A surface in (n+1) dimensions that represents the variations in the expected value of a response variable (see regression) as the values of n explanatory variables are varied. Usually the interest is in finding the combination that gives a global maximum (or minimum). One interactive procedure is the method of steepest ascent (or descent), in which, in a sequence of experiments, the points corresponding to the successive values of the explanatory variables are collinear and lie on the estimated line of greatest (or least) slope that passes through the origin of the original experimental design.
There are several specialist experimental designs for efficient experimentation near the supposed optimum. With n explanatory variables a central composite design consists of observations at vertices of a hypercube, together with repeated observations at the origin and observations on each axis at a distance c from the origin. If c=√n then all the non-central points are at the same distance from the origin and the design is an example of a rotatable design. A Box–Behnken design uses fewer observations by replacing the observations at the vertices by observations at the mid-points of the edges. See also factorial experiment; Nelder–Mead simplex method.
Subjects: Probability and Statistics.