Monte Carlo Response Surfaces: Difference between revisions
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=Using Response Surface Data= | |||
{{Response Surface Explanation}} | |||
=Response Surface Results= | =Response Surface Results= | ||
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{{ResponseSurface | {{ResponseSurface | ||
|link=http://files.charlesmartinreid.com/ExperimentalDesign/ | |link=http://files.charlesmartinreid.com/ExperimentalDesign/MCResponseSurface_Yp_x1_6dim_3deg.mat | ||
|comments1= | |comments1= | ||
|image= | |image=MCResponseSurface_Yp_x1_6dim_3deg.png | ||
|comments2= | |comments2= | ||
|statistics=<pre> | |statistics=<pre> | ||
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Degree of response surface is 3. | Degree of response surface is 3. | ||
MSE = 0. | MSE = 0.00101396 | ||
MSE DoF = 4951 | MSE DoF = 4951 | ||
L-inf norm resid = 0. | L-inf norm resid = 0.21757339 | ||
R^2 = 0. | R^2 = 0.97845992 | ||
adjusted R^2 = 0. | adjusted R^2 = 0.97809881 | ||
--------------------------------------------------- | --------------------------------------------------- | ||
</pre> | </pre> | ||
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{{ResponseSurface | {{ResponseSurface | ||
|link=http://files.charlesmartinreid.com/ExperimentalDesign/ | |link=http://files.charlesmartinreid.com/ExperimentalDesign/MCResponseSurface_Yp_x1_6dim_4deg.mat | ||
|comments1= | |comments1= | ||
|image= | |image=MCResponseSurface_Yp_x1_6dim_4deg.png | ||
|comments2= | |comments2= | ||
|statistics=<pre> | |statistics=<pre> | ||
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Degree of response surface is 4. | Degree of response surface is 4. | ||
MSE = 0. | MSE = 0.00025835 | ||
MSE DoF = 4825 | MSE DoF = 4825 | ||
L-inf norm resid = 0. | L-inf norm resid = 0.07439991 | ||
R^2 = 0. | R^2 = 0.99465134 | ||
adjusted R^2 = 0. | adjusted R^2 = 0.99441966 | ||
--------------------------------------------------- | --------------------------------------------------- | ||
</pre> | </pre> | ||
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Degree of response surface is 4. | Degree of response surface is 4. | ||
MSE = 0. | MSE = 0.00485155 | ||
MSE DoF = 4825 | MSE DoF = 4825 | ||
L-inf norm resid = 0. | L-inf norm resid = 0.22879783 | ||
R^2 = 0. | R^2 = 0.95819495 | ||
adjusted R^2 = 0. | adjusted R^2 = 0.95638412 | ||
--------------------------------------------------- | --------------------------------------------------- | ||
</pre> | </pre> | ||
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==Yp at X3== | ==Yp at X3== | ||
===Quadratic | ===Quadratic Response, 6 Dimensions=== | ||
{{ResponseSurface | {{ResponseSurface | ||
|link=http://files.charlesmartinreid.com/ExperimentalDesign/MCResponseSurface_Yp_x3_6dim_2deg.mat | |link=http://files.charlesmartinreid.com/ExperimentalDesign/MCResponseSurface_Yp_x3_6dim_2deg.mat | ||
|comments1= | |comments1= | ||
|image= | |image=MCResponseSurface_Yp_x3_6dim_2deg.png | ||
|comments2= | |||
|statistics=<pre> | |||
--------------------------------------------------- | |||
Response surface summary of information: | |||
Number of variables in response surface is 6. | |||
Number of terms in polynomial is 28. | |||
Degree of response surface is 2. | |||
MSE = 0.03700744 | |||
MSE DoF = 5007 | |||
L-inf norm resid = 0.50180157 | |||
R^2 = 0.72177521 | |||
adjusted R^2 = 0.72027490 | |||
--------------------------------------------------- | |||
</pre> | |||
|comments3= | |||
|text= | |||
}} | |||
===Quadratic Response, 2 Dimensions=== | |||
{{ResponseSurface | |||
|link=http://files.charlesmartinreid.com/ExperimentalDesign/MCResponseSurface_Yp_x3_2dim_2deg.mat | |||
|comments1=This is a response surface with the same data, but only regressing on the two visualized variables. | |||
