AL_RNAmf: Active Learning for Recursive Non-Additive Emulator
In RNAmf: Recursive Non-Additive Emulator for Multi-Fidelity Data
AL_RNAmf R Documentation
Active Learning for Recursive Non-Additive Emulator
Description
The function acquires the new point and fidelity level by maximizing one of the
four active learning criteria: ALM, ALC, ALD, or ALMC.
-
ALM (Active Learning MacKay): It calculates the ALM criterion \frac{\sigma^{*2}_l(\bm{x})}{\sum^l_{j=1}C_j},
where \sigma^{*2}_l(\bm{x}) is the posterior predictive variance
at each fidelity level l and C_j is the simulation cost at level j.
-
ALD (Active Learning Decomposition): It calculates the ALD criterion \frac{V_l(\bm{x})}{\sum^l_{j=1}C_j},
where V_l(\bm{x}) is the variance contribution of GP emulator
at each fidelity level l and C_j is the simulation cost at level j.
-
ALC (Active Learning Cohn): 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.
-
ALMC (Active Learning MacKay-Cohn): A hybrid approach.
It finds the optimal input location \bm{x}^*
by maximizing \sigma^{*2}_L(\bm{x}),
the posterior predictive variance at the highest-fidelity level L.
After selecting \bm{x}^*,
it finds the optimal fidelity level by maximizing ALC criterion at \bm{x}^*,
\text{argmax}_{l\in\{1,\ldots,L\}} \frac{\Delta \sigma_L^{2}(l,\bm{x}^*)}{\sum^l_{j=1}C_j},
where C_j is the simulation cost at level j.
A new point is acquired on Xcand. If Xcand=NULL and Xref=NULL, a new point is acquired on unit hypercube [0,1]^d.
For details, see Heo and Sung (2025, <\Sexpr[results=rd]{tools:::Rd_expr_doi("https://doi.org/10.1080/00401706.2024.2376173")}>).
Usage
AL_RNAmf(criterion = c("ALM", "ALC", "ALD", "ALMC"), fit,
Xref = NULL, Xcand = NULL, MC = FALSE, mc.sample = 100, cost = NULL,
use_optim = TRUE, parallel = FALSE, ncore = 1, trace = TRUE)
Arguments
criterion
character string specifying the active learning criterion to use. Must be one of "ALM", "ALD", "ALC", or "ALMC". Default is "ALM".
fit
object of class RNAmf.
Xref
vector or matrix of reference locations to approximate the integral of ALC. If Xref=NULL, 100 \times d points from 0 to 1 are generated by Latin hypercube design. Only used when criterion="ALC" or "ALMC". Default is NULL.
Xcand
vector or matrix of the candidate set for grid-based search. If use_optim=FALSE, the criterion is evaluated and optimized only on this set. If Xcand=NULL, 100 \times d points from 0 to 1 generated by Latin hypercube design (or Xref for ALC and ALMC) are used. Default is NULL.
MC
logical indicating whether to use Monte Carlo approximation to impute the posterior variance (for ALC/ALMC). If FALSE, posterior means are used. Default is FALSE.
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 an ascending order of positive values. Default is NULL.
use_optim
logical indicating whether to optimize the criterion using optim's gradient-based L-BFGS-B method. If TRUE, 5 \times d starting points are generated by Latin hypercube design for optimization. If FALSE, the point is selected from Xcand. Default is TRUE.
parallel
logical indicating whether to use parallel computation. Default is FALSE.
ncore
integer specifying the number of cores for parallel computation. Used only if parallel=TRUE. Default is 1.
trace
logical indicating whether to print the computational progress and time. Default is TRUE.
Details
For "ALC", or "ALMC", 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.
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 at
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
A list containing:
-
AL: The values of the selected criterion at the candidate points (if use_optim=FALSE) or the optimized value (if use_optim=TRUE). For ALMC, it returns the ALC scores for each level at the chosen point.
-
cost: A copy of the cost argument.
-
Xcand: A copy of the Xcand argument used.
-
chosen: A list containing the chosen fidelity level and the new input location Xnext.
-
time: The computation time in seconds.
