Calculate the Average Absolute Correlation

Description

AvgAbsCor returns the average absolute correlation of a matrix

Usage

AvgAbsCor(X)

Arguments

Value

If all inputs are logical, then the output will be a positive number indicating the average absolute correlation of input matrix. average absolute correlation = \frac{2 \sum_{i=1}^{k-1} \sum_{j=i+1}^{k}|q_{ij}|}{k(k-1)}

References

Georgiou, S. D. (2009) Orthogonal Latin hypercube designs from generalized orthogonal designs. Journal of Statistical Planning and Inference, 139, 1530-1540.

Examples

#create a toy LHD with 5 rows and 3 columns
toy=rLHD(n=5,k=3);toy

#Calculate the average absolute correlation of toy
AvgAbsCor(X=toy)

Fast Maximin Distance LHD

Description

FastMmLHD returns a n by k maximin distance LHD matrix generated by the construction method of Wang, L., Xiao, Q., and Xu, H. (2018)

Usage

FastMmLHD(n, k, method = "manhattan", t1 = 10)

Arguments

Value

If all inputs are logical, then the output will be a n by k maximin distance LHD under under the maximin L_1 distance criterion..

References

Wang, L., Xiao, Q., and Xu, H. (2018) Optimal maximin $L_1$-distance Latin hypercube designs based on good lattice point designs. The Annals of Statistics, 46(6B), 3741-3766.

Examples

#n by n design when 2n+1 is prime
try=FastMmLHD(8,8)
try
phi_p(try)   #calculate the phi_p of "try".

#n by n design when n+1 is prime
try2=FastMmLHD(12,12)
try2
phi_p(try2)   #calculate the phi_p of "try2".

#n by n-1 design when n is prime
try3=FastMmLHD(7,6)
try3
phi_p(try3)   #calculate the phi_p of "try3".

#General cases
try4=FastMmLHD(24,8)
try4
phi_p(try4)   #calculate the phi_p of "try4".

Genetic Algorithm for LHD

Description

GA returns a n by k LHD matrix generated by genetic algorithm (GA)

Usage

GA(
  n,
  k,
  m = 10,
  N = 10,
  pmut = 1/(k - 1),
  OC = "phi_p",
  p = 15,
  q = 1,
  maxtime = 5
)

Arguments

Value

If all inputs are logical, then the output will be a n by k LHD.

References

Liefvendahl, M., and Stocki, R. (2006) A study on algorithms for optimization of Latin hypercubes. Journal of Statistical Planning and Inference, 136, 3231-3247.

Examples

#generate a 5 by 3 maximin distance LHD with the default setting
try=GA(n=5,k=3)
try
phi_p(try)   #calculate the phi_p of "try".

#Another example
#generate a 8 by 4 nearly orthogonal LHD
try2=GA(n=8,k=4,OC="AvgAbsCor")
try2
AvgAbsCor(try2)  #calculate the average absolute correlation.

Good Lattice Point Design

Description

GLP returns a n by k design matrix generated by good lattice point (GLP)

Usage

GLP(n, k, h = sample(seq(from = 1, to = (n - 1)), k))

Arguments

Value

If all inputs are logical, then the output will be a n by k GLP design matrix.

References

Korobov, A.N. (1959) The approximate computation of multiple integrals. Dokl. Akad. Nauk SSSR, 124, 1207-1210.

Examples

#generate a 5 by 3 GLP design with the default setting
try=GLP(n=5,k=3)
try

#Another example
#generate a 8 by 4 GLP design with given h vector
try2=GLP(n=8,k=4,h=c(1,3,5,7))
try2

Particle Swarm Optimization for LHD

Description

LaPSO returns a n by k LHD matrix generated by particle swarm optimization algorithm (PSO)

Usage

LaPSO(
  n,
  k,
  m = 10,
  N = 10,
  SameNumP = 0,
  SameNumG = n/4,
  p0 = 1/(k - 1),
  OC = "phi_p",
  p = 15,
  q = 1,
  maxtime = 5
)

Arguments

Value

If all inputs are logical, then the output will be a n by k LHD. Here are some general suggestions about the parameters:

• SameNumP is approximately n/2 when SameNumG is 0.

• SameNumG is approximately n/4 when SameNumP is 0.

