interpolant {emulator} | R Documentation |
Calculates the a postiori distribution of results at a
point using the techniques outlined by Oakley. This function is the
primary function of the package. Function interpolant.quick()
gives the expectation of the emulator at a set of points, and function
interpolant()
gives the expectation and other information (such
as the variance) at a single point. Function int.qq()
gives a
quick-quick vectorized interpolant using certain timesaving assumptions.
interpolant(x, d, xold, Ainv=NULL, A=NULL, use.Ainv=TRUE, scales=NULL, pos.def.matrix=NULL, func=regressor.basis, give.full.list = FALSE, distance.function=corr, ...) interpolant.quick(x, d, xold, Ainv, scales=NULL, pos.def.matrix=NULL, func=regressor.basis, give.Z = FALSE, distance.function=corr, ...) int.qq(x, d, xold, Ainv, pos.def.matrix, func=regressor.basis)
x |
Point(s) at which estimation is desired. For
interpolant.quick() , argument x is a matrix
and an expectation is given for each row |
d |
vector of observations, one for each row of xold |
xold |
Matrix with rows corresponding to points at which the function is known |
A |
Correlation matrix A . If not given, it is calculated |
Ainv |
Inverse of correlation matrix A . Required by
interpolant.quick() and int.qq() . In interpolant() ,
using the default value of NULL results in Ainv being
calculated explicitly (which may be slow: see next argument for more
details) |
use.Ainv |
Boolean, with default TRUE meaning to use the
inverse matrix Ainv (and, if necessary, calculate it using
solve(.) ). This requires the not inconsiderable overhead of
inverting a matrix. If, however, Ainv is available, using
the default option is much faster than setting
use.Ainv=FALSE ; see below.
If FALSE , function interpolant() does not use
Ainv , but makes extensive use of solve(A,x) (mostly in
the form of quad.form.inv() calls). This option avoids the
overhead of inverting a matrix, but has non-negligible marginal costs.
If Ainv is not available, there is little to choose, in terms
of execution time, between calculating it explicitly (that is,
setting use.Ainv=TRUE ) and using solve(A,x) (ie
use.Ainv=TRUE ).
Note: if Ainv is given to the function, but
use.Ainv is FALSE , the code will do as requested and use
the slow solve(A,x) , which is probably not what you want |
func |
Function used to determine basis vectors, defaulting
to regressor.basis if not given |
give.full.list |
In interpolant() , Boolean variable with
TRUE meaning to return the whole list of a postiori
parameters as detailed on pp12-15 of Oakley, and default FALSE
meaning to return just the best estimate |
scales |
Vector of “roughness” lengths used to calculate
t(x) . Note that scales is needed twice: once to calculate
Ainv and once to calculate t(x) inside
interpolant (which is determined by calling corr
inside an apply() loop). A good place to start might be
scales=rep(1,ncol(xold)) |
pos.def.matrix |
A positive definite matrix that is used if
scales is not supplied. Note that precisely one of
scales and pos.def.matrix must be supplied |
give.Z |
In function interpolant.quick() , Boolean variable
with TRUE meaning to return the best estimate and the error,
and default FALSE meaning to return just the best estimate |
distance.function |
Function to compute distances between
points, defaulting to corr() . See corr.Rd for
details. Note that method=2 or method=3 is required
if a non-standard distance function is used |
... |
Further arguments passed to the distance function,
usually corr() |
If give.full.list
is TRUE, a list is return with components
betahat |
Standard MLE of the (linear) fit, given the observations |
prior |
Estimate for the prior |
sigmahat.square |
A postiori estimate for variance |
mstar.star |
A postiori expectation |
cstar |
a priori correlation of a point with itself |
cstar.star |
A postiori correlation of a point with itself |
Z |
Standard deviation (although the distribution is actually a t-distribution with n-q degrees of freedom) |
