straussDesign: Designs based on Strauss process

Description

Space-Filling Designs based on Strauss process

Usage

straussDesign(n,dimension, RND, alpha=0.5, repulsion=0.001, NMC=1000,
              constraints1D=0, repulsion1D=0.0001, seed=NULL)

Value

A list containing:

n: the number of experiments
dimension: the number $d$ of variables
design_init: the initial distribution of $n$ points $[0,1]^{d}$
radius: the radius of interaction
alpha: the potential power alpha
repulsion: the repulsion parameter $\gamma$
NMC: the number of iterations McMC
constraints1D: an integer indicating if constraints on the factorial axis are imposed. If its value is different from zero, a component repulsion1D containing the value of the repulsion parameter $\gamma$ in dimension 1 is added at the list.
design: the design of experiments in [0,1]$^{d}$
seed: the seed corresponding to the design

Arguments

n: the number of experiments
dimension: the number of input variables
RND: a real number which represents the radius of interaction
alpha: the potential power (default, fixed at 0.5)
repulsion: the repulsion parameter in the unit cube (gamma)
NMC: the number of McMC iterations (this number must be large to converge)
constraints1D: 1 to impose 1D projection constraints, 0 otherwise
repulsion1D: the repulsion parameter in 1D
seed: seed for the uniform generation of number

Author

J. Franco

Details

Strauss designs are Space-Filling designs initially defined from Strauss process: $$\pi (X) = k \gamma^{s(X)}$$ where $s(X)$ is is the number of pairs of points $(x^{i}, x^{j})$ of the design $X = \left( x^{1}, \ldots, x^{n} \right)$ that are separated by a distance no greater than the radius of interaction RND, $k$ is the normalizing constant and $\gamma$ is the repulsion parameter. This distribution corresponds to the particular case alpha=0.

For the general case, a stochastic simulation is used to construct a Markov chain which converges to a spatial density of points $\pi(X)$ described by the Strauss-Gibbs potential. In practice, the Metropolis-Hastings algorithm is implemented to simulate a distribution of points which converges to the stationary law: $$\pi (X) \propto exp(-U(X))$$ with a potentiel $U$ defined by: $$U(X) = \beta \sum_{1 \leq i < j \leq n} \varphi \left( \| x^{i}-x^{j} \| \right)$$ where $\beta = - \ln \gamma$, $\varphi (h) = \left( 1 - \frac{h}{RND} \right) ^{\alpha}$ if $h \leq$ RND and 0 otherwise.

The input parameters of straussDesign function can be interpreted as follows:

- RND is used to compute the number of pairs of points of the design separated by a distance no more than RND. A point is said "in interaction" with another if the spheres of radius RND/2 centered on these points intersect.

- alpha is the potential power $\alpha$. The case alpha=0 corresponds to Strauss process (0-1 potential).

- repulsion is equal to the $\gamma$ parameter of the Strauss process. Note that $\gamma$ belongs to ]0,1].

- constraints1D allows to specify some constraints into the margin. If constraints1D==1, two repulsion parameters are needed: one for the all space (repulsion) and the other for the 1D projection (repulsion1D). Default values are repulsion=0.001 and repulsion1D=0.001. Note that the value of the radius of interaction in the one-dimensional axis is not an input parameter and is automatically fixed at $0.75/n$.

References

J. Franco, X. Bay, B. Corre and D. Dupuy (2008) Planification d'experiences numeriques a partir du processus ponctuel de Strauss, https://hal.science/hal-00260701/fr/.

Examples

Run this code

## Strauss-Gibbs designs in dimension 2 (n=20 points)
S1 <- straussDesign(n=20, dimension=2, RND=0.2)
plot(S1$design, xlim=c(0,1), ylim=c(0,1))
theta <- seq(0,2*pi, by=2*pi/(100 - 1))
for(i in 1:S1$n){
   lines(S1$design[i,1]+S1$radius/2*cos(theta),
	     S1$design[i,2]+S1$radius/2*sin(theta), col='red')
}

## 2D-Strauss design
S2 <- straussDesign(n=20, dimension=2, RND=0.2, NMC=200,
	         constraints1D=0, alpha=0, repulsion=0.01)
plot(S2$design,xlim=c(0,1),ylim=c(0,1))

## 2D-Strauss designs with constraints on the axis
S3 <- straussDesign(n=20, dimension=2, RND=0.18, NMC=200, 
	         constraints1D=1, alpha=0.5, repulsion=0.1, repulsion1D=0.01)
plot(S3$design, xlim=c(0,1),ylim=c(0,1))
rug(S3$design[,1], side=1)
rug(S3$design[,2], side=2)

## Change the dimnames, adjust to range (-10, 10) and round to 2 digits
xDRDN(S3, letter="T", dgts=2, range=c(-10, 10))

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