gek (version 1.2.0)

sigma.gekm: Extract Process Standard Deviation

Description

Extract the estimated process standard deviation of a Kriging model with or without derivatives.

Usage

# S3 method for gekm
sigma(object, scale = FALSE, ...)

Value

The (scaled) estimated process standard deviation.

Arguments

object

an object of class "gekm".

scale

logical. Should the estimated process standard deviation be scaled? Default is FALSE, see ‘Details’.

...

further arguments, currently not used.

Author

Carmen van Meegen

Details

By default, the process variance \(\sigma^2\) is estimated using the maximum likelihood estimator, which uses nobs in the denominator, where nobs is the total number of observations used to fit the model. Note for gradient-enhanced Kriging: nobs = n + n * d with n and d being the number of response values and inputs, respectively.

Setting scale = TRUE replaces the denominator nobs with nobs - p - 2, where p is the number of regression coefficients. If the correlation parameters are known and weak priors are assumed for the hyperparameters (the regression coefficients and the process variance), i.e., \(f(\beta, \sigma^2) \propto \sigma^{-2}\), this leads to the Bayesian estimator of the process variance.

References

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. 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. tools:::Rd_expr_doi("10.1093/biomet/89.4.769").

O'Hagan, A. (1991). Bayes-Hermite Quadrature. Journal of Statistical Planning an Inference, 29(3):245--260. tools:::Rd_expr_doi("10.1016/0378-3758(91)90002-V").

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.

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. tools:::Rd_expr_doi("10.1016/S0098-3004(00)00016-9").

Santner, T. J., Williams, B. J., and Notz, W. I. (2018). The Design and Analysis of Computer Experiments. 2nd edition. Springer-Verlag.

Zimmermann, R. (2015). On the Condition Number Anomaly of Gaussian Correlation Matrices. Linear Algebra and its Applications, 466:512-–526. tools:::Rd_expr_doi("10.1016/j.laa.2014.10.038").

See Also

gekm for fitting a (gradient-enhanced) Kriging model.

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 (gradient-enhanced) Kriging model
km.1d <- gekm(y ~ x, data = dat, covtype = "gaussian", theta = 1)
gekm.1d <- gekm(y ~ x, data = dat, deriv = deri, covtype = "gaussian", theta = 1)

# Extact estimated process standard deviation
sigma(km.1d)
sigma(gekm.1d)					
sigma(gekm.1d, scale = TRUE)