Package 'lhs' reference manual

Title:	Latin Hypercube Samples
Description:	Provides a number of methods for creating and augmenting Latin Hypercube Samples and Orthogonal Array Latin Hypercube Samples.
Authors:	Rob Carnell [aut, cre]
Maintainer:	Rob Carnell <[email protected]>
License:	GPL-3
Version:	1.2.0
Built:	2024-10-28 05:21:59 UTC
Source:	https://github.com/bertcarnell/lhs

Augment a Latin Hypercube Design

Description

Augments an existing Latin Hypercube Sample, adding points to the design, while maintaining the latin properties of the design.

Usage

augmentLHS(lhs, m = 1)
augmentLHS(lhs, m = 1)

Arguments

`lhs`	The Latin Hypercube Design to which points are to be added. Contains an existing latin hypercube design with a number of rows equal to the points in the design (simulations) and a number of columns equal to the number of variables (parameters). The values of each cell must be between 0 and 1 and uniformly distributed
`m`	The number of additional points to add to matrix `lhs`

Details

Augments an existing Latin Hypercube Sample, adding points to the design, while maintaining the latin properties of the design. Augmentation is perfomed in a random manner.

The algorithm used by this function has the following steps. First, create a new matrix to hold the candidate points after the design has been re-partitioned into $(n+m)^{2}$ cells, where n is number of points in the original lhs matrix. Then randomly sweep through each column (1...k) in the repartitioned design to find the missing cells. For each column (variable), randomly search for an empty row, generate a random value that fits in that row, record the value in the new matrix. The new matrix can contain more filled cells than m unles $m = 2n$ , in which case the new matrix will contain exactly m filled cells. Finally, keep only the first m rows of the new matrix. It is guaranteed to have m full rows in the new matrix. The deleted rows are partially full. The additional candidate points are selected randomly due to the random search for empty cells.

Value

An n by k Latin Hypercube Sample matrix with values uniformly distributed on [0,1]

Author(s)

Rob Carnell

References

Stein, M. (1987) Large Sample Properties of Simulations Using Latin Hypercube Sampling. Technometrics. 29, 143–151.

Examples

set.seed(1234)
a <- randomLHS(4,3)
b <- augmentLHS(a, 2)
set.seed(1234)
a <- randomLHS(4,3)
b <- augmentLHS(a, 2)

Transformed Latin hypercube with a multivariate distribution

Description

A method to create a transformed Latin Hypercube sample where the marginal distributions can be correlated according to an arbitrary set of criteria contained in a minimized cost function

Usage

correlatedLHS(
  lhs,
  marginal_transform_function,
  cost_function,
  debug = FALSE,
  maxiter = 10000,
  ...
)
correlatedLHS(
  lhs,
  marginal_transform_function,
  cost_function,
  debug = FALSE,
  maxiter = 10000,
  ...
)

Arguments

`lhs`	a Latin hypercube sample that is uniformly distributed on the margins
`marginal_transform_function`	a function that takes Latin hypercube sample as the first argument and other passed-through variables as desired. `...` must be passed as a argument. For example, `f <- function(W, second_argument, ...)`. Must return a `matrix` or `data.frame`
`cost_function`	a function that takes a transformed Latin hypercube sample as the first argument and other passed-through variables as desired. `...` must be passed as a argument. For example, `f <- function(W, second_argument, ...)`
`debug`	Should debug messages be printed. Causes cost function output and iterations to be printed to aid in setting the maximum number of iterations
`maxiter`	the maximum number of iterations. The algorithm proceeds by swapping one variable of two points at a time. Each swap is an iteration.
`...`	Additional arguments to be passed through to the `marginal_transform_function` and `cost_function`

Value

a list of the Latin hypercube with uniform margins, the transformed Latin hypercube, and the final cost

Examples

correlatedLHS(lhs::randomLHS(30, 2),
  marginal_transform_function = function(W, ...) {
    W[,1] <- qnorm(W[,1], 1, 3)
    W[,2] <- qexp(W[,2], 2)
    return(W)
  },
  cost_function = function(W, ...) {
    (cor(W[,1], W[,2]) - 0.5)^2
  },
  debug = FALSE,
  maxiter = 1000)
correlatedLHS(lhs::randomLHS(30, 2),
  marginal_transform_function = function(W, ...) {
    W[,1] <- qnorm(W[,1], 1, 3)
    W[,2] <- qexp(W[,2], 2)
    return(W)
  },
  cost_function = function(W, ...) {
    (cor(W[,1], W[,2]) - 0.5)^2
  },
  debug = FALSE,
  maxiter = 1000)

