Response Surface Designs – MATLAB & Simulink

Quadratic response surfaces are simple models that provide a maximum or minimum without making additional assumptions about the form of the response. Quadratic models can be calibrated using full factorial designs with three or more levels for each factor, but these designs generally require more runs than necessary to accurately estimate model parameters. This section discusses designs for calibrating quadratic models that are much more efficient, using three or five levels for each factor, but not using all combinations of levels.

Central Composite Designs

Central composite designs (CCDs), also known as Box-Wilson designs,
are appropriate for calibrating full quadratic models. There are three
types of CCDs—circumscribed, inscribed, and faced—pictured
below:

Each design consists of a factorial design (the corners of a cube) together with
center and star points that allow
for estimation of second-order effects. For a full quadratic model with
n factors, CCDs have enough design points to estimate the
(n+2)(n+1)/2 coefficients.

The type of CCD used (the position of the factorial and star
points) is determined by the number of factors and by the desired
properties of the design. The following table summarizes some important
properties. A design is rotatable if the prediction variance
depends only on the distance of the design point from the center of
the design.

DesignRotatableFactor LevelsUses
Points Outside ±1 Accuracy of EstimatesCircumscribed (CCC)Yes5YesGood over entire design spaceInscribed (CCI)Yes5NoGood over central subset of design spaceFaced (CCF)No3NoFair over entire design space; poor for pure quadratic coefficients

Generate CCDs with the function ccdesign:

dCC = ccdesign(3,'type','circumscribed')
dCC =
   -1.0000   -1.0000   -1.0000
   -1.0000   -1.0000    1.0000
   -1.0000    1.0000   -1.0000
   -1.0000    1.0000    1.0000
    1.0000   -1.0000   -1.0000
    1.0000   -1.0000    1.0000
    1.0000    1.0000   -1.0000
    1.0000    1.0000    1.0000
   -1.6818         0         0
    1.6818         0         0
         0   -1.6818         0
         0    1.6818         0
         0         0   -1.6818
         0         0    1.6818
         0         0         0
         0         0         0
         0         0         0
         0         0         0
         0         0         0
         0         0         0
         0         0         0
         0         0         0
         0         0         0
         0         0         0

The repeated center point runs allow for a more uniform estimate
of the prediction variance over the entire design space.