predict.gekm: Predict Method for (Gradient-Enhanced) Kriging Model Fits
In gek: Gradient-Enhanced Kriging
predict.gekm R Documentation
Predict Method for (Gradient-Enhanced) Kriging Model Fits
Description
Predicted values and standard deviations based on a gekm model.
Usage
## S3 method for class 'gekm'
predict(object, newdata, ...)
Arguments
object
an object of class "gekm".
newdata
a data.frame containing the points where to perform predictions.
...
further arguments, currently not used.
Value
fit
predicted mean computed at newdata.
sd.fit
predicted standard deviation of predicted mean.
Author(s)
Carmen van Meegen
References
Cressie, N. A. C. (1993). Statistics for Spartial Data. John Wiley & Sons. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1002/9781119115151")}.
Koehler, J. and Owen, A. (1996). Computer Experiments. In Ghosh, S. and Rao, C. (eds.), Design and Analysis of Experiments, volume 13 of Handbook of Statistics, pp. 261–308. Elsevier Science. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/S0169-7161(96)13011-X")}.
Krige, D. G. (1951). A Statistical Approach to Some Basic Mine Valuation Problems on the Witwatersrand. Journal of the Southern African Institute of Mining and Metallurgy, 52(6):199–139.
Laurent, L., Le Riche, R., Soulier, B., and Boucard, PA. (2019). An Overview of Gradient-Enhanced Metamodels with Applications. Archives of Computational Methods in Engineering, 26(1):61–106. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/s11831-017-9226-3")}.
Martin, J. D. and Simpson, T. W. (2005). Use of Kriging Models to Approximate Deterministic Computer Models. AIAA Journal, 43(4):853–863. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.2514/1.8650")}.
Morris, M., Mitchell, T., and Ylvisaker, D. (1993). Bayesian Design and Analysis of Computer Experiments: Use of Derivatives in Surface Prediction. Technometrics, 35(3):243–255. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/00401706.1993.10485320")}.
Oakley, J. and O'Hagan, A. (2002). Bayesian Inference for the Uncertainty Distribution of Computer Model Outputs. Biometrika, 89(4):769–784. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1093/biomet/89.4.769")}.
O'Hagan, A., Kennedy, M. C., and Oakley, J. E. (1999). Uncertainty Analysis and Other Inference Tools for Complex Computer Codes. In Bayesian Statistics 6, Ed. J. M. Bernardo, J. O. Berger, A. P. Dawid and A .F. M. Smith, 503–524. Oxford University Press.
O'Hagan, A. (2006). Bayesian Analysis of Computer Code Outputs: A Tutorial. Reliability Engineering & System Safet, 91(10):1290–1300. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/j.ress.2005.11.025")}.
Park, J.-S. and Beak, J. (2001). Efficient Computation of Maximum Likelihood Estimators in a Spatial Linear Model with Power Exponential Covariogram. Computers & Geosciences, 27(1):1–7. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/S0098-3004(00)00016-9")}.
Ranjan, P., Haynes, R. and Karsten, R. (2011). A Computationally Stable Approach to Gaussian Process Interpolation of Deterministic Computer Simulation Data. Technometrics, 53:366–378. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1198/TECH.2011.09141")}.
Rasmussen, C. E. and Williams, C. K. I. (2006). Gaussian Processes for Machine Learning. The MIT Press. https://gaussianprocess.org/gpml/.
Ripley, B. D. (1981). Spatial Statistics. John Wiley & Sons. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1002/0471725218")}.
Sacks, J., Welch, W. J., Mitchell, T. J., and Wynn, H. P. (1989). Design and Analysis of Computer Experiments. Statistical Science, 4(4):409–423. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1214/ss/1177012413")}.
Santner, T. J., Williams, B. J., and Notz, W. I. (2018). The Design and Analysis of Computer Experiments. 2nd edition. Springer-Verlag.
Stein, M. L. (1999). Interpolation of Spatial Data: Some Theory for Kriging. Springer Series in Statistics. Springer-Verlag. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/978-1-4612-1494-6")}.
See Also
gekm for fitting a (gradient-enhanced) Kriging model.
tangents for drawing tangent lines.