Create a Galois field

Description

Create a Galois field

Usage

create_galois_field(q)
create_galois_field(q)

Arguments

`q`	The order of the Galois Field q = p^n

Value

a GaloisField object containing

n: q = p^n
p: The prime modulus of the field q=p^n
q: The order of the Galois Field q = p^n. q must be a prime power.
xton: coefficients of the characteristic polynomial where the first coefficient is on $x^0$, the second is on $x^1$ and so on
inv: An index for which row of poly (zero based) is the multiplicative inverse of this row. An NA indicates that this row of poly has no inverse. e.g. c(3, 4) means that row 4=3+1 is the inverse of row 1 and row 5=4+1 is the inverse of row 2
neg: An index for which row of poly (zero based) is the negative or additive inverse of this row. An NA indicates that this row of poly has no negative. e.g. c(3, 4) means that row 4=3+1 is the negative of row 1 and row 5=4+1 is the negative of row 2
root: An index for which row of poly (zero based) is the square root of this row. An NA indicates that this row of poly has no square root. e.g. c(3, 4) means that row 4=3+1 is the square root of row 1 and row 5=4+1 is the square root of row 2
plus: sum table of the Galois Field
times: multiplication table of the Galois Field
poly: rows are polynomials of the Galois Field where the entries are the coefficients of the polynomial where the first coefficient is on $x^0$, the second is on $x^1$ and so on

Examples

gf <- create_galois_field(4);
gf <- create_galois_field(4);

Create an orthogonal array Latin hypercube

Description

Create an orthogonal array Latin hypercube

Usage

create_oalhs(n, k, bChooseLargerDesign, bverbose)
create_oalhs(n, k, bChooseLargerDesign, bverbose)

Arguments

`n`	the number of samples or rows in the LHS (integer)
`k`	the number of parameters or columns in the LHS (integer)
`bChooseLargerDesign`	should a larger oa design be chosen than the n and k requested?
`bverbose`	should information be printed with execution

Value

a numeric matrix which is an orthogonal array Latin hypercube sample

Examples

set.seed(34)
A <- create_oalhs(9, 4, TRUE, FALSE)
B <- create_oalhs(9, 4, TRUE, FALSE)
set.seed(34)
A <- create_oalhs(9, 4, TRUE, FALSE)
B <- create_oalhs(9, 4, TRUE, FALSE)

Create an orthogonal array using the Addelman-Kempthorne algorithm.

Description

The addelkemp program produces OA( 2q^2, k, q, 2 ), k <= 2q+1, for odd prime powers q.

Usage

createAddelKemp(q, ncol, bRandom = TRUE)
createAddelKemp(q, ncol, bRandom = TRUE)

Arguments

`q`	the number of symbols in the array
`ncol`	number of parameters or columns
`bRandom`	should the array be randomized

Details

From Owen: An orthogonal array A is a matrix of n rows, k columns with every element being one of q symbols 0,...,q-1. The array has strength t if, in every n by t submatrix, the q^t possible distinct rows, all appear the same number of times. This number is the index of the array, commonly denoted lambda. Clearly, lambda*q^t=n. The notation for such an array is OA( n, k, q, t ).

Value

an orthogonal array

References

Owen, Art. Orthogonal Arrays for: Computer Experiments, Visualizations, and Integration in high dimensions. https://lib.stat.cmu.edu/designs/oa.c. 1994 S. Addelman and O. Kempthorne (1961) Annals of Mathematical Statistics, Vol 32 pp 1167-1176.

Examples

A <- createAddelKemp(3, 3, TRUE)
B <- createAddelKemp(3, 5, FALSE)
A <- createAddelKemp(3, 3, TRUE)
B <- createAddelKemp(3, 5, FALSE)

Create an orthogonal array using the Addelman-Kempthorne algorithm with `2q^3` rows.

