predict.RNAmf {RNAmf}

prediction of the RNAmf emulator with 2 or 3 fidelity levels.

Description

The function computes the posterior mean and variance of RNA models with two or three fidelity levels by fitted model using RNAmf_two_level or RNAmf_three_level.

Usage

## S3 method for class 'RNAmf'
predict(object, x, ...)

Arguments

object

a class RNAmf object fitted by RNAmf_two_level or RNAmf_three_level.

x

vector or matrix of new input locations to predict.

...

for compatibility with generic method predict.

Details

From the model fitted by RNAmf_two_level or RNAmf_three_level, the posterior mean and variance are calculated based on the closed form expression derived by a recursive fashion. The formulas depend on its kernel choices. For details, see Heo and Sung (2024, <doi:10.1080/00401706.2024.2376173>).

Value

• mu: vector of predictive posterior mean.

• sig2: vector of predictive posterior variance.

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

See Also

RNAmf_two_level or RNAmf_three_level for the model.

Examples

### two levels example ###
library(lhs)

### Perdikaris function ###
f1 <- function(x) {
  sin(8 * pi * x)
}

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

### RMSE ###
print(sqrt(mean((predy - f2(x))^2)))

### visualize the emulation performance ###
plot(x, predy,
  type = "l", lwd = 2, col = 3, # emulator and confidence interval
  ylim = c(-2, 1)
)
lines(x, predy + 1.96 * sqrt(predsig2 * length(y2) / (length(y2) - 2)), col = 3, lty = 2)
lines(x, predy - 1.96 * sqrt(predsig2 * length(y2) / (length(y2) - 2)), col = 3, lty = 2)

curve(f2(x), add = TRUE, col = 1, lwd = 2, lty = 2) # high fidelity function

points(X1, y1, pch = 1, col = "red") # low-fidelity design
points(X2, y2, pch = 4, col = "blue") # high-fidelity design

### three levels example ###
### Branin function ###
branin <- function(xx, l){
  x1 <- xx[1]
  x2 <- xx[2]
  if(l == 1){
    10*sqrt((-1.275*(1.2*x1+0.4)^2/pi^2+5*(1.2*x1+0.4)/pi+(1.2*x2+0.4)-6)^2 +
    (10-5/(4*pi))*cos((1.2*x1+0.4))+ 10) + 2*(1.2*x1+1.9) - 3*(3*(1.2*x2+2.4)-1) - 1 - 3*x2 + 1
  }else if(l == 2){
    10*sqrt((-1.275*(x1+2)^2/pi^2+5*(x1+2)/pi+(x2+2)-6)^2 +
    (10-5/(4*pi))*cos((x1+2))+ 10) + 2*(x1-0.5) - 3*(3*x2-1) - 1
  }else if(l == 3){
    (-1.275*x1^2/pi^2+5*x1/pi+x2-6)^2 + (10-5/(4*pi))*cos(x1)+ 10
  }
}

output.branin <- function(x, l){
  factor_range <- list("x1" = c(-5, 10), "x2" = c(0, 15))

  for(i in 1:length(factor_range)) x[i] <- factor_range[[i]][1] + x[i] * diff(factor_range[[i]])
  branin(x[1:2], l)
}

### training data ###
n1 <- 20; n2 <- 15; n3 <- 10

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

### generate initial nested design ###
X <- NestedX(c(n1, n2, n3), 2)
X1 <- X[[1]]
X2 <- X[[2]]
X3 <- X[[3]]

### n1, n2 and n3 might be changed from NestedX ###
### assign n1, n2 and n3 again ###
n1 <- nrow(X1)
n2 <- nrow(X2)
n3 <- nrow(X3)

y1 <- apply(X1,1,output.branin, l=1)
y2 <- apply(X2,1,output.branin, l=2)
y3 <- apply(X3,1,output.branin, l=3)

### n=10000 grid test data ###
x <- as.matrix(expand.grid(seq(0, 1, length.out = 100),seq(0, 1, length.out = 100)))

### fit an RNAmf ###
fit.RNAmf <- RNAmf_three_level(X1, y1, X2, y2, X3, y3, kernel = "sqex")

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

### RMSE ###
print(sqrt(mean((predy - apply(x,1,output.branin, l=3))^2)))

### visualize the emulation performance ###
x1 <- x2 <- seq(0, 1, length.out = 100)
oldpar <- par(mfrow=c(1,2))
image(x1, x2, matrix(apply(x,1,output.branin, l=3), ncol=100),
zlim=c(0,310), main="Branin function")
image(x1, x2, matrix(predy, ncol=100),
zlim=c(0,310), main="RNAmf prediction")
par(oldpar)

### predictive variance ###
print(predsig2)