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ODE for high order response surface model

Jul 1, 2015 3:28 PM
(6008 views)

Dear all:

I need to construct classical optimal designs (A-, D-, E- and G-optimality criteria) for a high order response surface model with continuous factors. My response surface model is a multidimensional polynomial and it has about 300 regression parameters to be determined.

My understanding is that exchange methods for this problem are not useful since the search process is very time consuming.

I would appreciate if you could let me know about the best algorithm for this problem to construct optimal designs efficiently. Is JMP capable of constructing optimal designs for my problem?

Best

SM

13 REPLIES 13

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Re: ODE for high order response surface model

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Re: ODE for high order response surface model

Ian:

Thanks for your reply. It seems that the highest power is 5 which would not be large enough for my model. Other than that, is there any way that I can enter my response model manually similar to what they do in Gosset (http://neilsloane.com/gosset/)?

Best,

SM

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Re: ODE for high order response surface model

Peter:

Great question! Please let me be more specific about my system. In fact, I have the both scenarios you mentioned. In one case, I have a large number of continuous factors with low order polynomials. Say I have 10 factors; x_1, …, x_10 \in [-1,1]. Let's say least square regression problem is given by:

u(x_1, …, x_10) = \sum_{i=1}^{i=P} a_i*y_i(x_1, …, x_10)

where u is the response, y_i is the basis function and a_i's are the regression coefficients to be determined. P is also the number of unknown regression coefficients. The second order y_i is given by:

y_i = 0.0541266*x_1 + 0.0541266*x_2 + 0.0541266*x_3 + 0.0541266*x_4 + 0.0541266*x_5 + 0.0541266*x_6 + 0.0541266*x_7 + 0.0541266*x_8 + 0.0541266*x_9 + 0.0541266*x_10 + 0.09375*x_1*x_2 + 0.09375*x_1*x_3 + 0.09375*x_1*x_4 + 0.09375*x_2*x_3 + 0.09375*x_1*x_5 + 0.09375*x_2*x_4 + 0.09375*x_1*x_6 + 0.09375*x_2*x_5 + 0.09375*x_3*x_4 + 0.09375*x_1*x_7 + 0.09375*x_2*x_6 + 0.09375*x_3*x_5 + 0.09375*x_1*x_8 + 0.09375*x_2*x_7 + 0.09375*x_3*x_6 + 0.09375*x_4*x_5 + 0.09375*x_1*x_9 + 0.09375*x_2*x_8 + 0.09375*x_3*x_7 + 0.09375*x_4*x_6 + 0.09375*x_1*x_10 + 0.09375*x_2*x_9 + 0.09375*x_3*x_8 + 0.09375*x_4*x_7 + 0.09375*x_5*x_6 + 0.09375*x_2*x_10 + 0.09375*x_3*x_9 + 0.09375*x_4*x_8 + 0.09375*x_5*x_7 + 0.09375*x_3*x_10 + 0.09375*x_4*x_9 + 0.09375*x_5*x_8 + 0.09375*x_6*x_7 + 0.09375*x_4*x_10 + 0.09375*x_5*x_9 + 0.09375*x_6*x_8 + 0.09375*x_5*x_10 + 0.09375*x_6*x_9 + 0.09375*x_7*x_8 + 0.09375*x_6*x_10 + 0.09375*x_7*x_9 + 0.09375*x_7*x_10 + 0.09375*x_8*x_9 + 0.09375*x_8*x_10 + 0.09375*x_9*x_10 + 0.104816*x_1^2 + 0.104816*x_2^2 + 0.104816*x_3^2 + 0.104816*x_4^2 + 0.104816*x_5^2 + 0.104816*x_6^2 + 0.104816*x_7^2 + 0.104816*x_8^2 + 0.104816*x_9^2 + 0.104816*x_10^2 - 0.318136