Robin K. S. Hankin
J. Oakley 2004. “Estimating percentiles of uncertain computer code outputs”. Applied Statistics, 53(1), pp89-93.
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)
# example has 10 observations on 6 dimensions. # function is just sum( (1:6)*x) where x=c(x_1, ... , x_2) data(toy) val <- toy real.relation <- function(x){sum( (0:6)*x )} H <- regressor.multi(val) d <- apply(H,1,real.relation) fish <- rep(1,6) fish[6] <- 4 A <- corr.matrix(val,scales=fish, power=2) Ainv <- solve(A) # now add some suitably correlated noise to d: d.noisy <- as.vector(rmvnorm(n=1, mean=d, 0.1*A)) names(d.noisy) <- names(d) # First try a value at which we know the answer (the first row of val): x.known <- as.vector(val[1,]) bayes.known <- interpolant(x.known, d, val, Ainv=Ainv, scales=fish, g=FALSE) print("error:") print(d[1]-bayes.known) # Now try the same value, but with noisy data: print("error:") print(d.noisy[1]-interpolant(x.known, d.noisy, val, Ainv=Ainv, scales=fish, g=FALSE)) #And now one we don't know: x.unknown <- rep(0.5 , 6) bayes.unknown <- interpolant(x.unknown, d.noisy, val, scales=fish, Ainv=Ainv,g=TRUE) ## [ compare with the "true" value of sum(0.5*0:6) = 10.5 ] # Just a quickie for int.qq(): int.qq(x=rbind(x.unknown,x.unknown+0.1),d.noisy,val,Ainv,pos.def.matrix=diag(fish)) ## (To find the best correlation lengths, use optimal.scales()) # Now we use the SAME dataset but a different set of basis functions. # Here, we use the functional dependence of # "A+B*(x[1]>0.5)+C*(x[2]>0.5)+...+F*(x[6]>0.5)". # Thus the basis functions will be c(1,x>0.5). # The coefficients will again be 1:6. # Basis functions: f <- function(x){c(1,x>0.5)} # (other examples might be # something like "f <- function(x){c(1,x>0.5,x[1]^2)}" # now create the data real.relation2 <- function(x){sum( (0:6)*f(x) )} d2 <- apply(val,1,real.relation2) # Define a point at which the function's behaviour is not known: x.unknown2 <- rep(1,6) # Thus real.relation2(x.unknown2) is sum(1:6)=21 # Now try the emulator: interpolant(x.unknown2, d2, val, Ainv=Ainv, scales=fish, g=TRUE)$mstar.star # Heh, it got it wrong! (we know that it should be 21) # Now try it with the correct basis functions: interpolant(x.unknown2, d2, val, Ainv=Ainv,scales=fish, func=f,g=TRUE)$mstar.star # That's more like it. # We can tell that the coefficients are right by: betahat.fun(val,Ainv,d2,func=f) # Giving c(0:6), as expected. # It's interesting to note that using the *wrong* basis functions # gives the *correct* answer when evaluated at a known point: interpolant(val[1,], d2, val, Ainv=Ainv,scales=fish, g=TRUE)$mstar.star real.relation2(val[1,]) # Which should agree. # Now look at Z. Define a function Z() which determines the # standard deviation at a point near a known point. Z <- function(o) { x <- x.known x[1] <- x[1]+ o interpolant(x, d.noisy, val, Ainv=Ainv, scales=fish, g=TRUE)$Z } Z(0) #should be zero because we know the answer (this is just Z at x.known) Z(0.1) #nonzero error. ## interpolant.quick() should give the same results faster, but one ## needs a matrix: u <- rbind(x.known,x.unknown) interpolant.quick(u, d.noisy, val, scales=fish, Ainv=Ainv,g=TRUE) # Now an example from climate science. "results.table" is a dataframe # of goldstein (a climate model) results. Each of its 100 rows shows a # point in parameter space together with certain key outputs from the # goldstein program. The following R code shows how we can set up an # emulator based on the first 27 goldstein runs, and use the emulator to # predict the output for the remaining 73 goldstein runs. The results # of the emulator are then plotted on a scattergraph showing that the # emulator is producing estimates that are close to the "real" goldstein # runs. data(results.table) data(expert.estimates) # Decide which column we are interested in: output.col <- 26 # extract the "important" columns: wanted.cols <- c(2:9,12:19) # Decide how many to keep; # 30-40 is about the most we can handle: wanted.row <- 1:27 # Values to use are the ones that appear in goin.test2.comments: val <- results.table[wanted.row , wanted.cols] # Now normalize val so that 0<results.table[,i]<1 is # approximately true for all i: normalize <- function(x){(x-mins)/(maxes-mins)} unnormalize <- function(x){mins + (maxes-mins)*x} mins <- expert.estimates$low maxes <- expert.estimates$high jj <- t(apply(val,1,normalize)) jj <- as.data.frame(jj) names(jj) <- names(val) val <- jj ## The value we are interested in is the 19th (or 20th or ... or 26th) column. d <- results.table[wanted.row , output.col] ## Now some scales, estimated earlier from the data using ## optimal.scales(): scales.optim <- exp(c( -2.917, -4.954, -3.354, 2.377, -2.457, -1.934, -3.395, -0.444, -1.448, -3.075, -0.052, -2.890, -2.832, -2.322, -3.092, -1.786)) A <- corr.matrix(val,scales=scales.optim, method=2, power=1.5) Ainv <- solve(A) print("and plot points used in optimization:") d.observed <- results.table[ , output.col] A <- corr.matrix(val,scales=scales.optim, method=2, power=1.5) Ainv <- solve(A) print("now plot all points:") design.normalized <- as.matrix(t(apply(results.table[,wanted.cols],1,normalize))) d.predicted <- interpolant.quick(design.normalized , d , val , Ainv=Ainv, scales=scales.optim, power=1.5) jj <- range(c(d.observed,d.predicted)) par(pty="s") plot(d.observed, d.predicted, pch=16, asp=1, xlim=jj,ylim=jj, xlab=expression(paste(temperature," (",{}^o,C,"), model" )), ylab=expression(paste(temperature," (",{}^o,C,"), emulator")) ) abline(0,1)