Composite Experimental Design - Revision history

Admin: Fix math formatting: remove trailing space in b_{ii} subscript, collapse multi-line $tags in polynomial ordering table to single lines (removes extraneous whitespace in rendered output) (via update-page on MediaWiki MCP Server)$

2026-06-15T00:01:49Z

Fix math formatting: remove trailing space in b_{ii} subscript, collapse multi-line <math> tags in polynomial ordering table to single lines (removes extraneous whitespace in rendered output) (via update-page on MediaWiki MCP Server)

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Admin: /* Quadratic Surface, 6 Dimensions */

2017-12-18T19:46:53Z

Quadratic Surface, 6 Dimensions

← Older revision		Revision as of 19:46, 18 December 2017
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	<math>		<math>
	\hat{y}(\boldsymbol{x}) = b_0 + \sum_{i=1}^{6} b_i x_i + \sum_{i < j} \sum_{j=1}^{6} b_{ij} x_i x_j + \sum_{i=1}^{6} ~~b_i~~ x_i^2		\hat{y}(\boldsymbol{x}) = b_0 + \sum_{i=1}^{6} b_i x_i + \sum_{i > j} \sum_{j=1}^{6} b_{ij} x_i x_j + \sum_{i=1}^{6} b_{ii }x_i^2
	</math>		</math>

Admin: /* Cubic Surface, 6 Dimensions: Trouble in Paradise */

2017-12-16T11:35:58Z

Cubic Surface, 6 Dimensions: Trouble in Paradise

← Older revision		Revision as of 11:35, 16 December 2017
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	===Cubic Surface, 6 Dimensions: Trouble in Paradise===		===Cubic Surface, 6 Dimensions: Trouble in Paradise===

	Download the response surface here: http://files.charlesmartinreid.com/ExperimentalDesign/ResponseSurface_6dim_3deg.mat		<s>Download the response surface here: http://files.charlesmartinreid.com/ExperimentalDesign/ResponseSurface_6dim_3deg.mat
	* contains 2 variables:		* contains 2 variables:
	** <code>response_surface</code> - structure containing information about the response surface (coefficient vector is in <code>response_surface.beta</code>)		** <code>response_surface</code> - structure containing information about the response surface (coefficient vector is in <code>response_surface.beta</code>)
	** <code>model</code> - this is a matrix containing the polynomial powers of each variable (variable order given in section above, [[#A Note on Coefficient and Variable Order]]; description of polynomial powers matrix given in section above, [[#Polynomial Powers Matrix]])		** <code>model</code> - this is a matrix containing the polynomial powers of each variable (variable order given in section above, [[#A Note on Coefficient and Variable Order]]; description of polynomial powers matrix given in section above, [[#Polynomial Powers Matrix]])</s>

	{\|border="0"		{\|border="0"

Admin: /* Quadratic Surface, 2 Dimensions */

2017-12-16T11:35:26Z

Quadratic Surface, 2 Dimensions

← Older revision		Revision as of 11:35, 16 December 2017
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	===Quadratic Surface, 2 Dimensions===		===Quadratic Surface, 2 Dimensions===

	Download the response surface here: http://files.charlesmartinreid.com/ExperimentalDesign/ResponseSurface_2dim_2deg.mat		<s>Download the response surface here: http://files.charlesmartinreid.com/ExperimentalDesign/ResponseSurface_2dim_2deg.mat
	* contains 2 variables:		* contains 2 variables:
	** <code>response_surface</code> - structure containing information about the response surface (coefficient vector is in <code>response_surface.beta</code>)		** <code>response_surface</code> - structure containing information about the response surface (coefficient vector is in <code>response_surface.beta</code>)
	** <code>model</code> - this is a matrix containing the polynomial powers of each variable (variable order given in section above, [[#A Note on Coefficient and Variable Order]]; description of polynomial powers matrix given in section above, [[#Polynomial Powers Matrix]])		** <code>model</code> - this is a matrix containing the polynomial powers of each variable (variable order given in section above, [[#A Note on Coefficient and Variable Order]]; description of polynomial powers matrix given in section above, [[#Polynomial Powers Matrix]])</s>

