Monte Carlo Response Surfaces: Difference between revisions

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(Created page with "=Response Surface Results= ==Yp at exit== ===Quadratic Surface, 6 Dimensions=== Download the response surface here: http://files.charlesmartinreid.com/ExperimentalDesign/MCRes...")
 
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=Response Surface Results=
=Response Surface Results=
==Yp at X1==
==Yp at X2==
==Yp at X3==
===Quadratic Surface, 6 Dimensions===
{{{ResponseSurface
|link=http://files.charlesmartinreid.com/ExperimentalDesign/MCResponseSurface_Yp_x3_6dim_2deg.mat
|comments1=
|polynomial_coefficient_vector=
|polynomial_powers_matrix=
|comments2=
|image=
|comments3=
|statistics=
|comments4=
|text=
}}}


==Yp at exit==
==Yp at exit==
Line 5: Line 37:
===Quadratic Surface, 6 Dimensions===
===Quadratic Surface, 6 Dimensions===


Download the response surface here: http://files.charlesmartinreid.com/ExperimentalDesign/MCResponseSurface_Yp_exit_6dim_2deg.mat
{{{ResponseSurface
 
* contains 2 variables:
** <code>response_surface</code> - structure containing information about the response surface (coefficient vector is in response_surface.beta)
** <code>model</code> - this is a matrix containing the polynomial powers of each variable (variable order given in section [[Composite Experimental Design#A Note on Coefficient and Variable Order]]; description of polynomial powers matrix given in section [[Composite Experimental Design#Polynomial Powers Matrix]])


A quadratic response surface was computed using all of the information from the Monte Carlo samples. There were 10,000 samples in total.
|link=http://files.charlesmartinreid.com/ExperimentalDesign/MCResponseSurface_Yp_exit_6dim_2deg.mat


The resulting polynomial coefficient vector is:
|comments1=A quadratic response surface was computed using all of the information from the Monte Carlo samples.  There were 10,000 samples in total.


<pre>
|polynomial_coefficient_vector=<pre>
b(1) = 175.9  
b(1) = 175.9  
b(2) = -55.47  
b(2) = -55.47  
Line 46: Line 74:
</pre>
</pre>


for the polynomial powers matrix:
|polynomial_powers_matrix=<pre>
 
<pre>
     0    0    0    0    0    0
     0    0    0    0    0    0
     0    0    0    0    0    1
     0    0    0    0    0    1
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</pre>
</pre>


The response surface, plotted with the non-visualized dimensions set to a constant (their means), is:
|comments2=


[[Image:MCResponseSurface_6dim_2deg.png|500px]]
|image=MCResponseSurface_Yp_out_6dim_2deg.png


and some important statistics are:
|comments3=


<pre>
|statistics=<pre>
---------------------------------------------------
---------------------------------------------------
Response surface summary of information:
Response surface summary of information:
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</pre>
</pre>


|comments4=
|text=
}}}


===Quadratic Surface, 2 Dimensions===
===Quadratic Surface, 2 Dimensions===


Download the response surface here: http://files.charlesmartinreid.com/ExperimentalDesign/MCResponseSurface_2dim_2deg.mat
{{{ResponseSurface
 
* contains 2 variables:
** <code>response_surface</code> - structure containing information about the response surface (coefficient vector is in response_surface.beta)
** <code>model</code> - this is a matrix containing the polynomial powers of each variable (variable order given in section [[Composite Experimental Design#A Note on Coefficient and Variable Order]]; description of polynomial powers matrix given in section [[Composite Experimental Design#Polynomial Powers Matrix]])


The same set of Monte Carlo samples was fit to a quadratic surface, but with 2 variables instead of 6.
|link=http://files.charlesmartinreid.com/ExperimentalDesign/MCResponseSurface_Yp_exit_2dim_2deg.mat


The resulting polynomial coefficient vector is:
|comments1=The same set of Monte Carlo samples was fit to a quadratic surface, but with 2 variables instead of 6.


<pre>
|polynomial_coefficient_vector=<pre>
b(1) = 0.2606  
b(1) = 0.2606  
b(2) = -0.01793  
b(2) = -0.01793  
Line 124: Line 151:
</pre>
</pre>


for the polynomial powers matrix:
|polynomial_powers_matrix=<pre>
 
<pre>
     0    0
     0    0
     0    1
     0    1
Line 135: Line 160:
</pre>
</pre>


This results in a response surface that looks similar to the 6-dimensional quadratic response surface:
|comments2=This results in a response surface that looks similar to the 6-dimensional quadratic response surface:


[[Image:MCResponseSurface_2dim_2deg.png|500px]]
|image=MCResponseSurface_Yp_out_2dim_2deg.png


The statistics show that the fit is better for the 2-dimensional surface than for the 6-dimensional surface. This, combined with the fact that he response surfaces look similar, means we can conclude that the additional dimensions are ''probably'' independent of the two visualized dimensions, or that they ave a minimal impact on the response.
|comments3=The statistics show that the fit is better for the 2-dimensional surface than for the 6-dimensional surface. This, combined with the fact that he response surfaces look similar, means we can conclude that the additional dimensions are ''probably'' independent of the two visualized dimensions, or that they ave a minimal impact on the response.


