Empirical Model-Building and Response Surfaces - Revision history

Admin: /* Chapter 13: Design Aspects of Variance, Bias, and Lack of Fit */

2011-07-20T12:56:29Z

Chapter 13: Design Aspects of Variance, Bias, and Lack of Fit

← Older revision		Revision as of 12:56, 20 July 2011
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	<math>\boldsymbol{y} = y_1, y_2, \dots, y_n</math>		<math>\boldsymbol{y} = y_1, y_2, \dots, y_n</math>

	<math>\boldsymbol{f(\xi)} = f(\boldsymbol{\xi_1}), f(\~~bolsymbol~~{\xi_2}), \dots, f(\boldsymbol{\xi_n})</math> (where the n different <math>\xi</math> are n observations of <math>\xi</math>, the true model is not:		<math>\boldsymbol{f(\xi)} = f(\boldsymbol{\xi_1}), f(\boldsymbol{\xi_2}), \dots, f(\boldsymbol{\xi_n})</math> (where the n different <math>\xi</math> are n observations of <math>\xi</math>, the true model is not:

	<math>		<math>
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	<math>		<math>
	\begin{array}{rcl}		\begin{array}{rcl}
	\mathbf{y} &=& \boldsymbol{\eta}(\~~bolsymbol~~{\xi}) + \boldsymbol{\epsilon} \\		\mathbf{y} &=& \boldsymbol{\eta}(\boldsymbol{\xi}) + \boldsymbol{\epsilon} \\
	&=& \mathbf{f}(\boldsymbol{\xi}) + \boldsymbol{\delta}(\boldsymbol{\xi}) + \boldsymbol{\epsilon}		&=& \mathbf{f}(\boldsymbol{\xi}) + \boldsymbol{\delta}(\boldsymbol{\xi}) + \boldsymbol{\epsilon}
	\end{array}		\end{array}
Line 1,197:		Line 1,197:

	<math>		<math>
	\boldsymbol{\delta}(\boldsymbol{\xi}) = \boldsymbol{\eta}(\~~boldysmbol~~{\xi}) - \mathbf{f}(\boldsymbol{\xi})		\boldsymbol{\delta}(\boldsymbol{\xi}) = \boldsymbol{\eta}(\boldsymbol{\xi}) - \mathbf{f}(\boldsymbol{\xi})
	</math>		</math>

Line 1,248:		Line 1,248:
	<math>		<math>
	\begin{array}{rcl}		\begin{array}{rcl}
			&=&
	\frac{		\frac{
	NE(\hat{y}_x - \eta_x)^2		NE(\hat{y}_x - \eta_x)^2
	}{		}{
	\sigma^2		\sigma^2
	} &=&		} \\
			&=&
	\frac{		\frac{
	NE \left[ \hat{y}_x - E( \hat{y}_x ) + E( \hat{y}_x ) - \eta_x \right]^2		NE \left[ \hat{y}_x - E( \hat{y}_x ) + E( \hat{y}_x ) - \eta_x \right]^2
	}{		}{
	\sigma^2		\sigma^2
	}		} \\
	&=&		&=&
	\frac{		\frac{
Line 1,276:		Line 1,278:
	<math>		<math>
	M_x = V_x + B_x		M_x = V_x + B_x
	<m/~~ath~~>		</math>

	standardized mean squared error <math>M_x</math> is equal to variance <math>V_x</math> plus squared bias <math>B_x</math> at x		standardized mean squared error <math>M_x</math> is equal to variance <math>V_x</math> plus squared bias <math>B_x</math> at x
Line 1,318:		Line 1,320:
	}}		}}

	===Regions of Interest: Weight Functions==		===Regions of Interest: Weight Functions===

	"Region of interest" discussion implies accuracy of prediction is uniform over interval R		"Region of interest" discussion implies accuracy of prediction is uniform over interval R
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	Parsimonious class of designs of this type: central composite arrangements in which a cube, consisting of two-level factorial with coded points <math>\pm 1</math>, or fraction of resolution <math>R \geq 5</math>, augmented by an added "star" with axial points at coded distance <math>\alpha</math> and by <math>n_0</math> added center points		Parsimonious class of designs of this type: central composite arrangements in which a cube, consisting of two-level factorial with coded points <math>\pm 1</math>, or fraction of resolution <math>R \geq 5</math>, augmented by an added "star" with axial points at coded distance <math>\alpha</math> and by <math>n_0</math> added center points

	(note that <math>n_c</math. is number of cube points, <math>n_s</math> is number of star points, <math>n_{0c}</math> number of center cube points, <math>n_{0s}</math> number of center star points, and thus <math>n_0 = n_{0c} + n_{0s}</math>)		(note that <math>n_c</math> is number of cube points, <math>n_s</math> is number of star points, <math>n_{0c}</math> number of center cube points, <math>n_{0s}</math> number of center star points, and thus <math>n_0 = n_{0c} + n_{0s}</math>)

