BACCO (version 1.0-50)

EK.eqn10.supp: Posterior mean of K

Description

Estimates the posterior mean of K as per equation 10 of KOH2001S, section 4.2

Usage

EK.eqn10.supp(X.dist, D1, D2, H1, H2, d, hbar.fun,
lower.theta,upper.theta, extractor, give.info, phi, ...)

Arguments

X.dist

Probability distribution of X, in the form of a two-element list. The first element is the mean (which should have name mean), and the second element is the variance matrix, which should be a positi

D1

Matrix whose rows are the code run points

D2

Matrix whose rows are field observation points

H1

Regression function for D1

H2

Regression function for D2

d

Vector of code outputs and field observations

hbar.fun

Function that gives expectation (with respect to X) of h1(x,theta) and h2(x) as per section 4.2

lower.theta

Lower integration limit for theta (NB: a vector)

upper.theta

Lower integration limit for theta (NB: a vector)

extractor

Extractor function; see extractor.toy() for an example

give.info

Boolean, with default FALSE meaning to return just the answer and TRUE to return the answer along with all output from both integrations as performed by adapt().

phi

Hyperparameters

...

Extra arguments passed to the integration function. If multidimensional (ie length(theta)>1), then the arguments are passed to adapt(); if one dimensional, they are passed to integrate().

Value

• Returns a scalar

Details

This function evaluates a numerical approximation to equation 10 of section 4.2 of the supplement.

Equation 10 integrates over the prior distribution of theta. If theta is a vector, multidimensional integration is necessary.

In the case of multidimensional integration, the eponymous adapt() is used. Note that, as of version 1.0-3, this is restricted to less than 20 dimensions---which is not checked for. Evaluation is slow, as multidimensional integration is hard (spot the understatement).

In the case of one dimensional integration---theta being a scalar---function integrate() of the stats package is used.

Note that equation 10 is conditional on the observed data and the hyperparameters.

References

M. C. Kennedy and A. O'Hagan 2001. Bayesian calibration of computer models. Journal of the Royal Statistical Society B, 63(3) pp425-464 M. C. Kennedy and A. O'Hagan 2001. Supplementary details on Bayesian calibration of computer models, Internal report, University of Sheffield. Available at http://www.shef.ac.uk/~st1ao/ps/calsup.ps R. K. S. Hankin 2005. Introducing BACCO, an R bundle for Bayesian analysis of computer code output, Journal of Statistical Software, 14(16)

Examples

1+1
# Not run because it takes R CMD check too long.
data(toys)
EK.eqn10.supp(X.dist=X.dist.toy, D1=D1.toy, D2=D2.toy, H1=H1.toy, H2=H2.toy, d=d.toy, hbar.fun=hbar.fun.toy, lower.theta=c(-3,-3,-3), upper.theta=c(3,3,3),extractor=extractor.toy, phi=phi.toy)