Yielding of a chemical product (Y) is a function of concentration of 3 ingredients (X1, X2, X3), temperature (T), and pH. I want to find a fitting model which relates Y with all the factors (X1, X2, X3, T, pH).
* T and pH are independently adjusted in each experiment (they do not depend on amount (or concentration) of ingredients and each other).
** I want to investigate the factors at 3 levels (3 concentrations of ingredients, 3 temperatures and 3 pHs).
*** There is no linear relation between Y and factors' levels.
Ideally, you plan the experiment by defining how man individual measurements you can make. Then make each of these measurements at a different combination of X[i], T, pH. I would, if possible, not use fixed values for the independent variables. Depending on the possible shape of the response curve, it might be better to select combinations of random values for the independent variables or to place a more or less homogeneous grid over the predictor space.
The relations between your predictors and the response is non-linear. The ideal scenario would now be to have an idea about the functional relationship and use this in the model. If not, you might at least have an idea if you will have for instance an exponential relationship, a reciprocal relationship, or an optimum-curve relationship (what might be simply modelled by a quadratic function). If it is cumbersome to think of functional relationships, then you might fit splines instead.
If -sadly- the factor levels are given and fixed, then ou won't be able to use functional relations in any reasonable way. In such a setting it will only be possible to go for a general linear model with fixed factors as predictors, so that you can make predictions just for the given factor-level combinations. (The other disadvantage is that factorial designg eat more degrees of freedom, but this won't be a major problem here, where you have just 3 levels per factor, so you will loose 2^5 = 32 d.f. compared to a functional model without using polynomes).
Dear Hamed, do you have a candidate model or not, this would be very helpful for further recommendations. In general, I addressed DoE for nonlinear systems (with unknown parameters that enter the equation linearly however) in a worst case approach. To make a long story short, in these cases one should focus on experiments with "extreme" stimuli, i.e. combinations of e.g. high concentrations (C)-high temp (T)-high pH, or high C-high T-low pH, combinations of these kinds. The reason is that under additive (e.g. Gaussian noise) uncertainties, these extreme experiments are optimal (in a worst case sense). Maybe you can provide some details on the structure of the model?
best
If you use a central composite design with star points (circumscribed), then you will be evaluating your factors at five levels, instead of three. This would give more information. If you are not interested in extreme combinations of concentrations and pH, temperature, then you can use box-behnken design. The good thing about box-behnken is you can do sequential analysis. Your choice of model will depend on the number of experiments you can perform based on resources and time, and the type of information you are looking for.
I suggest to use a neural network model for this data fitting. I can help you if you desire it.
based on the info provided you need so called mixture+process DoE. You have a mixture of three components X1, X2, X3 - they are mutually dependent, you need to keep their sum always 100% (or 1). The rest of the variables are so called "process variables". You can vary each of them independently. You say there is no linear dependence, so most probably you need either second order regression model or may be incomplete 3rd order. Pay attention that the regression model which contains "mixture" variables does not have "b0", if you know what I mean. More details in the works of Cornell (http://www.stat.ufl.edu/personnel/usrpages/cornell.shtml) or you can ask me - my PhD was related with this topic
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