	Before accepting utility of fitted equation, need to be reassured on two questions:		Before accepting utility of fitted equation, need to be reassured on two questions:
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	<math>		<math>
	\begin{array}{rcl}		\begin{array}{rcl}
	c_{31} &=& \frac{1}{36} \mathbf{x}_{111}~~^{\prime}~~ \mathbf{y} \\		c_{31} &=& \frac{1}{36} \mathbf{x}_{111}' \mathbf{y} \\
	&=& \frac{1}{3} \left[		&=&
			\frac{1}{3} \left[
	\frac{ \overline{y}_{\alpha} - \overline{y}_{-\alpha} }{ 2 \alpha }		\frac{ \overline{y}_{\alpha} - \overline{y}_{-\alpha} }{ 2 \alpha }
	- \frac{ \overline{y}_{1} - \overline{y}_{-1} }{ 2 }		- \frac{ \overline{y}_{1} - \overline{y}_{-1} }{ 2 }
	\right]		\right]
			\end{array}
	</math>		</math>

Admin: /* Bias Characteristics of the Design */

2011-07-20T12:48:17Z

Bias Characteristics of the Design

← Older revision		Revision as of 12:48, 20 July 2011
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	Alias or bias matrix A:		Alias or bias matrix A:

	<math>A = \left( Z_1^{\prime} Z_1 \~~left~~)^{-1} Z_1^{\prime} Z_2</math>		<math>A = \left( Z_1^{\prime} Z_1 \right)^{-1} Z_1^{\prime} Z_2</math>

	It shows that only the estimates of first order terms are biased by third-order terms		It shows that only the estimates of first order terms are biased by third-order terms

Admin: /* Estimation/Elimination of Block Differences */

2011-07-20T12:44:07Z

Estimation/Elimination of Block Differences

← Older revision		Revision as of 12:44, 20 July 2011
Line 850:		Line 850:

	<math>		<math>
	y = g ( \mathbf{x}, \boldsymbol{\beta} ) + x_B \~~detla~~ + \epsilon		y = g ( \mathbf{x}, \boldsymbol{\beta} ) + x_B \delta + \epsilon
	</math>		</math>

Admin: /* Competing effects of bias and variance */

2011-07-20T12:40:19Z

Competing effects of bias and variance

@@ Line 1,248: / Line 1,248: @@
 <math>
 \begin{array}{rcl}
-\frac{ NE(\hat{y}_x - \eta_x)^2 }{ \sigma^2 } &=& \frac{ NE \left[ \hat{y}_x - E( \hat{y}_x ) + E( \hat{y}_x} - \eta_x \right]^2 }{ \sigma^2 }
+\frac{
-&=& \frac{ NV(\hat{y}_x) }{ \sigma^2 } + \frac{ N\left[ E(\hat{y}_x) - \eta_x \right]^2 }{ \sigma^2 }
+  NE(\hat{y}_x - \eta_x)^2
 \end{array}
 </math>

Admin: /* Chapter 15: Practical Choice of a Response Surface Design */

2011-05-31T19:03:38Z

Chapter 15: Practical Choice of a Response Surface Design

@@ Line 1,673: / Line 1,673: @@
 =Chapter 15: Practical Choice of a Response Surface Design=
 {|class="wikitable" border=1
@@ Line 1,725: / Line 1,727: @@
 |}
 ==Sequential assembly==
@@ Line 1,796: / Line 1,796: @@
 Inductive criticism enhanced by possibility of being able to plot original data and residuals against variable parameter <math>\xi_1</math> for each individual level of <math>\xi_2</math>, or vice versa

Admin: /* Chapter 15: Practical Choice of a Response Surface Design */

2011-05-31T09:05:50Z

Chapter 15: Practical Choice of a Response Surface Design

← Older revision		Revision as of 09:05, 31 May 2011
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	\|-		\|-
	\|Allows check of fit		\|Allows check of fit
	\|* size of experimental region		\|
			* size of experimental region
	* smoothness of response function		* smoothness of response function
	* complexity of model		* complexity of model
Line 1,685:		Line 1,686:
	\|-		\|-
	\|Allows estimations of transformations		\|Allows estimations of transformations
	\|* lack of fit that could be corrected by transformation		\|
			* lack of fit that could be corrected by transformation

	\|-		\|-
	\|Permits sequential assembly		\|Permits sequential assembly
	\|* ability to perform runs sequentially		\|
			* ability to perform runs sequentially
	* ability to move in space of the variables		* ability to move in space of the variables

	\|-		\|-
	\|Can be run in blocks		\|Can be run in blocks
	\|* homogeneity of experimental materials		\|
			* homogeneity of experimental materials
	* state of control of the process		* state of control of the process

	\|-		\|-
	\|Provides independent estimate of error		\|Provides independent estimate of error
	\|* number of runs permissible		\|
			* number of runs permissible
	* possibility of large experimental error		* possibility of large experimental error
	* existance of reliable prior estimate of error		* existance of reliable prior estimate of error
Line 1,705:		Line 1,710:
	\|-		\|-
	\|Robustness of distribution of design points		\|Robustness of distribution of design points
	\|* ~~possiblity~~ of occasional aberrant runs and/or observations		\|
			* possibility of occasional aberrant runs and/or observations
	* nature of error function		* nature of error function

