LatticeDesign package

Description

Generate lattice-based space-filling designs with fill or separation distance properties. These include interleaved lattice-based minimax distance designs, interleaved lattice-based maximin distance designs, interleaved lattice-based designs with both fill and separation distance properties, (sliced) rotated sphere packing designs, and densest packing-based maximum projections designs.

Details

Important functions in this package are: InterleavedMinimaxD generates an interleaved lattice-based minimax distance design. InterleavedMaximinD generates an interleaved lattice-based maximin distance design. InterleavedFillSepD generates an interleaved lattice-based design with both fill and separation distance properties. DPMPD generates a densest packing-based maximum projection design. RSPD generates a rotated sphere packing design. SlicedRSPD generates a sliced rotated sphere packing design by partitioning one rotated sphere packing design. AdaptiveRSPD generates a sliced rotated sphere packing design by enlarging one rotated sphere packing design.

All those functions generate space-filling designs with fill or separation distance properties. Such designs are useful for accurate emulation of computer experiments, fitting nonparametric models and resource allocation. They are constructed from lattices, i.e., sets of points with group structures.

RSPD and DPMPD generate designs in two to eight dimensions with both unprojected and projective distance properties. Such designs are desirable when possibly the output value is insensitive to some variables. DPMPD can be seen as an upgrade of RSPD using new magic rotation matrices. Another distinction is that RSPD generates designs with better unprojected fill distance for nonboundary regions while DPMPD generates designs with better unprojected separation distance. RSPD and DPMPD construct designs by rescaling, rotating, translating and extracting the points of the lattice with asymptotically optimal fill and separation distance, respectively.

SlicedRSPD and AdaptiveRSPD generate sliced rotated sphere packing designs, i.e., a rotated sphere packing design that can be partitioned into several smaller rotated sphere packing designs. SlicedRSPD partitions one rotated sphere packing design. The generated designs are useful for computer experiments with a categorical variable, computer experiments from multiply resources and model validation. Alternatively, AdaptiveRSPD enlarges a smaller rotated sphere packing design, which is useful for adaptive design of computer experiments.

InterleavedMinimaxD generates designs in two to eight dimensions with low fill distance. InterleavedMaximinD generates designs with high separation distance. InterleavedFillSepD generates designs with both properties. InterleavedMaximinD and InterleavedFillSepD allow users to specify the relative importance of variables and is applicable to problems with any number of variables. These designs are useful for accurate emulation of computer experiments when the variables are almost equally important in predicting the output value or relatively accurate a priori guess on the variable importance is available. On the other hand, these designs are poor in projective distance properties and are thus not recommended when the output value is insensitive to many unknown variables.

Author(s)

Maintainer: Xu He <hexu@amss.ac.cn>

References

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

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

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

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

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

He, Xu (2021). "Lattice-based designs possessing quasi-optimal separation distance on all projections", Biometrika, 108(2): 443-454.

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.

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

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:

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.

Densest packing-based maximum projection designs

Description

Generates a densest packing-based maximum projection design.

Usage

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

Arguments

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:

References

He, Xu (2021). "Lattice-based designs possessing quasi-optimal separation distance on all projections", Biometrika, 108(2): 443-454.

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 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.

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.

Interleaved lattice-based fill and separation distance designs

Description

Generates an interleaved lattice-based design 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

Details

This function generates an interleaved lattice-based design 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 is much bigger than pfrom.

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

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:

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 minimax distance designs

Description

Generates an interleaved lattice-based minimax distance design.

Usage

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

Arguments

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:

References

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

Examples

InterleavedMinimaxD(p=2,n=20);

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. 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

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:

References

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

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 (2021). "Lattice-based designs possessing quasi-optimal separation distance on all projections", Biometrika, 108(2): 443-454.

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

Details

This function generates a rotated sphere packing design.

Value

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

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

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:

References

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

Examples

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