Description

The addelkemp3 program produces OA( 2*q^3, k, q, 2 ), k <= 2q^2+2q+1, for prime powers q. q may be an odd prime power, or q may be 2 or 4.

Usage

createAddelKemp3(q, ncol, bRandom = TRUE)
createAddelKemp3(q, ncol, bRandom = TRUE)

Arguments

`q`	the number of symbols in the array
`ncol`	number of parameters or columns
`bRandom`	should the array be randomized

Details

Value

an orthogonal array

References

Examples

A <- createAddelKemp3(3, 3, TRUE)
B <- createAddelKemp3(3, 5, FALSE)
A <- createAddelKemp3(3, 3, TRUE)
B <- createAddelKemp3(3, 5, FALSE)

Create an orthogonal array using the Addelman-Kempthorne algorithm with alternate strength with `2q^n` rows.

Description

The addelkempn program produces OA( 2*q^n, k, q, 2 ), k <= 2(q^n - 1)/(q-1)-1, for prime powers q. q may be an odd prime power, or q may be 2 or 4.

Usage

createAddelKempN(q, ncol, exponent, bRandom = TRUE)
createAddelKempN(q, ncol, exponent, bRandom = TRUE)

Arguments

`q`	the number of symbols in the array
`ncol`	number of parameters or columns
`exponent`	the exponent on q
`bRandom`	should the array be randomized

Details

Value

an orthogonal array

Examples

A <- createAddelKempN(3, 4, 3, TRUE)
B <- createAddelKempN(3, 4, 4, TRUE)
A <- createAddelKempN(3, 4, 3, TRUE)
B <- createAddelKempN(3, 4, 4, TRUE)

Create an orthogonal array using the Bose algorithm.

Description

The bose program produces OA( q^2, k, q, 2 ), k <= q+1 for prime powers q.

Usage

createBose(q, ncol, bRandom = TRUE)
createBose(q, ncol, bRandom = TRUE)

Arguments

`q`	the number of symbols in the array
`ncol`	number of parameters or columns
`bRandom`	should the array be randomized

Details

Value

an orthogonal array

References

Owen, Art. Orthogonal Arrays for: Computer Experiments, Visualizations, and Integration in high dimensions. https://lib.stat.cmu.edu/designs/oa.c. 1994 R.C. Bose (1938) Sankhya Vol 3 pp 323-338

Examples

A <- createBose(3, 3, FALSE)
B <- createBose(5, 4, TRUE)
A <- createBose(3, 3, FALSE)
B <- createBose(5, 4, TRUE)

Create an orthogonal array using the Bose-Bush algorithm.

Description

The bosebush program produces OA( 2q^2, k, q, 2 ), k <= 2q+1, for powers of 2, q=2^r.

Usage

createBoseBush(q, ncol, bRandom = TRUE)
createBoseBush(q, ncol, bRandom = TRUE)

Arguments

`q`	the number of symbols in the array
`ncol`	number of parameters or columns
`bRandom`	should the array be randomized

Details

Value

an orthogonal array

References

Owen, Art. Orthogonal Arrays for: Computer Experiments, Visualizations, and Integration in high dimensions. https://lib.stat.cmu.edu/designs/oa.c. 1994 R.C. Bose and K.A. Bush (1952) Annals of Mathematical Statistics, Vol 23 pp 508-524.

Examples

A <- createBoseBush(4, 3, FALSE)
B <- createBoseBush(8, 3, TRUE)
A <- createBoseBush(4, 3, FALSE)
B <- createBoseBush(8, 3, TRUE)

Create an orthogonal array using the Bose-Bush algorithm with alternate strength >= 3.

Description

The bosebushl program produces OA( lambda*q^2, k, q, 2 ), k <= lambda*q+1, for prime powers q and lambda > 1. Both q and lambda must be powers of the same prime.

Usage

createBoseBushl(q, ncol, lambda, bRandom = TRUE)
createBoseBushl(q, ncol, lambda, bRandom = TRUE)

Arguments

`q`	the number of symbols in the array
`ncol`	number of parameters or columns
`lambda`	the lambda of the BoseBush algorithm
`bRandom`	should the array be randomized

Details

Value

an orthogonal array

References

Examples

A <- createBoseBushl(3, 3, 3, TRUE)
B <- createBoseBushl(4, 4, 16, TRUE)
A <- createBoseBushl(3, 3, 3, TRUE)
B <- createBoseBushl(4, 4, 16, TRUE)

Create an orthogonal array using the Bush algorithm.

