crit_EI {hetGP}

Expected Improvement criterion

Description

Computes EI for minimization

Usage

crit_EI(x, model, cst = NULL, preds = NULL)

Arguments

x

matrix of new designs, one point per row (size n x d)

model

homGP or hetGP model, or their TP equivalents, including inverse matrices

cst

optional plugin value used in the EI, see details

preds

optional predictions at x to avoid recomputing if already done

Details

cst is classically the observed minimum in the deterministic case. In the noisy case, the min of the predictive mean works fine.

Note

This is a beta version at this point.

References

Mockus, J.; Tiesis, V. & Zilinskas, A. (1978). The application of Bayesian methods for seeking the extremum Towards Global Optimization, Amsterdam: Elsevier, 2, 2.

Vazquez E, Villemonteix J, Sidorkiewicz M, Walter E (2008). Global Optimization Based on Noisy Evaluations: An Empirical Study of Two Statistical Approaches, Journal of Physics: Conference Series, 135, IOP Publishing.

A. Shah, A. Wilson, Z. Ghahramani (2014), Student-t processes as alternatives to Gaussian processes, Artificial Intelligence and Statistics, 877–885.

Examples

## Optimization example
set.seed(42)


## Noise field via standard deviation
noiseFun <- function(x, coef = 1.1, scale = 1){
if(is.null(nrow(x)))
 x <- matrix(x, nrow = 1)
   return(scale*(coef + cos(x * 2 * pi)))
}

## Test function defined in [0,1]
ftest <- function(x){
if(is.null(nrow(x)))
x <- matrix(x, ncol = 1)
return(f1d(x) + rnorm(nrow(x), mean = 0, sd = noiseFun(x)))
}

n_init <- 10 # number of unique designs
N_init <- 100 # total number of points
X <- seq(0, 1, length.out = n_init)
X <- matrix(X[sample(1:n_init, N_init, replace = TRUE)], ncol = 1)
Z <- ftest(X)

## Predictive grid
ngrid <- 51
xgrid <- seq(0,1, length.out = ngrid)
Xgrid <- matrix(xgrid, ncol = 1)

model <- mleHetGP(X = X, Z = Z, lower = 0.001, upper = 1)

EIgrid <- crit_EI(Xgrid, model)
preds <- predict(x = Xgrid, model)

par(mar = c(3,3,2,3)+0.1)
plot(xgrid, f1d(xgrid), type = 'l', lwd = 1, col = "blue", lty = 3,
xlab = '', ylab = '', ylim = c(-8,16))
points(X, Z)
lines(Xgrid, preds$mean, col = 'red', lwd = 2)
lines(Xgrid, qnorm(0.05, preds$mean, sqrt(preds$sd2)), col = 2, lty = 2)
lines(Xgrid, qnorm(0.95, preds$mean, sqrt(preds$sd2)), col = 2, lty = 2)
lines(Xgrid, qnorm(0.05, preds$mean, sqrt(preds$sd2 + preds$nugs)), col = 3, lty = 2)
lines(Xgrid, qnorm(0.95, preds$mean, sqrt(preds$sd2 + preds$nugs)), col = 3, lty = 2)
par(new = TRUE)
plot(NA, NA, xlim = c(0, 1), ylim = c(0,max(EIgrid)), axes = FALSE, ylab = "", xlab = "")
lines(xgrid, EIgrid, lwd = 2, col = 'cyan')
axis(side = 4)
mtext(side = 4, line = 2, expression(EI(x)), cex = 0.8)
mtext(side = 2, line = 2, expression(f(x)), cex = 0.8)