|image=MCResponseSurface_Yp_x3_2dim_2deg.png | |||
|comments2= | |||
|statistics=<pre> | |||
--------------------------------------------------- | |||
Response surface summary of information: | |||
Number of variables in response surface is 2. | |||
Number of terms in polynomial is 6. | |||
Degree of response surface is 2. | |||
MSE = 0.03703099 | |||
MSE DoF = 5029 | |||
L-inf norm resid = 0.47547043 | |||
R^2 = 0.72037494 | |||
adjusted R^2 = 0.72009692 | |||
--------------------------------------------------- | |||
</pre> | |||
|comments3= | |||
|text= | |||
}} | |||
===Cubic Response, 6 Dimensions=== | |||
{{ResponseSurface | |||
|link=http://files.charlesmartinreid.com/ExperimentalDesign/MCResponseSurface_Yp_x3_6dim_3deg.mat | |||
|comments1= | |||
|image=MCResponseSurface_Yp_x3_6dim_3deg.png | |||
|comments2= | |||
|statistics=<pre> | |||
--------------------------------------------------- | |||
Response surface summary of information: | |||
Number of variables in response surface is 6. | |||
Number of terms in polynomial is 84. | |||
Degree of response surface is 3. | |||
MSE = 0.02076042 | |||
MSE DoF = 4951 | |||
L-inf norm resid = 0.46211638 | |||
R^2 = 0.84566717 | |||
adjusted R^2 = 0.84307989 | |||
--------------------------------------------------- | |||
</pre> | |||
|comments3= | |||
|text= | |||
}} | |||
===Quartic Response, 6 Dimensions=== | |||
{{ResponseSurface | |||
|link=http://files.charlesmartinreid.com/ExperimentalDesign/MCResponseSurface_Yp_x3_6dim_4deg.mat | |||
|comments1= | |||
|image=MCResponseSurface_Yp_x3_6dim_4deg.png | |||
|comments2= | |comments2= | ||
|statistics= | |statistics=<pre> | ||
--------------------------------------------------- | |||
Response surface summary of information: | |||
Number of variables in response surface is 6. | |||
Number of terms in polynomial is 210. | |||
Degree of response surface is 4. | |||
MSE = 0.01150170 | |||
MSE DoF = 4825 | |||
L-inf norm resid = 0.30974734 | |||
R^2 = 0.91667246 | |||
adjusted R^2 = 0.91306304 | |||
--------------------------------------------------- | |||
</pre> | |||
|comments3= | |comments3= | ||
|text= | |text= | ||
}} | |||
==Yp at exit== | ==Yp at exit== | ||
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===Quadratic Surface, 6 Dimensions=== | ===Quadratic Surface, 6 Dimensions=== | ||
{{ResponseSurface | |||
|link=http://files.charlesmartinreid.com/ExperimentalDesign/ | |link=http://files.charlesmartinreid.com/ExperimentalDesign/MCResponseSurface_Yp_out_6dim_2deg.mat | ||
|comments1=A quadratic response surface was computed using all of the information from the Monte Carlo samples. There were 10,000 samples in total. | |comments1=A quadratic response surface was computed using all of the information from the Monte Carlo samples. There were 10,000 samples in total. | ||
|image=MCResponseSurface_Yp_out_6dim_2deg.png | |image=MCResponseSurface_Yp_out_6dim_2deg.png | ||
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{{ResponseSurface | {{ResponseSurface | ||
|link=http://files.charlesmartinreid.com/ExperimentalDesign/ | |link=http://files.charlesmartinreid.com/ExperimentalDesign/MCResponseSurface_Yp_out_2dim_2deg.mat | ||
|comments1=The same set of Monte Carlo samples was fit to a quadratic surface, but with 2 variables instead of 6. This results in a response surface that looks similar to the 6-dimensional quadratic response surface: | |comments1=The same set of Monte Carlo samples was fit to a quadratic surface, but with 2 variables instead of 6. This results in a response surface that looks similar to the 6-dimensional quadratic response surface: | ||
|image=MCResponseSurface_Yp_out_2dim_2deg.png | |image=MCResponseSurface_Yp_out_2dim_2deg.png | ||
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{{ResponseSurface | {{ResponseSurface | ||