Examples
## 1-dimensional example: Oakley and O’Hagan (2002)
# Define test function and its gradient
f <- function(x) 5 + x + cos(x)
fGrad <- function(x) 1 - sin(x)
# Generate coordinates and calculate slopes
x <- seq(-5, 5, length = 5)
y <- f(x)
dy <- fGrad(x)
dat <- data.frame(x, y)
deri <- data.frame(x = dy)
# Fit Kriging model
km.1d <- gekm(y ~ x, data = dat, covtype = "gaussian", theta = 1)
# Fit gradient-enhanced Kriging model
gekm.1d <- gekm(y ~ x, data = dat, deriv = deri, covtype = "gaussian", theta = 1)
# Generate new data for prediction
newdat <- data.frame(x = seq(-6, 6, length = 100))
# Prediction for both models
pred.km.1d <- predict(km.1d, newdat)
pred.gekm.1d <- predict(gekm.1d, newdat)
# Draw curves
curve(f(x), from = -6, to = 6, lwd = 2)
lines(newdat$x, pred.km.1d$fit, type = "l", col = rgb(0.5, 0.5, 0.5), lwd = 2, lty = 2)
lines(newdat$x, pred.gekm.1d$fit, type = "l", col = rgb(0, 0.5, 1), lwd = 2, lty = 2)
# Add pointswise confidence intervals
lower.km.1d <- pred.km.1d$fit - qnorm(0.975) * pred.km.1d$sd
upper.km.1d <- pred.km.1d$fit + qnorm(0.975) * pred.km.1d$sd
lower.gekm.1d <- pred.gekm.1d$fit - qnorm(0.975) * pred.gekm.1d$sd
upper.gekm.1d <- pred.gekm.1d$fit + qnorm(0.975) * pred.gekm.1d$sd
polygon(c(newdat$x, rev(newdat$x)), c(lower.km.1d, rev(upper.km.1d)),
col = rgb(0.5, 0.5, 0.5, alpha = 0.1), border = NA)
polygon(c(newdat$x, rev(newdat$x)), c(lower.gekm.1d, rev(upper.gekm.1d)),
col = rgb(0, 0.5, 1, alpha = 0.1), border = NA)
# Add tangent lines and observations
tangents(x, y, dy, lwd = 2, col = "red")
points(x, y, pch = 16)
# Add legend
legend("topleft", lty = c(1, 2, 2), col = c("black", rgb(0.5, 0.5, 0.5), rgb(0, 0.5, 1)),
legend = c("Test function", "Kriging", "GEK"), lwd = 2, bty = "n")
## 2-dimensional example: Branin-Hoo function
# Generate a grid for training
n <- 5
x1 <- seq(-5, 10, length = n)
x2 <- seq(0, 15, length = n)
x <- expand.grid(x1 = x1, x2 = x2)
y <- branin(x)
dy <- braninGrad(x)
dat <- data.frame(x, y)
deri <- data.frame(dy)
# Fit Kriging model
km.2d <- gekm(y ~ ., data = dat)
km.2d
# Fit gradient-enhanced Kriging model
gekm.2d <- gekm(y ~ ., data = dat, deriv = deri)
gekm.2d
# Generate new data for prediction
n.grid <- 50
x1.grid <- seq(-5, 10, length = n.grid)
x2.grid <- seq(0, 15, length = n.grid)
newdat <- expand.grid(x1 = x1.grid, x2 = x2.grid)
# Prediction for both models and actual outcome
pred.km.2d <- predict(km.2d, newdat)
pred.gekm.2d <- predict(gekm.2d, newdat)
truth <- outer(x1.grid, x2.grid, function(x1, x2) branin(cbind(x1, x2)))
# Contour plots of predicted and actual output
par(mfrow = c(1, 3), oma = c(3.5, 3.5, 0, 0.2), mar = c(0, 0, 1.5, 0))
contour(x1.grid, x2.grid, truth, nlevels = 30,
levels = seq(0, 300, 10), main = "Branin-Hoo")
points(x, pch = 16)
contour(x1.grid, x2.grid, matrix(pred.km.2d$fit, nrow = n.grid, ncol = n.grid),
levels = seq(0, 300, 10), main = "Kriging", yaxt = "n")
points(x, pch = 16)
contour(x1.grid, x2.grid, matrix(pred.gekm.2d$fit, nrow = n.grid, ncol = n.grid),
levels = seq(0, 300, 10), main = "GEK", yaxt = "n")
points(x, pch = 16)
mtext(side = 1, outer = TRUE, line = 2.5, expression(x[1]))
mtext(side = 2, outer = TRUE, line = 2, expression(x[2]))
# Contour plots of predicted variance
par(mfrow = c(1, 2), oma = c(3.5, 3.5, 0, 0.2), mar = c(0, 0, 1.5, 0))
contour(x1.grid, x2.grid, matrix(pred.km.2d$sd.fit^2, nrow = n.grid, ncol = n.grid),
main = "Kriging variance")
points(x, pch = 16)
contour(x1.grid, x2.grid, matrix(pred.gekm.2d$sd.fit^2, nrow = n.grid, ncol = n.grid),
main = "GEK variance", yaxt = "n")
points(x, pch = 16)
mtext(side = 1, outer = TRUE, line = 2.5, expression(x[1]))
mtext(side = 2, outer = TRUE, line = 2, expression(x[2]))
gek documentation built on April 4, 2025, 12:35 a.m.
