| Title: | Designs of Computer Experiments |
|---|---|
| Description: | Space-Filling Designs and space-filling criteria (distance-based and uniformity-based), with emphasis to computer experiments; <doi:10.18637/jss.v065.i11>. |
| Authors: | Jessica Franco, Delphine Dupuy, Olivier Roustant, Patrice Kiener, Guillaume Damblin and Bertrand Iooss. |
| Maintainer: | Celine Helbert <[email protected]> |
| License: | GPL-3 |
| Version: | 1.10 |
| Built: | 2024-09-02 06:14:00 UTC |
| Source: | CRAN |
Space-Filling Designs (SFD) and space-filling criteria (distance-based and uniformity-based).
This package provides tools to create some specific Space-Filling Design (SFD) and to test their quality:
Latin Hypercube designs (randomized or centered)
Strauss SFD and Maximum entropy SFD, WSP designs
Optimal (low-discrepancy and maximin) Latin Hypercube desigsn by simulated annealing and genetic algorithms,
Orthogonal and Nearly Orthogonal Latin Hypercube designs,
Discrepancies criteria, distance measures,
Minimal spanning tree criteria,
Radial scanning statistic
Part of this work was conducted on 2006-2009 within the frame of the DICE (Deep Inside Computer Experiments) Consortium between ARMINES, Renault, EDF, IRSN, ONERA and TOTAL S.A. (http://dice.emse.fr/).
In this package, only Faure's sequence is implemented. Note that the randtoolbox package provides the following quasi random sequences: the Sobol sequence, the Halton (hence Van Der Corput) sequence and the Torus sequence (also known as Kronecker sequence). Note also that the lhs package provides other types of algorithms to compute optimized LHS.
J. Franco, D. Dupuy, O. Roustant, P. Kiener, G. Damblin and B. Iooss. Thanks to A. Jourdan for discussions about OA131.
Maintainer: Celine Helbert [email protected]
Cioppa T.M., Lucas T.W. (2007). Efficient nearly orthogonal and space-filling Latin hypercubes. Technometrics 49, 45-55.
Damblin G., Couplet M., and Iooss B. (2013). Numerical studies of space filling designs: optimization of Latin Hypercube Samples and subprojection properties, Journal of Simulation, 7:276-289, 2013.
De Rainville F.-M., Gagne C., Teytaud O., Laurendeau D. (2012). Evolutionary optimization of low-discrepancy sequences. ACM Transactions on Modeling and Computer Simulation (TOMACS), 22(2), 9.
Dupuy D., Helbert C., Franco J. (2015), DiceDesign and DiceEval: Two R-Packages for Design and Analysis of Computer Experiments, Journal of Statistical Software, 65(11), 1–38.
Fang K.-T., Li R. and Sudjianto A. (2006) Design and Modeling for Computer Experiments, Chapman & Hall.
Fang K-T., Liu M-Q., Qin H. and Zhou Y-D. (2018) Theory and application of uniform experimental designs. Springer.
Nguyen N.K. (2008) A new class of orthogonal Latinhypercubes, Statistics and Applications, Volume 6, issues 1 and 2, pp.119-123.
Owen A.B. (2020), On dropping the first Sobol point, https://arxiv.org/abs/2008.08051.
Roustant O., Franco J., Carraro L., Jourdan A. (2010), A radial scanning statistic for selecting space-filling designs in computer experiments, MODA-9 proceedings.
Santner T.J., Williams B.J. and Notz W.I. (2003) The Design and Analysis of Computer Experiments, Springer, 121-161.
# ********************** # Designs of experiments # ********************** # A maximum entropy design with 20 points in [0,1]^2 p <- dmaxDesign(20,2,0.9,200) plot(p$design,xlim=c(0,1),ylim=c(0,1)) # Change the dimnames, adjust to range (-10, 10) and round to 2 digits xDRDN(p, letter = "T", dgts = 2, range = c(-10, 10)) # ************************ # Criteria: L2-discrepancy # ************************ dp <- discrepancyCriteria(p$design,type=c('L2','C2')) # Coverage measure covp <- coverage(p$design) # ******************************* # Criteria: Minimal Spanning Tree # ******************************* mstCriteria(p$design,plot2d=TRUE) # **************************************************************** # Radial scanning statistic: Detection of defects of Sobol designs # **************************************************************** # requires randtoolbox package library(randtoolbox) # in 2D rss <- rss2d(design=sobol(n=20, dim=2), lower=c(0,0), upper=c(1,1), type="l", col="red") # in 8D. All pairs of dimensions are tried to detect the worst defect # (according to the specified goodness-of-fit statistic). d <- 8 n <- 10*d rss <- rss2d(design=sobol(n=n, dim=d), lower=rep(0,d), upper=rep(1,d), type="l", col="red") # avoid this defect with scrambling ? # 1. Faure-Tezuka scrambling (type "?sobol" for more details and options) rss <- rss2d(design=sobol(n=n, dim=d, scrambling=2), lower=rep(0,d), upper=rep(1,d), type="l", col="red") # 2. Owen scrambling rss <- rss2d(design=sobol(n=n, dim=d, scrambling=1), lower=rep(0,d), upper=rep(1,d), type="l", col="red")# ********************** # Designs of experiments # ********************** # A maximum entropy design with 20 points in [0,1]^2 p <- dmaxDesign(20,2,0.9,200) plot(p$design,xlim=c(0,1),ylim=c(0,1)) # Change the dimnames, adjust to range (-10, 10) and round to 2 digits xDRDN(p, letter = "T", dgts = 2, range = c(-10, 10)) # ************************ # Criteria: L2-discrepancy # ************************ dp <- discrepancyCriteria(p$design,type=c('L2','C2')) # Coverage measure covp <- coverage(p$design) # ******************************* # Criteria: Minimal Spanning Tree # ******************************* mstCriteria(p$design,plot2d=TRUE) # **************************************************************** # Radial scanning statistic: Detection of defects of Sobol designs # **************************************************************** # requires randtoolbox package library(randtoolbox) # in 2D rss <- rss2d(design=sobol(n=20, dim=2), lower=c(0,0), upper=c(1,1), type="l", col="red") # in 8D. All pairs of dimensions are tried to detect the worst defect # (according to the specified goodness-of-fit statistic). d <- 8 n <- 10*d rss <- rss2d(design=sobol(n=n, dim=d), lower=rep(0,d), upper=rep(1,d), type="l", col="red") # avoid this defect with scrambling ? # 1. Faure-Tezuka scrambling (type "?sobol" for more details and options) rss <- rss2d(design=sobol(n=n, dim=d, scrambling=2), lower=rep(0,d), upper=rep(1,d), type="l", col="red") # 2. Owen scrambling rss <- rss2d(design=sobol(n=n, dim=d, scrambling=1), lower=rep(0,d), upper=rep(1,d), type="l", col="red")
Compute the coverage measure
coverage(design)coverage(design)
design |
a matrix (or a data.frame) representing the design of experiments representing the design of experiments in the unit cube [0,1] |
The coverage criterion is defined by
where is the minimal distance between the point
and the other points of the design and is
the mean of the .
Note that for a regular mesh, cov=0. Then, a small value of cov means that the design is close to a regular grid.
A real number equal to the value of the coverage criterion for the design.
J. Franco
Gunzburer M., Burkdart J. (2004) Uniformity measures for point samples in hypercubes, https://people.sc.fsu.edu/~jburkardt/.
other distance criteria like meshRatio, phiP and mindist.
discrepancy measures provided by discrepancyCriteria.
dimension <- 2 n <- 40 X <- matrix(runif(n*dimension), n, dimension) coverage(X)dimension <- 2 n <- 40 X <- matrix(runif(n*dimension), n, dimension) coverage(X)
Compute discrepancy criteria.
discrepancyCriteria(design,type='all')discrepancyCriteria(design,type='all')
design |
a matrix (or a data.frame) corresponding to the design of experiments.
The discrepancy criteria are computed for a design in the unit cube [0,1] |
|||||||||||||||||
type |
type of discrepancies (single value or vector) to be computed:
|
The discrepancy measures how far a given distribution of points deviates
from a perfectly uniform one. Different L2 discrepancies are available in DiceDesign.
For example, if we denote by the volume of a subset of and the number of points of falling in , the discrepancy is:
where , and . The other L2-discrepancies are defined according to the same principle with different form from the subset .
