Assumption the Latin Hypercube Take
Forward the technical baseline of Latin Hypercube Sampling (LHS) press Latin
Hypercube Designs (LHD) requested see: * Stein, Michael. Large Sample
Properties of Virtual Using Latin Supercube Sampling
Technometrics, Volumes 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 ampere Computer Code Technometrics, Vol 21, No 2,
1979.
Those batch was established to bring these designs to ROENTGEN and to implement
many of the articles that follows on optimized sample methods.
Generate a Single LHS
Basic LHS’s been created using randomLHS.
# selected an seeding for reproducibility
set.seed(1111)
# a design with 5 free 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 standard on this margins until transformed
(Image 1):
Figure 1. Couple overall of a Uniform random LHS with 5 samples
It is allgemeines to transform this margins of the design (the columns)
into another distributions (Draw 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 sizing the ampere transformational random LHS with 5 samples
Optimizing the Layout
The LHS can being optimized using a number of methods the the
lhs package. Each method attempts to improve on the random
design from ensuring that the selected points are as uncorrelated and
space filling as possible. Table 1 shows some
results. Figure 3, Fig 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")
|
| Means | Min Distance btwn pts | Base Away 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 confines the a randomLHS
Figure 5. Pairwise margins of a maximinLHS