Examples
### 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]]
y1 <- f1(X1)
y2 <- f2(X2)
### n=100 uniform test data ###
x <- seq(0, 1, length.out = 100)
### fit an RNAmf ###
fit.RNAmf <- RNAmf(list(X1, X2), list(y1, y2), kernel = "sqex", constant=TRUE)
### 1. ALM Criterion ###
alm.RNAmf <- AL_RNAmf(criterion="ALM",
Xcand = x, fit=fit.RNAmf, cost = cost,
use_optim = FALSE, parallel = TRUE, ncore = 2)
print(alm.RNAmf$chosen)
### visualize ALM ###
oldpar <- par(mfrow = c(1, 2))
plot(x, alm.RNAmf$AL$ALM1,
type = "l", lty = 2,
xlab = "x", ylab = "ALM criterion at the low-fidelity level",
ylim = c(min(c(alm.RNAmf$AL$ALM1, alm.RNAmf$AL$ALM2)),
max(c(alm.RNAmf$AL$ALM1, alm.RNAmf$AL$ALM2))))
points(alm.RNAmf$chosen$Xnext,
alm.RNAmf$AL$ALM1[which(x == drop(alm.RNAmf$chosen$Xnext))],
pch = 16, cex = 1, col = "red")
plot(x, alm.RNAmf$AL$ALM2,
type = "l", lty = 2,
xlab = "x", ylab = "ALM criterion at the high-fidelity level",
ylim = c(min(c(alm.RNAmf$AL$ALM1, alm.RNAmf$AL$ALM2)),
max(c(alm.RNAmf$AL$ALM1, alm.RNAmf$AL$ALM2))))
par(oldpar)
### 2. ALD Criterion ###
ald.RNAmf <- AL_RNAmf(criterion="ALD",
Xcand = x, fit=fit.RNAmf, cost = cost,
use_optim = FALSE, parallel = TRUE, ncore = 2)
print(ald.RNAmf$chosen)
### visualize ALD ###
oldpar <- par(mfrow = c(1, 2))
plot(x, ald.RNAmf$AL$ALD1,
type = "l", lty = 2,
xlab = "x", ylab = "ALD criterion at the low-fidelity level",
ylim = c(min(c(ald.RNAmf$AL$ALD1, ald.RNAmf$AL$ALD2)),
max(c(ald.RNAmf$AL$ALD1, ald.RNAmf$AL$ALD2))))
points(ald.RNAmf$chosen$Xnext,
ald.RNAmf$AL$ALD1[which(x == drop(ald.RNAmf$chosen$Xnext))],
pch = 16, cex = 1, col = "red")
plot(x, ald.RNAmf$AL$ALD2,
type = "l", lty = 2,
xlab = "x", ylab = "ALD criterion at the high-fidelity level",
ylim = c(min(c(ald.RNAmf$AL$ALD1, ald.RNAmf$AL$ALD2)),
max(c(ald.RNAmf$AL$ALD1, ald.RNAmf$AL$ALD2))))
par(oldpar)
### 3. ALC Criterion ###
alc.RNAmf <- AL_RNAmf(criterion="ALC",
Xref = x, Xcand = x, fit=fit.RNAmf, cost = cost,
use_optim = FALSE, parallel = TRUE, ncore = 2)
print(alc.RNAmf$chosen)
### visualize ALC ###
oldpar <- par(mfrow = c(1, 2))
plot(x, alc.RNAmf$AL$ALC1,
type = "l", lty = 2,
xlab = "x", ylab = "ALC criterion augmented at the low-fidelity level",
ylim = c(min(c(alc.RNAmf$AL$ALC1, alc.RNAmf$AL$ALC2)),
max(c(alc.RNAmf$AL$ALC1, alc.RNAmf$AL$ALC2))))
points(alc.RNAmf$chosen$Xnext,
alc.RNAmf$AL$ALC1[which(x == drop(alc.RNAmf$chosen$Xnext))],
pch = 16, cex = 1, col = "red")
plot(x, alc.RNAmf$AL$ALC2,
type = "l", lty = 2,
xlab = "x", ylab = "ALC criterion augmented at the high-fidelity level",
ylim = c(min(c(alc.RNAmf$AL$ALC1, alc.RNAmf$AL$ALC2)),
max(c(alc.RNAmf$AL$ALC1, alc.RNAmf$AL$ALC2))))
par(oldpar)
### 4. ALMC Criterion ###
almc.RNAmf <- AL_RNAmf(criterion="ALMC",
Xref = x, Xcand = x, fit=fit.RNAmf, cost = cost,
use_optim = FALSE, parallel = TRUE, ncore = 2)
print(almc.RNAmf$chosen)
RNAmf documentation built on May 5, 2026, 9:06 a.m.