|link=http://files.charlesmartinreid.com/ExperimentalDesign/ | |link=http://files.charlesmartinreid.com/ExperimentalDesign/MCResponseSurface_Yp_out_6dim_3deg.mat | ||
|image= | |image=MCResponseSurface_Yp_out_6dim_3deg.png | ||
|statistics=<pre> | |statistics=<pre> | ||
--------------------------------------------------- | --------------------------------------------------- | ||
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}} | }} | ||
===Quartic | ===Quartic Surface, 6 Dimensions=== | ||
{{ResponseSurface | {{ResponseSurface | ||
|link=http://files.charlesmartinreid.com/ExperimentalDesign/ | |link=http://files.charlesmartinreid.com/ExperimentalDesign/MCResponseSurface_Yp_out_6dim_4deg.mat | ||
|image=MCResponseSurface_Yp_out_6dim_4deg.png | |image=MCResponseSurface_Yp_out_6dim_4deg.png | ||
|statistics=<pre> | |statistics=<pre> | ||
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Degree of response surface is 4. | Degree of response surface is 4. | ||
MSE = 0. | MSE = 0.01467238 | ||
MSE DoF = | MSE DoF = 4825 | ||
L-inf norm resid = 0. | L-inf norm resid = 0.34656843 | ||
R^2 = 0. | R^2 = 0.89710012 | ||
adjusted R^2 = 0. | adjusted R^2 = 0.89264291 | ||
--------------------------------------------------- | --------------------------------------------------- | ||
</pre> | </pre> | ||
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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. | ||
}} | }} | ||
{{ExperimentalDesign}} | |||
Latest revision as of 21:42, 13 July 2011
Using Response Surface Data
Once you download the .mat file associated with a response surface, you will see two variables:
model- matrix of size $ N_{vars} \times N_{polynomial terms} $
- number of columns is equal to the number of input variables $ N_{vars} $
- number of rows is equal to the number of terms in the polynomial response surface $ N_{polynomial terms} $
- variable order is as follows:
- Mass flowrate $ \dot{m} $
- Reaction rate $ k(T) $
- Mixing length $ L_{mix} $
- Measurement location 1 $ z_1 $
- Measurement location 2 $ z_2 $
- Measurement location 3 $ z_3 $
response_surface- cell object that results from Matlab's
regstats()function - polynomial coefficients corresponding to each row of
modelobject are contained inresponse_surface.beta - covariance matrix is stored in
response_surface.covb - R-squared and adjusted R-squared values stored in
response_surface.rsquareandresponse_surface.adjrsquare, respectively - mean square error is contained in
response_surface.mse
- cell object that results from Matlab's
See the Matlab regstats() help page for more information: http://www.mathworks.com/help/toolbox/stats/regstats.html
Example of how to use this information to compute the value of a response surface is here: EvaluateResponseSurface.m
Response Surface Results
Yp at X1
Quadratic Response, 6 Dimensions
Download the response surface here: http://files.charlesmartinreid.com/ExperimentalDesign/MCResponseSurface_Yp_x1_6dim_2deg.mat
The following is a visualization of the response surface (non-visualized dimensions are kept constant at their mean value):
Some key statistics for this response surface are given below:
--------------------------------------------------- Response surface summary of information: Number of variables in response surface is 6. Number of terms in polynomial is 28. Degree of response surface is 2. MSE = 0.00388068 MSE DoF = 5007 L-inf norm resid = 0.45155983 R^2 = 0.91662780 adjusted R^2 = 0.91617822 ---------------------------------------------------
Quadratic Response, 2 Dimensions
Download the response surface here: http://files.charlesmartinreid.com/ExperimentalDesign/MCResponseSurface_Yp_x1_2dim_2deg.mat
This is a response surface with the same data, but only regressing on the two visualized variables.
