Monte Carlo Response Surfaces
From charlesreid1
Response Surface Results
Yp at X1
Yp at X2
Yp at X3
Quadratic Surface, 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):
[[Image:|500px]]
Some key statistics for this response surface are given below:
}
Yp at exit
Quadratic Surface, 6 Dimensions
{
Download the response surface here: http://files.charlesmartinreid.com/ExperimentalDesign/MCResponseSurface_Yp_exit_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_exit_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_exit_6dim_3deg.mat
The following is a visualization of the response surface (non-visualized dimensions are kept constant at their mean value):
File:MCResponseSurface Yp exit 6dim 3deg.png
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 Response Surface
Download the response surface here: http://files.charlesmartinreid.com/ExperimentalDesign/MCResponseSurface_Yp_exit_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 = 0.02069806 MSE DoF = 9790 L-inf norm resid = 0.37829408 R^2 = 0.85452284 adjusted R^2 = 0.85141715 ---------------------------------------------------
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.