Description

The bush program produces OA( q^3, k, q, 3 ), k <= q+1 for prime powers q.

Usage

createBush(q, ncol, bRandom = TRUE)
createBush(q, ncol, bRandom = TRUE)

Arguments

`q`	the number of symbols in the array
`ncol`	number of parameters or columns
`bRandom`	should the array be randomized

Details

Value

an orthogonal array

References

Owen, Art. Orthogonal Arrays for: Computer Experiments, Visualizations, and Integration in high dimensions. https://lib.stat.cmu.edu/designs/oa.c. 1994 K.A. Bush (1952) Annals of Mathematical Statistics, Vol 23 pp 426-434

Examples

A <- createBush(3, 3, FALSE)
B <- createBush(4, 5, TRUE)
A <- createBush(3, 3, FALSE)
B <- createBush(4, 5, TRUE)

Create an orthogonal array using the Bush algorithm with alternate strength.

Description

The busht program produces OA( q^t, k, q, t ), k <= q+1, t>=3, for prime powers q.

Usage

createBusht(q, ncol, strength, bRandom = TRUE)
createBusht(q, ncol, strength, bRandom = TRUE)

Arguments

`q`	the number of symbols in the array
`ncol`	number of parameters or columns
`strength`	the strength of the array to be created
`bRandom`	should the array be randomized

Details

Value

an orthogonal array

References

Examples

set.seed(1234)
A <- createBusht(3, 4, 2, TRUE)
B <- createBusht(3, 4, 3, FALSE)
G <- createBusht(3, 4, 3, TRUE)
set.seed(1234)
A <- createBusht(3, 4, 2, TRUE)
B <- createBusht(3, 4, 3, FALSE)
G <- createBusht(3, 4, 3, TRUE)

Latin Hypercube Sampling with a Genetic Algorithm

Description

Draws a Latin Hypercube Sample from a set of uniform distributions for use in creating a Latin Hypercube Design. This function attempts to optimize the sample with respect to the S optimality criterion through a genetic type algorithm.

Usage

geneticLHS(
  n = 10,
  k = 2,
  pop = 100,
  gen = 4,
  pMut = 0.1,
  criterium = "S",
  verbose = FALSE
)
geneticLHS(
  n = 10,
  k = 2,
  pop = 100,
  gen = 4,
  pMut = 0.1,
  criterium = "S",
  verbose = FALSE
)

Arguments

`n`	The number of partitions (simulations or design points or rows)
`k`	The number of replications (variables or columns)
`pop`	The number of designs in the initial population
`gen`	The number of generations over which the algorithm is applied
`pMut`	The probability with which a mutation occurs in a column of the progeny
`criterium`	The optimality criterium of the algorithm. Default is `S`. `Maximin` is also supported
`verbose`	Print informational messages. Default is `FALSE`

Details

Latin hypercube sampling (LHS) was developed to generate a distribution of collections of parameter values from a multidimensional distribution. A square grid containing possible sample points is a Latin square iff there is only one sample in each row and each column. A Latin hypercube is the generalisation of this concept to an arbitrary number of dimensions. When sampling a function of k variables, the range of each variable is divided into n equally probable intervals. n sample points are then drawn such that a Latin Hypercube is created. Latin Hypercube sampling generates more efficient estimates of desired parameters than simple Monte Carlo sampling.

This program generates a Latin Hypercube Sample by creating random permutations of the first n integers in each of k columns and then transforming those integers into n sections of a standard uniform distribution. Random values are then sampled from within each of the n sections. Once the sample is generated, the uniform sample from a column can be transformed to any distribution by using the quantile functions, e.g. qnorm(). Different columns can have different distributions.

S-optimality seeks to maximize the mean distance from each design point to all the other points in the design, so the points are as spread out as possible.

