DiceDesign (version 1.10)

phiP: phiP criterion

Description

Compute the \(\phi_p\) criterion (strongly linked to mindist criterion)

Usage

phiP(design, p=50)

Value

A real number equal to the value of the \(\phi_p\) criterion for the design.

Arguments

design

a matrix (or a data.frame) corresponding to the design of experiments.

p

the "p" in the Lp norm which is taken

Author

G. Damblin & B.Iooss

Details

The \(\phi_p\) criterion is defined by the \(L_p\) norm of the sum of the inverses of the design inter-point euclidean distances: $$\phi_{p}=\left[\sum_{i,j=1\ldots N,i<j}\,\,d_{ij}^{-p}\right]^{\frac{1}{p}}$$

A higher value corresponds to a more regular scaterring of design points.

When \(p\) tends to infinity, optimizing a design with \(\phi_p\) is equivalent to optimizing a design with mindist.

References

Damblin G., Couplet M., and Iooss B. (2013). Numerical studies of sapce filling designs: optimization of Latin Hypercube Samples and subprojection properties, Journal of Simulation, 7:276-289, 2013.

Fang K.-T., Li R. and Sudjianto A. (2006). Design and Modeling for Computer Experiments, Chapman & Hall.

Pronzato, L. and Muller, W. (2012). Design of computer experiments: space filling and beyond, Statistics and Computing, 22:681-701.

See Also

geometric criterion (mindist)

Examples

dimension <- 2
n <- 40
X <- matrix(runif(n*dimension), n, dimension)
phiP(X)