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| AL_RNAmf | R Documentation |
Active Learning for Recursive Non-Additive Emulator
Description
The function acquires the new point and fidelity level by maximizing one of the four active learning criteria: ALM, ALC, ALD, or ALMC.
-
ALM (Active Learning MacKay): It calculates the ALM criterion
\frac{\sigma^{*2}_l(\bm{x})}{\sum^l_{j=1}C_j}, where\sigma^{*2}_l(\bm{x})is the posterior predictive variance at each fidelity levellandC_jis the simulation cost at levelj. -
ALD (Active Learning Decomposition): It calculates the ALD criterion
\frac{V_l(\bm{x})}{\sum^l_{j=1}C_j}, whereV_l(\bm{x})is the variance contribution of GP emulator at each fidelity levellandC_jis the simulation cost at levelj. -
ALC (Active Learning Cohn): 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}, wheref_Lis the highest-fidelity simulation code,\sigma_L^{*2}(\bm{\xi})is the posterior variance off_L(\bm{\xi}),C_jis the simulation cost at fidelity levelj, 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 tol. The integration is approximated by MC integration using uniform reference samples. -
ALMC (Active Learning MacKay-Cohn): A hybrid approach. It finds the optimal input location
\bm{x}^*by maximizing\sigma^{*2}_L(\bm{x}), the posterior predictive variance at the highest-fidelity levelL. After selecting\bm{x}^*, it finds the optimal fidelity level by maximizing ALC criterion at\bm{x}^*,\text{argmax}_{l\in\{1,\ldots,L\}} \frac{\Delta \sigma_L^{2}(l,\bm{x}^*)}{\sum^l_{j=1}C_j}, whereC_jis the simulation cost at levelj.
A new point is acquired on Xcand. If Xcand=NULL and Xref=NULL, a new point is acquired on unit hypercube [0,1]^d.
For details, see Heo and Sung (2025, <\Sexpr[results=rd]{tools:::Rd_expr_doi("https://doi.org/10.1080/00401706.2024.2376173")}>).
Usage
AL_RNAmf(criterion = c("ALM", "ALC", "ALD", "ALMC"), fit,
Xref = NULL, Xcand = NULL, MC = FALSE, mc.sample = 100, cost = NULL,
use_optim = TRUE, parallel = FALSE, ncore = 1, trace = TRUE)
Arguments
criterion |
character string specifying the active learning criterion to use. Must be one of |
fit |
object of class |
Xref |
vector or matrix of reference locations to approximate the integral of ALC. If |
Xcand |
vector or matrix of the candidate set for grid-based search. If |
MC |
logical indicating whether to use Monte Carlo approximation to impute the posterior variance (for ALC/ALMC). If |
mc.sample |
a number of MC samples generated for the imputation through MC approximation. Default is |
cost |
vector of the costs for each level of fidelity. If |
use_optim |
logical indicating whether to optimize the criterion using |
parallel |
logical indicating whether to use parallel computation. Default is |
ncore |
integer specifying the number of cores for parallel computation. Used only if |
trace |
logical indicating whether to print the computational progress and time. Default is |
Details
For "ALC", or "ALMC", 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.
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 at
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
A list containing:
-
AL: The values of the selected criterion at the candidate points (ifuse_optim=FALSE) or the optimized value (ifuse_optim=TRUE). ForALMC, it returns the ALC scores for each level at the chosen point. -
cost: A copy of thecostargument. -
Xcand: A copy of theXcandargument used. -
chosen: A list containing the chosen fidelityleveland the new input locationXnext. -
time: The computation time in seconds.
