calibrator (version 1.2-9)

cov.p5.supp: Covariance function for posterior distribution of z

Description

Covariance function for posterior distribution of \(z(\cdot)\) conditional on estimated hyperparameters and calibration parameters \(\theta\).

Usage

Cov.eqn9.supp(x, xdash=NULL, theta, d, D1, D2, H1, H2, phi)
cov.p5.supp  (x, xdash=NULL, theta, d, D1, D2, H1, H2, phi)

Value

Returns a matrix of covariances

Arguments

x

first point, or (Cov.eqn9.supp()) a matrix whose rows are the points of interest

xdash

The second point, or (Cov.eqn9.supp()) a matrix whose rows are the points of interest. The default of NULL means to use xdash=x

theta

Parameters. For Cov.eqn9.supp(), supply a vector which will be interpreted as a single point in parameter space. For cov.p5.supp(), supply a matrix whose rows will be interpreted as points in parameter space

d

Observed values

D1

Code run design matrix

D2

Observation points of real process

H1

Basis function for D1

H2

Basis function for D2

phi

Hyperparameters

Author

Robin K. S. Hankin

Details

Evaluates the covariance function: the last formula on page 5 of the supplement. The two functions documented here are vectorized differently.

Function Cov.eqn9.supp() takes matrices for arguments x and xdash and a single vector for theta. Evaluation is thus taken at a single, fixed value of theta. The function returns a matrix whose rows correspond to rows of x and whose columns correspond to rows of xdash.

Function cov.p5.supp() takes a vector for arguments x and xdash and a matrix for argument theta whose rows are the points in parameter space. A vector V, with elements corresponding to the rows of argument theta is returned: $$V[i] = \mbox{cov}\left(z(x),z(x')|\theta_i\right)$$

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.tonyohagan.co.uk/academic/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

data(toys)
x <- rbind(x.toy,x.toy+1,x.toy,x.toy,x.toy)
rownames(x) <- letters[1:5]
xdash <- rbind(x*2,x.toy)
rownames(xdash) <- LETTERS[1:6]

Cov.eqn9.supp(x=x,xdash=xdash,theta=theta.toy,d=d.toy,D1=D1.toy,
    D2=D2.toy,H1=H1.toy,H2=H2.toy,phi=phi.toy)

phi.true <- phi.true.toy(phi=phi.toy)

Cov.eqn9.supp(x=x,xdash=xdash,theta=theta.toy,d=d.toy,D1=D1.toy,
     D2=D2.toy,H1=H1.toy,H2=H2.toy,phi=phi.true)


# Now try a sequence of thetas:
cov.p5.supp(x=x.toy,theta=t.vec.toy,d=d.toy,D1=D1.toy,D2=D2.toy,
    H1=H1.toy,H2=H2.toy,phi=phi.toy)