Maximin Latin Hypercube Sample

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).

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

Value

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

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.

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

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

See also

[randomLHS()], [geneticLHS()], [improvedLHS()] and [optimumLHS()] to generate Latin Hypercube Samples. [optAugmentLHS()], [optSeededLHS()], and [augmentLHS()] to modify and augment existing designs.

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")