Package 'LatticeDesign'

Sliced rotated sphere packing designs by enlarging a design

Description

Generates a sliced rotated sphere packing design by enlarging one rotated sphere packing design.

Usage

AdaptiveRSPD(p=2,n,w=100)

Arguments

p	Number of dimensions, must be an integer greater than one.
n	Number of points of the small design, must be a positive integer.
w	Number of rotation matrices to try.

Details

This function generates a small rotated sphere packing design and the candidate points for enlarging it.

Value

The value returned from the function is a list containing the following components:

Design	The generated design.
candidates	The candidate points to add.
generator	The generator matrix.
rotation	The rotation matrix.
delta	The value of parameter delta.
Theta	The value of parameter Theta.
l	The value of parameter l.
FillDistance	The fill distance of the design for the nonboundary region.

References

He, Xu (2018). "Sliced rotated sphere packing designs", Technometrics, 61(1): 66-76.

Examples

AdaptiveRSPD(p=2,n=50,w=100)

---

Continuous choices

Description

These data sets give the solutions found from the continuous optimization algorithm. Data sets CCs2, CCs3, CCs4, CCs5, CCs6, CCs7, and CCs8 give the choices in 2, 3, 4, 5, 6, 7, and 8 dimensions, respectively.

Usage

data(CCs2);
data(CCs3);
data(CCs4);
data(CCs5);
data(CCs6);
data(CCs7);
data(CCs8);

Format

Matrices containing the choices.

References

He, Xu (2024). "Efficient Kriging using interleaved lattice-based designs with low fill and high separation distance properties", SIAM/ASA Journal on Uncertainty Quantification, 12(4): 1113-1134.

---

Coprime mumbers

Description

Find all potive numbers that is coprime with n and lower than n/2.

Usage

CoprimeNumbers(n);

Arguments

n	A positive integer.

Details

This function finds all potive numbers that is coprime with n and lower than n/2.

Value

The value returned from the function gives the vector containing all potive numbers that are coprime with n and lower than n/2.

References

Xu He, Junpeng Gong, and Zhaohui Li (2025) "optimized and regularly repeated lattice-based Latin hypercube designs for large-scale computer experiment", <arXiv:2506.04582v1>

Examples

CoprimeNumbers(20);

---

Criterion value of a lattice-based Latin hypercube design

Description

Computes the criterion value of a lattice-based Latin hypercube design.

Usage

WSL(n,v);
WPL(n,v);
WDL(n,v);
bWSL(n,v);
bWFL(n,v);
bWDL(n,v);
WSL2(n,v);
WFL2(n,v);
WDL2(n,v);

Arguments

n	Number of points.
v	Generator vector of the lattice-based Latin hypercube design.

Details

These functions compute the criterion value of a lattice-based Latin hypercube design. WSL, WPL, and WDL give the wrap-around reciprocal separation distance, wrap-around projective separation distance, and wrap-around discrepancy, respectively, for lattice-based Latin hypercube designs. bWSL, bWFL, and bWDL give the bivariate wrap-around reciprocal separation distance, bivariate wrap-around fill distance, and bivariate wrap-around discrepancy, respectively, for lattice-based Latin hypercube designs. WSL2, WFL2, and WDL2 give the wrap-around reciprocal separation distance, wrap-around fill distance, and wrap-around discrepancy, respectively, for two-dimensional lattice-based Latin hypercube designs.

Value

The value returned from the function gives the criterion value of the lattice-based Latin hypercube design.

References

Xu He, Junpeng Gong, and Zhaohui Li (2025) "optimized and regularly repeated lattice-based Latin hypercube designs for large-scale computer experiment", <arXiv:2506.04582v1>

Examples

WSL(100,c(3,7,13));
WPL(100,c(3,7,13));
WDL(100,c(3,7,13));
WSL2(100,c(3,7));
WSL2(100,c(3,13));
WSL2(100,c(13,7));
WFL2(100,c(3,7));
WDL2(101,c(3,7));
bWSL(100,c(3,7,13));
bWFL(100,c(3,7,13));
bWDL(101,c(3,7,13));

---

Densest packing-based maximum projection designs

Description

Generates a densest packing-based maximum projection design.