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| predict.gekm | R Documentation |
Predict Method for (Gradient-Enhanced) Kriging Model Fits
Description
Predicted values and standard deviations based on a gekm model.
Usage
## S3 method for class 'gekm'
predict(object, newdata, ...)
Arguments
object |
an object of class |
newdata |
a |
... |
further arguments, currently not used. |
Value
fit |
predicted mean computed at newdata. |
sd.fit |
predicted standard deviation of predicted mean. |
Author(s)
Carmen van Meegen
References
Cressie, N. A. C. (1993). Statistics for Spartial Data. John Wiley & Sons. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1002/9781119115151")}.
Koehler, J. and Owen, A. (1996). Computer Experiments. In Ghosh, S. and Rao, C. (eds.), Design and Analysis of Experiments, volume 13 of Handbook of Statistics, pp. 261–308. Elsevier Science. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/S0169-7161(96)13011-X")}.
Krige, D. G. (1951). A Statistical Approach to Some Basic Mine Valuation Problems on the Witwatersrand. Journal of the Southern African Institute of Mining and Metallurgy, 52(6):199–139.
Laurent, L., Le Riche, R., Soulier, B., and Boucard, PA. (2019). An Overview of Gradient-Enhanced Metamodels with Applications. Archives of Computational Methods in Engineering, 26(1):61–106. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/s11831-017-9226-3")}.
Martin, J. D. and Simpson, T. W. (2005). Use of Kriging Models to Approximate Deterministic Computer Models. AIAA Journal, 43(4):853–863. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.2514/1.8650")}.
Morris, M., Mitchell, T., and Ylvisaker, D. (1993). Bayesian Design and Analysis of Computer Experiments: Use of Derivatives in Surface Prediction. Technometrics, 35(3):243–255. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/00401706.1993.10485320")}.
Oakley, J. and O'Hagan, A. (2002). Bayesian Inference for the Uncertainty Distribution of Computer Model Outputs. Biometrika, 89(4):769–784. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1093/biomet/89.4.769")}.
O'Hagan, A., Kennedy, M. C., and Oakley, J. E. (1999). Uncertainty Analysis and Other Inference Tools for Complex Computer Codes. In Bayesian Statistics 6, Ed. J. M. Bernardo, J. O. Berger, A. P. Dawid and A .F. M. Smith, 503–524. Oxford University Press.
O'Hagan, A. (2006). Bayesian Analysis of Computer Code Outputs: A Tutorial. Reliability Engineering & System Safet, 91(10):1290–1300. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/j.ress.2005.11.025")}.
Park, J.-S. and Beak, J. (2001). Efficient Computation of Maximum Likelihood Estimators in a Spatial Linear Model with Power Exponential Covariogram. Computers & Geosciences, 27(1):1–7. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/S0098-3004(00)00016-9")}.
Ranjan, P., Haynes, R. and Karsten, R. (2011). A Computationally Stable Approach to Gaussian Process Interpolation of Deterministic Computer Simulation Data. Technometrics, 53:366–378. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1198/TECH.2011.09141")}.
Rasmussen, C. E. and Williams, C. K. I. (2006). Gaussian Processes for Machine Learning. The MIT Press. https://gaussianprocess.org/gpml/.
Ripley, B. D. (1981). Spatial Statistics. John Wiley & Sons. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1002/0471725218")}.
Sacks, J., Welch, W. J., Mitchell, T. J., and Wynn, H. P. (1989). Design and Analysis of Computer Experiments. Statistical Science, 4(4):409–423. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1214/ss/1177012413")}.
Santner, T. J., Williams, B. J., and Notz, W. I. (2018). The Design and Analysis of Computer Experiments. 2nd edition. Springer-Verlag.
Stein, M. L. (1999). Interpolation of Spatial Data: Some Theory for Kriging. Springer Series in Statistics. Springer-Verlag. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/978-1-4612-1494-6")}.
See Also
gekm for fitting a (gradient-enhanced) Kriging model.
tangents for drawing tangent lines.