For the second case, I have only two factors x_1 and x_2 \in [-1,1] with a high order polynomial, i.e., y_i is given by:

y_i = 3.11846*x_1 + 3.11846*x_2 + 5.5*(1.875*x_1 - 8.75*x_1^3 + 7.875*x_1^5)*(1.875*x_2 - 8.75*x_2^3 + 7.875*x_2^5) + 3.3541*(1.5*x_2^2 - 0.5)*(4.375*x_1^4 - 3.75*x_1^2 + 0.375) + 3.3541*(1.5*x_1^2 - 0.5)*(4.375*x_2^4 - 3.75*x_2^2 + 0.375) + 3.1225*x_2*(6.5625*x_1^2 - 19.6875*x_1^4 + 14.4375*x_1^6 - 0.3125) + 3.1225*x_1*(6.5625*x_2^2 - 19.6875*x_2^4 + 14.4375*x_2^6 - 0.3125) - 4.7697*(1.5*x_2 - 2.5*x_2^3)*(6.5625*x_1^2 - 19.6875*x_1^4 + 14.4375*x_1^6 - 0.3125) - 4.7697*(1.5*x_1 - 2.5*x_1^3)*(6.5625*x_2^2 - 19.6875*x_2^4 + 14.4375*x_2^6 - 0.3125) + 2.87228*x_2*(1.875*x_1 - 8.75*x_1^3 + 7.875*x_1^5) + 2.87228*x_1*(1.875*x_2 - 8.75*x_2^3 + 7.875*x_2^5) + 3.77492*x_2*(2.46094*x_1 - 36.0938*x_1^3 + 140.766*x_1^5 - 201.094*x_1^7 + 94.9609*x_1^9) + 3.77492*x_1*(2.46094*x_2 - 36.0938*x_2^3 + 140.766*x_2^5 - 201.094*x_2^7 + 94.9609*x_2^9) - 4.38748*(1.5*x_2 - 2.5*x_2^3)*(1.875*x_1 - 8.75*x_1^3 + 7.875*x_1^5) - 4.38748*(1.5*x_1 - 2.5*x_1^3)*(1.875*x_2 - 8.75*x_2^3 + 7.875*x_2^5) + 1.5*x_1*x_2 - 2.29129*x_2*(1.5*x_1 - 2.5*x_1^3) - 2.29129*x_1*(1.5*x_2 - 2.5*x_2^3) + 4.5*(4.375*x_1^4 - 3.75*x_1^2 + 0.375)*(4.375*x_2^4 - 3.75*x_2^2 + 0.375) + 4.03113*(1.5*x_2^2 - 0.5)*(6.5625*x_1^2 - 19.6875*x_1^4 + 14.4375*x_1^6 - 0.3125) + 4.03113*(1.5*x_1^2 - 0.5)*(6.5625*x_2^2 - 19.6875*x_2^4 + 14.4375*x_2^6 - 0.3125) + 3.5*(1.5*x_1 - 2.5*x_1^3)*(1.5*x_2 - 2.5*x_2^3) + 3.7081*(1.5*x_2^2 - 0.5)*(1.875*x_1 - 8.75*x_1^3 + 7.875*x_1^5) + 3.7081*(1.5*x_1^2 - 0.5)*(1.875*x_2 - 8.75*x_2^3 + 7.875*x_2^5) + 3.57071*x_2*(54.1406*x_1^4 - 9.84375*x_1^2 - 93.8438*x_1^6 + 50.2734*x_1^8 + 0.273438) + 3.57071*x_1*(54.1406*x_2^4 - 9.84375*x_2^2 - 93.8438*x_2^6 + 50.2734*x_2^8 + 0.273438) + 1.93649*x_1*(1.5*x_2^2 - 0.5) + 1.93649*x_2*(1.5*x_1^2 - 0.5) - 2.95804*(1.5*x_1 - 2.5*x_1^3)*(1.5*x_2^2 - 0.5) - 2.95804*(1.5*x_2 - 2.5*x_2^3)*(1.5*x_1^2 - 0.5) - 3.3541*x_2*(2.1875*x_1 - 19.6875*x_1^3 + 43.3125*x_1^5 - 26.8125*x_1^7) - 3.3541*x_1*(2.1875*x_2 - 19.6875*x_2^3 + 43.3125*x_2^5 - 26.8125*x_2^7) + 5.12348*(1.5*x_2 - 2.5*x_2^3)*(2.1875*x_1 - 19.6875*x_1^3 + 43.3125*x_1^5 - 26.8125*x_1^7) + 5.12348*(1.5*x_1 - 2.5*x_1^3)*(2.1875*x_2 - 19.6875*x_2^3 + 43.3125*x_2^5 - 26.8125*x_2^7) + 5.40833*(4.375*x_2^4 - 3.75*x_2^2 + 0.375)*(6.5625*x_1^2 - 19.6875*x_1^4 + 14.4375*x_1^6 - 0.3125) + 5.40833*(4.375*x_1^4 - 3.75*x_1^2 + 0.375)*(6.5625*x_2^2 - 19.6875*x_2^4 + 14.4375*x_2^6 - 0.3125) + 4.97494*(1.875*x_1 - 8.75*x_1^3 + 7.875*x_1^5)*(4.375*x_2^4 - 3.75*x_2^2 + 0.375) + 4.97494*(1.875*x_2 - 8.75*x_2^3 + 7.875*x_2^5)*(4.375*x_1^4 - 3.75*x_1^2 + 0.375) + 4.60977*(1.5*x_2^2 - 0.5)*(54.1406*x_1^4 - 9.84375*x_1^2 - 93.8438*x_1^6 + 50.2734*x_1^8 + 0.273438) + 4.60977*(1.5*x_1^2 - 0.5)*(54.1406*x_2^4 - 9.84375*x_2^2 - 93.8438*x_2^6 + 50.2734*x_2^8 + 0.273438) + 2.5*(1.5*x_1^2 - 0.5)*(1.5*x_2^2 - 0.5) + 2.59808*x_2*(4.375*x_1^4 - 3.75*x_1^2 + 0.375) + 2.59808*x_1*(4.375*x_2^4 - 3.75*x_2^2 + 0.375) - 4.33013*(1.5*x_2^2 - 0.5)*(2.1875*x_1 - 19.6875*x_1^3 + 43.3125*x_1^5 - 26.8125*x_1^7) - 4.33013*(1.5*x_1^2 - 0.5)*(2.1875*x_2 - 19.6875*x_2^3 + 43.3125*x_2^5 - 26.8125*x_2^7) + 18.6023*x_1^2 - 51.7429*x_1^3 + 18.6023*x_2^2 - 186.095*x_1^4 - 51.7429*x_2^3 + 235.976*x_1^5 - 186.095*x_2^4 + 638.9*x_1^6 + 235.976*x_2^5 - 386.351*x_1^7 + 638.9*x_2^6 - 875.481*x_1^8 - 386.351*x_2^7 + 206.963*x_1^9 - 875.481*x_2^8 + 413.407*x_1^10 + 206.963*x_2^9 + 413.407*x_2^10 - 3.96863*(1.5*x_1 - 2.5*x_1^3)*(4.375*x_2^4 - 3.75*x_2^2 + 0.375) - 3.96863*(1.5*x_2 - 2.5*x_2^3)*(4.375*x_1^4 - 3.75*x_1^2 + 0.375) - 0.6201