Usage

DPMPD(p,n,rotation="magic",w=100)

Arguments

p	Number of dimensions, must be an integer greater than one and no higher than eight.
n	Number of points, must be an integer greater than one.
rotation	Optional, whether to use magic rotation matrices (for p=2,3,4,6,8, recommended) or random rotation matrices.
w	Number of rotation matrices to try.

Details

This function generates a densest packing-based maximum projection design in two to eight dimensions. For p=2,4,8 with rotation="magic", the designs are generated following the Biometrika paper "Lattice-based designs possessing quasi-optimal separation distance on all projections". For p=3,6 with rotation="magic", the designs are generated following the arXiv paper "Lattice-based designs with quasi-uniform projections". For other p or rotation!="magic", the designs are generated from random rotations.

Value

The value returned from the function is a list containing the following components:

Design	The generated design.
ProjectiveSeparationDistance	The projective separation distance of the generated design, from one-dimensional projections to the unprojected design.

References

He, Xu (2020). "Lattice-based designs possessing quasi-optimal separation distance on all projections", Biometrika, accepted, DOI:10.1093/biomet/asaa057.

He, Xu (2018). "Lattice-based designs with quasi-uniform projections", arXiv:1709.02062v2.

Examples

DPMPD(p=4,n=200,w=100)

---

Generator matrices of standard interleaved lattices, treating dimension permuted lattices as different lattices

Description

These data sets give the generator matrices of standard interleaved lattices, treating dimension permuted lattices as different lattices. Data sets GeneratorMatrices2, GeneratorMatrices3, GeneratorMatrices4, and GeneratorMatrices5 give the matrices in 2, 3, 4, and 5, dimensions, respectively.

Usage

data(GeneratorMatrices2);
data(GeneratorMatrices3);
data(GeneratorMatrices4);
data(GeneratorMatrices5);

Format

Matrices containing generator matrices.

References

He, Xu (2019). "Interleaved lattice-based maximin distance designs", Biometrika, 106(2): 453-464.

---

Generator matrices of standard interleaved lattices, treating dimension permuted lattices as the same lattice

Description

These data sets give the generator matrices of standard interleaved lattices, treating dimension permuted lattices as the same lattice. Data sets GMs2, GMs3, GMs4, GMs5, GMs6, GMs7, and GMs8 give the matrices in 2, 3, 4, 5, 6, 7, and 8 dimensions, respectively.

Usage

data(GMs2);
data(GMs3);
data(GMs4);
data(GMs5);
data(GMs6);
data(GMs7);
data(GMs8);

Format

Matrices containing generator matrices.

References

He, Xu (2017). "Interleaved lattice-based minimax distance designs", Biometrika, 104(3): 713-725.

---

Interleaved lattice-based fill and separation distance designs

Description

Generates an interleaved lattice-based designs with low fill and high separation distance properties.

Usage

InterleavedFillSepD(p,n,w=rep(1,p),pfrom=p,a=1/2,nmin=floor(n*.8),
  nmax=ceiling(n*1.2),coefF=-4,coefS=1,msC=0,NL=10,NP=100,NJ=10,NS=100);

Arguments

p	Number of dimensions.
n	Targeted number of points, must be an integer greater than one.
w	Optional, weights of the dimensions.
pfrom	Optional, number of dimensions designs are generated and supplemented from, no more than p and no more than 8.
a	Optional, translation parameter, with a=0 for uniform design and a=1 for pushing the points to the boundary.
nmin	Optional, minimal acceptable number of points, no less than n.
nmax	Optional, maximal acceptable number of points, no greater than n.
coefF	Optional, coefficient of r_F in the criterion.
coefS	Optional, coefficient of r_S in the criterion.
msC	Optional, maximal allowed sum of projuct weights for pairs of binary aliased dimensions.
NL	Optional, maximum number of lattices to try.
NP	Optional, maximum number of dimension permutations to try.
NJ	Optional, maximum number of discretization choices for each continuous lattice and s combination.
NS	Optional, maximum number of choices to supplement from.