Among all the possibilities, discrepancyCriteria implements only the L2 discrepancies because it can be expressed analytically even for high dimension.
Centered L2-discrepancy is computed using the analytical expression done by Hickernell (1998). The user will refer to Pleming and Manteufel (2005) to have more details about the wrap around discrepancy.
A list containing the L2-discrepancies of the design.
J. Franco, D. Dupuy & B. Iooss
Fang K.T, Li R. and Sudjianto A. (2006) Design and Modeling for Computer Experiments, Chapman & Hall.
Fang K-T., Liu M-Q., Qin H. and Zhou Y-D. (2018) Theory and application of uniform experimental designs. Springer.
Franco J. (2008) Planification d'experiences numerique en phase exploratoire pour la simulation des phenomenes complexes, PhD thesis, Ecole Nationale Superieure des Mines de Saint Etienne.
Hickernell F.J. (1998) A generalized discrepancy and quadrature error bound. Mathematics of Computation, 67, 299-322.
Pleming J.B. and Manteufel R.D. (2005) Replicated Latin Hypercube Sampling, 46th Structures, Structural Dynamics & Materials Conference, 16-21 April 2005, Austin (Texas) – AIAA 2005-1819.
distance criteria (coverage, meshRatio,
mindist and phiP)
dimension <- 2 n <- 40 X <- matrix(runif(n*dimension), n, dimension) discrepancyCriteria(X)dimension <- 2 n <- 40 X <- matrix(runif(n*dimension), n, dimension) discrepancyCriteria(X)
The objective is to produce low-discrepancy LHS. ESE is a powerful genetic algorithm to produce space-filling designs. It has been adapted here to main discrepancy criteria.
discrepESE_LHS(design, T0=0.005*discrepancyCriteria(design,type='C2')[[1]], inner_it=100, J=50, it=2, criterion="C2")discrepESE_LHS(design, T0=0.005*discrepancyCriteria(design,type='C2')[[1]], inner_it=100, J=50, it=2, criterion="C2")
design |
a matrix (or a data.frame) corresponding to the design of experiments. |
T0 |
The initial temperature of the ESE algorithm |
inner_it |
The number of iterations for inner loop |
J |
The number of new proposed LHS inside the inner loop |
it |
The number of iterations for outer loop |
criterion |
The criterion to be optimized. One can choose three different L2-discrepancies: the C2 (centered) discrepancy ("C2"), the L2-star discrepancy ("L2star") and the W2 (wrap-around) discrepancy ("W2") |
This function implements a stochastic algorithm (ESE) to produce optimized LHS. It is based on Jin et al works (2005). Here, it has been adapted to some discrepancy criteria taking into account new ideas about the revaluations of discrepancy value after a LHS elementary perturbation (in order to avoid computing all terms in the discrepancy formulas).
A list containing:
InitialDesign |
the starting design |
T0 |
the initial temperature of the ESE algorithm |
inner_it |
the number of iterations for inner loop |
J |
the number of new proposed LHS inside the inner loop |
it |
the number of iterations for outer loop |
criterion |
the criterion to be optimized |
design |
the matrix of the final design (low-discrepancy LHS) |
critValues |
vector of criterion values along the iterations |
tempValues |
vector of temperature values along the iterations |
probaValues |
vector of acceptation probability values along the iterations |
G.Damblin & B. Iooss
Damblin G., Couplet M., and Iooss B. (2013). Numerical studies of space filling designs: optimization of Latin Hypercube Samples and subprojection properties, Journal of Simulation, 7:276-289, 2013.
M. Morris and J. Mitchell (1995) Exploratory designs for computational experiments. Journal of Statistical Planning and Inference, 43:381-402.
R. Jin, W. Chen and A. Sudjianto (2005) An efficient algorithm for constructing optimal design of computer experiments. Journal of Statistical Planning and Inference, 134:268-287.
Latin Hypercube Sample(lhsDesign),
discrepancy criteria(discrepancyCriteria),
geometric criterion (mindistphiP),
optimization (maximinSA_LHS, maximinESE_LHS, discrepSA_LHS)
## Not run: dimension <- 2 n <- 10 X <- lhsDesign(n, dimension)$design Xopt <- discrepESE_LHS(X, T0=0.005*discrepancyCriteria(X, type='C2')[[1]], inner_it=100, J=50, it=2) plot(Xopt$design) plot(Xopt$critValues, type="l") ## End(Not run)## Not run: dimension <- 2 n <- 10 X <- lhsDesign(n, dimension)$design Xopt <- discrepESE_LHS(X, T0=0.005*discrepancyCriteria(X, type='C2')[[1]], inner_it=100, J=50, it=2) plot(Xopt$design) plot(Xopt$critValues, type="l") ## End(Not run)
The objective is to produce low-discrepancy LHS. SA is an efficient algorithm to produce space-filling designs. It has been adapted here to main discrepancy criteria.
discrepSA_LHS(design, T0=10, c=0.95, it=2000, criterion="C2", profile="GEOM", Imax=100)discrepSA_LHS(design, T0=10, c=0.95, it=2000, criterion="C2", profile="GEOM", Imax=100)
design |
a matrix (or a data.frame) corresponding to the design of experiments |
T0 |
The initial temperature |
c |
A constant parameter regulating how the temperature goes down |
it |
The number of iterations |
criterion |
The criterion to be optimized. One can choose three different L2-discrepancies: the C2 (centered) discrepancy ("C2"), the L2-star discrepancy ("L2star") and the W2 (wrap-around) discrepancy ("W2") |
profile |
The temperature down-profile, purely geometric called "GEOM", geometrical according to the Morris algorithm called "GEOM_MORRIS" or purely linear called "LINEAR" |
Imax |
A parameter given only if you choose the Morris down-profile. It adjusts the number of iterations without improvement before a new elementary perturbation |
This function implements a classical routine to produce optimized LHS. It is based on the work of Morris and Mitchell (1995). They have proposed a SA version for LHS optimization according to mindist criterion. Here, it has been adapted to some discrepancy criteria taking in account new ideas about the reevaluations of a discrepancy value after a LHS elementary perturbation (in order to avoid computing all terms in the discrepancy formulas).
A list containing:
InitialDesign |
the starting design |
T0 |
the initial temperature of the SA algorithm |
c |
the constant parameter regulating how the temperature goes down |
it |
the number of iterations |
criterion |
the criterion to be optimized |
profile |
the temperature down-profile |
Imax |
The parameter given in the Morris down-profile |
design |
the matrix of the final design (low-discrepancy LHS) |
critValues |
vector of criterion values along the iterations |
tempValues |
vector of temperature values along the iterations |
probaValues |
vector of acceptation probability values along the iterations |
G. Damblin & B. Iooss
Damblin G., Couplet M., and Iooss B. (2013). Numerical studies of space filling designs: optimization of Latin Hypercube Samples and subprojection properties, Journal of Simulation, 7:276-289, 2013.
M. Morris and J. Mitchell (1995) Exploratory designs for computationnal experiments. Journal of Statistical Planning and Inference, 43:381-402.
R. Jin, W. Chen and A. Sudjianto (2005) An efficient algorithm for constructing optimal design of computer experiments. Journal of Statistical Planning and Inference, 134:268-287.
Latin Hypercube Sample(lhsDesign),discrepancy criteria(discrepancyCriteria), geometric criterion (mindistphiP), optimization (maximinSA_LHS,maximinESE_LHS ,discrepESE_LHS)
dimension <- 2 n <- 10 X <- lhsDesign(n, dimension)$design ## Optimize the LHS with C2 criterion Xopt <- discrepSA_LHS(X, T0=10, c=0.99, it=2000, criterion="C2") plot(Xopt$design) plot(Xopt$critValues, type="l") ## Optimize the LHS with C2 criterion and GEOM_MORRIS profile ## Not run: Xopt2 <- discrepSA_LHS(X, T0=10, c=0.99, it=1000, criterion="C2", profile="GEOM_MORRIS") plot(Xopt2$design) ## End(Not run)dimension <- 2 n <- 10 X <- lhsDesign(n, dimension)$design ## Optimize the LHS with C2 criterion Xopt <- discrepSA_LHS(X, T0=10, c=0.99, it=2000, criterion="C2") plot(Xopt$design) plot(Xopt$critValues, type="l") ## Optimize the LHS with C2 criterion and GEOM_MORRIS profile ## Not run: Xopt2 <- discrepSA_LHS(X, T0=10, c=0.99, it=1000, criterion="C2", profile="GEOM_MORRIS") plot(Xopt2$design) ## End(Not run)
Space-Filling Designs with experiments based on covariance matrix in
[0,1].
dmaxDesign(n, dimension, range, niter_max=1000, seed=NULL)dmaxDesign(n, dimension, range, niter_max=1000, seed=NULL)
n |
number of experiments |
dimension |
number of variables |
range |
range of variogram |
niter_max |
number of iterations |
seed |
seed used to generate uniform design |
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 :
|
if |
|
if |
where is the distance between and
, denotes the range of the variogram and
is a spherical variogram:
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 |
J. Franco
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.
