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Admin: /* Response Surface Models (Multivariate) */

2011-06-30T22:35:22Z

Response Surface Models (Multivariate)

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	==Response Surface Models (Multivariate)==		==Response Surface Models (Multivariate)==

			{{Main\|Response Surface Methodology}}

	Response surfaces may be created in a number of different ways:		Response surfaces may be created in a number of different ways:
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	* others???		* others???




	=See Also=		=See Also=

Admin: Created page with "=Linear Model Classification= It is useful to describe various classifications of linear models to better understand how each type of linear model (e.g. [[Response Surface Metho..."

2011-06-30T22:34:53Z

Created page with "=Linear Model Classification= It is useful to describe various classifications of linear models to better understand how each type of linear model (e.g. [[Response Surface Metho..."

New page

=Linear Model Classification=

It is useful to describe various classifications of linear models to better understand how each type of linear model (e.g. [[Response Surface Methodology]]) fits into the "big picture".

A very helpful guide, given by Matlab, that describes and illustrates various regression analysis techniques: http://www.mathworks.com/help/toolbox/stats/bq_676m-2.html

==Generalized Linear Models==

"Generalized linear models" are linear models that can account for arbitrary numbers of inputs and outputs. These models assume errors are Gaussian, use statistics and create statistical models for data analysis.

General form:

<math>
\mathbf{Y} = \mathbf{B X} + \mathbf{U}
</math>

where:

<math>Y</math> is the response of the function of interest

<math>\mathbf{B}</math> is the coefficient vector/matrix for the linear model coefficients

<math>\mathbf{X}</math> are the inputs to the function

<math>\mathbf{U}</math> is the error

General linear model information:
* http://en.wikipedia.org/wiki/Multivariate_regression_model
* Nelder and Wedderburn (1972)

Matlab functions:
* glmfit
** http://www.mathworks.com/help/toolbox/stats/glmfit.html
** Carries out the regression

* glmval
** http://www.mathworks.com/help/toolbox/stats/glmval.html
** Evaluates the value of a generalized linear model

==Multiple Linear Regression==

"Multiple linear regression" is a model for one response variable ("y"), and multiple predictor variables ("X").

Linear regression information:
* http://en.wikipedia.org/wiki/Linear_regression
* http://www.mathworks.com/help/toolbox/stats/mvregress.html
* Mason Ch. 15

Generic multiple linear regression model looks like:

<math>
y_i = \beta_0 + \beta_1 x_{i1} + \beta_2 x_{i2} + \dots + \beta_{k} x_{ik} + e_i \quad i=1 \dots n
</math>

for <math>n</math> responses (and all responses treated independently). Here, <math>y_i</math> is the <math>i^{th}</math> response from the "real" (i.e. unknown) process, which is perfectly represented using the polynomial above.

It is desirable to create a surrogate model that creates as good an approximation as possible of the above polynomial.

This "real", unknown polynomial can also be written (similar to ANOVA model):

<math>
y_i = \mu_i + e_i
</math>

where

<math>
\mu_i = \beta_0 + \beta_1 x_{i1} + \beta_2 x_{i2} + \dots + \beta_{k} x_{ik}
</math>

If each of the coefficients <math>\beta_j</math> are approximated (using linear algebra, e.g. least squares), the mean can be approximated:

<math>
\hat{\mu}_i = \beta_0 + \beta_1 x_{i1} + \beta_2 x_{i2} + \dots + \beta_{k} x_{ik}
</math>

and using that, the "real" polynomial responses <math>y_i</math> can be approximated with surrogate (or predicted) polynomial responses <math>\hat{y}_i</math>:

<math>
\hat{y}_i = \hat{\mu}_i
</math>

==Multivariate Linear Regression==

"Multivariate linear regression" broadens multiple linear regression to account for more than one response variable ("Y").

Multivariate regression/analysis information:
* http://en.wikipedia.org/wiki/Multivariate_analysis

==Polynomial Models (Univariate)==

Polynomial models can be used to fit a univariate function of a single input paramter, e.g. <math>y(x)</math>

This can be done using the following Matlab functions:
* polyfit
** http://www.mathworks.com/help/techdoc/ref/polyfit.html
** Fits a polynomial of a given degree to a set of inputs x and outputs y

* polyval
** http://www.mathworks.com/help/techdoc/ref/polyval.html
** Evaluates the value of a given polynomial model at given input variable value or values

* polyconf
** http://www.mathworks.com/help/toolbox/stats/polyconf.html
** Can be used to construct confidence intervals for polynomial models

==Response Surface Models (Multivariate)==

Response surfaces may be created in a number of different ways:

* regress

* regstats

* SurfaceFit (Statistics toolbox)

* others???

=See Also=

Pages for different types of linear models:
* [[Response Surface Methodology]]

[[Category:Experimental Design]]

Linear Models - Revision history

Admin at 16:31, 16 December 2017

Admin at 16:31, 16 December 2017

Admin: /* Response Surface Models (Multivariate) */

Admin: Created page with "=Linear Model Classification= It is useful to describe various classifications of linear models to better understand how each type of linear model (e.g. [[Response Surface Metho..."