EI {DiceOptim} | R Documentation |

## Analytical expression of the Expected Improvement criterion

### Description

Computes the Expected Improvement at current location. The current minimum of the observations can be replaced by an arbitrary value (plugin), which is usefull in particular in noisy frameworks.

### Usage

```
EI(
x,
model,
plugin = NULL,
type = "UK",
minimization = TRUE,
envir = NULL,
proxy = FALSE
)
```

### Arguments

`x` |
a vector representing the input for which one wishes to calculate EI, |

`model` |
an object of class |

`plugin` |
optional scalar: if provided, it replaces the minimum of the current observations, |

`type` |
"SK" or "UK" (by default), depending whether uncertainty related to trend estimation has to be taken into account, |

`minimization` |
logical specifying if EI is used in minimiziation or in maximization, |

`envir` |
an optional environment specifying where to assign intermediate values for future gradient calculations. Default is NULL. |

`proxy` |
an optional Boolean, if TRUE EI is replaced by the kriging mean (to minimize) |

### Value

The expected improvement, defined as

```
EI(x) := E[( min(Y(X)) -
Y(x))^{+} | Y(X)=y(X)],
```

where X is the current design of experiments and Y is the random process assumed to have generated the objective function y. If a plugin is specified, it replaces

`min(Y(X))`

in the previous formula.

### Author(s)

David Ginsbourger

Olivier Roustant

Victor Picheny

### References

D.R. Jones, M. Schonlau, and W.J. Welch (1998), Efficient global
optimization of expensive black-box functions, *Journal of Global
Optimization*, 13, 455-492.

J. Mockus (1988), *Bayesian Approach to Global Optimization*. Kluwer
academic publishers.

T.J. Santner, B.J. Williams, and W.J. Notz (2003), *The design and
analysis of computer experiments*, Springer.

M. Schonlau (1997), *Computer experiments and global optimization*,
Ph.D. thesis, University of Waterloo.

### See Also

### Examples

```
set.seed(123)
##########################################################################
### EI SURFACE ASSOCIATED WITH AN ORDINARY KRIGING MODEL ####
### OF THE BRANIN FUNCTION KNOWN AT A 9-POINTS FACTORIAL DESIGN ####
##########################################################################
# a 9-points factorial design, and the corresponding response
d <- 2; n <- 9
design.fact <- expand.grid(seq(0,1,length=3), seq(0,1,length=3))
names(design.fact)<-c("x1", "x2")
design.fact <- data.frame(design.fact)
names(design.fact)<-c("x1", "x2")
response.branin <- apply(design.fact, 1, branin)
response.branin <- data.frame(response.branin)
names(response.branin) <- "y"
# model identification
fitted.model1 <- km(~1, design=design.fact, response=response.branin,
covtype="gauss", control=list(pop.size=50,trace=FALSE), parinit=c(0.5, 0.5))
# graphics
n.grid <- 12
x.grid <- y.grid <- seq(0,1,length=n.grid)
design.grid <- expand.grid(x.grid, y.grid)
#response.grid <- apply(design.grid, 1, branin)
EI.grid <- apply(design.grid, 1, EI,fitted.model1)
z.grid <- matrix(EI.grid, n.grid, n.grid)
contour(x.grid,y.grid,z.grid,25)
title("Expected Improvement for the Branin function known at 9 points")
points(design.fact[,1], design.fact[,2], pch=17, col="blue")
```

*DiceOptim*version 2.1.1 Index]