dmaxDesign: Maximum Entropy Designs

Description

Space-Filling Designs with $n$ experiments based on covariance matrix in [0,1]$^d$.

Usage

dmaxDesign(n, dimension, range, niter_max=1000, seed=NULL)

Value

A list with components:

n: the number of points
design: the design of experiments
dimension: the number of variables
range: the range of the variogram
niter_mx: the number of iterations
design_init: the initial distribution
det_init: the value of the determinant for the initial distribution
det_end: the value of the determinant at the end of the procedure
seed: the value of the seed

Arguments

n: number of experiments
dimension: number of variables
range: range of variogram
niter_max: number of iterations
seed: seed used to generate uniform design

Author

J. Franco

Details

Maximum entropy design is a kind of optimal design based on Shannon's definition of entropy as the amount of information. Originally, maximum entropy sampling was proposed by Shewry and Wynn (1987). The goal of the design is to maximize the entropy defined as the determinant of the correlation matrix using a Fedorov-Mitchell exchange algorithm.

The spatial correlation matrix is defined by $C=\left( \rho_{ij} \right)$:

$\rho_{ij}=1-\gamma\left(h_{ij}\right)$	if $h_{ij}\leq a$,
$\rho_{ij}=0$	if $h_{ij}>a$,

where $h_{ij}$ is the distance between $x_{i}$ and $x_{j}$, $a$ denotes the range of the variogram and $\gamma$ is a spherical variogram: $$\gamma(h) = 1.5 \frac{h}{a}- 0.5\left(\frac{h}{a}\right)^3 \textnormal{ for } h \leq a$$

References

Currin C., Mitchell T., Morris M. and Ylvisaker D. (1991) Bayesian Prediction of Deterministic Functions With Applications to the Design and Analysis of Computer Experiments, American Statistical Association, 86, 416, 953-963.

Shewry, M. C. and Wynn and H. P. (1987) Maximum entropy sampling, Journal of Applied Statistics 14, 165-170.

Examples

Run this code

n <- 20
dimension <- 2
range <-0.9
niter_max <- 200
out <- dmaxDesign(n, dimension, range, niter_max)

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

Run the code above in your browser using DataLab

\(\rho_{ij}=1-\gamma\left(h_{ij}\right)\)	if \(h_{ij}\leq a\),
\(\rho_{ij}=0\)	if \(h_{ij}>a\),