I know that this is not a standard approach in optimal design of experiments. In fact, my system is described by a computer simulation rather than a real world experiment. That is why I need to have higher order terms.

Is it even computationally feasible to construct an optimal design for such high order polynomials?

Best,

SM

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Re: ODE for high order response surface model

SM,

Since you are working with computer simulations have you considered Space Filling designs? With a response surface design you will likely get multiple center point conditions and with computer simulations you only need one point per condition due to the deterministic nature of the simulation. That is unless you are forcing variation. From past experience computer simulations are highly non-linear and using a space filing design will help you better understand and ultimately model that non-linearity.

Gaussian Process and Neural Net models are typically better suited to handle the modeling for computer simulations to build your surrogate model. Neural Nets are typically faster than Gaussian Process especially if you are up past 100 simulation conditions. RS models can be nice if they can fit your data because they are usually simpler than either Gaussian or NN.

The Fast Flexible Filling design with MaxPro optimality will give you a good option versus a response surface design especially for a design with higher dimensions.

Best,

Bill

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Re: ODE for high order response surface model

Bill:

Thank you for your reply. I have tested Latin Hypercube designs and they work well for my problem. In fact, I am not just trying to find an optimal design for my problem, I want to see how different optimality criteria affect the accuracy of the solution and how they are compared for example with LH designs.

I am a newbie in ODE, so, my major question would be: is it even feasible to construct D-, A-, E, and G-optimal designs for the model described in reply to Peter's answer?

Best,

SM

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Re: ODE for high order response surface model

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Re: ODE for high order response surface model

Peter:

The model with 300 regression coefficients would be my extreme case. The model I provided (y_i) has 66 unknowns. What about this model? Is it still too large and challenging? I have to add that I need to find optimal deigns with a number of runs from one to ten times larger than the number of unknowns.

Best,

SM

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Re: ODE for high order response surface model