Details

This function generate an interleaved lattice-based designs with low fill and high separation distance properties in p dimensions and around n points, following the algorithm provided in the paper "Efficient Kriging using interleaved lattice-based designs with low fill and high separation distance properties".

Value

The value returned from the function is a matrix containing the generated design. Remark that no qualified design might be found if (a) both nmax and nmin are to close to n, (b) both n and msC are small, or (c) both pfrom and msC are small while p-pfrom is big.

References

He, Xu (2024). "Efficient Kriging using interleaved lattice-based designs with low fill and high separation distance properties", SIAM/ASA Journal on Uncertainty Quantification, 12(4): 1113-1134.

Examples

InterleavedFillSepD(p=2,n=20);

---

Interleaved lattice-based maximin distance designs

Description

Generates an interleaved lattice-based maximin distance design.

Usage

InterleavedMaximinD(p,n,weight=rep(1,p));
InterleavedMaximinDAlg1(p,n,weight=rep(1,p));
InterleavedMaximinDAlg2(p,n,weight=rep(1,p));
InterleavedMaximinDAlg3(p,n,weight=rep(1,p));

Arguments

p	Number of dimensions, must be an integer greater than one.
n	Targeted number of points, must be an integer greater than one.
weight	Optional, the weights used in the distance measure, higher for more important variable.

Details

This function generates an interleaved lattice-based maximin distance design in p dimensions and at least n points, following the algorithms provided in the paper "Interleaved lattice-based maximin distance designs". Function InterleavedMaximinD uses the recommended algorithm provided in the paper. Functions InterleavedMaximinDAlg1, InterleavedMaximinDAlg2, and InterleavedMaximinDAlg3 use Algorithm 1, 2, and 3, respectively. For InterleavedMaximinDAlg1, p must be no greater than 5. For InterleavedMaximinDAlg3, p must be greater than 8.

Value

The value returned from the function is a list containing the following components:

Design	The generated design.
SeparationDistance	The separation distance of the generated design.
m	The actual number of points of the generated design.
DesignTransformed	The generated design that is transformed to the rectangular design space given the weights.
weight	The weight used in the distance measure, higher for more important variable.
s_vector	The numbers of distinct levels of the generated design.
L01	The base design.

References

He, Xu (2019). "Interleaved lattice-based maximin distance designs", Biometrika, 106(2): 453-464.

Examples

InterleavedMaximinD(p=3,n=10,weight=rep(1,3));
InterleavedMaximinDAlg1(p=3,n=10);
InterleavedMaximinDAlg2(p=6,n=10);
InterleavedMaximinDAlg3(p=9,n=257);

---

Interleaved lattice-based maximin distance designs for mixed continuous, ordinal, and binary variables

Description

Generates an interleaved lattice-based maximin distance design for mixed continuous, ordinal, and binary variables.

Usage

InterleavedMaximinDMixVars(p,n,discrete_dims,ordinal_levels,weight=rep(1,p));
ILMmDMixVarsAlg6(p,n,discrete_dims,ordinal_levels,weight=rep(1,p));
ILMmDMixVarsAlg8(p,n,discrete_dims,ordinal_levels,weight=rep(1,p),N=1,ReturnAll=0);
ILMmDMixVarsAlg9(p,n,discrete_dims,ordinal_levels,weight=rep(1,p),pfrom=8);

Arguments

p	Number of dimensions, must be an integer greater than one.
n	Targeted number of points, must be an integer greater than one.
discrete_dims	Index of discrete variables in 1:p.
ordinal_levels	Allowable levels of ordinal variables by a matrix.
weight	Optional, the weights used in the distance measure, higher for more important variable.
N	Number of results saved.
ReturnAll	Whether or not outputing all designs with the highest separation distance, regardless of their sample size.
pfrom	Number of dimensions to supplement from.

