View source for Monte Carlo Response Surfaces

=Response Surface Results=

==Yp at exit==

===Quadratic Surface, 6 Dimensions===

Download the response surface here: http://files.charlesmartinreid.com/ExperimentalDesign/MCResponseSurface_Yp_exit_6dim_2deg.mat

* 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.

The resulting polynomial coefficient vector is:

<pre>
b(1) = 175.9 
b(2) = -55.47 
b(3) = -59.65 
b(4) = -75.2 
b(5) = 1.12 
b(6) = 0.1184 
b(7) = -44.47 
b(8) = 4.065 
b(9) = -0.8079 
b(10) = 5.412 
b(11) = -10.95 
b(12) = 25.88 
b(13) = 21.01 
b(14) = -0.1112 
b(15) = 0.2904 
b(16) = -0.3872 
b(17) = -0.003535 
b(18) = -0.03342 
b(19) = -0.009973 
b(20) = -0.01873 
b(21) = 0.0003347 
b(22) = -0.0003496 
b(23) = 21.21 
b(24) = 16.26 
b(25) = 21.64 
b(26) = -0.5508 
b(27) = 0.0131 
b(28) = -10.72
</pre>

for the polynomial powers matrix:

<pre>
     0     0     0     0     0     0
     0     0     0     0     0     1
     0     0     0     0     1     0
     0     0     0     1     0     0
     0     0     1     0     0     0
     0     1     0     0     0     0
     1     0     0     0     0     0
     0     0     0     0     0     2
     0     0     0     0     1     1
     0     0     0     0     2     0
     0     0     0     1     0     1
     0     0     0     1     1     0
     0     0     0     2     0     0
     0     0     1     0     0     1
     0     0     1     0     1     0
     0     0     1     1     0     0
     0     0     2     0     0     0
     0     1     0     0     0     1
     0     1     0     0     1     0
     0     1     0     1     0     0
     0     1     1     0     0     0
     0     2     0     0     0     0
     1     0     0     0     0     1
     1     0     0     0     1     0
     1     0     0     1     0     0
     1     0     1     0     0     0
     1     1     0     0     0     0
     2     0     0     0     0     0
</pre>

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

[[Image:MCResponseSurface_6dim_2deg.png|500px]]

and some important statistics are:

<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 =			 0.04265621 
MSE DoF = 		 5007 

L-inf norm resid = 	 0.53414457 

R^2 =			 0.68956066 
adjusted R^2 =		 0.68788663 
---------------------------------------------------
</pre>


===Quadratic Surface, 2 Dimensions===

Download the response surface here: http://files.charlesmartinreid.com/ExperimentalDesign/MCResponseSurface_2dim_2deg.mat

* 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.

The resulting polynomial coefficient vector is:

<pre>
b(1) = 0.2606 
b(2) = -0.01793 
b(3) = 0.0367 
b(4) = -0.003453 
b(5) = 0.0003092 
b(6) = -0.0003485 
</pre>

for the polynomial powers matrix:

<pre>
     0     0
     0     1
     1     0
     0     2
     1     1
     2     0
</pre>

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

[[Image:MCResponseSurface_2dim_2deg.png|500px]]

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>
---------------------------------------------------
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 
---------------------------------------------------
</pre>

===Cubic Surface, 6 Dimensions===

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

* 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 polynomial coefficient vector is:

<pre>
b(1) = 9.4335e+04 
b(2) = -7.1360e+04 
b(3) = -1.3930e+04 
b(4) = 2.4439e+04 
b(5) = 6.3177e+01 
b(6) = -7.3399e-01 
b(7) = -4.7962e+04 
b(8) = 2.5084e+04 
b(9) = 1.2428e+04 
b(10) = 9.7792e+02 
b(11) = -9.2480e+03 
b(12) = -5.1276e+03 
b(13) = -5.7739e+03 
b(14) = -1.3153e+02 
b(15) = 1.5852e+01 
b(16) = -1.7894e+01 
b(17) = 7.7344e-01 
b(18) = -1.1606e+00 
b(19) = -1.8691e+00 
b(20) = 5.6333e+00 
b(21) = 5.4130e-02 
b(22) = -3.5192e-03 
b(23) = 1.7109e+03 
b(24) = -1.8055e+03 
b(25) = -6.1622e+03 
b(26) = 9.2918e+01 
b(27) = 2.2992e+00 
b(28) = 2.4321e+04 
b(29) = -3.2712e+03 
b(30) = -2.0094e+03 
b(31) = -8.4970e+02 
b(32) = 6.7127e+02 
b(33) = 5.4902e+02 
b(34) = 1.6649e+03 
b(35) = 3.9418e+01 
b(36) = -8.6106e+01 
b(37) = 2.8060e+03 
b(38) = 8.4698e+02 
b(39) = 4.0922e+01 
b(40) = -4.1602e+01 
b(41) = 3.6996e+01 
b(42) = 2.7639e+01 
b(43) = -3.6318e+01 
b(44) = 1.9221e+01 
b(45) = -1.3503e-01 
b(46) = 4.1899e-01 
b(47) = 3.5445e-02 
b(48) = 3.4742e-03 
b(49) = 1.0020e-01 
b(50) = 9.6863e-01 
b(51) = 1.0226e-01 
b(52) = -1.0614e+00 
b(53) = -7.4942e-01 
b(54) = -6.7971e-01 
b(55) = -2.5682e-02 
b(56) = 1.6590e-03 
b(57) = 6.0159e-04 
b(58) = -3.3300e-04 
b(59) = 6.4587e-04 
b(60) = 4.1078e-04 
b(61) = 8.1751e-04 
b(62) = -4.3508e-06 
b(63) = 1.0649e-05 
b(64) = 1.0716e+03 
b(65) = -3.1054e+02 
b(66) = -9.7221e+02 
b(67) = 2.0289e+03 
b(68) = -9.5553e+02 
b(69) = 2.5068e+02 
b(70) = -1.2136e+01 
b(71) = -2.9474e+00 
b(72) = -8.2761e+00 
b(73) = -5.5199e-01 
b(74) = -1.3527e-01 
b(75) = -2.5765e-01 
b(76) = -6.1860e-01 
b(77) = 4.1224e-03 
b(78) = -3.8697e-04 
b(79) = -1.8981e+03 
b(80) = 1.5007e+03 
b(81) = 5.7488e+02 
b(82) = -1.3082e+01 
b(83) = -3.1649e-01 
b(84) = -3.6841e+03 
</pre>

