewing is how I make a living and it's some thing I really appreciate.<br><br>My blog post; [http://www.friscowebsolutions.com/UserProfile/tabid/379/userId/203/Default.aspx extended car warranty]

	~~After the designed experiment is performed, [[linear regression]] is used, sometimes iteratively,~~ to ~~obtain results. Coded variables are often used when constructing this design.~~

	~~==Implementation==~~

	~~The design consists of three distinct sets of experimental runs:~~

	~~# A [[factorial design\|factorial]] (perhaps [[Fractional factorial design\|fractional]]) design in the factors studied, each having two levels;~~
	~~# A set of ''center points'', experimental runs whose values of each factor are the medians of the values used in the factorial portion~~. ~~This point~~ is ~~often replicated in order to improve the precision of the experiment;~~
	~~# A set of~~ '~~'axial points'', experimental runs identical to the centre points except~~ for ~~one factor, which will take on values both below and above the median of the two factorial levels, and typically both outside their range~~. ~~All factors are varied in this way.~~

	~~==Design matrix ==~~
	~~The design matrix for a central composite design experiment involving ''k'' factors~~ is ~~derived from~~ a ~~matrix, '''d''', containing the following three different parts corresponding to the three types of experimental runs:~~
	~~#The matrix '''F''' obtained from the factorial experiment. The factor levels are scaled so that its entries are coded as +1~~ and ~~−1.~~
	~~#The matrix '''C''' from the center points, denoted in coded variables as (0,0,0,...,0), where there are ''k'' zeros.~~
	~~#A matrix '''E''' from the axial points, with 2''k'~~' ~~rows. Each factor is sequentially placed at ±α and all other factors are at zero. The value of α is determined by the designer; while arbitrary,~~ some ~~values may give the design desirable properties~~. ~~This part would look like:~~

	:<~~math~~> ~~\bold E = \begin{bmatrix}~~
	~~\alpha & 0 & 0 & \cdots & \cdots & \cdots & 0 \\~~
	~~{ - \alpha } & 0 & 0 & \cdots & \cdots & \cdots & 0 \\~~
	~~0 & \alpha & 0 & \cdots & \cdots & \cdots & 0 \\~~
	~~0 & { - \alpha } & 0 & \cdots & \cdots & \cdots & 0 \\~~
	~~\vdots & {} & {} & {} & {} & {} & \vdots \\~~
	~~0 & 0 & 0 & 0 & \cdots & \cdots & \alpha \\~~
	~~0 & 0 & 0 & 0 & \cdots & \cdots & { - \alpha } \\~~
	~~\end{bmatrix}.~~ <~~/math~~>

	~~Then '''d''' is the vertical concatenation:~~
	~~:<math> \bold d = \begin{bmatrix} \bold F \\ \bold C \\ \bold E~~
	~~\end{bmatrix}. </math>~~

	~~The design matrix '''X''' used in linear regression is the horizontal concatenation of a column of 1s (intercept), '''d''', and all elementwise products of a pair of columns of '''d'''~~:

	:<math>\bold X = \begin{bmatrix} \bold 1 & \bold d & \bold d(1)\times\bold d(2) & \bold d(1)\times\bold d(3) & \cdots & \bold d(k-1)\times\bold d(k) & \bold d(1)^2 &\bold d(2)^2 &\cdots & \bold d(k)^2 \end{bmatrix}, </~~math>~~

	~~where '''d'''(''i'') represents the ''i''<sup>th<~~/~~sup> column in '''d'''~~.

	~~=== Choosing α ===~~

	There are many different methods to select a useful value of α. Let ''F'' be the number of points due to the factorial design and ''T'' = 2''k'' + ''n'', the number of additional points, where ''n'' is the number of central points in the design. ~~Common values are as follows (Myers, 1971):~~
	~~#'''Orthogonal design:''': <math>\alpha = (Q\times F~~/~~4)^{1~~/~~4}\,\!<~~/~~math>, where <math> Q = (\sqrt{F + T} -\sqrt{F})^2 <~~/~~math>;~~
	~~#'''Rotatable design''': α = ''F''<sup>1~~/4</~~sup> (the design implemented by [[MATLAB]]’s ''ccdesign'' function)~~.