The following is a visualization of the response surface (non-visualized dimensions are kept constant at their mean value):
Some key statistics for this response surface are given below:
--------------------------------------------------- Response surface summary of information: Number of variables in response surface is 2. Number of terms in polynomial is 6. Degree of response surface is 2. MSE = 0.00389885 MSE DoF = 5029 L-inf norm resid = 0.43695247 R^2 = 0.91586950 adjusted R^2 = 0.91578585 ---------------------------------------------------
Cubic Response, 6 Dimensions
Download the response surface here: http://files.charlesmartinreid.com/ExperimentalDesign/MCResponseSurface_Yp_x1_6dim_3deg.mat
The following is a visualization of the response surface (non-visualized dimensions are kept constant at their mean value):
Some key statistics for this response surface are given below:
--------------------------------------------------- Response surface summary of information: Number of variables in response surface is 6. Number of terms in polynomial is 84. Degree of response surface is 3. MSE = 0.00101396 MSE DoF = 4951 L-inf norm resid = 0.21757339 R^2 = 0.97845992 adjusted R^2 = 0.97809881 ---------------------------------------------------
Quartic Response, 6 Dimensions
Download the response surface here: http://files.charlesmartinreid.com/ExperimentalDesign/MCResponseSurface_Yp_x1_6dim_4deg.mat
The following is a visualization of the response surface (non-visualized dimensions are kept constant at their mean value):
Some key statistics for this response surface are given below:
--------------------------------------------------- Response surface summary of information: Number of variables in response surface is 6. Number of terms in polynomial is 210. Degree of response surface is 4. MSE = 0.00025835 MSE DoF = 4825 L-inf norm resid = 0.07439991 R^2 = 0.99465134 adjusted R^2 = 0.99441966 ---------------------------------------------------
Yp at X2
Quadratic Response, 6 Dimensions
Download the response surface here: http://files.charlesmartinreid.com/ExperimentalDesign/MCResponseSurface_Yp_x2_6dim_2deg.mat
The following is a visualization of the response surface (non-visualized dimensions are kept constant at their mean value):
Some key statistics for this response surface are given below:
--------------------------------------------------- Response surface summary of information: Number of variables in response surface is 6. Number of terms in polynomial is 28. Degree of response surface is 2. MSE = 0.02215980 MSE DoF = 5007 L-inf norm resid = 0.47094014 R^2 = 0.80184973 adjusted R^2 = 0.80078121 ---------------------------------------------------
Quadratic Response, 2 Dimensions
Download the response surface here: http://files.charlesmartinreid.com/ExperimentalDesign/MCResponseSurface_Yp_x2_2dim_2deg.mat
This is a response surface with the same data, but only regressing on the two visualized variables.
The following is a visualization of the response surface (non-visualized dimensions are kept constant at their mean value):
Some key statistics for this response surface are given below:
--------------------------------------------------- Response surface summary of information: Number of variables in response surface is 2. Number of terms in polynomial is 6. Degree of response surface is 2. MSE = 0.02219639 MSE DoF = 5029 L-inf norm resid = 0.43596913 R^2 = 0.80065039 adjusted R^2 = 0.80045219 ---------------------------------------------------
Cubic Response, 6 Dimensions
Download the response surface here: http://files.charlesmartinreid.com/ExperimentalDesign/MCResponseSurface_Yp_x2_6dim_3deg.mat
The following is a visualization of the response surface (non-visualized dimensions are kept constant at their mean value):
Some key statistics for this response surface are given below:
--------------------------------------------------- Response surface summary of information: Number of variables in response surface is 6. Number of terms in polynomial is 84. Degree of response surface is 3. MSE = 0.01047521 MSE DoF = 4951 L-inf norm resid = 0.33881196 R^2 = 0.90737957 adjusted R^2 = 0.90582686 ---------------------------------------------------
Quartic Response, 6 Dimensions
Download the response surface here: http://files.charlesmartinreid.com/ExperimentalDesign/MCResponseSurface_Yp_x2_6dim_4deg.mat
The following is a visualization of the response surface (non-visualized dimensions are kept constant at their mean value):
Some key statistics for this response surface are given below:
--------------------------------------------------- Response surface summary of information: Number of variables in response surface is 6. Number of terms in polynomial is 210. Degree of response surface is 4. MSE = 0.00485155 MSE DoF = 4825 L-inf norm resid = 0.22879783 R^2 = 0.95819495 adjusted R^2 = 0.95638412 ---------------------------------------------------
Yp at X3
Quadratic Response, 6 Dimensions
Download the response surface here: http://files.charlesmartinreid.com/ExperimentalDesign/MCResponseSurface_Yp_x3_6dim_2deg.mat
The following is a visualization of the response surface (non-visualized dimensions are kept constant at their mean value):
Some key statistics for this response surface are given below:
--------------------------------------------------- Response surface summary of information: Number of variables in response surface is 6. Number of terms in polynomial is 28. Degree of response surface is 2. MSE = 0.03700744 MSE DoF = 5007 L-inf norm resid = 0.50180157 R^2 = 0.72177521 adjusted R^2 = 0.72027490 ---------------------------------------------------
Quadratic Response, 2 Dimensions
Download the response surface here: http://files.charlesmartinreid.com/ExperimentalDesign/MCResponseSurface_Yp_x3_2dim_2deg.mat
This is a response surface with the same data, but only regressing on the two visualized variables.