Examples
## 1-dimensional example: Oakley and O’Hagan (2002)
# Define test function and its gradient
f <- function(x) 5 + x + cos(x)
fGrad <- function(x) 1 - sin(x)
# Generate coordinates and calculate slopes
x <- seq(-5, 5, length = 5)
y <- f(x)
dy <- fGrad(x)
dat <- data.frame(x, y)
deri <- data.frame(x = dy)
# Fit Kriging model
km.1d <- gekm(y ~ x, data = dat, covtype = "gaussian", theta = 1)
# Fit gradient-enhanced Kriging model
gekm.1d <- gekm(y ~ x, data = dat, deriv = deri, covtype = "gaussian", theta = 1)
# Generate new data for prediction
newdat <- data.frame(x = seq(-6, 6, length = 100))
# Prediction for both models
pred.km.1d <- predict(km.1d, newdat)
pred.gekm.1d <- predict(gekm.1d, newdat)
# Draw curves
curve(f(x), from = -6, to = 6, lwd = 2)
lines(newdat$x, pred.km.1d$fit, type = "l", col = rgb(0.5, 0.5, 0.5), lwd = 2, lty = 2)
lines(newdat$x, pred.gekm.1d$fit, type = "l", col = rgb(0, 0.5, 1), lwd = 2, lty = 2)
# Add pointswise confidence intervals
lower.km.1d <- pred.km.1d$fit - qnorm(0.975) * pred.km.1d$sd
upper.km.1d <- pred.km.1d$fit + qnorm(0.975) * pred.km.1d$sd
lower.gekm.1d <- pred.gekm.1d$fit - qnorm(0.975) * pred.gekm.1d$sd
upper.gekm.1d <- pred.gekm.1d$fit + qnorm(0.975) * pred.gekm.1d$sd
polygon(c(newdat$x, rev(newdat$x)), c(lower.km.1d, rev(upper.km.1d)),
col = rgb(0.5, 0.5, 0.5, alpha = 0.1), border = NA)
polygon(c(newdat$x, rev(newdat$x)), c(lower.gekm.1d, rev(upper.gekm.1d)),
col = rgb(0, 0.5, 1, alpha = 0.1), border = NA)
# Add tangent lines and observations
tangents(x, y, dy, lwd = 2, col = "red")
points(x, y, pch = 16)
# Add legend
legend("topleft", lty = c(1, 2, 2), col = c("black", rgb(0.5, 0.5, 0.5), rgb(0, 0.5, 1)),
legend = c("Test function", "Kriging", "GEK"), lwd = 2, bty = "n")
## 2-dimensional example: Branin-Hoo function
# Generate a grid for training
n <- 5
x1 <- seq(-5, 10, length = n)
x2 <- seq(0, 15, length = n)
x <- expand.grid(x1 = x1, x2 = x2)
y <- branin(x)
dy <- braninGrad(x)
dat <- data.frame(x, y)
deri <- data.frame(dy)
# Fit Kriging model
km.2d <- gekm(y ~ ., data = dat)
km.2d
# Fit gradient-enhanced Kriging model
gekm.2d <- gekm(y ~ ., data = dat, deriv = deri)
gekm.2d
# Generate new data for prediction
n.grid <- 50
x1.grid <- seq(-5, 10, length = n.grid)
x2.grid <- seq(0, 15, length = n.grid)
newdat <- expand.grid(x1 = x1.grid, x2 = x2.grid)
# Prediction for both models and actual outcome
pred.km.2d <- predict(km.2d, newdat)
pred.gekm.2d <- predict(gekm.2d, newdat)
truth <- outer(x1.grid, x2.grid, function(x1, x2) branin(cbind(x1, x2)))
# Contour plots of predicted and actual output
par(mfrow = c(1, 3), oma = c(3.5, 3.5, 0, 0.2), mar = c(0, 0, 1.5, 0))
contour(x1.grid, x2.grid, truth, nlevels = 30,
levels = seq(0, 300, 10), main = "Branin-Hoo")
points(x, pch = 16)
contour(x1.grid, x2.grid, matrix(pred.km.2d$fit, nrow = n.grid, ncol = n.grid),
levels = seq(0, 300, 10), main = "Kriging", yaxt = "n")
points(x, pch = 16)
contour(x1.grid, x2.grid, matrix(pred.gekm.2d$fit, nrow = n.grid, ncol = n.grid),
levels = seq(0, 300, 10), main = "GEK", yaxt = "n")
points(x, pch = 16)
mtext(side = 1, outer = TRUE, line = 2.5, expression(x[1]))
mtext(side = 2, outer = TRUE, line = 2, expression(x[2]))
# Contour plots of predicted variance
par(mfrow = c(1, 2), oma = c(3.5, 3.5, 0, 0.2), mar = c(0, 0, 1.5, 0))
contour(x1.grid, x2.grid, matrix(pred.km.2d$sd.fit^2, nrow = n.grid, ncol = n.grid),
main = "Kriging variance")
points(x, pch = 16)
contour(x1.grid, x2.grid, matrix(pred.gekm.2d$sd.fit^2, nrow = n.grid, ncol = n.grid),
main = "GEK variance", yaxt = "n")
points(x, pch = 16)
mtext(side = 1, outer = TRUE, line = 2.5, expression(x[1]))
mtext(side = 2, outer = TRUE, line = 2, expression(x[2]))
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