Details

These functions generate an interleaved lattice-based maximin distance design for mixed continuous, ordinal, and binary variables in p dimensions and at most n points, following the algorithms provided in the paper "Maximin distance designs for mixed continuous, ordinal, and binary variables". Function InterleavedMaximinDMixVars uses the recommended algorithm provided in the paper. Functions ILMmDMixVarsAlg6, ILMmDMixVarsAlg8, and ILMmDMixVarsAlg9 use Algorithms 6, 8, and 9, respectively. For ILMmDMixVarsAlg6, p must be no greater than 5. For ILMmDMixVarsAlg8, p must be no greater than 8.

Value

The value returned from the function is a list containing the following components:

Design	The generated design.
SeparationDistance	The separation distance of the generated design.
m	The actual number of points of the generated design.
DesignTransformed	The generated design that is transformed to the rectangular design space given the weights.
weight	The weight used in the distance measure, higher for more important variable.
s_vector	The numbers of distinct levels of the generated design.
L01	The base design.

References

Lan, Hui and He, Xu (2025). "maximin distance designs for mixed continuous, ordinal, and binary variables", arXiv:2507.23405v1.

Examples

ordinal_levels_inpaper <- list(
  Q1 = c(0, 1),
  Q2 = c(0, 0.5, 1),
  Q3 = c(0, 0.1, 0.3, 0.6, 1),
  Q4 = c(0, 0.25, 0.5, 0.75, 1),
  Q5 = c(0, 0.2, 0.3, 0.5, 0.7, 1),
  Q6 = c(0, 0.2, 0.4, 0.6, 0.8, 1),
  Q7 = c(0, 0.1, 0.3, 0.5, 0.7, 0.9, 1),
  Q8 = c(0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1)
)

discrete_dims <- c(2,4)
ordinal_levels <- list(
  v1 = ordinal_levels_inpaper$Q6,
  v2 = ordinal_levels_inpaper$Q7
)
res <- InterleavedMaximinDMixVars(p=4,n=50,discrete_dims,ordinal_levels,weight=(3/4)^(0:3));
res$Design

discrete_dims <- c()
ordinal_levels <- c()
res <- InterleavedMaximinDMixVars(p=3,n=50,discrete_dims,ordinal_levels,weight=rep(1,3));
res$Design

discrete_dims <- c(1,2,3)
ordinal_levels <- list(
  v1 = ordinal_levels_inpaper$Q8,
  v2 = ordinal_levels_inpaper$Q3,
  v3 = ordinal_levels_inpaper$Q8
)
res <- ILMmDMixVarsAlg6(p=3,n=30,discrete_dims,ordinal_levels);
res$Design

discrete_dims <- c(3,4,5)
ordinal_levels <- list(
  v1 = ordinal_levels_inpaper$Q1,
  v2 = ordinal_levels_inpaper$Q5,
  v3 = ordinal_levels_inpaper$Q8
)
res <- ILMmDMixVarsAlg8(p=5,n=40,discrete_dims,ordinal_levels);
res$Design

discrete_dims <- c(4:10)
ordinal_levels <- list(
  v1 = ordinal_levels_inpaper$Q1,
  v2 = ordinal_levels_inpaper$Q1,
  v3 = ordinal_levels_inpaper$Q1,
  v4 = ordinal_levels_inpaper$Q2,
  v5 = ordinal_levels_inpaper$Q3,
  v6 = ordinal_levels_inpaper$Q4,
  v7 = ordinal_levels_inpaper$Q5
)
res <- ILMmDMixVarsAlg9(p=10,n=60,discrete_dims,ordinal_levels,pfrom=5);
res$Design

---

Interleaved lattice-based minimax distance designs

Description

Generates an interleaved lattice-based minimax distance design.

Usage

InterleavedMinimaxD(p,n,maxdissimilarity=2*p);

Arguments

p	Number of dimensions, must be an integer between 2 and 8.
n	Targeted number of points, must be an integer greater than one.
maxdissimilarity	Optional, the maximum dissimilarity allowed for the number of levels.

