discrepSA_LHS {DiceDesign} | R Documentation |

## Simulated annealing (SA) routine for Latin Hypercube Sample (LHS) optimization via L2-discrepancy criteria

### Description

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.

### Usage

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

### Arguments

`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 |

### Details

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).

### Value

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 |

### Author(s)

G. Damblin & B. Iooss

### References

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.

### See Also

Latin Hypercube Sample(`lhsDesign`

),discrepancy criteria(`discrepancyCriteria`

), geometric criterion (`mindist`

`phiP`

), optimization (`maximinSA_LHS`

,`maximinESE_LHS`

,`discrepESE_LHS`

)

### Examples

```
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)
```

*DiceDesign*version 1.10 Index]