tt.fun: Integrals needed in KOH2001

Description

Calculates the three integrals needed for V, under the restrictions specified in the KOH2001 supplement

Usage

tt.fun(D1, extractor, x.i, x.j,  test.for.symmetry=FALSE, method=1, phi)
ht.fun(x.i, x.j, D1, extractor,  Edash.theta,  H1, fast.but.opaque=TRUE,
x.star=NULL, t.vec=NULL, phi) 
hh.fun(x.i, x.j,  H1, E.theta,  phi)
t.fun(x, D1, extractor,  phi)

Arguments

Matrix of code run points

regression basis functions for D1

extractor

Function to extract x.star and t.vec from D1

Lat and long of a point in t.fun() (eg D2[1,])

x.i

Lat and long of first point (eg D2[1,])

x.j

Lat and long of second point (eg D2[2,])

theta

parameters

Edash.theta

Function to return expectation of H with respect to the alternative distribution of $\theta$; Edash.theta.toy is the example for the toy dataset

E.theta

Function to return expectation of H with respect to $\theta$

test.for.symmetry

In tt.fun(), Boolean with TRUE meaning to calculate each element of $C$ explicitly. If FALSE, then calculate only the elements of $C$ that lie on or over the diagonal and use the fact that $C$ is symmetric to calculate the other matrix elements. For $n$ observations, this means $n(n+1)/2$ evaluations, compared with $n^2$ for the full case.

Set this argument to TRUE only when debugging, or testing accuracy.

fast.but.opaque

In ht.fun(), Boolean with default TRUE meaning to pass some precalculated results as arguments, to save time. Set this argument to FALSE only when debugging.

x.star

In ht.fun(), value of $x^*$ (required only if fast.but.opaque is TRUE)

t.vec

In ht.fun(), value of $t$ (required only if fast.but.opaque is TRUE)

method

In tt.fun(), zero means use the old method and nonzero means use the new method. The new method is faster, but the code is harder to understand. The two methods should give identical results.

phi

Hyperparameters

Value

Each function returns a matrix as described in KOH2001

Details

The four functions return integrals representing means taken over theta. To wit:

Function tt.fun() evaluates $$\int t(x_j,\theta)t(x_i,\theta)^Tp(\theta)d\theta$$ and is used in V.fun(). Note that this function is symmetric in $x_i$ and $x_j$.
Function ht.fun() evaluates $$\int h_1(x_j,\theta)t(x_i,\theta)^Tp(\theta)d\theta$$ and is used in V.fun(). Note that this function is not symmetric in $x_i$ and $x_j$.
Function hh.fun() evaluates $$\int h_1(x_j,\theta)h_1(x_i,\theta)^Tp(\theta)d\theta$$ and is used in V.fun(). Note that this function is symmetric in $x_i$ and $x_j$.
Function t.fun() evaluates $$\int t(x_i,\theta)^Tp(\theta)d\theta= \int c_1\left( (x_i,\theta),(x_j^*,t_j)\right)p(\theta)\,d\theta $$ using the formula $$ \sigma_1^2\left|I+2V_\theta\Omega_x\right|^{-1/2} \exp\left\{ -\left(x_i-x_j^*\right)^T\Omega_x\left(x_i-x_j^*\right) \right\}\times \exp\left\{ -\left(m_\theta-t_j\right)^T \left(2V_\theta+\Omega_t^{-1}\right)^{-1} \left(m_\theta-t_j\right)\right\}. $$ It is used in Ez_eq7.supp(). NB: do not confuse this function with tee(), which is different.

These functions are not generally of much interest to the end user; they are called by V.fun(). They are defined separately as a debugging aid, and to simplify the structure of V.fun().

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

Run this code

# NOT RUN {
data(toys)

tt.fun(D1=D1.toy, extractor=extractor.toy, x.i=D2.toy[1,],
    x.j=D2.toy[2,],  phi=phi.toy)

ht.fun(x.i=D2.toy[1,], x.j=D2.toy[2,], D1=D1.toy,
    extractor=extractor.toy, 
    Edash.theta=Edash.theta.toy, H1=H1.toy, fast.but.opaque=FALSE, phi=phi.toy)

ht.fun(x.i=D2.toy[1,], x.j=D2.toy[2,], D1=D1.toy,
    extractor=extractor.toy, 
    Edash.theta=Edash.theta.toy, H1=H1.toy, fast.but.opaque=TRUE,
    x.star=extractor.toy(D1.toy)$x.star, t.vec=extractor.toy(D1.toy)$t.vec,
    phi=phi.toy)



hh.fun(x.i=D2.toy[1,], x.j=D2.toy[2,],
    H1=H1.toy, E.theta=E.theta.toy,  phi=phi.toy)

t.fun(x=x.toy, D1=D1.toy, extractor=extractor.toy, phi=phi.toy)
# }

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