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))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))
Create a factorial design with = pow(levels,dimension) experiments
in [0,1].
factDesign(dimension, levels)factDesign(dimension, levels)
dimension |
an integer given the number of input variables |
levels |
an integer given the number of levels |
It is possible to take a different number of levels for
any factor. In this case, the argument levels should be a
vector.
factDesign returns a list containing all the
input arguments detailed before, plus the following components:
n |
the number of experiments |
design |
the design of experiments |
G. Pujol and J. Franco
## First example g1 <- factDesign(2, 7) plot(g1$design, xlim=c(0,1), ylim=c(0,1)) ## Second example g2 <- factDesign(2, c(2,7)) plot(g2$design, xlim=c(0,1), ylim=c(0,1)) ## Change the dimnames, adjust to range (-10, 10) and round to 2 digits xDRDN(g1, letter = "T", dgts = 2, range = c(-10, 10)) xDRDN(g2, letter = "T", dgts = 2, range = c(-10, 10))## First example g1 <- factDesign(2, 7) plot(g1$design, xlim=c(0,1), ylim=c(0,1)) ## Second example g2 <- factDesign(2, c(2,7)) plot(g2$design, xlim=c(0,1), ylim=c(0,1)) ## Change the dimnames, adjust to range (-10, 10) and round to 2 digits xDRDN(g1, letter = "T", dgts = 2, range = c(-10, 10)) xDRDN(g2, letter = "T", dgts = 2, range = c(-10, 10))
Generate a Faure sequence with experiments in [0,1]
or other domains (see the details) where is the first prime number equal
or larger than and is an exponent, usually 2.
faureprimeDesign(dimension, u = 2, range = c(0, -1))faureprimeDesign(dimension, u = 2, range = c(0, -1))
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. |
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 , the design is orthogonal and has resolution
4. It is a perfect grid on each pair of
variables where is the first prime number equal or
larger than the dimension .
Six domain ranges are predefined and cover most applications:
c(0, 0) corresponds to .
c(1, 1) corresponds to .
c(0, 1) corresponds to .
c(0,-1) corresponds to .
c(-1,-1) corresponds to .
c(-1, 1) corresponds to .
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
P. Kiener
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.
## 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$m1m1 ; colnames(mat) <- LETTERS[1:5] fml <- ~ (A+B+C+D+E)^2+I(A^2)+I(B^2)+I(C^2)+I(D^2)+I(E^2) mmm <- model.matrix(fml, data = as.data.frame(mat))[,-1] ; tail(mmm) cmm <- round(cov2cor(crossprod(mmm)), 3) ; cmm lattice::levelplot(cmm[, ncol(cmm):1], at = seq(-1, 1, length.out = 10), col.regions = rev(grDevices::hcl.colors(9, "PuOr")))## Range c(0,-1) returns the design produced by runif.faure() plan1 <- runif.faure(n = 24, dimension = 5)$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")))
Simple (random) Latin Hypercube Design (randomized or centered) with experiments in [0,1].
lhsDesign(n, dimension, randomized=TRUE, seed=NULL)lhsDesign(n, dimension, randomized=TRUE, seed=NULL)
n |
number of experiments |
dimension |
number of variables |
randomized |
TRUE for randomized LHS; FALSE for centered LHS |
seed |
seed used to generate the random permutations and perturbations |
This program builds a Latin Hypercube Design (LHD), also called a Latin Hypercube Sample (LHS), on the space [0,1] (with uniform probability measures). LHD aims at ensuring that each variable has its whole range well scanned: the range of each variable is divided into equally probable stratas. Each stratum of each variable contains only one point of the LHD. Centered LHD is obtained by choosing for each point the center of the corresponding case, while randomized LHD is obtained by adding random perturbations inside each point case.
Once the sample is generated, the uniform sample from a column can be transformed to any distribution by using the quantile functions.
A list with components:
n |
the number of points |
dimension |
the number of variables |
design |
the design of experiments |
randomized |
the type of LHD |
seed |
the value of the seed |
B. Iooss
McKay M., Conover W. and Beckman R. (1979) A comparison of three methods for selecting values of input variables in the analysis of output from a computer code, Technometrics, 21, 2, 239-245.
Stein M. (1987) Large sample properties of simulations using Latin hypercube sampling, Technometrics, 29, 143-151.
LHD optimization (maximinSA_LHS,discrepSA_LHS,maximinESE_LHS ,discrepESE_LHS)
n <- 20 dimension <- 2 out <- lhsDesign(n, dimension) out$design ## Change the dimnames, adjust to range (-10, 10) and round to 2 digits xDRDN(out, letter = "T", dgts = 2, range = c(-10, 10))n <- 20 dimension <- 2 out <- lhsDesign(n, dimension) out$design ## Change the dimnames, adjust to range (-10, 10) and round to 2 digits xDRDN(out, letter = "T", dgts = 2, range = c(-10, 10))
The objective is to produce maximin LHS. ESE is a powerful genetic algorithm allowing to produce space-filling designs.
maximinESE_LHS(design, T0=0.005*phiP(design,p=50), inner_it=100, J=50, it=1, p=50)maximinESE_LHS(design, T0=0.005*phiP(design,p=50), inner_it=100, J=50, it=1, p=50)
design |
a matrix (or a data.frame) corresponding to the design of experiments. |
T0 |
The initial temperature of the ESE algorithm |
inner_it |
The number of iterations for inner loop |
J |
The number of new proposed LHS inside the inner loop |
it |
The number of iterations for outer loop |
p |
power required in phiP criterion |
This function implements a stochastic algorithm (ESE) to produce optimized LHS. It is based on Jin et al works (2005).
A list containing:
InitialDesign |
the starting design |
T0 |
the initial temperature of the ESE algorithm |
inner_it |
the number of iterations for inner loop |
J |
the number of new proposed LHS inside the inner loop |
it |
the number of iterations for outer loop |
p |
power required in phiP criterion |
design |
the matrix of the final design (maximin LHS) |
critValues |
vector of criterion values along the iterations |
tempValues |
vector of temperature values along the iterations |
probaValues |
vector of acceptation probability values along the iterations |
G. Damblin & B. Iooss
Damblin G., Couplet M., and Iooss B. (2013). Numerical studies of space filling designs: optimization of Latin Hypercube Samples and subprojection properties, Journal of Simulation, 7:276-289, 2013.
M. Morris and J. Mitchell (1995) Exploratory designs for computationnal experiments. Journal of Statistical Planning and Inference, 43:381-402.
R. Jin, W. Chen and A. Sudjianto (2005) An efficient algorithm for constructing optimal design of computer experiments. Journal of Statistical Planning and Inference, 134:268-287.
Pronzato, L. and Muller, W. (2012). Design of computer experiments: space filling and beyond, Statistics and Computing, 22:681-701.
