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")
Table 1. Sample results and metrics of various LHS algorithms
| 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 4. Pairwise margins of a optimumLHS
Figure 5. Pairwise margins of a maximinLHS