Examples
### 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]]
y1 <- f1(X1)
y2 <- f2(X2)
### n=100 uniform test data ###
x <- seq(0, 1, length.out = 100)
### fit an RNAmf ###
fit.RNAmf <- RNAmf(list(X1, X2), list(y1, y2), kernel = "sqex", constant=TRUE)
### 1. ALM Criterion ###
alm.RNAmf <- AL_RNAmf(criterion="ALM",
Xcand = x, fit=fit.RNAmf, cost = cost,
use_optim = FALSE, parallel = TRUE, ncore = 2)
print(alm.RNAmf$chosen)
### visualize ALM ###
oldpar <- par(mfrow = c(1, 2))
plot(x, alm.RNAmf$AL$ALM1,
type = "l", lty = 2,
xlab = "x", ylab = "ALM criterion at the low-fidelity level",
ylim = c(min(c(alm.RNAmf$AL$ALM1, alm.RNAmf$AL$ALM2)),
max(c(alm.RNAmf$AL$ALM1, alm.RNAmf$AL$ALM2))))
points(alm.RNAmf$chosen$Xnext,
alm.RNAmf$AL$ALM1[which(x == drop(alm.RNAmf$chosen$Xnext))],
pch = 16, cex = 1, col = "red")
plot(x, alm.RNAmf$AL$ALM2,
type = "l", lty = 2,
xlab = "x", ylab = "ALM criterion at the high-fidelity level",
ylim = c(min(c(alm.RNAmf$AL$ALM1, alm.RNAmf$AL$ALM2)),
max(c(alm.RNAmf$AL$ALM1, alm.RNAmf$AL$ALM2))))
par(oldpar)
### 2. ALD Criterion ###
ald.RNAmf <- AL_RNAmf(criterion="ALD",
Xcand = x, fit=fit.RNAmf, cost = cost,
use_optim = FALSE, parallel = TRUE, ncore = 2)
print(ald.RNAmf$chosen)
### visualize ALD ###
oldpar <- par(mfrow = c(1, 2))
plot(x, ald.RNAmf$AL$ALD1,
type = "l", lty = 2,
xlab = "x", ylab = "ALD criterion at the low-fidelity level",
ylim = c(min(c(ald.RNAmf$AL$ALD1, ald.RNAmf$AL$ALD2)),
max(c(ald.RNAmf$AL$ALD1, ald.RNAmf$AL$ALD2))))
points(ald.RNAmf$chosen$Xnext,
ald.RNAmf$AL$ALD1[which(x == drop(ald.RNAmf$chosen$Xnext))],
pch = 16, cex = 1, col = "red")
plot(x, ald.RNAmf$AL$ALD2,
type = "l", lty = 2,
xlab = "x", ylab = "ALD criterion at the high-fidelity level",
ylim = c(min(c(ald.RNAmf$AL$ALD1, ald.RNAmf$AL$ALD2)),
max(c(ald.RNAmf$AL$ALD1, ald.RNAmf$AL$ALD2))))
par(oldpar)
### 3. ALC Criterion ###
alc.RNAmf <- AL_RNAmf(criterion="ALC",
Xref = x, Xcand = x, fit=fit.RNAmf, cost = cost,
use_optim = FALSE, parallel = TRUE, ncore = 2)
print(alc.RNAmf$chosen)
### visualize ALC ###
oldpar <- par(mfrow = c(1, 2))
plot(x, alc.RNAmf$AL$ALC1,
type = "l", lty = 2,
xlab = "x", ylab = "ALC criterion augmented at the low-fidelity level",
ylim = c(min(c(alc.RNAmf$AL$ALC1, alc.RNAmf$AL$ALC2)),
max(c(alc.RNAmf$AL$ALC1, alc.RNAmf$AL$ALC2))))
points(alc.RNAmf$chosen$Xnext,
alc.RNAmf$AL$ALC1[which(x == drop(alc.RNAmf$chosen$Xnext))],
pch = 16, cex = 1, col = "red")
plot(x, alc.RNAmf$AL$ALC2,
type = "l", lty = 2,
xlab = "x", ylab = "ALC criterion augmented at the high-fidelity level",
ylim = c(min(c(alc.RNAmf$AL$ALC1, alc.RNAmf$AL$ALC2)),
max(c(alc.RNAmf$AL$ALC1, alc.RNAmf$AL$ALC2))))
par(oldpar)
### 4. ALMC Criterion ###
almc.RNAmf <- AL_RNAmf(criterion="ALMC",
Xref = x, Xcand = x, fit=fit.RNAmf, cost = cost,
use_optim = FALSE, parallel = TRUE, ncore = 2)
print(almc.RNAmf$chosen)
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