RNAmf (version 1.0.1)

ALC_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)

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.

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, <tools:::Rd_expr_doi("https://doi.org/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: \(\text{argmax}_{l\in\{1,\ldots,L\}; \bm{x} \in \Omega} \frac{\Delta \sigma_L^{2}(l,\bm{x})}{\sum^l_{j=1}C_j}\).

Examples

# \donttest{
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)# }