• p0 * (k - 1) = 1 to 2 is often sufficient. So p0 = 1/(k - 1) to 2/(k - 1).

References

Chen, R.-B., Hsieh, D.-N., Hung, Y., and Wang, W. (2013) Optimizing Latin hypercube designs by particle swarm. Stat. Comput., 23, 663-676.

Examples

#generate a 5 by 3 maximin distance LHD with the default setting
try=LaPSO(n=5,k=3)
try
phi_p(try)   #calculate the phi_p of "try".

#Another example
#generate a 8 by 4 nearly orthogonal LHD
try2=LaPSO(n=8,k=4,OC="AvgAbsCor")
try2
AvgAbsCor(try2)  #calculate the average absolute correlation.

Calculate the Maximum Absolute Correlation

Description

MaxAbsCor returns the maximum absolute correlation of a matrix

Usage

MaxAbsCor(X)

Arguments

Value

If all inputs are logical, then the output will be a positive number indicating maximum absolute correlation. maximum absolute correlation = max_{ij} |q_{ij}|

References

Georgiou, S. D. (2009) Orthogonal Latin hypercube designs from generalized orthogonal designs. Journal of Statistical Planning and Inference, 139, 1530-1540.

Examples

#create a toy LHD with 5 rows and 3 columns
toy=rLHD(n=5,k=3);toy

#Calculate the maximum absolute correlation of toy
MaxAbsCor(X=toy)

Calculate the Maximum Projection Criterion

Description

MaxProCriterion returns the maximum projection criterion of an LHD

Usage

MaxProCriterion(X)

Arguments

Value

If all inputs are logical, then the output will be a positive number indicating maximum projection criterion. maximum projection criterion = \Bigg{ \frac{1}{{n \choose 2}} \sum_{i=1}^{n-1} \sum_{j=i+1}^{n} \frac{1}{\Pi_{l=1}^{k}(x_{il}-x_{jl})^2} \Bigg}^{1/k}

References

Joseph, V. R., Gul, E., and Ba, S. (2015) Maximum projection designs for computer experiments. Biometrika, 102, 371-380.

Examples

#create a toy LHD with 5 rows and 3 columns
toy=rLHD(n=5,k=3);toy

#Calculate the maximum projection criterion of toy
MaxProCriterion(X=toy)

Transfer an Orthogonal Array (OA) into an LHD

Description

OA2LHD transfers an OA into an LHD with corresponding size

Usage

OA2LHD(OA)

Arguments

Value

If the input is logical, then the output will be an LHD whose sizes are the same as input OA. The assumption is that the elements of OAs must be positive.

References

Tang, B. (1993) Orthogonal-array-based latin hypercubes. Journal of the Americal Statistical Association, 88, 1392-1397.

Examples

#create an OA(9,2,3,2)
OA=matrix(c(rep(1:3,each=3),rep(1:3,times=3)),ncol=2,nrow=9,byrow = FALSE);OA

#Transfer the "OA" above into a LHD according to Tang (1993)
tryOA=OA2LHD(OA)
OA;tryOA

Orthogonal-Array-Based Simulated Annealing

Description

OASA returns an LHD matrix generated by orthogonal-array-based simulated annealing algorithm (OASA)

Usage

OASA(
  OA,
  N = 10,
  T0 = 10,
  rate = 0.1,
  Tmin = 1,
  Imax = 5,
  OC = "phi_p",
  p = 15,
  q = 1,
  maxtime = 5
)

Arguments

Value

If all inputs are logical, then the output will be an LHD whose sizes are the same as input OA. The assumption is that the elements of OAs must be positive.

References

Leary, S., Bhaskar, A., and Keane, A. (2003) Optimal orthogonal-array-based latin hypercubes. Journal of Applied Statistics, 30, 585-598.

Examples

#create an OA(9,2,3,2)
OA=matrix(c(rep(1:3,each=3),rep(1:3,times=3)),ncol=2,nrow=9,byrow = FALSE)

#Use above "OA" as the input OA to generate a 9 by 2 maximin distance LHD
#with the default setting
try=OASA(OA=OA)
try
phi_p(try)   #calculate the phi_p of "try".