Latin Hypercube Sample (lhsDesign),
discrepancy criteria (discrepancyCriteria),
geometric criterion (mindist, phiP),
optimization (maximinSA_LHS, discrepESE_LHS, discrepSA_LHS)
dimension <- 2 n <- 10 X <- lhsDesign(n, dimension)$design Xopt <- maximinESE_LHS(X, T0=0.005*phiP(X), inner_it=100, J=50, it=2) plot(Xopt$design) plot(Xopt$critValues, type="l")dimension <- 2 n <- 10 X <- lhsDesign(n, dimension)$design Xopt <- maximinESE_LHS(X, T0=0.005*phiP(X), inner_it=100, J=50, it=2) plot(Xopt$design) plot(Xopt$critValues, type="l")
The objective is to produce maximin LHS. SA is an efficient algorithm to produce space-filling designs.
maximinSA_LHS(design, T0=10, c=0.95, it=2000, p=50, profile="GEOM", Imax=100)maximinSA_LHS(design, T0=10, c=0.95, it=2000, p=50, profile="GEOM", Imax=100)
design |
a matrix (or a data.frame) corresponding to the design of experiments |
T0 |
The initial temperature of the SA algorithm |
c |
A constant parameter regulating how the temperature goes down |
it |
The number of iterations |
p |
power required in phiP criterion |
profile |
The temperature down-profile, purely geometric called "GEOM", geometrical according to the Morris algorithm called "GEOM_MORRIS" or purely linear called "LINEAR" |
Imax |
A parameter given only if you choose the Morris down-profile. It adjusts the number of iterations without improvement before a new elementary perturbation |
This function implements a classical routine to produce optimized LHS. It is based on the work of Morris and Mitchell (1995). They have proposed a SA version for LHS optimization according to mindist criterion. Here, it has been adapted to the phiP criterion. It has been shown (Pronzato and Muller, 2012, Damblin et al., 2013) that optimizing phiP is more efficient to produce maximin designs than optimizing mindist. When tends to infinity, optimizing a design with phi_p is equivalent to optimizing a design with mindist.
A list containing:
InitialDesign |
the starting design |
T0 |
the initial temperature of the SA algorithm |
c |
the constant parameter regulating how the temperature goes down |
it |
the number of iterations |
p |
power required in phiP criterion |
profile |
the temperature down-profile |
Imax |
The parameter given in the Morris down-profile |
design |
the matrix of the final design (maximin LHS) |
critValues |
vector of criterion values along the iterations |
tempValues |
vector of temperature values along the iterations |
probaValues |
vector of acceptation probability values along the iterations |
G. Damblin & B. Iooss
Damblin G., Couplet M., and Iooss B. (2013). Numerical studies of space filling designs: optimization of Latin Hypercube Samples and subprojection properties, Journal of Simulation, 7:276-289, 2013.
M. Morris and J. Mitchell (1995) Exploratory designs for computationnal experiments. Journal of Statistical Planning and Inference, 43:381-402.
R. Jin, W. Chen and A. Sudjianto (2005) An efficient algorithm for constructing optimal design of computer experiments. Journal of Statistical Planning and Inference, 134:268-287.
Pronzato, L. and Muller, W. (2012). Design of computer experiments: space filling and beyond, Statistics and Computing, 22:681-701.
Latin Hypercube Sample (lhsDesign),
discrepancy criteria (discrepancyCriteria),
geometric criterion (mindist, phiP),
optimization (discrepSA_LHS, maximinESE_LHS, discrepESE_LHS)
dimension <- 2 n <- 10 X <- lhsDesign(n ,dimension)$design Xopt <- maximinSA_LHS(X, T0=10, c=0.99, it=2000) plot(Xopt$design) plot(Xopt$critValues, type="l") plot(Xopt$tempValues, type="l") ## Not run: Xopt <- maximinSA_LHS(X, T0=10, c=0.99, it=1000, profile="GEOM_MORRIS") ## End(Not run)dimension <- 2 n <- 10 X <- lhsDesign(n ,dimension)$design Xopt <- maximinSA_LHS(X, T0=10, c=0.99, it=2000) plot(Xopt$design) plot(Xopt$critValues, type="l") plot(Xopt$tempValues, type="l") ## Not run: Xopt <- maximinSA_LHS(X, T0=10, c=0.99, it=1000, profile="GEOM_MORRIS") ## End(Not run)
The meshRatio criterion is the ratio between the
maximimum and the minimum distance between two points of the experimental design.
meshRatio(design)meshRatio(design)
design |
a matrix (or a data.frame) representing the design of experiments in the unit cube [0,1] |
The meshRatio criterion is defined by
where denotes the minimal distance between the point
and the other points of the design.
Note that for a regular mesh, meshRatio=1.
A real number equal to the value of the meshRatio criterion for the design.
J. Franco
Gunzburer M. and Burkdart J. (2004), Uniformity measures for point samples in hypercubes, https://people.sc.fsu.edu/~jburkardt/.
Other distance criteria like meshRatio, phiP and mindist.
Discrepancy measures provided by discrepancyCriteria.
dimension <- 2 n <- 40 X <- matrix(runif(n*dimension), n, dimension) meshRatio(X)dimension <- 2 n <- 40 X <- matrix(runif(n*dimension), n, dimension) meshRatio(X)
Compute the mindist criterion (also called maximin)
mindist(design)mindist(design)
design |
a matrix (or a data.frame) representing the design of experiments in the unit cube [0,1] |
The mindist criterion is defined by
where is the minimal distance between the point
and the other points of the design.
A higher value corresponds to a more regular scaterring of design points.
A real number equal to the value of the mindist criterion for the design.
J. Franco
Gunzburer M., Burkdart J. (2004), Uniformity measures for point samples in hypercubes, https://people.sc.fsu.edu/~jburkardt/.
Jonshon M.E., Moore L.M. and Ylvisaker D. (1990), Minmax and maximin distance designs, J. of Statis. Planning and Inference, 26, 131-148.
Chen V.C.P., Tsui K.L., Barton R.R. and Allen J.K. (2003), A review of design and modeling in computer experiments, Handbook of Statistics, 22, 231-261.
other distance criteria like meshRatio and phiP,
discrepancy measures provided by discrepancyCriteria.
dimension <- 2 n <- 40 X <- matrix(runif(n*dimension), n, dimension) mindist(X)dimension <- 2 n <- 40 X <- matrix(runif(n*dimension), n, dimension) mindist(X)
Compute both the mean and the standard deviation of the Minimal Spanning Tree (MST)
mstCriteria(design, plot2d="FALSE")mstCriteria(design, plot2d="FALSE")
design |
a matrix (or a data.frame) corresponding to the design of experiments. |
plot2d |
an argument for visualizing the mst of a 2d design |
In our context, a MST is a tree whose the sum of the lengthes of the edges is minimal. Even if unicity does not hold, the overall length is stable. The mean and the standard deviation of the lengthes of the edges are usually derived to analyze the geometric profile of the design. A large mean and a small standard deviation characterize a so-called quasi-periodic design.
A list containing two components:
tree |
a list containing the MST: each component of it contains a vector with all vertices which are connected with the experiment corresponding to the number of the components |
stats |
vector with both the mean and the standard deviation values of the lengthes of the edges |
G. Damblin & B. Iooss
Damblin G., Couplet M., and Iooss B. (2013). Numerical studies of space filling designs: optimization of Latin hypercube samples and subprojection properties, Journal of Simulation, 7:276-289, 2013.
Dussert, C., Rasigni, G., Rasigni, M., and Palmari, J. (1986). Minimal spanning tree: A new approach for studying order and disorder. Physical Review B, 34(5):3528-3531.
Franco J. (2008). Planification d'experiences numerique en phase exploratoire pour la simulation des phenomenes complexes, PhD thesis, Ecole Nationale Superieure des Mines de Saint Etienne.
Franco, J., Vasseur, O., Corre, B., and Sergent, M. (2009). Minimum spanning tree: A new approach to assess the quality of the design of computer experiments. Chemometrics and Intelligent Laboratory Systems, 97:164-169.
Prim, R.C. (1957). Shortest connection networks and some generalizations, in Bell System Technical Journal 36:1389-1401.
dimension <- 2 n <- 40 X <- matrix(runif(n*dimension), n, dimension) mstCriteria(X, plot2d=TRUE)dimension <- 2 n <- 40 X <- matrix(runif(n*dimension), n, dimension) mstCriteria(X, plot2d=TRUE)
This function generates a NOLH design of dimension 2 to 29 and normalizes it to the selected range. The design is extracted from Cioppa's NOLHdesigns list.
nolhDesign(dimension, range = c(0, 1))nolhDesign(dimension, range = c(0, 1))
dimension |
number of input variables |
range |
the scale (min and max) of the inputs. Range (0, 0) and (1, 1) are special cases and call integer ranges |
A list with components:
n |
the number of lines/experiments |
dimension |
the number of columns/input variables |
design |
the design of experiments |
T.M. Cioppa for the designs. P. Kiener for the R code.
Cioppa's list NOLHdesigns. Other NOLH and OLH designs: nolhdrDesign, olhDesign.
