MuFiMeshGP (version 0.0.1)

IMSPE_AL: IMSPE optimal design point search

Description

Search for the best next design point according to the IMSPE criterion given a current MuFiMeshGP model fit.

Usage

IMSPE_AL(
  object,
  t.min,
  t.max,
  cost.func,
  cost.new = 0,
  gr = FALSE,
  gr_cost.func = NULL,
  DesCand = NULL,
  Wijs = NULL,
  Hijs = NULL,
  control = list(multi.start.n = 20, maxit = 20, DesStart = NULL, seed = NULL, ncores =
    1)
)

Value

a list with:

• x: the optimal input parameter, a vector.

• t: the optimal tuning parameter, a scalar.

• value: the IMSPE reduction at the optimal design location.

• new: whether the optimal tuning parameter defines a new fidelity level.

• id: the index of the optimal design location if DesCand is used.

Arguments

object

Current MuFiMeshGP model fit.

t.min, t.max

Lower and upper bounds on the fidelity space for the search.

cost.func

Function that maps the tunable parameter t to the corresponding cost running a simulation at that fidelity level. For example, function(t) 1/t^2.

cost.new

(optional) Cost of running a new simulation at a new fidelity level, scalar.

gr

whether the gradient should be used in the optimization of the IMSPE. (Not recommended due to numerical errors)

gr_cost.func

If grad is TRUE, the user needs to specify the gradient of the cost function, as a function.

DesCand

Design candidates to evaluate from.

Wijs, Hijs

(optional) Matrices from previous IMSPE search to obtain faster computation through matrix decomposition.

control

list of arguments udes for the optimization.

Examples

# Example code

f <- function(x, t){
  x <- c(x)
  return(exp(-1.4*x)*cos(3.5*pi*x)+sin(40*x)/10*t^2)
}

set.seed(1)
X <- matrix(runif(15,0,1), ncol = 1)
tt <- runif(15,0.5,2)

Y <- f(c(X), tt)

fit.mufimeshgp <- MuFiMeshGP(X, tt, Y)

xx <- matrix(seq(0,1,0.01), ncol = 1)
ftrue <- f(xx, 0)

# predict
pred.mufimeshgp <- predict(fit.mufimeshgp, xx, rep(0,101))

mu <- pred.mufimeshgp$mean
s <- pred.mufimeshgp$sd
lower <- mu + qnorm(0.025)*s
upper <- mu + qnorm(0.975)*s

# plot

oldpar <- par(mfrow = c(1,2))
plot(xx, ftrue, "l", ylim = c(-1,1.3), ylab = "y", xlab = "x")
lines(c(xx), mu, col = "blue")
lines(c(xx), lower, col = "blue", lty = 2)
lines(c(xx), upper, col = "blue", lty = 2)
points(c(X), Y, col = "red")

### RMSE ###
print(sqrt(mean((ftrue - mu))^2))

best <- IMSPE_AL(fit.mufimeshgp, 0.5, 2, function(t) return(1 / t^2))
new.Y <- f(best$x, best$t)
fit.mufimeshgp <- update(fit.mufimeshgp, best$x, best$t, new.Y)

pred.mufimeshgp <- predict(fit.mufimeshgp, xx, rep(0, 101))
mu <- pred.mufimeshgp$mean
s <- pred.mufimeshgp$sd
lower <- mu + qnorm(0.025)*s
upper <- mu + qnorm(0.975)*s

plot(xx, ftrue, "l", ylim = c(-1,1.3), ylab = "y", xlab = "x")
lines(c(xx), mu, col = "blue")
lines(c(xx), lower, col = "blue", lty = 2)
lines(c(xx), upper, col = "blue", lty = 2)
points(c(X), Y, col = "red")
points(c(best$x), new.Y, col = "green")
par(oldpar)

### RMSE ###
print(sqrt(mean((ftrue - mu))^2))