<pre>
|statistics=<pre>
---------------------------------------------------
---------------------------------------------------
Response surface summary of information:
Response surface summary of information:
Line 157: Line 182:
---------------------------------------------------
---------------------------------------------------
</pre>
</pre>
|comments4=
|text=
}}}


===Cubic Surface, 6 Dimensions===
===Cubic Surface, 6 Dimensions===


Download the response surface here: http://files.charlesmartinreid.com/ExperimentalDesign/MCResponseSurface_6dim_3deg.mat
{{{ResponseSurface


* contains 2 variables:
|link=http://files.charlesmartinreid.com/ExperimentalDesign/MCResponseSurface_Yp_exit_6dim_3deg.mat
** <code>response_surface</code> - structure containing information about the response surface (coefficient vector is in response_surface.beta)
** <code>model</code> - this is a matrix containing the polynomial powers of each variable (variable order given in section [[Composite Experimental Design#A Note on Coefficient and Variable Order]]; description of polynomial powers matrix given in section [[Composite Experimental Design#Polynomial Powers Matrix]])


The polynomial coefficient vector is:
|comments1=


<pre>
|polynomial_coefficient_vector=<pre>
b(1) = 9.4335e+04  
b(1) = 9.4335e+04  
b(2) = -7.1360e+04  
b(2) = -7.1360e+04  
Line 255: Line 284:
</pre>
</pre>


for the polynomial powers matrix:
|polynomial_powers_matrix=<pre>
 
<pre>
     0    0    0    0    0    0
     0    0    0    0    0    0
     0    0    0    0    0    1
     0    0    0    0    0    1
Line 344: Line 371:
</pre>
</pre>


and the response surface looks like:
|comments2=


[[Image:MCResponseSurface_6dim_3deg.png|500px]]
|image=MCResponseSurface_Yp_exit_6dim_3deg.png


Some important statistics are:
|comments3=


<pre>
|statistics=<pre>
---------------------------------------------------
---------------------------------------------------
Response surface summary of information:
Response surface summary of information:
Line 367: Line 394:
</pre>
</pre>


|comments4=


|text=
}}}


===Quartic Response Surface===
===Quartic Response Surface===


This response surface is available for download here: http://files.charlesmartinreid.com/ExperimentalDesign/MCResponseSurface_6dim_4deg.mat
{{{ResponseSurface


For the sake of brevity, the full coefficients and powers matrix won't be printed here (they are included in the response surface file above).
|link=http://files.charlesmartinreid.com/ExperimentalDesign/MCResponseSurface_Yp_exit_6dim_4deg.mat


The plot and relevant statistics are given here:
|comments1=For the sake of brevity, the full coefficients and powers matrix won't be printed here (they are included in the response surface file above).


[[Image:MCResponseSurface_6dim_4deg.png|500px]]
|polynomial_coefficient_vector=(not included)


<pre>
|polynomial_powers_matrix=(not included)
 
|image=MCResponseSurface_Yp_out_6dim_4deg.png
 
|statistics=<pre>
---------------------------------------------------
---------------------------------------------------
Response surface summary of information:
Response surface summary of information:
Line 396: Line 431:
</pre>
</pre>


It is clear that despite having a high-degree polynomial with a large number (210) of coefficients, the polynomial fit is still quite poor, and increasing the degree of the polynomial does not greatly increase the polynomial's fit to the data.
|comments4=It is clear that despite having a high-degree polynomial with a large number (210) of coefficients, the polynomial fit is still quite poor, and increasing the degree of the polynomial does not greatly increase the polynomial's fit to the data.


With the [[Composite Experimental Design#Computing Response Surface|composite design response surface]], the (reduced) third degree polynomial fit all of the data points exactly, and yielded 0 mean square error and an r-squared value of 1.0.  However, this is because there were only 45 sample points, and almost as many polynomial coefficients - 37.
With the [[Composite Experimental Design#Computing Response Surface|composite design response surface]], the (reduced) third degree polynomial fit all of the data points exactly, and yielded 0 mean square error and an r-squared value of 1.0.  However, this is because there were only 45 sample points, and almost as many polynomial coefficients - 37.
}}}

Revision as of 23:37, 3 July 2011

Response Surface Results

Yp at X1

Yp at X2

Yp at X3

Quadratic Surface, 6 Dimensions

link=http://files.charlesmartinreid.com/ExperimentalDesign/MCResponseSurface_Yp_x3_6dim_2deg.mat


Yp at exit

Quadratic Surface, 6 Dimensions

link=http://files.charlesmartinreid.com/ExperimentalDesign/MCResponseSurface_Yp_exit_6dim_2deg.mat


Quadratic Surface, 2 Dimensions

link=http://files.charlesmartinreid.com/ExperimentalDesign/MCResponseSurface_Yp_exit_2dim_2deg.mat


Cubic Surface, 6 Dimensions

link=http://files.charlesmartinreid.com/ExperimentalDesign/MCResponseSurface_Yp_exit_6dim_3deg.mat


Quartic Response Surface

link=http://files.charlesmartinreid.com/ExperimentalDesign/MCResponseSurface_Yp_exit_6dim_4deg.mat


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