## Classical normalizations nolhDesign(8, range = c(1, 1)) nolhDesign(8, range = c(0, 0)) nolhDesign(8, range = c(0, 1)) nolhDesign(8, range = c(-1, 1)) ## Change the dimnames, adjust to range (-10, 10) and round to 2 digits xDRDN(nolhDesign(8), letter = "T", dgts = 2, range = c(-10, 10)) ## A list of designs lapply(5:9, function(n) nolhDesign(n, range = c(-1, 1))$design)## Classical normalizations nolhDesign(8, range = c(1, 1)) nolhDesign(8, range = c(0, 0)) nolhDesign(8, range = c(0, 1)) nolhDesign(8, range = c(-1, 1)) ## Change the dimnames, adjust to range (-10, 10) and round to 2 digits xDRDN(nolhDesign(8), letter = "T", dgts = 2, range = c(-10, 10)) ## A list of designs lapply(5:9, function(n) nolhDesign(n, range = c(-1, 1))$design)
A list of the NOLH designs for 2 to 29 input variables proposed by Cioppa in 2007. These designs combine a latin structure, orthogonality between the main terms and the interactions (+ squares) and reduced correlations between the interactions (+ squares).
This list combines the Excel spreadsheets published by Sanchez (see Source). It is used internally by the function nolhDesign which provides various normalizations.
NOLHdesignsNOLHdesigns
A list of 5 matrices representing designs of experiments for 8 to 29 input variables:
nolh2_7: 2 to 7 input variables, 17 experiments.
nolh8_11: 8 to 11 input variables, 33 experiments.
nolh12_16: 12 to 16 input variables, 65 experiments.
nolh17_22: 17 to 22 input variables, 129 experiments.
nolh23_29: 23 to 29 input variables, 257 experiments.
T.M. Cioppa for the designs. P. Kiener for the R code.
Sanchez, S. M. (2011). NOLHdesigns in Excel file. Available online at https://nps.edu/web/seed/software-downloads/
Cioppa T.M., Lucas T.W. (2007). Efficient nearly orthogonal and space-filling Latin hypercubes. Technometrics 49, 45-55.
Kleijnen, J.P.C., Sanchez S.M., T.W. Lucas and Cioppa T. M.. A user's guide to the brave new world of designing simulation experiments. INFORMS Journal on Computing 17(3): 263-289.
Ye, K. Q. (1998). Orthogonal Latin hypercubes and their application in computer experiments. J. Amer. Statist. Asso. 93, 1430- 1439.
The main function nolhDesign. De Rainville's NOLH design list: NOLHDRdesigns.
## data(NOLHdesigns) ## all matrices names(NOLHdesigns) lapply(NOLHDRdesigns, tail, 2) ## The first matrix/design NOLHdesigns[["nolh2_7"]]## data(NOLHdesigns) ## all matrices names(NOLHdesigns) lapply(NOLHDRdesigns, tail, 2) ## The first matrix/design NOLHdesigns[["nolh2_7"]]
This function generates a NOLH design of dimension 2 to 29 and normalizes it to the selected range. From 2 to 7 input variables, the design is extracted from Cioppa's NOLHdesigns list and from 8 to 29 input variables it is extracted from De Rainville's NOLHDRdesigns list.
nolhdrDesign(dimension, range = c(0, 1))nolhdrDesign(dimension, range = c(0, 1))
dimension |
number of input variables |
range |
the scale (min and max) of the inputs. Range (0, 0) and (1, 1) are special cases and call integer ranges |
A list with components:
n |
the number of lines/experiments |
dimension |
the number of columns/input variables |
design |
the design of experiments |
T.M. Cioppa and F.-M. De Rainville for the designs. P. Kiener for the R code.
De Rainville's list NOLHDRdesigns. Other NOLH or OLH designs: nolhDesign, olhDesign.
## Classical normalizations nolhdrDesign(8, range = c(1, 1)) nolhdrDesign(8, range = c(0, 1)) nolhdrDesign(8, range = c(0, 0)) nolhdrDesign(8, range = c(-1, 1)) ## Change the dimnames, adjust to range (-10, 10) and round to 2 digits xDRDN(nolhdrDesign(8), letter = "T", dgts = 2, range = c(-10, 10)) ## A list of designs lapply(5:9, function(n) nolhdrDesign(n, range = c(-1, 1))$design)## Classical normalizations nolhdrDesign(8, range = c(1, 1)) nolhdrDesign(8, range = c(0, 1)) nolhdrDesign(8, range = c(0, 0)) nolhdrDesign(8, range = c(-1, 1)) ## Change the dimnames, adjust to range (-10, 10) and round to 2 digits xDRDN(nolhdrDesign(8), letter = "T", dgts = 2, range = c(-10, 10)) ## A list of designs lapply(5:9, function(n) nolhdrDesign(n, range = c(-1, 1))$design)
A list of the NOLH designs for 8 to 29 input variables proposed by De Rainville in 2012. These designs are said to be an improvment of Cioppa's NOLH designs as they have the same structure but better dispersion measures like the discrepancy.
This list combines the csv files published by De Rainville (see Source), centered and normalized to integer values. It is used internally by the function nolhdrDesign which provides various normalizations.
NOLHDRdesignsNOLHDRdesigns
A list of 22 matrices representing designs of experiments for 8 to 29 input variables:
nolhdr08 to nolhdr11: 8, 9, 10, 11 input variables, 33 experiments.
nolhdr12 to nolhdr16: 12, 13, 14, 15, 16 input variables, 65 experiments.
nolhdr17 to nolhdr22: 17, 18, 19, 20, 21, 22 input variables, 129 experiments.
nolhdr23 to nolhdr29: 23, 24, 25, 26, 27, 28, 29 input variables, 257 experiments.
F.-M. De Rainville for the designs. P. Kiener for the R code.
Main website: http://qrand.gel.ulaval.ca/
The python source code: https://github.com/fmder/pynolh/
The python package: https://pypi.org/project/pynolh/.
De Rainville F.-M., Gagne C., Teytaud O., Laurendeau D. (2012). Evolutionary optimization of low-discrepancy sequences. ACM Transactions on Modeling and Computer Simulation (TOMACS), 22(2), 9.
Cioppa T.M., Lucas T.W. (2007). Efficient nearly orthogonal and space-filling Latin hypercubes. Technometrics 49, 45-55.
The main function nolhdrDesign. Cioppa's NOLH design list: NOLHdesigns.
## data(NOLHDRdesigns) ## all matrices names(NOLHDRdesigns) lapply(NOLHDRdesigns, tail, 2) ## The first matrix/design NOLHDRdesigns[["nolhdr08"]]## data(NOLHDRdesigns) ## all matrices names(NOLHDRdesigns) lapply(NOLHDRdesigns, tail, 2) ## The first matrix/design NOLHDRdesigns[["nolhdr08"]]
A 3-dimensional linear orthogonal array (OA) of strength 2 with 49 points. The design points are equally spaced into 2 dimensional coordinate planes. However by construction, such OAs satisfy a linear relation, here: x1 + 3*x2 + x3 = 0 (mod 7). As a consequence, the design points are contained in parallel planes orthogonal to (1,3,1). Actually, they are also contained in parallel planes orthogonal to other directions, as (2,-1,2) or (3,2,3), since the congruence relation leads to 2*x1 - x2 + 2*x3 = 0 (mod 7) or 3*x1 + 2*x2 + 3*x3 = 0 (mod 7). For instance, they are contained in 4 parallel planes orthogonal to (2,-1,2).
data(OA131)data(OA131)
A data frame with 49 observations on the following 3 variables.
x1first coordinate
x2second coordinate
x3third coordinate
Roustant O., Franco J., Carraro L., Jourdan A. (2010), A radial scanning statistic for selecting space-filling designs in computer experiments, MODA-9 proceedings.
data(OA131) # centering and reducing to [0,1]^3 OA <- (OA131 + 0.5)/7 pairs(OA, xlim=c(0,1), ylim=c(0,1)) ## Not run: library(lattice) cloud(x3~x1+x2, data=OA, xlim=c(0,1), ylim=c(0,1), zlim=c(0,1), screen = list(z = 50, x = -70, y = 0)) ## End(Not run)data(OA131) # centering and reducing to [0,1]^3 OA <- (OA131 + 0.5)/7 pairs(OA, xlim=c(0,1), ylim=c(0,1)) ## Not run: library(lattice) cloud(x3~x1+x2, data=OA, xlim=c(0,1), ylim=c(0,1), zlim=c(0,1), screen = list(z = 50, x = -70, y = 0)) ## End(Not run)
This design is obtained by adding a uniform noise to each coordinate of the orthogonal array OA131.