	~~==References==~~
	~~Myers, Raymond H. ''Response Surface Methodology''. Boston: Allyn and Bacon, Inc., 1971~~
	~~{{Experimental design}}~~
	~~{{Statistics}}~~
	~~[[Category:Design of experiments]]~~
	~~[[Category:Educational research]]~~
	~~[[Category:Social sciences methodology]]~~
	~~[[Category:Evaluation methods]]~~
	~~[[Category:Mathematical optimization]~~]

en>ChrisGualtieri: /* Uses */General Fixes using AWB

2013-10-23T23:45:57Z

Uses: General Fixes using AWB

← Older revision		Revision as of 01:45, 24 October 2013
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	~~Myrtle Benny~~ is ~~how I'm called and I really feel comfy when people~~ use ~~the~~ complete ~~title~~. ~~Hiring~~ is ~~my profession~~. ~~California~~ is ~~our beginning location~~. ~~To do aerobics~~ is a factor that I'~~m completely addicted to~~.<br><br>~~my web page~~ :: ~~[http~~://~~xrambo~~.~~com~~/~~user~~/~~NEme xrambo~~.~~com~~]		In [[statistics]], a '''central composite design''' is an experimental design, useful in [[response surface methodology]], for building a second order (quadratic) model for the [[response variable]] without needing to use a complete three-level [[factorial experiment]].

			After the designed experiment is performed, [[linear regression]] is used, sometimes iteratively, to obtain results. Coded variables are often used when constructing this design.

			==Implementation==

			The design consists of three distinct sets of experimental runs:

			# A [[factorial design\|factorial]] (perhaps [[Fractional factorial design\|fractional]]) design in the factors studied, each having two levels;
			# A set of ''center points'', experimental runs whose values of each factor are the medians of the values used in the factorial portion. This point is often replicated in order to improve the precision of the experiment;
			# A set of ''axial points'', experimental runs identical to the centre points except for one factor, which will take on values both below and above the median of the two factorial levels, and typically both outside their range. All factors are varied in this way.

			==Design matrix ==
			The design matrix for a central composite design experiment involving ''k'' factors is derived from a matrix, '''d''', containing the following three different parts corresponding to the three types of experimental runs:
			#The matrix '''F''' obtained from the factorial experiment. The factor levels are scaled so that its entries are coded as +1 and −1.
			#The matrix '''C''' from the center points, denoted in coded variables as (0,0,0,...,0), where there are ''k'' zeros.
			#A matrix '''E''' from the axial points, with 2''k'' rows. Each factor is sequentially placed at ±α and all other factors are at zero. The value of α is determined by the designer; while arbitrary, some values may give the design desirable properties. This part would look like:

			:<math> \bold E = \begin{bmatrix}
			\alpha & 0 & 0 & \cdots & \cdots & \cdots & 0 \\
			{ - \alpha } & 0 & 0 & \cdots & \cdots & \cdots & 0 \\
			0 & \alpha & 0 & \cdots & \cdots & \cdots & 0 \\
			0 & { - \alpha } & 0 & \cdots & \cdots & \cdots & 0 \\
			\vdots & {} & {} & {} & {} & {} & \vdots \\
			0 & 0 & 0 & 0 & \cdots & \cdots & \alpha \\
			0 & 0 & 0 & 0 & \cdots & \cdots & { - \alpha } \\
			\end{bmatrix}. </math>

			Then '''d''' is the vertical concatenation:
			:<math> \bold d = \begin{bmatrix} \bold F \\ \bold C \\ \bold E
			\end{bmatrix}. </math>

			The design matrix '''X''' used in linear regression is the horizontal concatenation of a column of 1s (intercept), '''d''', and all elementwise products of a pair of columns of '''d''':

			:<math>\bold X = \begin{bmatrix} \bold 1 & \bold d & \bold d(1)\times\bold d(2) & \bold d(1)\times\bold d(3) & \cdots & \bold d(k-1)\times\bold d(k) & \bold d(1)^2 &\bold d(2)^2 &\cdots & \bold d(k)^2 \end{bmatrix}, </math>

			where '''d'''(''i'') represents the ''i''<sup>th</sup> column in '''d'''.

			=== Choosing α ===

			There are many different methods to select a useful value of α. Let ''F'' be the number of points due to the factorial design and ''T'' = 2''k'' + ''n'', the number of additional points, where ''n'' is the number of central points in the design. Common values are as follows (Myers, 1971):
			#'''Orthogonal design:''': <math>\alpha = (Q\times F/4)^{1/4}\,\!</math>, where <math> Q = (\sqrt{F + T} -\sqrt{F})^2 </math>;
			#'''Rotatable design''': α = ''F''<sup>1/4</sup> (the design implemented by [[MATLAB]]’s ''ccdesign'' function).

			==References==
			Myers, Raymond H. ''Response Surface Methodology''. Boston: Allyn and Bacon, Inc., 1971
			{{Experimental design}}
			{{Statistics}}
			[[Category:Design of experiments]]
			[[Category:Educational research]]
			[[Category:Social sciences methodology]]
			[[Category:Evaluation methods]]
			[[Category:Mathematical optimization]]

en>Citation bot: [344]+: issue.

2011-05-19T15:00:01Z

[344]+: issue.

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Myrtle Benny is how I'm called and I really feel comfy when people use the complete title. Hiring is my profession. California is our beginning location. To do aerobics is a factor that I'm completely addicted to.<br><br>my web page :: [http://xrambo.com/user/NEme xrambo.com]