Theory of Latin Hypercube Sampling
For the technical basis of Latin Hypercube Sampling (LHS) and Latin
Hypercube Designs (LHD) please see: * Stein, Michael. Large Sample
Properties of Simulations Using Latin Hypercube Sampling
Technometrics, Vol 28, No 2, 1987. * McKay, MD, et.al. A Comparison
of Three Methods for Selecting Values of Input Variables in the Analysis
of Output from a Computer Code Technometrics, Vol 21, No 2,
1979.
This package was created to bring these designs to R and to implement
many of the articles that followed on optimized sampling methods.
Create a Simple LHS
Basic LHS’s are created using randomLHS.
# set the seed for reproducibility
set.seed(1111)
# a design with 5 samples from 4 parameters
A <- randomLHS(5, 4)
A
#> [,1] [,2] [,3] [,4]
#> [1,] 0.6328827 0.48424369 0.1678234 0.1974741
#> [2,] 0.2124960 0.88111537 0.6069217 0.4771109
#> [3,] 0.1277885 0.64327868 0.3612360 0.9862456
#> [4,] 0.8935830 0.27182878 0.4335808 0.6052341
#> [5,] 0.5089423 0.02269382 0.8796676 0.2036678
In general, the LHS is uniform on the margins until transformed
(Figure 1):
Figure 1. Two dimensions of a Uniform random LHS with 5 samples
It is common to transform the margins of the design (the columns)
into other distributions (Figure 2)
B <- matrix(nrow = nrow(A), ncol = ncol(A))
B[,1] <- qnorm(A[,1], mean = 0, sd = 1)
B[,2] <- qlnorm(A[,2], meanlog = 0.5, sdlog = 1)
B[,3] <- A[,3]
B[,4] <- qunif(A[,4], min = 7, max = 10)
B
#> [,1] [,2] [,3] [,4]
#> [1,] 0.33949794 1.5848575 0.1678234 7.592422
#> [2,] -0.79779049 5.3686737 0.6069217 8.431333
#> [3,] -1.13690757 2.3803237 0.3612360 9.958737
#> [4,] 1.24581019 0.8982639 0.4335808 8.815702
#> [5,] 0.02241694 0.2228973 0.8796676 7.611003
Figure 2. Two dimensions of a transformed random LHS with 5 samples
Optimizing the Design
The LHS can be optimized using a number of methods in the
lhs package. Each method attempts to improve on the random
design by ensuring that the selected points are as uncorrelated and
space filling as possible. Table 1 shows some
results. Figure 3, Figure 4, and
Figure 5 show corresponding plots.
set.seed(101)
A <- randomLHS(30, 10)
A1 <- optimumLHS(30, 10, maxSweeps = 4, eps = 0.01)
A2 <- maximinLHS(30, 10, dup = 5)
A3 <- improvedLHS(30, 10, dup = 5)
A4 <- geneticLHS(30, 10, pop = 1000, gen = 8, pMut = 0.1, criterium = "S")
A5 <- geneticLHS(30, 10, pop = 1000, gen = 8, pMut = 0.1, criterium = "Maximin")
|
| Method | Min Distance btwn pts | Mean Distance btwn pts | Max
Correlation btwn pts :—–|:—–:|:—–:|:—–: randomLHS | 0.6346585 |
1.2913235 | 0.5173006 optimumLHS | 0.8717797 | 1.3001892 | 0.1268209
maximinLHS | 0.595395 | 1.2835191 | 0.2983643 improvedLHS | 0.6425673 |
1.2746711 | 0.5711527 geneticLHS (S) | 0.8340751 | 1.3026543 | 0.3971539
geneticLHS (Maximin) | 0.8105733 | 1.2933412 | 0.5605546 |
Figure 3. Pairwise margins of a randomLHS
Figure 5. Pairwise margins of a maximinLHS