data(OA131_scrambled)data(OA131_scrambled)
A data frame with 49 observations on the following 3 variables.
x1first coordinate
x2second coordinate
x3third coordinate
Roustant O., Franco J., Carraro L., Jourdan A. (2010), A radial scanning statistic for selecting space-filling designs in computer experiments, MODA-9 proceedings.
data(OA131) data(OA131_scrambled) pairs(OA131, xlim=c(0,1), ylim=c(0,1)) pairs(OA131_scrambled, xlim=c(0,1), ylim=c(0,1))data(OA131) data(OA131_scrambled) pairs(OA131, xlim=c(0,1), ylim=c(0,1)) pairs(OA131_scrambled, xlim=c(0,1), ylim=c(0,1))
Generate the Orthogonal Latin Hypercube (OLH) designs proposed by Nguyen in 2008. These OLHs have a latin structure, an orthogonality between the main terms and the interactions (+ squares) and low correlations between the interactions (+ squares). Very larges matrices can be obtained as the number of input variables and hence the number of lines is unconstrained. When the number of input variables is a power of 2, OLHs have columns and lines (experiments). A vertical truncature is applied when the number of input variables is not a power of 2. Various normalizations can be applied.
olhDesign(dimension, range = c(0, 1))olhDesign(dimension, range = c(0, 1))
dimension |
number of input variables |
range |
the scale (min and max) of the inputs. Ranges (0, 0) and (1, 1) are special cases and call integer ranges |
A list with components:
n |
the number of lines/experiments |
dimension |
the number of columns/input variables |
design |
the design of experiments |
N.K. Nguyen for the algorithm. P. Kiener for the recursive R code.
Nguyen N.K. (2008) A new class of orthogonal Latinhypercubes, Statistics and Applications, Volume 6, issues 1 and 2, pp.119-123.
Cioppa's and De Rainville's NOLH designs: nolhDesign, nolhdrDesign.
## Classical normalizations olhDesign(4, range = c(0, 0)) olhDesign(4, range = c(1, 1)) olhDesign(4, range = c(0, 1)) olhDesign(4, range = c(-1, 1)) ## Change the dimnames, adjust to range (-10, 10) and round to 2 digits xDRDN(olhDesign(4), letter = "T", dgts = 2, range = c(-10, 10)) ## A list of designs lapply(1:5, function(n) olhDesign(n, range = c(-1, 1))$design)## Classical normalizations olhDesign(4, range = c(0, 0)) olhDesign(4, range = c(1, 1)) olhDesign(4, range = c(0, 1)) olhDesign(4, range = c(-1, 1)) ## Change the dimnames, adjust to range (-10, 10) and round to 2 digits xDRDN(olhDesign(4), letter = "T", dgts = 2, range = c(-10, 10)) ## A list of designs lapply(1:5, function(n) olhDesign(n, range = c(-1, 1))$design)
Compute the criterion (strongly linked to mindist criterion)
phiP(design, p=50)phiP(design, p=50)
design |
a matrix (or a data.frame) corresponding to the design of experiments. |
p |
the "p" in the Lp norm which is taken |
The criterion is defined by the norm of the sum of the inverses of the design inter-point euclidean distances:
A higher value corresponds to a more regular scaterring of design points.
When tends to infinity, optimizing a design with is equivalent to optimizing a design with mindist.
A real number equal to the value of the criterion for the design.
G. Damblin & B.Iooss
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.
geometric criterion (mindist)
dimension <- 2 n <- 40 X <- matrix(runif(n*dimension), n, dimension) phiP(X)dimension <- 2 n <- 40 X <- matrix(runif(n*dimension), n, dimension) phiP(X)
For a 2-dimensional design, the 2D radial scanning statistic (RSS) scans angularly the domain. In each direction, it compares the distribution of projected points to their theoretical distribution under the assumption that all design points are drawn from uniform distribution. For a d-dimensional design, all pairs of dimensions are scanned.The RSS detects the defects of low discrepancy sequences or orthogonal arrays, and can be used for selecting space-filling designs.
rss2d(design, lower, upper, gof.test.type="greenwood", gof.test.stat=NULL, transform=NULL, n.angle=360, graphics=1, trace=TRUE, lines.lwd = 1, lines.lty = "dotted", ...)rss2d(design, lower, upper, gof.test.type="greenwood", gof.test.stat=NULL, transform=NULL, n.angle=360, graphics=1, trace=TRUE, lines.lwd = 1, lines.lty = "dotted", ...)
design |
a matrix or data.frame containing the d-dimensional design of experiments. The row no. i contains the values of the d input variables corresponding to simulation no. i |
lower |
the domain lower boundaries. |
upper |
the domain upper boundaries. |
gof.test.type |
an optional character indicating the kind of statistical test to be used to test the goodness-of-fit of the design projections to their theoretical distribution. Several tests are available, see |
gof.test.stat |
an optional number equal to the goodness-of-fit statistic at level |
transform |
an optional character indicating what type of transformation should be applied before testing uniformity. Only one choice available "spacings", that lead to over-detection. Default - and recommended - is |
n.angle |
an optional number indicating the number of angles used. Default is |
graphics |
an optional integer indicating whether a graph should be produced. If negative, no graph is produced. If superior to 2, the RSS curve only is plotted in the worst 2D coordinate subspace (corr. to the worst value of statistic). If 1 (default), the design is also added, with its projections onto the worst (oblique) axis. |
trace |
an optional boolean. Turn it to |
lines.lwd |
optional number specifying the width of the straight lines involved in the graphical outputs (axis and projections) |
lines.lty |
optional character string specifying the type of the straight lines involved in the graphical outputs (axis and projections) |
... |
optional graphical parameters of plot function, to draw the RSS curve. |
a list with components:
global.stat |
a matrix containing the values of the global statistic (equal to the maximum of statistic values over the RSS curve) for all pairs of dimensions. |
worst.case |
the worst pair of dimensions, that is the one that gives the worst value of |
worst.dir |
the worst direction, that is the one that gives the worst value of the global statistic in the coordinate plane defined by |
stat |
a vector of length |
angle |
a vector of length |
curve |
a |
gof.test.stat |
the threshold at significance level 0.05 for the specified goodness-of-fit statistic. It is equal to the radius of the circle superimposed on the RSS figure. |
O. Roustant
Roustant O., Franco J., Carraro L., Jourdan A. (2010), A radial scanning statistic for selecting space-filling designs in computer experiments, MODA-9 proceedings.
D Agostino R.B., Stephens M.A. (1986), Goodness-of-fit techniques, CRC Press, New York.
unif.test.statistic, unif.test.quantile, rss3d
## Detection of defects of Sobol designs ## requires randtoolbox package library(randtoolbox) ## In 2D rss <- rss2d(design=sobol(n=20, dim=2), lower=c(0,0), upper=c(1,1), type="l", col="red") ## In 8D ## All pairs of dimensions are tried to detect the worst defect ## (according to the specified goodness-of-fit statistic). d <- 8 n <- 10*d rss <- rss2d(design=sobol(n=n, dim=d), lower=rep(0,d), upper=rep(1,d), type="l", col="red") ## Avoid this defect with scrambling? ## 1. Faure-Tezuka scrambling (type "?sobol" for more details and options) rss <- rss2d(design=sobol(n=n, dim=d, scrambling=2), lower=rep(0,d), upper=rep(1,d), type="l", col="red") ## 2. Owen scrambling rss <- rss2d(design=sobol(n=n, dim=d, scrambling=1), lower=rep(0,d), upper=rep(1,d), type="l", col="red")## Detection of defects of Sobol designs ## requires randtoolbox package library(randtoolbox) ## In 2D rss <- rss2d(design=sobol(n=20, dim=2), lower=c(0,0), upper=c(1,1), type="l", col="red") ## In 8D ## All pairs of dimensions are tried to detect the worst defect ## (according to the specified goodness-of-fit statistic). d <- 8 n <- 10*d rss <- rss2d(design=sobol(n=n, dim=d), lower=rep(0,d), upper=rep(1,d), type="l", col="red") ## Avoid this defect with scrambling? ## 1. Faure-Tezuka scrambling (type "?sobol" for more details and options) rss <- rss2d(design=sobol(n=n, dim=d, scrambling=2), lower=rep(0,d), upper=rep(1,d), type="l", col="red") ## 2. Owen scrambling rss <- rss2d(design=sobol(n=n, dim=d, scrambling=1), lower=rep(0,d), upper=rep(1,d), type="l", col="red")
For a 3-dimensional design, the 3D radial scanning statistic (RSS) scans angularly the domain. In each direction, it compares the distribution of projected points to their theoretical distribution under the assumption that all design points are drawn from uniform distribution. For a d-dimensional design, all triplets of dimensions are scanned. The RSS detects the defects of low discrepancy sequences or orthogonal arrays, and can be used for selecting space-filling designs.
