Optimal Composite
Response Surface Designs
Here we consider the design space is k-dimensional ball (k-ball), i.e. 
. Fix the first-order design and find added points by D-optimal criterion. Under these
assumptions, we study the D-optimal composite
designs and minimal-point second-order designs.
Methodology
For the
cases of two and three factors, the close forms of |X’X| are
computed, and the optimal added supports are found by maximizing these
determinants directly. For the cases of more than 3 factors, the determinant of
the information matrix is too complicate to write down its close form. Then a
simulated annealing algorithm with a MCMC sampling method is proposed to get
the corresponding numerical results. In this works, the number of center point
is assumed to be 1.
For
simplicity, we let 
, where p=k(k-1)/2
is the number of rotation angles. The |X’X| is the function of θi, i=1, …, p, d(θi). Since the objective function is d(θi) that we want
to maximize, we denote a density
,
where T(t) is the
“temperature” at time t and is a
decreasing function from initial temperature, T(0) > 0, to 0+. The SA algorithm is described in the following:
Step1. Initialize ay any arbitrary configuration θ(0) and temperature level T(0).
Step2. Sample θ from πT(t)(θ) by a MCMC method.
Step3. Increase t to t+1.
For more information about SA algorithm, please see Chapter 10 in Liu
(2001). The MCMC method applied for sampling 
is the systematic
scan Gibbs sampler as follows:
1. Select the initial
value θ(0) = (
).
2. At the sth
iteration of Gibbs sampler, we sample 
from the conditional
distribution,
where θi=(θ1, θ2,
…, θi-1, θi+1, …θp).
We
summarize the algorithm that we use to maximize |X’X|, and this
algorithm is called the Best Angle (BA) sampler:
Step1. Select the initial angles, 
, i=1, …, p.
Step2. Run Nt
iterations of Gibbs sampler to sample θ(s) fromπT(t)(θ(s)) and at each iteration of
Gibbs sampler, for i=1, … , p, draw 
fromπT(t)( θi|) by inversion method.
Step3. Set t to t+1.
1. D-optimal
composite designs
2. Minimal-point
second-order designs