	The response surface resulting from the regression of ''only'' the two dimensions visualized (of the same form, but lower in dimension) results in a polynomial coefficient vector of:		The response surface resulting from the regression of ''only'' the two dimensions visualized (of the same form, but lower in dimension) results in a polynomial coefficient vector of:

Admin: /* Quadratic Surface, 6 Dimensions */

2017-12-16T11:35:05Z

Quadratic Surface, 6 Dimensions

← Older revision		Revision as of 11:35, 16 December 2017
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	===Quadratic Surface, 6 Dimensions===		===Quadratic Surface, 6 Dimensions===

	Download the response surface here: http://files.charlesmartinreid.com/ExperimentalDesign/ResponseSurface_6dim_2deg.mat		<s>Download the response surface here: http://files.charlesmartinreid.com/ExperimentalDesign/ResponseSurface_6dim_2deg.mat
	* contains 2 variables:		* contains 2 variables:
	** <code>response_surface</code> - structure containing information about the response surface (coefficient vector is in <code>response_surface.beta</code>)		** <code>response_surface</code> - structure containing information about the response surface (coefficient vector is in <code>response_surface.beta</code>)
	** <code>model</code> - this is a matrix containing the polynomial powers of each variable (variable order given in section above, [[#A Note on Coefficient and Variable Order]]; description of polynomial powers matrix given in section above, [[#Polynomial Powers Matrix]])		** <code>model</code> - this is a matrix containing the polynomial powers of each variable (variable order given in section above, [[#A Note on Coefficient and Variable Order]]; description of polynomial powers matrix given in section above, [[#Polynomial Powers Matrix]])</s>

	A quadratic response surface for <math>y_{p,exit}</math>, a quadratic function of 6 input parameters of the form:		A quadratic response surface for <math>y_{p,exit}</math>, a quadratic function of 6 input parameters of the form:

Admin: /* Polynomial Powers Matrix */

2017-12-16T11:34:41Z

Polynomial Powers Matrix

← Older revision		Revision as of 11:34, 16 December 2017
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	The polynomial powers matrix for an N-variable polynomial of M degrees is a <math>T \times N</math> matrix, where T is the number of different polynomial terms that can exist for a polynomial with N variables and M degrees.		The polynomial powers matrix for an N-variable polynomial of M degrees is a <math>T \times N</math> matrix, where T is the number of different polynomial terms that can exist for a polynomial with N variables and M degrees.

	The function <code>allVL1</code> ( available for download here: http://files.charlesmartinreid.com/ExperimentalDesign/allVL1.m ) creates this matrix of permutations. Alternatively, the <code>regstats</code> and <code>x2fx</code> Matlab functions will automatically generate their own versions of this matrix (albeit in a different order than from the <code>allLV1</code> function).		The function <code>allVL1</code> (<s>available for download here: http://files.charlesmartinreid.com/ExperimentalDesign/allVL1.m</s>) creates this matrix of permutations. Alternatively, the <code>regstats</code> and <code>x2fx</code> Matlab functions will automatically generate their own versions of this matrix (albeit in a different order than from the <code>allLV1</code> function).

