Designs based on Strauss process
Space-Filling Designs based on Strauss process
straussDesign(n,dimension, RND, alpha=0.5, repulsion=0.001, NMC=1000,
constraints1D=0, repulsion1D=0.0001, seed=NULL)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 |
Strauss designs are Space-Filling designs initially defined from Strauss process:
π (X) = k gamma^s(X)
where s(X) is is the number of pairs of points (xi,xj) of the design X = ( x1, ..., xn \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 π(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:
π(X) = k exp(-U(X))
with a potentiel U defined by:
beta Sum_{i<j} phi(|| xi-xj ||)
where beta = -ln(gamma), phi (h) = (1-h/RND)^{alpha} if h <= 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.
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 γ |
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 |
design |
the design of experiments in [0,1]^{d} |
seed |
the seed corresponding to the design |
J. Franco
J. Franco, X. Bay, B. Corre and D. Dupuy (2008) Planification d'experiences numeriques a partir du processus ponctuel de Strauss, https://hal.archives-ouvertes.fr/hal-00260701/fr/.
## 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))Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.