scales.likelihood {emulator}

Likelihood of roughness parameters

Description

Gives the a postiori likelihood for the roughness parameters as a function of the observations.

Usage

scales.likelihood(pos.def.matrix = NULL, scales = NULL, xold,
use.Ainv = TRUE, d, give_log=TRUE, func = regressor.basis)

Arguments

pos.def.matrix

Positive definite matrix used for the distance metric

scales

If the positive definite matrix is diagonal, scales specifies the diagonal elements. Specify exactly one of pos.def.matrix or scales (ie not both)

xold

Points at which code has been run

use.Ainv

Boolean, with default TRUE meaning to calculate A^{-1} explicitly and use it. Setting to FALSE means to use methods (such as quad.form.inv()) which do not require inverting the A matrix. Although one should avoid inverting a matrix if possible, in practice there does not appear to be much difference in execution time for the two methods

d

Observations in the form of a vector with entries corresponding to the rows of xold

give_log

Boolean, with default TRUE meaning to return the logarithm of the likelihood (ie the support) and FALSE meaning to return the likelihood itself

func

Function used to determine basis vectors, defaulting to regressor.basis if not given

Details

This function returns the likelihood function defined in Oakley's PhD thesis, equation 2.37. Maximizing this likelihood to estimate the roughness parameters is an alternative to the leave-out-one method on the interpolant() helppage; both methods perform similarly.

The value returned is

\left(\hat{\sigma}\right)^{-(n-q)/2} \left|A\right|^{-1/2}\cdot\left|H^TA^{-1}H\right|^{-1/2}.

Value

Returns the likelihood or support.

Note

This function uses a Boolean flag, use.Ainv, to determine whether A has to be inverted or not. Compare the other strategy in which separate functions, eg foo() and foo.A(), are written. An example would be betahat.fun().

Author(s)

Robin K. S. Hankin

References

• J. Oakley 1999. Bayesian uncertainty analysis for complex computer codes, PhD thesis, University of Sheffield.

• J. Oakley and A. O'Hagan, 2002. Bayesian Inference for the Uncertainty Distribution of Computer Model Outputs, Biometrika 89(4), pp769-784

• R. K. S. Hankin 2005. Introducing BACCO, an R bundle for Bayesian analysis of computer code output, Journal of Statistical Software, 14(16)

See Also

optimal.scales

Examples

 data(toy)
 val <- toy

 #define a real relation
 real.relation <- function(x){sum( (0:6)*x )}

 #Some scales:
 fish <- rep(1,6)
 fish[6] <- 4
 A <- corr.matrix(val,scales=fish)
 Ainv <- solve(A)

 # Gaussian process noise:
 H <- regressor.multi(val)
 d <- apply(H,1,real.relation)
 d.noisy <- as.vector(rmvnorm(n=1,mean=d, 0.1*A))

 # Compare likelihoods with true values and another value:
 scales.likelihood(scales=rep(1,6),xold=toy,d=d.noisy)
 scales.likelihood(scales=fish    ,xold=toy,d=d.noisy)


 # Verify that use.Ainv does not affect the numerical result:
u.true  <- scales.likelihood(scales=rep(1,6),xold=toy,d=d.noisy,use.Ainv=TRUE)
u.false <- scales.likelihood(scales=rep(1,6),xold=toy,d=d.noisy,use.Ainv=FALSE)
print(c(u.true, u.false))  # should be identical up to numerical accuracy


 # Now use optim():
 f <- function(fish){scales.likelihood(scales=exp(fish), xold=toy, d=d.noisy)}
 e <-
optim(log(fish),f,method="Nelder-Mead",control=list(trace=0,maxit=10,fnscale=
-1))
best.scales <- exp(e$par)