Admin: /* Cubic Surface, 6 Dimensions: Trouble in Paradise */

2011-07-03T08:08:11Z

Cubic Surface, 6 Dimensions: Trouble in Paradise

@@ Line 497: / Line 497: @@
 </pre>
-where the order for the first 28 terms is the same as for the quadratic models above, and the remaining 9 terms are in the order given in the table to the right.
+where the order for the first 28 terms is the same as for the quadratic models above, and the remaining 9 terms are in the order given in the table to the right. This makes the polynomial powers matrix:
 |style="vertical-align:top; width:20%;"|
@@ Line 558: / Line 598: @@
 [[Image:CompositeResponseSurface_6Dim_3Deg.png|500px]]
-Note the range of the response: this is clearly a fishy response surface (maximum predicted <math>y_p</math> is on the order of 1000???).  However, looking at the statistics shows that ''the polynomial creates a perfect fit''!!!
+Note the range of the response: this is clearly a fishy response surface (maximum predicted <math>y_p</math> is on the order of 1000???).  However, looking at the statistics shows that ''the polynomial creates a perfect fit''!
 <pre>

Admin: /* Quadratic Surface, 2 Dimensions */

2011-07-03T08:05:35Z

Quadratic Surface, 2 Dimensions

@@ Line 400: / Line 400: @@
 b(5) = -0.0145
 b(6) = -0.0009
 </pre>

Admin: /* Quadratic Surface, 6 Dimensions */

2011-07-03T08:01:36Z

Quadratic Surface, 6 Dimensions

@@ Line 327: / Line 327: @@
 b(27) = 1.1544e-02
 b(28) = 4.1922e+02
 </pre>

Admin: /* Computing Response Surface */

2011-07-03T06:29:11Z

Computing Response Surface

← Older revision		Revision as of 06:29, 3 July 2011
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	The resulting response surface, holding all other parameters constant at their mean value, looks like:		The resulting response surface, holding all other parameters constant at their mean value, looks like:

	[[Image:~~CompositeResponseSurface_Dim6_Deg2~~.png\|500px]]		[[Image:CompositeResponseSurface_6Dim_2Deg.png\|500px]]

	Some key statistics for the response surface are given here:		Some key statistics for the response surface are given here:
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	It also results in the following response surface:		It also results in the following response surface:

	[[Image:~~CompositeResponseSurface_Dim2_Deg2~~.png\|500px]]		[[Image:CompositeResponseSurface_2Dim_2Deg.png\|500px]]

	This surface has the following statistics:		This surface has the following statistics:
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	This model presents an interesting problem. The 6-dimensional response surface that results, plotted in 2 dimensions (again using mean values for non-visualized dimensions), looks like this:		This model presents an interesting problem. The 6-dimensional response surface that results, plotted in 2 dimensions (again using mean values for non-visualized dimensions), looks like this:

	[[Image:~~CompositeResponseSurface_Dim6_Deg3~~.png\|500px]]		[[Image:CompositeResponseSurface_6Dim_3Deg.png\|500px]]

	Note the range of the response: this is clearly a fishy response surface (maximum predicted <math>y_p</math> is on the order of 1000???). However, looking at the statistics shows that ''the polynomial creates a perfect fit''!!!		Note the range of the response: this is clearly a fishy response surface (maximum predicted <math>y_p</math> is on the order of 1000???). However, looking at the statistics shows that ''the polynomial creates a perfect fit''!!!

Composite Experimental Design - Revision history

Admin: Fix math formatting: remove trailing space in b_{ii} subscript, collapse multi-line tags in polynomial ordering table to single lines (removes extraneous whitespace in rendered output) (via update-page on MediaWiki MCP Server)

Admin: /* Quadratic Surface, 6 Dimensions */

Admin: /* Cubic Surface, 6 Dimensions: Trouble in Paradise */

Admin: /* Quadratic Surface, 2 Dimensions */

Admin: /* Quadratic Surface, 6 Dimensions */

Admin: /* Polynomial Powers Matrix */

Admin: /* Cubic Surface, 6 Dimensions: Trouble in Paradise */

Admin: /* Quadratic Surface, 2 Dimensions */

Admin: /* Quadratic Surface, 6 Dimensions */

Admin: /* Computing Response Surface */

Admin: Fix math formatting: remove trailing space in b_{ii} subscript, collapse multi-line $tags in polynomial ordering table to single lines (removes extraneous whitespace in rendered output) (via update-page on MediaWiki MCP Server)$