Design of Experiments Nested FCD

[#1]

Hey,

I want to make a nested face-centered CCD design (like that shown in the link below pages 20-22). I tried using custom design with 5 levels discrete variable but the generated design doesn't cover the design space as in the example. i also used a regular 2 level design and then tried to augment it but i couldn't figure out the proper way to do that. what should i do?

i have 2 response variables and 4 design variables. the regular design, with factorial points, center points, and axial points, is not enough to describe the nonlinearity so i want an extra level of complexity in some variable to make them have more level. this means that instead of a variable having levels of (-1,0,1) i want it to have levels of  (-1,-0.5,0,0.5,1). also, of course, i can't do a full factorial design because at 5 levels it will require 625 runs and this is not feasible in my case.

link to nested FCD: http://diginole.lib.fsu.edu/islandora/object/fsu:175634/datastream/PDF/view

Thank you for your help,

Diaa

[#2]

I'm a bit confused by your question(s) and your post title.  You are not describing a Nested design (e.g., where one or more factors are dependent on others in the experiment or the levels for one factor are dependent on levels of another (hierarchical).  Face centered designs are 3-level designs, TMK, and are used when the treatment combinations at the corners are not possible.  They also do not require center points, but those can be added for other reasons. I gave it a try and attached a JMP file.

[#3]

Thank you for your response. You're correct, I didn't mean nested as in the common meaning of hierarchical. Landman (an aerodynamicist) tried to use CCD to describe the change in the lift with different variables (let's say 3 variables). He had a run schedule of (2^3 factorial points +2*3 axial points + plus 2 center points). his experiment showed that lift is highly nonlinear with a variable number 1, so he decided to add more points. so now variable 1 has 5 levels instead of the initial 2. how can i make something like that in jmp? and how can i control the levels instead of jmp just assuming midpoints?

Thank you for your jmp file. the number of runs is very small and doesn't cover the design space. i know i can increase it in the user-specified number of runs section, but do you know how can i visualize it ( a cube plot or anything similar) and add points where i want?

Thank you for your time and sorry for the confusion. I would also appreciate it if you have a reference that can help me understand the differences between different DOE designs and the corresponding advantages and disadvantages of each one, Thanks again.

the reference that named that technique nested FCD is in the link in the main post.

[#4]

You said, "Thank you for your jmp file. the number of runs is very small and doesn't cover the design space. i know i can increase it in the user-specified number of runs section, but do you know how can i visualize it ( a cube plot or anything similar) and add points where i want?"

How would you know when it covers the design space? How would you know a good place to observe when you see it?

The optimal design (i.e., best set of conditions to be observed) depends on the model. You say that you expect the response to be non-linear. If that claim is true, I doubt that you are able to identify the best combination of factor levels. (I have the same doubt about everyone, including myself.) For example, take the simple case of a single continuous factor. The non-linearity of the response requires a cubic polynomial model. That model requires four levels of the factor. Most people would spread them evenly across the factor range: -1, -1/3, +1/3, +1. Those levels can be used to fit the cubic polynomial model, but they are not the optimal levels. Exactly where are those optimal levels?

I recommend that you use a method that is built to find the exact optimal levels for a linear model of any order: custom design. In this case, you could entertain higher order polynomial functions as linear models if you believe that the non-linearity is that complex. (Complexity of response dictates the complexity of the model of it.) My point is that you should not pick the points, you should pick the model and let JMP find the points.

If you are wrong, you can always augment the initial design with custom design and continue. Repeat as necessary.

You could also use a Space Filling design as used with computer simulations in the absence of a stochastic component in the response. A good choice for a model is the Gaussian Process interpolator. The response can include random variance, too. GP models can be used to predict the response locally and obtain optimum factor levels. You can start with a modest design and augment the space filling design if you feel that the non-linearity is not fully represented.

I would not use a visualization tool to find the best observation points. I would use a design of experiments tool. I would use a visualization tool to understand the model.

[#5 · Accepted solution]

I asked Bradley Jones about your request. Here is his suggestion. He might have more to add later.

1. Make a Box-Behnken design on the levels -0.5, 0, and 0.5 while leaving out the center runs. This requires 24 runs.
2. Now make a Face-Centered CCD on the levels -1, 0, and 1 keeping the two center runs. This requires 26 runs.
3. Combine the two designs for a 50 run design where each factor has 5 levels.

I believe this design has the structure the user wants.

[#6]

Thank you loads for your help. I really appreciate you taking the time to explain. I tried Bradley's idea and it works for me. I attached an example with a random response for anyone who has a similar question in the future. Thank you again.

[#7]

I'm glad you have a design to your liking.  can't imagine actually running that nor the potential complexity of the model it would create.  I prefer a more sequential approach (as Box recommends) and potentially add the parts of the sequential designs to estimate the non-linear model terms.  It doesn't seem practical to create that much information on 4 factors without considering any noise (since you mentioned lift, ambient conditions surely have an effect, consistency of the design factors dimensionally, etc.)?  Also, If the surface you are mapping is "centered" (I assume you already know these factors are the most important design factors and already have an idea of "where", in terms of level setting, the experiment should take place.then you don't need that much information at the periphery.

Best of luck in your journey.