	\|-		\|-
	\|Number of runs ~~requried~~		\|Number of runs required
	\|* cost of making runs		\|
			* cost of making runs

	\|-		\|-
	\|Simplicity of data pattern		\|Simplicity of data pattern
	\|* need to visualize data to motivate model		\|
			* need to visualize data to motivate model

	\|}		\|}

Admin: /* Chapter 15: */

2011-05-31T09:04:45Z

Chapter 15:

← Older revision		Revision as of 09:04, 31 May 2011
Line 1,672:		Line 1,672:
	Rotatability: logical extension of orthogonality		Rotatability: logical extension of orthogonality

	=Chapter 15: =		=Chapter 15: Practical Choice of a Response Surface Design=

			{\|class="wikitable" border=1
			!Characteristics of design
			!Relevant experimental circumstances
			\|-
			\|Allows check of fit
			\|* size of experimental region
			* smoothness of response function
			* complexity of model

			\|-
			\|Allows estimations of transformations
			\|* lack of fit that could be corrected by transformation

			\|-
			\|Permits sequential assembly
			\|* ability to perform runs sequentially
			* ability to move in space of the variables

			\|-
			\|Can be run in blocks
			\|* homogeneity of experimental materials
			* state of control of the process

			\|-
			\|Provides independent estimate of error
			\|* number of runs permissible
			* possibility of large experimental error
			* existance of reliable prior estimate of error

			\|-
			\|Robustness of distribution of design points
			\|* possiblity of occasional aberrant runs and/or observations
			* nature of error function

			\|-
			\|Number of runs requried
			\|* cost of making runs

			\|-
			\|Simplicity of data pattern
			\|* need to visualize data to motivate model

			\|}

			have to take account of stuff on the right to determine relative importance of stuff on left

			==Sequential assembly==

			Many examples of designs used sequentially, e.g. using steepest ascent with first-order designs, then finding sufficiently promising region, then creating second-order model inside that region

			Illustration: three-phase sequential construction of design
			* I: regular simplex (4 of 8 cube corners) and 2 center points
			* II: complementary simplex (remaining 4 cube corners) and 2 (additional) center points
			* III: six axial points (star)

			Phase I: orthogonal first-order design, checks for overall curvature (via contrast of average response of center points with average response on cube)

			If first-order effects were large compared with their standard errors, and no serious curvature exhibited, then would explore indicated direction of steepest ascent

			If doubts about adequacy of first-degree polynomial model, then Step II would be performed

			Steps (I+II) still first order, but they give much better indication of detecting lack of fit due to additional information permitting estimation of two-parameter interactions

			If first-order terms dominate, move to more promising region

			If first-order effects small, and two-factor interactions dominate or strong evidence of curvature, move to Step III (second order)

			==Other Factors==

			===Robustness===

			Minimization of effects of wild observations (not sure waht that means, exactly)

			===Number of Runs===

			Experimental design should focus experimental effort on whatever is judged important in the context

			Suppose that in addition to estimating <math>p</math> parameters of assumed model form, need <math>f \geq 0</math> contrasts needed to check adequacy of fit, <math>b \geq 0</math> further contrasts needed for blocking, and estimate of experimental error needed having <math>e \geq 0</math> degrees of freedom

			To obtain independent estimates of all items of interest, require a design with at least <math>p + f + b + e</math> runs

			Relative importance of checking fit, blocking, and obtaining error estimate will differ in different situations

			Minimum value of N runs will correspondingly differ

			Corresponding minimum design will only be adequate if <math>\sigma^2</math> is below a critical value

			When <math>\sigma^2</math> larger, designs larger than minimum design needed to obtain estimates of sufficient precision

			Even with <math>\sigma^2</math> small, designs where <math>N > p</math> not wasteful

			Depends on whether additional degrees of freedom are genuinely used to achieve the experimenter's objectives

			===Simple Design Patterns===

			Question: why use design patterns, instead of randomized designs?

			Statistician's task:

			1. Inductive criticism

			2. Deductive estimation

			(2) involves deducing consequences of given assumptions, in the light of the data obtained; this is easily done with randomized designs

			(1) involves two questions:

			a) what function should be fitted in the first place?

			b) how to examine residuals from fitted function to understand deviations from initial model? (relation to the predictor variables, how to find appropriate model modification)

			Factorial/composite designs use patterns of experimental points allowing such comparisons to be made

			Inductive criticism enhanced by possibility of being able to plot original data and residuals against variable parameter <math>\xi_1</math> for each individual level of <math>\xi_2</math>, or vice versa

Admin: /* Chapter 14 */

2011-05-31T08:34:53Z

Chapter 14

@@ Line 1,635: / Line 1,635: @@
 . Center points, number <math>n_0</math>, where <math>n_{0c}</math> in cube, <math>n_{0s}</math> in star
-=Chapter 14=
+=Chapter 14: Variance-Optimal Designs=
 =Chapter 15: =

Admin: /* Chapter 13 */

2011-05-31T08:20:42Z

Chapter 13

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Admin at 04:50, 31 May 2011

2011-05-31T04:50:34Z

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