DiceDesign (version 1.10)

faureprimeDesign: A special case of the low discrepancy Faure sequence

Description

Generate a Faure sequence with \(n=p^u-1\) experiments in [0,1]\(^d\) or other domains (see the details) where \(p\) is the first prime number equal or larger than \(d\) and \(u\) is an exponent, usually 2.

Usage

faureprimeDesign(dimension, u = 2, range = c(0, -1))

Value

faureprimeDesign returns a list with the following components:

• design: the design of experiments

• n: the number of experiments

• dimension: the dimension

• prime: the prime number

• u: the exponent

Arguments

dimension

the number of variables (< 199)

u

the exponent applied to the prime number

range

the scale (min and max) of the inputs. See the details for the six predefined ranges.

Author

P. Kiener

Details

This is a special case of runif.faure where the number of generated points depends exclusively on the dimension and the selected exponent. For the exponent \(u=2\), the design is orthogonal and has resolution 4. It is a perfect grid \((p-1)(p+1)\) on each pair of variables where \(p\) is the first prime number equal or larger than the dimension \(d\).

Six domain ranges are predefined and cover most applications:

• c(0, 0) corresponds to \([0, n]^d\).

• c(1, 1) corresponds to \([1-n, n-1]^d = [2-p^u, p^u -2]^d\).

• c(0, 1) corresponds to \([0, 1]^d\).

• c(0,-1) corresponds to \([p^{-u}, 1-p^{-u}]^d\).

• c(-1,-1) corresponds to \([-1+2p^{-u}, 1-2p^{-u}]^d\).

• c(-1, 1) corresponds to \([-1, 1]^d\).

References

Faure H. (1982), Discrepance de suites associees a un systeme de numeration (en dimension s), Acta Arith., 41, 337-351.

Owen A.B. (2020), On dropping the first Sobol point, https://arxiv.org/abs/2008.08051.

Examples

## Range c(0,-1) returns the design produced by runif.faure()
plan1 <- runif.faure(n = 24, dimension = 5)$design ; plan1
plan2 <- faureprimeDesign(dimension = 5, range = c(0,-1))$design ; plan2
all.equal(plan1, plan2, tolerance = 1e-15)

## Range c(0,0) returns the original sequence of integers.
## The first (p-1) lines are on the first diagonal.
## The remaining lines are LHSs grouped in p-1 blocks of p rows.
d <- p <- 5
plan <- faureprimeDesign(dimension = d, range = c(0,0))$design ; plan
apply(plan, 2, sort)

## A regular grid (p-1)x(p+1) rotated by a small angle
pairs(plan)

plot(plan[,1], plan[,2], las = 1)
points(plan[1:(p-1),1], plan[1:(p-1),2], pch = 17, cex = 1.6)
abline(v = plan[1:(p-1),1], col = 4)

## Designs of dimensions 24x5 in various ranges
lstrg <- list(p0p0 = c(0,0),  p1p1 = c(1,1),   p0p1 = c(0,1),
              p0m1 = c(0,-1), m1m1 = c(-1,-1), m1p1 = c(-1,1))
lst <- lapply(lstrg, function(rg) faureprimeDesign(
              dimension = 5, u = 2, range = rg)$design)
lapply(lst, tail)
sapply(lst, range)

## The odd designs (p1m1, m1m1, m1p1) are orthogonal and have resolution 4.
library(lattice)

mat <- lst$m1m1 ; colnames(mat) <- LETTERS[1:5]
fml <- ~ (A+B+C+D+E)^2+I(A^2)+I(B^2)+I(C^2)+I(D^2)+I(E^2)
mmm <- model.matrix(fml, data = as.data.frame(mat))[,-1] ; tail(mmm)
cmm <- round(cov2cor(crossprod(mmm)), 3) ; cmm
lattice::levelplot(cmm[, ncol(cmm):1], at = seq(-1, 1, length.out = 10),
                   col.regions = rev(grDevices::hcl.colors(9, "PuOr")))