#Another example
#generate a 9 by 2 nearly orthogonal LHD
try2=OASA(OA=OA,OC="MaxAbsCor")
try2
MaxAbsCor(try2)  #calculate the maximum absolute correlation.

Orthogonal Latin Hypercube Design

Description

OLHD.B2001 returns a n by k orthogonal Latin hypercube design generated by the construction method of Butler (2001)

Usage

OLHD.B2001(n, k)

Arguments

Value

If all inputs are logical, then the output will be a n by k orthogonal LHD.

References

Butler, N.A. (2001) Optimal and orthogonal Latin hypercube designs for computer experiments. Biometrika, 88(3), 847-857.

Examples

#create an orthogonal LHD with n=11 and k=5
OLHD.B2001(n=11,k=5)

#create an orthogonal LHD with n=7 and k=6
OLHD.B2001(n=7,k=6)

Orthogonal Latin Hypercube Design

Description

OLHD.C2007 returns a 2^m+1 by m+{m-1 \choose 2} orthogonal Latin hypercube design generated by the construction method of Cioppa and Lucas (2007)

Usage

OLHD.C2007(m)

Arguments

Value

If all inputs are logical, then the output will be an orthogonal LHD with the following run size: n=2^m+1 and the following factor size: k=m+{m-1 \choose 2}.

References

Cioppa, T.M., and Lucas, T.W. (2007) Efficient nearly orthogonal and space-filling Latin hypercubes. Technometrics, 49(1), 45-55.

Examples

#create an orthogonal LHD with m=4. So n=2^m+1=17 and k=4+3=7
OLHD.C2007(m=4)

#create an orthogonal LHD with m=5. So n=2^m+1=33 and k=5+6=11
OLHD.C2007(m=5)

Orthogonal Latin Hypercube Design

Description

OLHD.L2009 returns a n^2 by 2fp orthogonal Latin hypercube design generated by the construction method of Lin et al. (2009)

Usage

OLHD.L2009(OLHD, OA)

Arguments

Value

If all inputs are logical, e,g. a n by p orthogonal Latin hypercube design and an OA(n^2,2f,n,2) orthogonal array, then the output will be an orthogonal Latin hypercube design with the following run size: n^2 and the following factor size: 2fp.

References

Lin, C.D., Mukerjee, R., and Tang, B. (2009) Construction of orthogonal and nearly orthogonal Latin hypercubes. Biometrika, 96(1), 243-247.

Examples

#create a 5 by 2 OLHD
OLHD=OLHD.C2007(m=2)

#create an OA(25,6,5,2)
OA=matrix(c(2,2,2,2,2,1,2,1,5,4,3,5,3,2,1,5,4,5,1,5,4,3,2,5,
4,1,3,5,2,3,1,2,3,4,5,2,1,3,5,2,4,3,1,1,1,1,1,1,4,3,2,1,5,5,
5,5,5,5,5,1,4,4,4,4,4,1,3,1,4,2,5,4,3,3,3,3,3,1,3,5,2,4,1,3,
3,4,5,1,2,2,5,4,3,2,1,5,2,3,4,5,1,2,2,5,3,1,4,4,1,4,2,5,3,4,
4,2,5,3,1,4,2,4,1,3,5,3,5,3,1,4,2,4,5,2,4,1,3,3,5,1,2,3,4,2,
4,5,1,2,3,2),ncol=6,nrow=25,byrow=TRUE)

#Construct a 25 by 12 OLHD
OLHD.L2009(OLHD,OA)

Orthogonal Latin Hypercube Design

Description

OLHD.S2010 returns a r2^{C+1}+1 or r2^{C+1} by 2^C orthogonal Latin hypercube design generated by the construction method of Sun et al. (2010)

Usage

OLHD.S2010(C, r, type = "odd")

Arguments

Value

If all inputs are logical, then the output will be an orthogonal LHD with the following run size: n=r2^{C+1}+1 or n=r2^{C+1} and the following factor size: k=2^C.

References

Sun, F., Liu, M.Q., and Lin, D.K. (2010) Construction of orthogonal Latin hypercube designs with flexible run sizes. Journal of Statistical Planning and Inference, 140(11), 3236-3242.

