ALC_RNAmf {RNAmf}

find the next point by ALC criterion

Description

The function acquires the new point by the Active learning Cohn (ALC) criterion. It calculates the ALC criterion \frac{\Delta \sigma_L^{2}(l,\bm{x})}{\sum^l_{j=1}C_j} = \frac{\int_{\Omega} \sigma_L^{*2}(\bm{\xi})-\tilde{\sigma}_L^{*2}(\bm{\xi};l,\bm{x}){\rm{d}}\bm{\xi}}{\sum^l_{j=1}C_j}, where f_L is the highest-fidelity simulation code, \sigma_L^{*2}(\bm{\xi}) is the posterior variance of f_L(\bm{\xi}), C_j is the simulation cost at fidelity level j, and \tilde{\sigma}_L^{*2}(\bm{\xi};l,\bm{x}) is the posterior variance based on the augmented design combining the current design and a new input location \bm{x} at each fidelity level lower than or equal to l. The integration is approximated by MC integration using uniform reference samples.

A new point is acquired on Xcand. If Xcand=NULL and Xref=NULL, a new point is acquired on unit hypercube [0,1]^d.

Usage

ALC_RNAmf(Xref = NULL, Xcand = NULL, fit, mc.sample = 100,
cost = NULL, optim = TRUE, parallel = FALSE, ncore = 1, trace=TRUE)

Arguments

Xref

vector or matrix of reference location to approximate the integral of ALC. If Xref=NULL, 100 \times d points from 0 to 1 are generated by Latin hypercube design. Default is NULL.

Xcand

vector or matrix of candidate set which could be added into the current design only when optim=FALSE. Xcand is the set of the points where ALC criterion is evaluated. If Xcand=NULL, Xref is used. Default is NULL. See details.

fit

object of class RNAmf.

mc.sample

a number of mc samples generated for the imputation through MC approximation. Default is 100.

cost

vector of the costs for each level of fidelity. If cost=NULL, total costs at all levels would be 1. cost is encouraged to have a ascending order of positive value. Default is NULL.

optim

logical indicating whether to optimize AL criterion by optim's gradient-based L-BFGS-B method. If optim=TRUE, 5 \times d starting points are generated by Latin hypercube design for optimization. If optim=FALSE, AL criterion is optimized on the Xcand. Default is TRUE.

parallel

logical indicating whether to compute the AL criterion in parallel or not. If parallel=TRUE, parallel computation is utilized. Default is FALSE.

ncore

a number of core for parallel. It is only used if parallel=TRUE. Default is 1.

trace

logical indicating whether to print the computational time for each step. If trace=TRUE, the computation time for each step is printed. Default is TRUE.

Details

Xref plays a role of \bm{\xi} to approximate the integration. To impute the posterior variance based on the augmented design \tilde{\sigma}_L^{*2}(\bm{\xi};l,\bm{x}), MC approximation is used. Due to the nested assumption, imputing y^{[s]}_{n_s+1} for each 1\leq s\leq l by drawing samples from the posterior distribution of f_s(\bm{x}^{[s]}_{n_s+1}) based on the current design allows to compute \tilde{\sigma}_L^{*2}(\bm{\xi};l,\bm{x}). Inverse of covariance matrix is computed by the Sherman-Morrison formula. For details, see Heo and Sung (2024, <doi:10.1080/00401706.2024.2376173>).

To search for the next acquisition \bm{x^*} by maximizing AL criterion, the gradient-based optimization can be used by optim=TRUE. Firstly, \tilde{\sigma}_L^{*2}(\bm{\xi};l,\bm{x}) is computed on the 5 \times d number of points. After that, the point minimizing \tilde{\sigma}_L^{*2}(\bm{\xi};l,\bm{x}) serves as a starting point of optimization by L-BFGS-B method. Otherwise, when optim=FALSE, AL criterion is optimized only on Xcand.

The point is selected by maximizing the ALC criterion: \text{argmax}_{l\in\{1,\ldots,L\}; \bm{x} \in \Omega} \frac{\Delta \sigma_L^{2}(l,\bm{x})}{\sum^l_{j=1}C_j}.

Value

• ALC: list of ALC criterion integrated on Xref when each data point on Xcand is added at each level l if optim=FALSE. If optim=TRUE, ALC returns NULL.

• cost: a copy of cost.

• Xcand: a copy of Xcand.

• chosen: list of chosen level and point.

• time: a scalar of the time for the computation.

Examples

library(lhs)
library(doParallel)
library(foreach)

### simulation costs ###
cost <- c(1, 3)

### 1-d Perdikaris function in Perdikaris, et al. (2017) ###
# low-fidelity function
f1 <- function(x) {
  sin(8 * pi * x)
}

# high-fidelity function
f2 <- function(x) {
  (x - sqrt(2)) * (sin(8 * pi * x))^2
}

### training data ###
n1 <- 13
n2 <- 8

### fix seed to reproduce the result ###
set.seed(1)

### generate initial nested design ###
X <- NestedX(c(n1, n2), 1)
X1 <- X[[1]]
X2 <- X[[2]]

### n1 and n2 might be changed from NestedX ###
### assign n1 and n2 again ###
n1 <- nrow(X1)
n2 <- nrow(X2)

y1 <- f1(X1)
y2 <- f2(X2)

### n=100 uniform test data ###
x <- seq(0, 1, length.out = 100)

### fit an RNAmf ###
fit.RNAmf <- RNAmf_two_level(X1, y1, X2, y2, kernel = "sqex")

### predict ###
predy <- predict(fit.RNAmf, x)$mu
predsig2 <- predict(fit.RNAmf, x)$sig2

### active learning with optim=TRUE ###
alc.RNAmf.optim <- ALC_RNAmf(
  Xref = x, Xcand = x, fit.RNAmf, cost = cost,
  optim = TRUE, parallel = TRUE, ncore = 2
)
print(alc.RNAmf.optim$time) # computation time of optim=TRUE

### active learning with optim=FALSE ###
alc.RNAmf <- ALC_RNAmf(
  Xref = x, Xcand = x, fit.RNAmf, cost = cost,
  optim = FALSE, parallel = TRUE, ncore = 2
)
print(alc.RNAmf$time) # computation time of optim=FALSE

### visualize ALC ###
oldpar <- par(mfrow = c(1, 2))
plot(x, alc.RNAmf$ALC$ALC1,
  type = "l", lty = 2,
  xlab = "x", ylab = "ALC criterion augmented at the low-fidelity level",
  ylim = c(min(c(alc.RNAmf$ALC$ALC1, alc.RNAmf$ALC$ALC2)),
           max(c(alc.RNAmf$ALC$ALC1, alc.RNAmf$ALC$ALC2)))
)
plot(x, alc.RNAmf$ALC$ALC2,
  type = "l", lty = 2,
  xlab = "x", ylab = "ALC criterion augmented at the high-fidelity level",
  ylim = c(min(c(alc.RNAmf$ALC$ALC1, alc.RNAmf$ALC$ALC2)),
           max(c(alc.RNAmf$ALC$ALC1, alc.RNAmf$ALC$ALC2)))
)
points(alc.RNAmf$chosen$Xnext,
  alc.RNAmf$ALC$ALC2[which(x == drop(alc.RNAmf$chosen$Xnext))],
  pch = 16, cex = 1, col = "red"
)
par(oldpar)