scales.likelihood: 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)

Value

Returns the likelihood or support.

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

Author

Robin K. S. Hankin

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

${(\hat{σ})}^{- (n - q) / 2} {| A |}^{- 1 / 2} \cdot {| H^{T} A^{- 1} H |}^{- 1 / 2} .$

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)

Examples

Run this code

 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)

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