Examples

#create an orthogonal LHD with C=3, r=3, type="odd".
#So n=3*2^4+1=49 and k=2^3=8
OLHD.S2010(C=3,r=3,type="odd")

#create an orthogonal LHD with C=3, r=3, type="even".
#So n=3*2^4=48 and k=2^3=8
OLHD.S2010(C=3,r=3,type="even")

Orthogonal Latin Hypercube Design

Description

OLHD.Y1998 returns a 2^m+1 by 2m-2 orthogonal Latin hypercube design generated by the construction method of Ye (1998)

Usage

OLHD.Y1998(m)

Arguments

Value

If all inputs are logical, then the output will be an orthogonal LHD with the following run size: n=2^m+1 and the following factor size: k=2m-2.

References

Ye, K.Q. (1998) Orthogonal column Latin hypercubes and their application in computer experiments. Journal of the American Statistical Association, 93(444), 1430-1439.

Examples

#create an orthogonal LHD with m=3. So n=2^m+1=9 and k=2*m-2=4
OLHD.Y1998(m=3)

#create an orthogonal LHD with m=4. So n=2^m+1=17 and k=2*m-2=6
OLHD.Y1998(m=4)

Simulated Annealing for LHD

Description

SA returns a n by k LHD matrix generated by simulated annealing algorithm (SA)

Usage

SA(
  n,
  k,
  N = 10,
  T0 = 10,
  rate = 0.1,
  Tmin = 1,
  Imax = 5,
  OC = "phi_p",
  p = 15,
  q = 1,
  maxtime = 5
)

Arguments

Value

If all inputs are logical, then the output will be a n by k LHD.

References

Morris, M.D., and Mitchell, T.J. (1995) Exploratory designs for computer experiments. Journal of Statistical Planning and Inference, 43, 381-402.

Examples

#generate a 5 by 3 maximin distance LHD with the default setting
try=SA(n=5,k=3)
try
phi_p(try)   #calculate the phi_p of "try".

#Another example
#generate a 8 by 4 nearly orthogonal LHD
try2=SA(n=8,k=4,OC="AvgAbsCor")
try2
AvgAbsCor(try2)  #calculate the average absolute correlation.

Simulated Annealing for LHD with Multi-objective Optimization Approach

Description

SA2008 returns a n by k LHD matrix generated by simulated annealing algorithm with multi-objective optimization approach

Usage

SA2008(
  n,
  k,
  N = 10,
  T0 = 10,
  rate = 0.1,
  Tmin = 1,
  Imax = 5,
  OC = "phi_p",
  p = 15,
  q = 1,
  maxtime = 5
)

Arguments

Value

If all inputs are logical, then the output will be a n by k LHD. This modified simulated annealing algorithm reduces columnwise correlations and maximizes minimum distance between design points simultaneously, with a cost of more computational complexity.

References

Joseph, V.R., and Hung, Y. (2008) Orthogonal-maximin Latin hypercube designs. Statistica Sinica, 18, 171-186.

Examples

#generate a 5 by 3 maximin distance LHD with the default setting
try=SA2008(n=5,k=3)
try
phi_p(try)   #calculate the phi_p of "try".

#Another example
#generate a 8 by 4 nearly orthogonal LHD
try2=SA2008(n=8,k=4,OC="AvgAbsCor")
try2
AvgAbsCor(try2)  #calculate the average absolute correlation.

Sliced Latin Hypercube Design (SLHD)

Description

SLHD returns a n by k LHD matrix generated by improved two-stage algorithm

Usage

SLHD(
  n,
  k,
  t = 1,
  N = 10,
  T0 = 10,
  rate = 0.1,
  Tmin = 1,
  Imax = 3,
  OC = "phi_p",
  p = 15,
  q = 1,
  stage2 = FALSE,
  maxtime = 5
)

Arguments

Value

If all inputs are logical, then the output will be a n by k LHD. As mentioned from the original paper, the first stage plays a much more important role since it optimizes the slice level. More resources should be given to the first stage if computational budgets are limited. Let m=n/t, where m is the number of rows for each slice, if (m)^k >> n, the second stage becomes optional. That is the reason why we add a stage2 parameter to let users decide if the second stage is needed.

References

Ba, S., Myers, W.R., and Brenneman, W.A. (2015) Optimal Sliced Latin Hypercube Designs. Technometrics, 57, 479-487.