rss3d(design, lower, upper, gof.test.type = "greenwood", gof.test.stat = NULL, transform = NULL, n.angle = 60, graphics = 1, trace = TRUE)rss3d(design, lower, upper, gof.test.type = "greenwood", gof.test.stat = NULL, transform = NULL, n.angle = 60, graphics = 1, trace = TRUE)
design |
a matrix or data.frame containing the d-dimensional design of experiments. The row no. i contains the values of the d input variables corresponding to simulation no. i |
lower |
the domain lower boundaries. |
upper |
the domain upper boundaries. |
gof.test.type |
an optional character indicating the kind of statistical test to be used to test the goodness-of-fit of the design projections to their theoretical distribution. Several tests are available, see |
gof.test.stat |
an optional number equal to the goodness-of-fit statistic at level |
transform |
an optional character indicating what type of transformation should be applied before testing uniformity. Only one choice available "spacings", that lead to over-detection. Default - and recommended - is |
n.angle |
an optional number indicating the number of angles used. Default is |
graphics |
an optional integer indicating whether a graph should be produced. If negative, no graph is produced. Otherwise (default), the design is plotted in the worst 3D coordinate subspace (corr. to the worst value of statistic), with its projections onto the worst (oblique) axis. |
trace |
an optional boolean. Turn it to |
The RSS surface is continuous. However for computational purposes, a discretization is used. The default discretization step is tunable with n.angle.
a list with components:
global.stat |
an array containing the values of the global statistic (equal to the maximum of statistic values over the RSS surface) for all triplets of dimensions. |
print.out |
the same as |
worst.case |
the worst triplet of dimensions, that is the one that gives the worst value of |
worst.dir |
the worst direction, that is the one that gives the worst value of the statistic in the coordinate 3D subspace defined by |
stat |
a matrix of size |
angle |
a matrix of size |
gof.test.stat |
the threshold at significance level 0.05 for the specified goodness-of-fit statistic. |
O. Roustant
Roustant O., Franco J., Carraro L., Jourdan A. (2010), A radial scanning statistic for selecting space-filling designs in computer experiments, MODA-9 proceedings.
D Agostino R.B., Stephens M.A. (1986), Goodness-of-fit techniques, CRC Press, New York.
unif.test.statistic, unif.test.quantile, rss2d
## An orthogonal array in 3D data(OA131) ## centering the design points of this 7-levels design OA <- (OA131 + 0.5)/7 ## 2D projections onto coordinate axis pairs(OA, xlim=c(0,1), ylim=c(0,1)) ## Now let us look at the 3D properties with the 3D RSS (requires the rgl package) rss <- rss3d(OA, lower=c(0,0,0), upper=c(1,1,1)) ## The worst direction detected is nearly proportional to (2,-1,2) ## (type "?OA131" for explanations about this linear orthogonal array) print(rss$worst.dir) ## Now, scramble this design ## X <- (OA131 + matrix(runif(49*3, 49, 3)))/7 ## or load the design obtained this way data(OA131_scrambled) OA2 <- OA131_scrambled ## No feature is detected by the 2D RSS: rss <- rss2d(OA2, lower=c(0,0,0), upper=c(1,1,1)) ## 4 clusters are detected by the 3D RSS: rss <- rss3d(OA2, lower=c(0,0,0), upper=c(1,1,1)) ## Defect detection of 8D Sobol sequences ## All triplets of dimensions are tried to detect the worst defect ## (according to the specified goodness-of-fit statistic). ## requires randtoolbox library to generate the Sobol sequence ## Not run: library(randtoolbox) d <- 8 n <- 10*d rss <- rss3d(design=sobol(n=n, dim=d), lower=rep(0,d), upper=rep(1,d)) ## End(Not run)## An orthogonal array in 3D data(OA131) ## centering the design points of this 7-levels design OA <- (OA131 + 0.5)/7 ## 2D projections onto coordinate axis pairs(OA, xlim=c(0,1), ylim=c(0,1)) ## Now let us look at the 3D properties with the 3D RSS (requires the rgl package) rss <- rss3d(OA, lower=c(0,0,0), upper=c(1,1,1)) ## The worst direction detected is nearly proportional to (2,-1,2) ## (type "?OA131" for explanations about this linear orthogonal array) print(rss$worst.dir) ## Now, scramble this design ## X <- (OA131 + matrix(runif(49*3, 49, 3)))/7 ## or load the design obtained this way data(OA131_scrambled) OA2 <- OA131_scrambled ## No feature is detected by the 2D RSS: rss <- rss2d(OA2, lower=c(0,0,0), upper=c(1,1,1)) ## 4 clusters are detected by the 3D RSS: rss <- rss3d(OA2, lower=c(0,0,0), upper=c(1,1,1)) ## Defect detection of 8D Sobol sequences ## All triplets of dimensions are tried to detect the worst defect ## (according to the specified goodness-of-fit statistic). ## requires randtoolbox library to generate the Sobol sequence ## Not run: library(randtoolbox) d <- 8 n <- 10*d rss <- rss3d(design=sobol(n=n, dim=d), lower=rep(0,d), upper=rep(1,d)) ## End(Not run)
Generate a Faure sequence with experiments in [0,1].
runif.faure(n, dimension)runif.faure(n, dimension)
n |
the number of experiments |
dimension |
the number of variables (<100) |
A quasirandom or low discrepancy sequence, such as the Faure, Halton, Hammersley, Niederreiter or Sobol sequences, is "less random" than a pseudorandom number sequence, but more useful for such tasks as approximation of integrals in higher dimensions, and in global optimization. This is because low discrepancy sequences tend to sample space "more uniformly" than random numbers.
see randtoolbox or fOptions packages for other low discrepancy sequences.
runif.halton returns a list containing all the
input arguments detailed before, plus the following component:
design |
the design of experiments |
J. Franco
Faure H. (1982), Discrepance de suites associees a un systeme de numeration (en dimension s), Acta Arith., 41, 337-351
f <- runif.faure(20,2) plot(f$design, xlim=c(0,1), ylim=c(0,1)) xDRDN(f, letter="T", dgts=2, range=c(-10, 10))f <- runif.faure(20,2) plot(f$design, xlim=c(0,1), ylim=c(0,1)) xDRDN(f, letter="T", dgts=2, range=c(-10, 10))
This function scales the values of the design points to values comprised in [0,1]. The scaling can be made by the Rosenblatt transformation (uniformization by applying the empirical cumulative distribution function) or by translating the design from maximum and minimum values (given for each variable).
scaleDesign(design, min=NULL, max=NULL, uniformize=FALSE)scaleDesign(design, min=NULL, max=NULL, uniformize=FALSE)
design |
a matrix (or a data.frame) corresponding to the design of experiments to scale |
min |
the vector of minimal bounds of each design variable. If not given, the minimal value of each variable is taken |
max |
the vector of maximal bounds of each design variable. If not given, the maximal value of each variable is taken |
uniformize |
boolean: TRUE to use the Rosenblatt transformation (the min and max vectors are useless in this case). If FALSE (default value), the translation from max and min values is applied |
A list containing:
design |
the scaled design |
min |
the vector of minimal bounds that has been used |
max |
the vector of maximal bounds that has been used |
uniformize |
the value of this boolean argument |
InitialDesign |
the starting design |
B. Iooss
d <- 2 n <- 100 x <- matrix(rnorm(d*n), ncol=d) xscale1 <- scaleDesign(x, uniformize=FALSE) xscale2 <- scaleDesign(x, uniformize=TRUE) par(mfrow=c(1,2)) plot(xscale1$design) ; plot(xscale2$design)d <- 2 n <- 100 x <- matrix(rnorm(d*n), ncol=d) xscale1 <- scaleDesign(x, uniformize=FALSE) xscale2 <- scaleDesign(x, uniformize=TRUE) par(mfrow=c(1,2)) plot(xscale1$design) ; plot(xscale2$design)
Space-Filling Designs based on Strauss process
straussDesign(n,dimension, RND, alpha=0.5, repulsion=0.001, NMC=1000, constraints1D=0, repulsion1D=0.0001, seed=NULL)straussDesign(n,dimension, RND, alpha=0.5, repulsion=0.001, NMC=1000, constraints1D=0, repulsion1D=0.0001, seed=NULL)
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 |
Strauss designs are Space-Filling designs initially defined from Strauss process:
where is is the number of pairs of points of the design that are separated by a distance no greater than the radius of interaction RND, is the normalizing constant and 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 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:
with a potentiel defined by:
where , if 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 . The case alpha=0 corresponds to Strauss process (0-1 potential).