The following is a visualization of the response surface (non-visualized dimensions are kept constant at their mean value):
Some key statistics for this response surface are given below:
--------------------------------------------------- Response surface summary of information: Number of variables in response surface is 2. Number of terms in polynomial is 6. Degree of response surface is 2. MSE = 0.03703099 MSE DoF = 5029 L-inf norm resid = 0.47547043 R^2 = 0.72037494 adjusted R^2 = 0.72009692 ---------------------------------------------------
Cubic Response, 6 Dimensions
Download the response surface here: http://files.charlesmartinreid.com/ExperimentalDesign/MCResponseSurface_Yp_x3_6dim_3deg.mat
The following is a visualization of the response surface (non-visualized dimensions are kept constant at their mean value):
Some key statistics for this response surface are given below:
--------------------------------------------------- Response surface summary of information: Number of variables in response surface is 6. Number of terms in polynomial is 84. Degree of response surface is 3. MSE = 0.02076042 MSE DoF = 4951 L-inf norm resid = 0.46211638 R^2 = 0.84566717 adjusted R^2 = 0.84307989 ---------------------------------------------------
Quartic Response, 6 Dimensions
Download the response surface here: http://files.charlesmartinreid.com/ExperimentalDesign/MCResponseSurface_Yp_x3_6dim_4deg.mat
The following is a visualization of the response surface (non-visualized dimensions are kept constant at their mean value):
Some key statistics for this response surface are given below:
--------------------------------------------------- Response surface summary of information: Number of variables in response surface is 6. Number of terms in polynomial is 210. Degree of response surface is 4. MSE = 0.01150170 MSE DoF = 4825 L-inf norm resid = 0.30974734 R^2 = 0.91667246 adjusted R^2 = 0.91306304 ---------------------------------------------------
Yp at exit
Quadratic Surface, 6 Dimensions
Download the response surface here: http://files.charlesmartinreid.com/ExperimentalDesign/MCResponseSurface_Yp_out_6dim_2deg.mat
A quadratic response surface was computed using all of the information from the Monte Carlo samples. There were 10,000 samples in total.
The following is a visualization of the response surface (non-visualized dimensions are kept constant at their mean value):
Some key statistics for this response surface are given below:
--------------------------------------------------- Response surface summary of information: Number of variables in response surface is 6. Number of terms in polynomial is 28. Degree of response surface is 2. MSE = 0.04265621 MSE DoF = 5007 L-inf norm resid = 0.53414457 R^2 = 0.68956066 adjusted R^2 = 0.68788663 ---------------------------------------------------
Quadratic Surface, 2 Dimensions
Download the response surface here: http://files.charlesmartinreid.com/ExperimentalDesign/MCResponseSurface_Yp_out_2dim_2deg.mat
The same set of Monte Carlo samples was fit to a quadratic surface, but with 2 variables instead of 6. This results in a response surface that looks similar to the 6-dimensional quadratic response surface:
The following is a visualization of the response surface (non-visualized dimensions are kept constant at their mean value):
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.
Some key statistics for this response surface are given below:
--------------------------------------------------- Response surface summary of information: Number of variables in response surface is 2. Number of terms in polynomial is 6. Degree of response surface is 2. MSE = 0.04267264 MSE DoF = 5029 L-inf norm resid = 0.50344056 R^2 = 0.68807653 adjusted R^2 = 0.68776641 ---------------------------------------------------
Cubic Surface, 6 Dimensions
Download the response surface here: http://files.charlesmartinreid.com/ExperimentalDesign/MCResponseSurface_Yp_out_6dim_3deg.mat
The following is a visualization of the response surface (non-visualized dimensions are kept constant at their mean value):
Some key statistics for this response surface are given below:
--------------------------------------------------- Response surface summary of information: Number of variables in response surface is 6. Number of terms in polynomial is 84. Degree of response surface is 3. MSE = 0.02514706 MSE DoF = 4951 L-inf norm resid = 0.51330364 R^2 = 0.81903400 adjusted R^2 = 0.81600023 ---------------------------------------------------
Quartic Surface, 6 Dimensions
Download the response surface here: http://files.charlesmartinreid.com/ExperimentalDesign/MCResponseSurface_Yp_out_6dim_4deg.mat
The following is a visualization of the response surface (non-visualized dimensions are kept constant at their mean value):
Some key statistics for this response surface are given below:
--------------------------------------------------- Response surface summary of information: Number of variables in response surface is 6. Number of terms in polynomial is 210. Degree of response surface is 4. MSE = 0.01467238 MSE DoF = 4825 L-inf norm resid = 0.34656843 R^2 = 0.89710012 adjusted R^2 = 0.89264291 ---------------------------------------------------
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 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.
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