R: A special case of the low discrepancy Faure sequence

faureprimeDesign {DiceDesign} | R Documentation

## 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))

### 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.

### 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.

### 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

### Author(s)

P. Kiener

### 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")))

[Package DiceDesign version 1.9 Index]