Details

These functions generate an interleaved lattice-based minimax distance design in p dimensions and at most n points, following the algorithm provided in the paper "Interleaved lattice-based minimax distance designs".

Value

The value returned from the function is a list containing the following components:

Design	The generated design.
TargetFillDistance	The target fill distance, an estimate of the fill distance.
ActualSize	The actual number of points of the generated design.
s_vector	The numbers of distinct levels of the generated design.
L01	The base design.

References

He, Xu (2017). "Interleaved lattice-based minimax distance designs", Biometrika, 104(3): 713-725.

Examples

InterleavedMinimaxD(p=2,n=20);

---

Optimal generator vector for Lattice-based Latin hypercube designs

Description

Generates the optimal generator vector for lattice-based Latin hypercube designs.

Usage

LLHD(n,d,criterion="WS",T=1000,nstart=max(c(floor(T/(length(CoprimeNumbers(n))*d*5)),1)));

Arguments

n	Number of points, must be an integer greater than one.
d	Number of dimensions, must be an integer greater than one.
criterion	Optional, the criterion used in optimization, which can be "WS", "WP", "WD", "bWS", "bWF", or "bWD".
T	Optional, number of iterations.
nstart	Optional, number of random starts.

Details

This function generates the optimal generator vector for lattice-based Latin hypercube designs, following the algorithms provided in the paper "optimized and regularly repeated lattice-based Latin hypercube designs for large-scale computer experiment".

Value

The value returned from the function gives a generator vector for lattice-based Latin hypercube designs.

References

Xu He, Junpeng Gong, and Zhaohui Li (2025) "optimized and regularly repeated lattice-based Latin hypercube designs for large-scale computer experiment", <arXiv:2506.04582v1>

Examples

set.seed(85)
v <- LLHD(n=20,d=3,criterion="WS")
design <- LLHDpoints(n=20,v=v,delta=sample(x=1:20,size=3,replace=TRUE))
design

---

Points of lattice-based Latin hypercube designs

Description

Generates the points of a lattice-based Latin hypercube design.

Usage

LLHDpoints(n,v,delta);

Arguments

n	Number of points, must be an integer greater than one.
v	Generator vector.
delta	Translation parameter.

Details

This function gives the points of a lattice-based Latin hypercube design, following the formulas provided in the paper "optimized and regularly repeated lattice-based Latin hypercube designs for large-scale computer experiment".

Value

The value returned from the function gives the design matrix for the lattice-based Latin hypercube design.

References

Xu He, Junpeng Gong, and Zhaohui Li (2025) "optimized and regularly repeated lattice-based Latin hypercube designs for large-scale computer experiment", <arXiv:2506.04582v1>

Examples

set.seed(85)
v <- LLHD(n=20,d=3,criterion="WS")
design <- LLHDpoints(n=20,v=v,delta=sample(x=1:20,size=3,replace=TRUE))
design

---

Vertexes of a polytope giving halfspace definition

Description

Computes the radius, widths, and vertexes of a polytope giving halfspace definition. The program is a R shell of LRS (v.5.1a with lrsmp.h), a reverse search vertex enumeration program/CH package in C which is developed by David Avis <http://cgm.cs.mcgill.ca/~avis/C/lrs.html>. Consider the problem of Ax<=b, where A is an n*p matrix, x is a p-vector, and b is an n-vector. Please make sure that the solution of x is nonempty and bounded. Then the nonequalities give the halfspace definition of a polytope. Also make sure that A and b are rational numbers.

Usage

LRS(numerator,denominator);

Arguments

numerator	The numerators of cbind(b,A), an n*(p+1) matrix of integer numbers.
denominator	The denominators of cbind(b,A), an n*(p+1) matrix of integer numbers.

Details

This function computes the radius, widths, and vertexes of a polytope giving halfspace definition. It is used in constructing interleaved lattice-based minimax distance designs. Currently only tested when the maximum values of numerators and denominators are below 2^20. If the nonequalities are not defined by rational numbers, round-up to small rational numbers is needed before calling the function. The computation is slow for large p but very fast for slow p. Avoid redundant nonequalities may accelerate the calculation.