Examples

#generate a 5 by 3 maximin distance LHD with the default setting
try=SLHD(n=5,k=3)
try
phi_p(try)   #calculate the phi_p of "try".

#generate a 5 by 3 maximin distance LHD with stage II
#let stage2=TRUE and other input are the same as above
try2=SLHD(n=5,k=3,stage2=TRUE)
try2
phi_p(try2)   #calculate the phi_p of "try2".

#Another example
#generate a 8 by 4 nearly orthogonal LHD
try3=SLHD(n=8,k=4,OC="AvgAbsCor",stage2=TRUE)
try3
AvgAbsCor(try3)  #calculate the average absolute correlation.

Williams Transformation

Description

WT returns a matrix after implementing the Williams transformation

Usage

WT(X, baseline = 1)

Arguments

Value

If all inputs are logical, then the output will be a matrix whose sizes are the same as input matrix.

References

Williams, E. J. (1949) Experimental designs balanced for the estimation of residual effects of treatments. Australian Journal of Chemistry, 2, 149-168.

Examples

#create a toy LHD with 5 rows and 3 columns
toy=rLHD(n=5,k=3);toy
toy2=toy-1;toy2  #make elements of "toy" become 0,1,2,3,4

#Implementing Williams transformation on both toy and toy2:
#The result shows that "WT" function is able to detect the
#elements of input matrix and make adjustments.
WT(toy)
WT(toy2)

#Change the baseline
WT(toy,baseline=5)
WT(toy,baseline=10)

Calculate the Inter-site Distance

Description

dij returns the inter-site distance of two design points of an LHD

Usage

dij(X, i, j, q = 1)

Arguments

Value

If all inputs are logical, then the output will be a positive number indicating the distance. dij = \left\{ \sum_{k=1}^{m} \vert x_{ik}-x_{jk}\vert ^q \right\}^{1/q}

Examples

#create a toy LHD with 5 rows and 3 columns
toy=rLHD(n=5,k=3);toy

#Calculate the inter-site distance of the 2nd and the 4th row of toy (with default q)
dij(X=toy,i=2,j=4)

#Calculate the inter-site distance of the 2nd and the 4th row of toy (with q=2)
dij(X=toy,i=2,j=4,q=2)

Exchange two random elements in a matrix

Description

exchange returns a new matrix by switching two randomly selected elements from a user-defined matrix

Usage

exchange(X, j, type = "col")

Arguments

Value

If all inputs are logical, then the output will be a new design matrix after the exchange.

Examples

#create a toy LHD with 5 rows and 3 columns
toy=rLHD(n=5,k=3);toy

#Choose the first column of toy and exchange two randomly selected elements.
try.col=exchange(X=toy,j=1,type="col")
toy;try.col

#Choose the first row of toy and exchange two randomly selected elements.
try.row=exchange(X=toy,j=1,type="row")
toy;try.row

Calculate the phi_p Criterion

Description

phi_p returns the phi_p criterion of an LHD

Usage

phi_p(X, p = 15, q = 1)

Arguments

Value

If all inputs are logical, then the output will be a positive number indicating phi_p. \phi_p = (\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}dij^{-p})^{1/p}, where dij = \left\{ \sum_{k=1}^{m} \vert x_{ik}-x_{jk}\vert ^q \right\}^{1/q}

References

Jin, R., Chen, W., and Sudjianto, A. (2005) An efficient algorithm for constructing optimal design of computer experiments. Journal of Statistical Planning and Inference, 134, 268-287.

Examples

#create a toy LHD with 5 rows and 3 columns
toy=rLHD(n=5,k=3);toy

#Calculate the phi_p criterion of toy with default setting
phi_p(X=toy)

#Calculate the phi_p criterion of toy with p=50 and q=2
phi_p(X=toy,p=50,q=2)

Generate a random Latin Hypercube Design (LHD)

Description

rLHD returns a random n by k Latin hypercube design matrix

Usage

rLHD(n, k)

Arguments

Value

If all inputs are positive integer, then the output will be a n by k design matrix.

Examples

#create a toy LHD with 5 rows and 3 columns
toy=rLHD(n=5,k=3);toy

#another example with 9 rows and 2 columns
rLHD(9,2)