# SLHD

##### Sliced Latin Hypercube Design (SLHD)

`SLHD`

returns an LHD matrix generated by "improved two-stage algorithm" from Ba et al. (2015).

##### Usage

```
SLHD(
n,
k,
t = 1,
N = 10,
T0 = 10,
rate = 0.1,
Tmin = 1,
Imax = 3,
OC = "phi_p",
p = 15,
q = 1,
stage2 = FALSE
)
```

##### Arguments

- n
A positive integer.

- k
A positive integer.

- t
A positive integer.

- N
A positive integer.

- T0
A positive number.

- rate
A positive percentage.

- Tmin
A positive number.

- Imax
A positive integer.

- OC
An optimality criterion.

- p
A positive integer.

- q
The default is set to be 1, and it could be either 1 or 2.

- stage2
The default is set to be FALSE, and it could be either FALSE or TRUE.

##### Details

`n`

stands for the number of rows (or run size).`k`

stands for the number of columns (or the number of factors).`t`

stands for the number of slices.`n`

/`t`

must be an integer, that is, n is divisible by t.`t`

must not exceed`k`

when`n`

is 9 or larger, and`t`

must be smaller than`k`

when`n`

is smaller than 9. Otherwise, the funtion will never stop. The default is set to be 1.`N`

stands for the number of iterations. The default is set to be 10.`T0`

stands for the user-defined initial temperature. The default is set to be 10.`rate`

stands for temperature decrease rate, and it should be in (0,1). For example, rate=0.25 means the temperature decreases by 25% each time. The default is set to be 10%.`Tmin`

stands for the minimium temperature allowed. When current temperature becomes smaller or equal to`Tmin`

, the stopping criterion for current loop is met. The default is set to be 1.`Imax`

stands for the maximum perturbations the algorithm will try without improvements before temperature is reduced. For the computation complexity consideration,`Imax`

is recommended to be smaller or equal to 3, which is the default setting.`OC`

stands for the optimality criterion, the default setting is "phi_p", and it could be one of the following: "phi_p", "AvgAbsCor", "MaxAbsCor", "MaxProCriterion".`p`

is the parameter in the phi_p formula, and`p`

is prefered to be large. The default is set to be 15.If

`q`

is 1 (the default setting),`dij`

is the rectangular distance. If`q`

is 2,`dij`

is the Euclidean distance.If

`stage2`

is FALSE (the default setting),`SLHD`

will only implement the first stage of the algorithm. If`stage2`

is TRUE,`SLHD`

will implement the whole algorithm.

##### Value

If all inputs are logical, then the output will be a `n`

by `k`

LHD.

##### Note

As mentioned from the original paper, the first stage plays a much more important role since it optimizes the slice level. More resources should be given to the first stage if computational budgets are limited. Let m=n/t, where m is the number of rows for each slice, if (m)^k >> n, the second stage becomes optional. That is the reason why we add a `stage2`

parameter to let users decide if the second stage is needed.

##### References

Ba, S., Myers, W.R., and Brenneman, W.A. (2015) Optimal Sliced Latin Hypercube Designs. *Technometrics*, **57**, 479-487.

##### Examples

```
# NOT RUN {
#generate a 5 by 3 maximin distance LHD with the default setting
trySLHD1=SLHD(n=5,k=3)
trySLHD1
phi_p(trySLHD1) #calculate the phi_p of "trySLHD1".
#generate a 5 by 3 maximin distance LHD with stage II
#let stage2=TRUE and other input are the same as above
trySLHD2=SLHD(n=5,k=3,stage2=TRUE)
trySLHD2
phi_p(trySLHD2) #calculate the phi_p of "trySLHD2".
#Another example
#generate a 8 by 4 nearly orthogonal LHD
trySLHD3=SLHD(n=8,k=4,OC="AvgAbsCor",stage2=TRUE)
trySLHD3
AvgAbsCor(trySLHD3) #calculate the average absolute correlation.
# }
```

*Documentation reproduced from package LHD, version 1.1.0, License: MIT + file LICENSE*