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4. Process Modeling
4.3. Data Collection for Process Modeling

4.3.3.

What are some general design principles for process modeling?
Six Principles of Experiment Design Applied to Process Modeling There are six principles of experiment design as applied to process modeling:
  1. Capacity for Primary Model
  2. Capacity for Alternative Model
  3. Minimal Variance of Coefficients
  4. Sample where the Variation Is (Non-Constant Variance Case)
  5. Sample where the Variation Is (Steep Curve Case)
  6. Replication
  7. Randomization
We discuss each in detail below.
Capacity for Primary Model For your best-guess model, make sure that the design has the capacity for estimating the coefficients of that model. For a simple example of this, if you are fitting a quadratic, then make sure you have at least three distinct horixontal axis points.
Capacity for Alternative Model If your best-guess model happens to be incorrect, make sure that the design has the capacity to estimate the coefficients of your best-guess back-up alternative model (which means implicitly that you should have already identified such a back-up alternative model). For a simple example, if you suspect (but are not positive) that a linear model is appropriate, then it is best to employ a globally robust design (say, four points at each extreme, and three points in the middle, for a ten point design) as opposed to the locally optimal design (such as five points at each extreme). The locally optimal design will provide a best fit to the line, but have no capacity to fit a quadratic. The globally robust design will provide a good (though not optimal) fit to the line and additionally provide a good (though not optimal) fit to the quadratic.
Minimal Variance of Coefficients For a given model, make sure the design has the property of minimizing the variation of the least squares estimated coefficients. This is a general principle which is always in effect but which in practice is hard to put in place for many models beyond the simpler 1-factor models. For more complicated 1-factor models, and for most multi-factor models, the expressions for the variance of the least squares estimates, although available, are complicated and assume more than the analyst typically knows. The net result is that this principle, though important, is harder to apply beyond the simple cases.
Sample Where the Variation Is (Non Constant Variance Case) Regardless of the simplicity or complexity of the model, there are situations where certain regions of the curve are noisier than others. A simple case is where there is a linear relationship between X and Y but the recording device is proportional rather than absolute and so larger values of Y are intrinsically noisier than smaller values of Y. In such cases, sampling where the variation is means to have more replicated points in those regions which are noisier. The practical answer to how many such replicated points there should be is

where is the theoretical standard deviation for that given region of the curve. Usually is estimated by a-priori guesses for what the local standard deviations are.

Sample Where the Variation Is (Steep Curve Case) A common occurance for non-linear models is for some regions of the curve to be steeper than others. For example, in fitting an exponential model (small X yielding large Y, and large Y yielding small X) it is often the case that the Y data in the steep region is intrinsically noisier than the Y data in the relatively flat regions. The reason for this is that commonly the X values themselves have a bit of noise and this X-noise gets translated into larger Y-noise in the steep sections than in the shallow sections. In such cases, where we know the shape of the response curve well enough to identify step-versus-shallow regions, it is often a good idea to sample more heavily in the steep regions than in shallow regions. A practical rule of thumb for where to position the X values in such situations is to
  1. sketch out your best-guess for what the resulting curve will be;
  2. partition the vertical (that is the Y) axis into N equi-spaced points (where N = the total number of data points that you can afford);
  3. draw horizontal lines from each vertical axis point to where it hits the sketched-in curve.
  4. drop a vertical projection line from the curve intersection point to the horizontal axis.
This will be the recommended X values to use in the design.

The above rough procedure for an exponentially decreasing curve would thus yield a logarithmic preponderance of points in the steep region of the curve and relatively few points in the flatter part of the curve.

Replication If affordable, replication should be part of every design. Replication allows us to compute an model-free estimate of the process standard deviation. Such an estimate may then be used as a criterion in an objective goodness of fit test to assess whether a given model is adequate (it makes no sense to have a model which is so complicated that the model-dependent residual standard deviation is smaller than the model-free residual standard deviation). We want to fit signal--why fit noise? Such an objective goodness of fit F test can be employed only if the design has built-in replication. Some replication is essential; replication at every point is ideal.
Randomization Just because the X's have some natural ordering, does not mean that the data should be collected in the same order as the X's. Some aspect of randomization should enter into every experiment, and experiments for process modeling is no exception. Thus if your are sampling ten points of a curve, the ten Y values should not be collected by sequentially stepping through the X values from the smallest to the largest. If you do so, and if some extraneous drifting or wear occurs in the machine, the operator, the environment, the measuring device, etc., then that drift will unwittingly contaminate the Y values and in turn contaminate the final fit. To minimize the effect of such potential drift, it is best to randomize (use random number tables) the sequence of the X values. This will not make the drift go away, but it will spread the contaminatory drift effect fairly over the entire curve, realistically inflating the variation of the fitted values, and providing some mechanism after-the-fact (at the residual analysis model validation stage) of uncovering or discovering such a drift. If you do not randomize the run sequence, you give up your ability to detect such a drift if it occurs.
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