Genetic Algorithm:

Generate pop random latin hypercube designs of size n by k
Calculate the S optimality measure of each design
Keep the best design in the first position and throw away half of the rest of the population
Take a random column out of the best matrix and place it in a random column of each of the other matricies, and take a random column out of each of the other matricies and put it in copies of the best matrix thereby causing the progeny
For each of the progeny, cause a genetic mutation pMut percent of the time. The mutation is accomplished by swtching two elements in a column

Value

An n by k Latin Hypercube Sample matrix with values uniformly distributed on [0,1]

Author(s)

Rob Carnell

References

Stocki, R. (2005) A method to improve design reliability using optimal Latin hypercube sampling Computer Assisted Mechanics and Engineering Sciences 12, 87–105.

Stein, M. (1987) Large Sample Properties of Simulations Using Latin Hypercube Sampling. Technometrics. 29, 143–151.

Examples

set.seed(1234)
A <- geneticLHS(4, 3, 50, 5, .25)
set.seed(1234)
A <- geneticLHS(4, 3, 50, 5, .25)

Get version information for all libraries in the lhs package

Description

Get version information for all libraries in the lhs package

Usage

get_library_versions()
get_library_versions()

Value

a character string containing the versions

Examples

get_library_versions()
get_library_versions()

Improved Latin Hypercube Sample

Description

Draws a Latin Hypercube Sample from a set of uniform distributions for use in creating a Latin Hypercube Design. This function attempts to optimize the sample with respect to an optimum euclidean distance between design points.

Usage

improvedLHS(n, k, dup = 1)
improvedLHS(n, k, dup = 1)

Arguments

`n`	The number of partitions (simulations or design points or rows)
`k`	The number of replications (variables or columns)
`dup`	A factor that determines the number of candidate points used in the search. A multiple of the number of remaining points than can be added.

Details

This program generates a Latin Hypercube Sample by creating random permutations of the first n integers in each of k columns and then transforming those integers into n sections of a standard uniform distribution. Random values are then sampled from within each of the n sections. Once the sample is generated, the uniform sample from a column can be transformed to any distribution byusing the quantile functions, e.g. qnorm(). Different columns can have different distributions.

This function attempts to optimize the sample with respect to an optimum euclidean distance between design points.

$Optimum distance = frac{n}{n^{\frac{1.0}{k}}}$

Value

An n by k Latin Hypercube Sample matrix with values uniformly distributed on [0,1]

References

Beachkofski, B., Grandhi, R. (2002) Improved Distributed Hypercube Sampling American Institute of Aeronautics and Astronautics Paper 1274.

This function is based on the MATLAB program written by John Burkardt and modified 16 Feb 2005 https://people.math.sc.edu/Burkardt/m_src/ihs/ihs.html

Examples

set.seed(1234)
A <- improvedLHS(4, 3, 2)
set.seed(1234)
A <- improvedLHS(4, 3, 2)

Maximin Latin Hypercube Sample

Description

Draws a Latin Hypercube Sample from a set of uniform distributions for use in creating a Latin Hypercube Design. This function attempts to optimize the sample by maximizing the minium distance between design points (maximin criteria).

Usage

maximinLHS(
  n,
  k,
  method = "build",
  dup = 1,
  eps = 0.05,
  maxIter = 100,
  optimize.on = "grid",
  debug = FALSE
)
maximinLHS(
  n,
  k,
  method = "build",
  dup = 1,
  eps = 0.05,
  maxIter = 100,
  optimize.on = "grid",
  debug = FALSE
)

Arguments

`n`	The number of partitions (simulations or design points or rows)
`k`	The number of replications (variables or columns)
`method`	`build` or `iterative` is the method of LHS creation. `build` finds the next best point while constructing the LHS. `iterative` optimizes the resulting sample on [0,1] or sample grid on [1,N]
`dup`	A factor that determines the number of candidate points used in the search. A multiple of the number of remaining points than can be added. This is used when `method="build"`
`eps`	The minimum percent change in the minimum distance used in the `iterative` method
`maxIter`	The maximum number of iterations to use in the `iterative` method
`optimize.on`	`grid` or `result` gives the basis of the optimization. `grid` optimizes the LHS on the underlying integer grid. `result` optimizes the resulting sample on [0,1]
`debug`	prints additional information about the process of the optimization

Details

Here, values are added to the design one by one such that the maximin criteria is satisfied.

Value

An n by k Latin Hypercube Sample matrix with values uniformly distributed on [0,1]

References

Stein, M. (1987) Large Sample Properties of Simulations Using Latin Hypercube Sampling. Technometrics. 29, 143–151.

