fssf_f {FSSF}

Generate fully-sequential maximin designs inside a unit hypercube.

Description

Produces a random fully-sequential design (a nested sequence of designs with points added one at a time) inside a unit hypercube such that the design points are as far away from each other as possible (Shang and Apley, 2019). The "f" stands for "forward", since the algorithm begins with the smallest design and adds points one at a time.

Usage

	fssf_f(d, nMax, N=-1, ScaleVector = NULL, Dinit = NULL)

Arguments

d

The dimension of the design space.

nMax

The largest design size required by the user.

N

Size of the candidate set used to generate the design points. -1 corresponds to the default setting, and the candidate set size will be calculated as 1000 \times d + 2 \times nMax. Using large N will make the design more space-filling, but will slow down the program.

ScaleVector

Array of the lengthscale parameters of different inputs. Default is NULL, which corresponds to the ScaleVector being a unit vector of length d. When ScaleVector is not NULL, for instance, ScaleVector is (\theta_1, \cdots, \theta_d), the distance between point (x_1, \cdots, x_d) and point (y_1, \cdots, y_d) will be computed as \sum_{j=1}^{d} {\Big( \frac{y_j - x_j}{\theta_j} \Big)}^2

Dinit

Numerical Matrix with n_{init} rows and d columns, where n_{init} is a user-specified parameter. This is an optional initial design with size n_{init} provided by the user. Default is NULL, which corresponds to no initial design. If Dinit is not NULL, then the algorithm will select nMax additional design points taking into account of this initial design.

Details

The fssf_f uses a similar idea as proposed by Kennard and Stone(1969); modifications have been made to improve the space-filling performance as well as efficiency.

Value

A nMax \times d matrix with the i^{th} row corresponding to the i^{th} design point.

Author(s)

Boyang Shang [e-mail]
Daniel W. Apley [e-mail]

References

Shang, B. and Apley, D.W. (2019), "Large-Scale Fully-Sequential Space-filling Algorithms for Computer Experiments", Journal of Quality Technology (in press). doi:10.1080/00224065.2019.1705207.

Kennard, R.W. and Stone, L.A. (1969). "Computer aided design of experiments". Technometrics 11.1, pp. 137-148.

Examples

##Generate a design using the fssf_f function with no scaling vector and no initial design.
Design <- fssf_f(d=2, nMax = 320)
plot(Design[,1], Design[,2])

##Generate a design using the fssf_f function with scaling vector and no initial design.
d = 2
n = 100
ScaleVector = c(1.0, 20.0)*0.5
Design = fssf_f(d = d, nMax = n, ScaleVector = ScaleVector)
plot(Design[,1], Design[,2])

##Generate a design using the fssf_f function with a scaling vector and with an initial design
d = 2
n = 100
Dinit =  fssf_f(d=2, nMax = 40)
ScaleVector = c(1.0, 20.0)*0.5
Design = fssf_f(d = d, nMax = n, ScaleVector = ScaleVector, Dinit = Dinit)
plot(Design[,1], Design[,2])
points(Dinit[,1], Dinit[,2], col="red")