discrepSA_LHS: Simulated annealing (SA) routine for Latin Hypercube Sample...
In DiceDesign: Designs of Computer Experiments

Description Usage Arguments Details Value Author(s) References See Also Examples

The objective is to produce low-discrepancy LHS. SA is an efficient algorithm to produce space-filling designs. It has been adapted here to main discrepancy criteria.

1	discrepSA_LHS(design, T0=10, c=0.95, it=2000, criterion="C2", profile="GEOM", Imax=100)

`design`	a matrix (or a data.frame) corresponding to the design of experiments
`T0`	The initial temperature
`c`	A constant parameter regulating how the temperature goes down
`it`	The number of iterations
`criterion`	The criterion to be optimized. One can choose three different L2-discrepancies: the C2 (centered) discrepancy ("C2"), the L2-star discrepancy ("L2star") and the W2 (wrap-around) discrepancy ("W2")
`profile`	The temperature down-profile, purely geometric called "GEOM", geometrical according to the Morris algorithm called "GEOM_MORRIS" or purely linear called "LINEAR"
`Imax`	A parameter given only if you choose the Morris down-profile. It adjusts the number of iterations without improvement before a new elementary perturbation

This function implements a classical routine to produce optimized LHS. It is based on the work of Morris and Mitchell (1995). They have proposed a SA version for LHS optimization according to mindist criterion. Here, it has been adapted to some discrepancy criteria taking in account new ideas about the reevaluations of a discrepancy value after a LHS elementary perturbation (in order to avoid computing all terms in the discrepancy formulas).

A list containing:

`InitialDesign`	the starting design
`T0`	the initial temperature of the SA algorithm
`c`	the constant parameter regulating how the temperature goes down
`it`	the number of iterations
`criterion`	the criterion to be optimized
`profile`	the temperature down-profile
`Imax`	The parameter given in the Morris down-profile
`design`	the matrix of the final design (low-discrepancy LHS)
`critValues`	vector of criterion values along the iterations
`tempValues`	vector of temperature values along the iterations
`probaValues`	vector of acceptation probability values along the iterations

G. Damblin & B. Iooss

Damblin G., Couplet M., and Iooss B. (2013). Numerical studies of space filling designs: optimization of Latin Hypercube Samples and subprojection properties, Journal of Simulation, 7:276-289, 2013.

M. Morris and J. Mitchell (1995) Exploratory designs for computationnal experiments. Journal of Statistical Planning and Inference, 43:381-402.

R. Jin, W. Chen and A. Sudjianto (2005) An efficient algorithm for constructing optimal design of computer experiments. Journal of Statistical Planning and Inference, 134:268-287.

Latin Hypercube Sample(lhsDesign),discrepancy criteria(discrepancyCriteria), geometric criterion (mindistphiP), optimization (maximinSA_LHS,maximinESE_LHS ,discrepESE_LHS)

dimension <- 2
n <- 10
X <- lhsDesign(n, dimension)$design

## Optimize the LHS with C2 criterion
Xopt <- discrepSA_LHS(X, T0=10, c=0.99, it=2000, criterion="C2")
plot(Xopt$design)
plot(Xopt$critValues, type="l")

## Optimize the LHS with C2 criterion and GEOM_MORRIS profile
## Not run: 
Xopt2 <- discrepSA_LHS(X, T0=10, c=0.99, it=1000, criterion="C2", profile="GEOM_MORRIS")
plot(Xopt2$design)

## End(Not run)