- repulsion is equal to the parameter of the Strauss process. Note that 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 .
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 |
J. Franco
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/.
## 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))## 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))
Computes the quantile of a uniformity test at a given significance level (see available tests and levels below).
unif.test.quantile(type, n, alpha)unif.test.quantile(type, n, alpha)
type |
a character indicating which test is used. The choices are the following: "greenwood", "qm" (for Quesenberry-Miller), "ks" (Kolmogorov-Smirnov), "cvm" (Cramer-Von Mises) and "V" (D+ + D- from Kolmogorov-Smirnov). |
n |
an integer equal to the sample size. |
alpha |
a real number equal to significance level. At present stage, only four values are available: 0.1, 0.05, 0.025 and 0.01. |
Modified statistics are used. For alpha = 0.05, the quantile is (see D Agostino and Stephens, 1986, section 4.4.): 1.358/(sqrt(n) + 0.12 + 0.11/sqrt(n)) for Kolmogorov-Smirnov and 0.461/(1+1/n) + 0.4/n - 0.6/n^2 for Cramer-von Mises. When the design size is < 20, the corrected value seems to be a good approximation, but the non asymptotical value should be preferred.
A real number equal to the quantile of the specified test at significance level alpha for n observations.
O. Roustant
D Agostino R.B., Stephens M.A. (1986), Goodness-of-fit techniques, CRC Press, New York.
unif.test.statistic, rss2d, rss3d
Computes the statistic of a uniformity test (see available tests below).
unif.test.statistic(x, type, transform=NULL)unif.test.statistic(x, type, transform=NULL)
x |
a vector containing the sample values. |
type |
a character indicating which test is used. The choices are the following: "greenwood", "qm" (for Quesenberry-Miller), "ks" (Kolmogorov-Smirnov), "cvm" (Cramer-Von Mises) and "V" (D+ + D- from Kolmogorov-Smirnov). |
transform |
an optional character indicating what type of transformation should be applied before testing uniformity. Default is NULL. |
A real number equal to the statistic of the specified test.
O. Roustant
D Agostino R.B., Stephens M.A. (1986), Goodness-of-fit techniques, CRC Press, New York.
This function unscales the values of a scaled design (values in [0,1]). The unscaling can be made by the inverse Rosenblatt transformation (by applying the empirical quantile function given by another design) or by translating the design from maximum and minimum values (given for each variable).
unscaleDesign(design, min=NULL, max=NULL, uniformize=FALSE, InitialDesign=NULL)unscaleDesign(design, min=NULL, max=NULL, uniformize=FALSE, InitialDesign=NULL)
design |
a matrix (or a data.frame) corresponding to the design of experiments to unscale |
min |
the vector of minimal bounds of each design variable |
max |
the vector of maximal bounds of each design variable |
uniformize |
boolean: TRUE to use the inverse Rosenblatt transformation (the min and max vectors are useless in this case). If FALSE (default value), the translation from max and min values is applied |
InitialDesign |
If the inverse Rosenblatt transformation is applied (uniformize = TRUE): a matrix (or a data.frame) corresponding to the design which gives the empirical quantiles |
A list containing:
design |
the unscaled design |
min |
the vector of minimal bounds that has been used |
max |
the vector of maximal bounds that has been used |
uniformize |
the value of this boolean argument |
B. Iooss
d <- 2 n <- 100 x <- matrix(rnorm(d*n), ncol=d) xscale <- scaleDesign(x, uniformize=TRUE) xunscale1 <- unscaleDesign(xscale$design, uniformize=TRUE, InitialDesign=x) xunscale2 <- unscaleDesign(xscale$design, min=c(min(x[,1]), min(x[,2])), max = c(max(x[,1]), max(x[,2]))) par(mfrow=c(2,2)) plot(x) ; plot(xscale$design) plot(xunscale1$design) ; plot(xunscale2$design)d <- 2 n <- 100 x <- matrix(rnorm(d*n), ncol=d) xscale <- scaleDesign(x, uniformize=TRUE) xunscale1 <- unscaleDesign(xscale$design, uniformize=TRUE, InitialDesign=x) xunscale2 <- unscaleDesign(xscale$design, min=c(min(x[,1]), min(x[,2])), max = c(max(x[,1]), max(x[,2]))) par(mfrow=c(2,2)) plot(x) ; plot(xscale$design) plot(xunscale1$design) ; plot(xunscale2$design)
The WSP (Wooton, Sergent, Phan-Tan-Luu) algorithm is an iterative algorithm based on suppression of some experiments from an initial design in each step. WSP leads to a space filling design
wspDesign(design, dmin, init = "center")wspDesign(design, dmin, init = "center")
design |
a matrix (or a data.frame) corresponding to the design of experiments |
dmin |
a minimum bound for mindist value of the final design |
init |
defines the initialization point (input coordinates) of the algorithm: "center" (default value) takes the central point of the domain "random" takes a random point inside the domain |
WSP enables to create a design D which is such that mindist(D)>dmin. However, it cannot assess the number of experiments. Similarly to straussDesign function, WSP is a powerful algorithm to construct space filling designs in high dimension
A list containing:
InitialDesign |
the starting design |
dmin |
minimum bound for mindist value of the final design |
design |
the matrix of the final design |
ResidualDesign |
the matrix of the residual design (points of InitialDesign not in design) |
G. Damblin & B. Iooss
J. Santiago, M. Claeys-Bruno, M.Sergent (2012). Construction of space filling designs using WSP algorithm for high dimensional spaces, Chenometrics and Intelligent Laboratory Systems, 113:26-31.
dimension <- 2 n <- 100 X <- matrix(runif(n*dimension), n, dimension) m <- wspDesign(X, 0.1) plot(m$design) xDRDN(m, letter = "T", dgts = 2, range = c(-10, 10))dimension <- 2 n <- 100 X <- matrix(runif(n*dimension), n, dimension) m <- wspDesign(X, 0.1) plot(m$design) xDRDN(m, letter = "T", dgts = 2, range = c(-10, 10))
Extract a design contained in a list (i.e. with a item), adjust the range, give it dimnames and finally round the values to a certain number of digits. Colnames will look like (A,B,C), (X1,X2,X3), (X01,X02,X03), (X001,X002,X003).
xDRDN(obj, width = 1, letter = "X", dgts = NULL, range = NULL)xDRDN(obj, width = 1, letter = "X", dgts = NULL, range = NULL)
obj |
a list that contains a |
width |
the digit width in colnames (to write for instance X1, X01, X001). If 0, colnames are filled with capital and small letters (without letters |
letter |
the generic letter used in colnames |
dgts |
the number of digits to which the design is rounded |
range |
a vector |
A rounded matrix or a data.frame with appropriate dimnames and an adjusted range.
xDRDN(lhsDesign(5, 12)) xDRDN(lhsDesign(5, 12), width = 2, letter = "V", dgts = 2, range = c(-10, 10)) head(xDRDN(olhDesign(50, range = c(1,1)), width = 0, letter = "Z"), 3) head(xDRDN(olhDesign(51, range = c(1,1)), width = 0, letter = "Z"), 3)xDRDN(lhsDesign(5, 12)) xDRDN(lhsDesign(5, 12), width = 2, letter = "V", dgts = 2, range = c(-10, 10)) head(xDRDN(olhDesign(50, range = c(1,1)), width = 0, letter = "Z"), 3) head(xDRDN(olhDesign(51, range = c(1,1)), width = 0, letter = "Z"), 3)