This function is motivated by the MATLAB program written by John Burkardt and modified 16 Feb 2005 https://people.math.sc.edu/Burkardt/m_src/ihs/ihs.html

Examples

set.seed(1234)
A1 <- maximinLHS(4, 3, dup=2)
A2 <- maximinLHS(4, 3, method="build", dup=2)
A3 <- maximinLHS(4, 3, method="iterative", eps=0.05, maxIter=100, optimize.on="grid")
A4 <- maximinLHS(4, 3, method="iterative", eps=0.05, maxIter=100, optimize.on="result")
set.seed(1234)
A1 <- maximinLHS(4, 3, dup=2)
A2 <- maximinLHS(4, 3, method="build", dup=2)
A3 <- maximinLHS(4, 3, method="iterative", eps=0.05, maxIter=100, optimize.on="grid")
A4 <- maximinLHS(4, 3, method="iterative", eps=0.05, maxIter=100, optimize.on="result")

Create a Latin hypercube from an orthogonal array

Description

Create a Latin hypercube from an orthogonal array

Usage

oa_to_oalhs(n, k, oa)
oa_to_oalhs(n, k, oa)

Arguments

`n`	the number of samples or rows in the LHS (integer)
`k`	the number of parameters or columns in the LHS (integer)
`oa`	the orthogonal array to be used as the basis for the LHS (matrix of integers) or data.frame of factors

Value

a numeric matrix which is a Latin hypercube sample

Examples

oa <- createBose(3, 4, TRUE)
B <- oa_to_oalhs(9, 4, oa)
oa <- createBose(3, 4, TRUE)
B <- oa_to_oalhs(9, 4, oa)

Optimal Augmented Latin Hypercube Sample

Description

Augments an existing Latin Hypercube Sample, adding points to the design, while maintaining the latin properties of the design. This function attempts to add the points to the design in an optimal way.

Usage

optAugmentLHS(lhs, m = 1, mult = 2)
optAugmentLHS(lhs, m = 1, mult = 2)

Arguments

`lhs`	The Latin Hypercube Design to which points are to be added
`m`	The number of additional points to add to matrix `lhs`
`mult`	`m*mult` random candidate points will be created.

Details

S-optimality seeks to maximize the mean distance from each design point to all the other points in the design, so the points are as spread out as possible.

Value

An n by k Latin Hypercube Sample matrix with values uniformly distributed on [0,1]

References

Stein, M. (1987) Large Sample Properties of Simulations Using Latin Hypercube Sampling. Technometrics. 29, 143–151.

Examples

set.seed(1234)
a <- randomLHS(4,3)
b <- optAugmentLHS(a, 2, 3)
set.seed(1234)
a <- randomLHS(4,3)
b <- optAugmentLHS(a, 2, 3)

Optimum Latin Hypercube Sample

Description

Draws a Latin Hypercube Sample from a set of uniform distributions for use in creating a Latin Hypercube Design. This function uses the Columnwise Pairwise (CP) algorithm to generate an optimal design with respect to the S optimality criterion.

Usage

optimumLHS(n = 10, k = 2, maxSweeps = 2, eps = 0.1, verbose = FALSE)
optimumLHS(n = 10, k = 2, maxSweeps = 2, eps = 0.1, verbose = FALSE)

Arguments

`n`	The number of partitions (simulations or design points or rows)
`k`	The number of replications (variables or columns)
`maxSweeps`	The maximum number of times the CP algorithm is applied to all the columns.
`eps`	The optimal stopping criterion. Algorithm stops when the change in optimality measure is less than eps*100% of the previous value.
`verbose`	Print informational messages

Details

S-optimality seeks to maximize the mean distance from each design point to all the other points in the design, so the points are as spread out as possible.

This function uses the CP algorithm to generate an optimal design with respect to the S optimality criterion.

Value

An n by k Latin Hypercube Sample matrix with values uniformly distributed on [0,1]

References

Stocki, R. (2005) A method to improve design reliability using optimal Latin hypercube sampling Computer Assisted Mechanics and Engineering Sciences 12, 87–105.

