straussDesign {DiceDesign} | R Documentation |

## 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)
```

### 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 |

### 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`

.

### Value

A list containing:

`n` |
the number of experiments |

`dimension` |
the number |

`design_init` |
the initial distribution of |

`radius` |
the radius of interaction |

`alpha` |
the potential power alpha |

`repulsion` |
the repulsion parameter |

`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] |

`seed` |
the seed corresponding to the design |

### Author(s)

J. Franco

### 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

```
## 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))
```

*DiceDesign*version 1.10 Index]