Value

The value returned from the function is a list containing the following components:

Radius	The maximum L2 distance of vertexes to the origin.
MaxValue	The maximum k-dimensional value of the vertexes, for k from 1 to p.
Vertexes	The vertexes of the polytope.

References

Avis, David. LRS, http://cgm.cs.mcgill.ca/~avis/C/lrs.html.

He, Xu (2017). "Interleaved lattice-based minimax distance designs", Biometrika, 104(3): 713-725.

See Also

InterleavedMinimaxD.

Examples

num = matrix(0,5,3)
den = matrix(1,5,3)
num[1,2] = -1;  den[1,2] = 2; 
num[1,1] = 1;   den[1,1] = 8;
num[2,3] = -1;
num[2,1] = 1;   den[2,1] = 2;
num[3,2] = -1;  den[3,2] = 4;
num[3,3] = -1;  den[3,3] = 2;
num[3,1] = 5;   den[3,1] = 32;
num[4,2] = 1; 
num[4,1] = 0;
num[5,3] = 1;
num[5,1] = 0; 
LRS(num,den)

---

Projective separation distance of a design

Description

Computes the projective separation distance of a design.

Usage

ProjSepD(design);

Arguments

design

The experimental design, must be a matrix whose rows indicate experimental runs.

Details

This function computes the squared projective separation distance of a design.

Value

The value returned from the function gives the squared one-dimensional, two-dimensional, ..., (p-1)-dimensional projective separation distances, and the unprojected separation distance, where p is the number of dimensions of the design.

References

He, Xu (2020). "Lattice-based designs possessing quasi-optimal separation distance on all projections", Biometrika, accepted, DOI:10.1093/biomet/asaa057.

Examples

design = rbind(1:3,c(41,1.2,1.3),c(5.4,5.48,5.7),c(4.3,2.3,2));
ProjSepD(design);

---

Rotated sphere packing designs

Description

Generates a rotated sphere packing design.

Usage

RSPD(p=2,n,rotation="magic",w=100)

Arguments

p	Number of dimensions, must be an integer greater than one.
n	Number of points, must be a positive integer.
rotation	Optional, whether to use the magic rotation matrix (for p=2, recommended) or random rotation matrices.
w	Number of rotation matrices to try, fixed to 1 when p=2 and rotation="magic".

Details

This function generates a rotated sphere packing design.

Value

The value returned from the function is a list containing the following components:

Design	The generated design.
generator	The generator matrix.
rotation	The rotation matrix.
delta	The value of parameter delta.
Theta	The value of parameter Theta.
l	The value of parameter l.
FillDistance	The fill distance of the design for the nonboundary region.

References

He, Xu (2017). "Rotated sphere packing designs", Journal of the American Statistical Association, 112(520): 1612-1622.

Examples

RSPD(p=2,n=50,rotation="magic",w=100)

---

Sliced rotated sphere packing designs by partitioning a design

Description

Generates a sliced rotated sphere packing design by partitioning one rotated sphere packing design.

Usage

SlicedRSPD(p=2,n,rotation="magic",w=100)

Arguments

p	Number of dimensions, must be an integer greater than one.
n	Number of points, must be a positive integer.
rotation	Optional, whether to use magic rotation matrices (for p=2, recommended) or random rotation matrices.
w	Number of rotation matrices to try.

Details

This function generates a rotated sphere packing design and the slice indexes of points.

Value

The value returned from the function is a list containing the following components:

Design	The generated design.
slices	The slice indexes of design points.
generator	The generator matrix.
rotation	The rotation matrix.
delta	The value of parameter delta.
Theta	The value of parameter Theta.
l	The value of parameter l.
FillDistance	The fill distance of the design for the nonboundary region.

References

He, Xu (2019). "Sliced rotated sphere packing designs", Technometrics, 61(1): 66-76.

Examples

SlicedRSPD(p=2,n=50,rotation="magic",w=100)

---