Examples

A <- optimumLHS(4, 3, 5, .05)
A <- optimumLHS(4, 3, 5, .05)

Optimum Seeded Latin Hypercube Sample

Description

Augments an existing Latin Hypercube Sample, adding points to the design, while maintaining the latin properties of the design. This function then uses the columnwise pairwise (CP) algoritm to optimize the design. The original design is not necessarily maintained.

Usage

optSeededLHS(seed, m = 0, maxSweeps = 2, eps = 0.1, verbose = FALSE)
optSeededLHS(seed, m = 0, maxSweeps = 2, eps = 0.1, verbose = FALSE)

Arguments

`seed`	The number of partitions (simulations or design points)
`m`	The number of additional points to add to the seed matrix `seed`. default value is zero. If m is zero then the seed design is optimized.
`maxSweeps`	The maximum number of times the CP algorithm is applied to all the columns.
`eps`	The optimal stopping criterion
`verbose`	Print informational messages

Details

Augments an existing Latin Hypercube Sample, adding points to the design, while maintaining the latin properties of the design. This function then uses the CP algoritm to optimize the design. The original design is not necessarily maintained.

Value

An n by k Latin Hypercube Sample matrix with values uniformly distributed on [0,1]

References

Stein, M. (1987) Large Sample Properties of Simulations Using Latin Hypercube Sampling. Technometrics. 29, 143–151.

Examples

  set.seed(1234)
  a <- randomLHS(4,3)
  b <- optSeededLHS(a, 2, 2, .1)
set.seed(1234)
  a <- randomLHS(4,3)
  b <- optSeededLHS(a, 2, 2, .1)

Multiplication in polynomial representation

Description

Multiplication in polynomial representation

Usage

poly_prod(p, n, xton, p1, p2)
poly_prod(p, n, xton, p1, p2)

Arguments

`p`	modulus
`n`	length of polynomials
`xton`	characteristic polynomial vector for the field (x to the n power)
`p1`	polynomial vector 1
`p2`	polynomial vector 2

Value

the product of p1 and p2

Examples

gf <- create_galois_field(4)
a <- poly_prod(gf$p, gf$n, gf$xton, c(1, 0), c(0, 1))
stopifnot(all(a == c(0, 1)))
gf <- create_galois_field(4)
a <- poly_prod(gf$p, gf$n, gf$xton, c(1, 0), c(0, 1))
stopifnot(all(a == c(0, 1)))

Addition in polynomial representation

Description

Addition in polynomial representation

Usage

poly_sum(p, n, p1, p2)
poly_sum(p, n, p1, p2)

Arguments

`p`	modulus
`n`	length of polynomial 1 and 2
`p1`	polynomial vector 1
`p2`	polynomial vector 2

Value

the sum of p1 and p2

Examples

gf <- create_galois_field(4)
a <- poly_sum(gf$p, gf$n, c(1, 0), c(0, 1))
stopifnot(all(a == c(1, 1)))
gf <- create_galois_field(4)
a <- poly_sum(gf$p, gf$n, c(1, 0), c(0, 1))
stopifnot(all(a == c(1, 1)))

Convert polynomial to integer in <code>0..q-1</code>

Description

Convert polynomial to integer in <code>0..q-1</code>

Usage

poly2int(p, n, poly)
poly2int(p, n, poly)

Arguments

`p`	modulus
`n`	the length of poly
`poly`	the polynomial vector

Value

an integer

Examples

gf <- create_galois_field(4)
stopifnot(poly2int(gf$p, gf$n, c(0, 0)) == 0)
gf <- create_galois_field(4)
stopifnot(poly2int(gf$p, gf$n, c(0, 0)) == 0)

Quantile Transformations

Description

A collection of functions that transform the margins of a Latin hypercube sample in multiple ways

Usage

qfactor(p, fact)

qinteger(p, a, b)

qdirichlet(X, alpha)
qfactor(p, fact)

qinteger(p, a, b)

qdirichlet(X, alpha)

Arguments

`p`	a vector of LHS samples on (0,1)
`fact`	a factor or categorical variable. Ordered and un-ordered variables are allowed.
`a`	a minimum integer
`b`	a maximum integer
`X`	multiple columns of an LHS sample on (0,1)
`alpha`	Dirichlet distribution parameters. All `alpha >= 1` The marginal mean probability of the Dirichlet distribution is given by `alpha[i] / sum(alpha)`

Details

qdirichlet is not an exact quantile function since the quantile of a multivariate distribution is not unique. qdirichlet is also not the independent quantiles of the marginal distributions since those quantiles do not sum to one. qdirichlet is the quantile of the underlying gamma functions, normalized. This is the same procedure that is used to generate random deviates from the Dirichlet distribution therefore it will produce transformed Latin hypercube samples with the intended distribution.

q_factor divides the [0,1] interval into nlevel(fact) equal sections and assigns values in those sections to the factor level.