for the polynomial powers matrix:

<pre>
     0     0     0     0     0     0
     0     0     0     0     0     1
     0     0     0     0     1     0
     0     0     0     1     0     0
     0     0     1     0     0     0
     0     1     0     0     0     0
     1     0     0     0     0     0
     0     0     0     0     0     2
     0     0     0     0     1     1
     0     0     0     0     2     0
     0     0     0     1     0     1
     0     0     0     1     1     0
     0     0     0     2     0     0
     0     0     1     0     0     1
     0     0     1     0     1     0
     0     0     1     1     0     0
     0     0     2     0     0     0
     0     1     0     0     0     1
     0     1     0     0     1     0
     0     1     0     1     0     0
     0     1     1     0     0     0
     0     2     0     0     0     0
     1     0     0     0     0     1
     1     0     0     0     1     0
     1     0     0     1     0     0
     1     0     1     0     0     0
     1     1     0     0     0     0
     2     0     0     0     0     0
     0     0     0     0     0     3
     0     0     0     0     1     2
     0     0     0     0     2     1
     0     0     0     0     3     0
     0     0     0     1     0     2
     0     0     0     1     1     1
     0     0     0     1     2     0
     0     0     0     2     0     1
     0     0     0     2     1     0
     0     0     0     3     0     0
     0     0     1     0     0     2
     0     0     1     0     1     1
     0     0     1     0     2     0
     0     0     1     1     0     1
     0     0     1     1     1     0
     0     0     1     2     0     0
     0     0     2     0     0     1
     0     0     2     0     1     0
     0     0     2     1     0     0
     0     0     3     0     0     0
     0     1     0     0     0     2
     0     1     0     0     1     1
     0     1     0     0     2     0
     0     1     0     1     0     1
     0     1     0     1     1     0
     0     1     0     2     0     0
     0     1     1     0     0     1
     0     1     1     0     1     0
     0     1     1     1     0     0
     0     1     2     0     0     0
     0     2     0     0     0     1
     0     2     0     0     1     0
     0     2     0     1     0     0
     0     2     1     0     0     0
     0     3     0     0     0     0
     1     0     0     0     0     2
     1     0     0     0     1     1
     1     0     0     0     2     0
     1     0     0     1     0     1
     1     0     0     1     1     0
     1     0     0     2     0     0
     1     0     1     0     0     1
     1     0     1     0     1     0
     1     0     1     1     0     0
     1     0     2     0     0     0
     1     1     0     0     0     1
     1     1     0     0     1     0
     1     1     0     1     0     0
     1     1     1     0     0     0
     1     2     0     0     0     0
     2     0     0     0     0     1
     2     0     0     0     1     0
     2     0     0     1     0     0
     2     0     1     0     0     0
     2     1     0     0     0     0
     3     0     0     0     0     0
</pre>

and the response surface looks like:

[[Image:MCResponseSurface_6dim_3deg.png|500px]]

Some important statistics are:

<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 =			 0.02514706 
MSE DoF = 		 4951 

L-inf norm resid = 	 0.51330364 

R^2 =			 0.81903400 
adjusted R^2 =		 0.81600023 
---------------------------------------------------
</pre>



===Quartic Response Surface===

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

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).

The plot and relevant statistics are given here:

[[Image:MCResponseSurface_6dim_4deg.png|500px]]

<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 =			 0.02069806 
MSE DoF = 		 9790 

L-inf norm resid = 	 0.37829408 

R^2 =			 0.85452284 
adjusted R^2 =		 0.85141715 
---------------------------------------------------
</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.

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.