Value

the transformed column or columns

Examples

X <- randomLHS(20, 7)
Y <- as.data.frame(X)
Y[,1] <- qnorm(X[,1], 2, 0.5)
Y[,2] <- qfactor(X[,2], factor(LETTERS[c(1,3,5,7,8)]))
Y[,3] <- qinteger(X[,3], 5, 17)
Y[,4:6] <- qdirichlet(X[,4:6], c(2,3,4))
Y[,7] <- qfactor(X[,7], ordered(LETTERS[c(1,3,5,7,8)]))
X <- randomLHS(20, 7)
Y <- as.data.frame(X)
Y[,1] <- qnorm(X[,1], 2, 0.5)
Y[,2] <- qfactor(X[,2], factor(LETTERS[c(1,3,5,7,8)]))
Y[,3] <- qinteger(X[,3], 5, 17)
Y[,4:6] <- qdirichlet(X[,4:6], c(2,3,4))
Y[,7] <- qfactor(X[,7], ordered(LETTERS[c(1,3,5,7,8)]))

Construct a random Latin hypercube design

Description

randomLHS(4,3) returns a 4x3 matrix with each column constructed as follows: A random permutation of (1,2,3,4) is generated, say (3,1,2,4) for each of K columns. Then a uniform random number is picked from each indicated quartile. In this example a random number between .5 and .75 is chosen, then one between 0 and .25, then one between .25 and .5, finally one between .75 and 1.

Usage

randomLHS(n, k, preserveDraw = FALSE)
randomLHS(n, k, preserveDraw = FALSE)

Arguments

`n`	the number of rows or samples
`k`	the number of columns or parameters/variables
`preserveDraw`	should the draw be constructed so that it is the same for variable numbers of columns?

Value

a Latin hypercube sample

Examples

a <- randomLHS(5, 3)
a <- randomLHS(5, 3)

Create a Random Sample of Uniform Integers

Description

Create a Random Sample of Uniform Integers

Usage

runifint(n = 1, min_int = 0, max_int = 1)
runifint(n = 1, min_int = 0, max_int = 1)

Arguments

`n`	The number of samples
`min_int`	the minimum integer `x >= min_int`
`max_int`	the maximum integer `x <= max_int`

Value

the sample sample of size n

Package 'lhs'

Help Index

Augment a Latin Hypercube Design

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Transformed Latin hypercube with a multivariate distribution

Description

Usage

Arguments

Value

Examples

Create a Galois field

Description

Usage

Arguments

Value

Examples

Create an orthogonal array Latin hypercube

Description

Usage

Arguments

Value

Examples

Create an orthogonal array using the Addelman-Kempthorne algorithm.

Description

Usage

Arguments

Details

Value

References

See Also

Examples

Create an orthogonal array using the Addelman-Kempthorne algorithm with 2q^3 rows.

Description

Usage

Arguments

Details

Value

References

See Also

Examples

Create an orthogonal array using the Addelman-Kempthorne algorithm with alternate strength with 2q^n rows.

Description

Usage

Arguments

Details

Value

See Also

Examples

Create an orthogonal array using the Bose algorithm.

Description

Usage

Arguments

Details

Value

References

See Also

Examples

Create an orthogonal array using the Bose-Bush algorithm.

Description

Usage

Arguments

Details

Value

References

See Also

Examples

Create an orthogonal array using the Bose-Bush algorithm with alternate strength >= 3.

Description

Usage

Arguments

Details

Value

Create an orthogonal array using the Addelman-Kempthorne algorithm with `2q^3` rows.

Create an orthogonal array using the Addelman-Kempthorne algorithm with alternate strength with `2q^n` rows.