Stage 1,2 and 3 optimization on toy dataset

Perform O'Hagan's three stage optimization on the toy dataset. Function stage1() and stage2() find the optimal values for the hyperparameters and stage3() finds the optimal values for the three parameters.

stage1(D1, y, H1,  maxit,  trace=100, method="Nelder-Mead",
      directory = ".", do.filewrite=FALSE, do.print=TRUE,, lognormally.distributed=FALSE, include.prior=TRUE, phi)
stage2(D1, D2, H1, H2, y, z, maxit, trace=100, method = "Nelder-Mead",
      directory = ".", do.filewrite=FALSE, do.print=TRUE,  extractor,, E.theta, Edash.theta, isotropic=FALSE,
      lognormally.distributed = FALSE, include.prior = TRUE,
      use.standin = FALSE, rho.eq.1 = TRUE, phi) 
stage3(D1, D2, H1, H2, d, maxit, trace=100, method="Nelder-Mead",
      directory = ".", do.filewrite=FALSE, do.print=TRUE,
      include.prior = TRUE, lognormally.distributed=FALSE,
      theta.start=NULL, phi)

Maximum number of iterations as passed to optim()


Amount of information displayed, as passed to optim()


Matrix whose rows are points at which code output is known


Matrix whose rows are points at which observations were made


Regressor basis functions for D1 and D2


Code outputs. Toy example is y.toy


Observations. Toy example is z.toy


Data vector consisting of the code runs and observations


extractor function for D1


Expectation WRT theta, and dashed theta. Toy examples are E.theta.toy() and Edash.theta.toy()

Function to create hyperparameters; passed (in stage1() and stage2()) to phi.change(). Toy version is


Method argument passed to optim(); qv


Boolean variable with default TRUE meaning to include the prior distribution in the optimization process and FALSE meaning to use an uniformative prior (effectively uniform support). This variable is passed to p.eqn4.supp() for stage1(), p.page4() for stage2(), and p.eqn8.supp() for stage3()


Boolean with TRUE meaning to use a lognormal distn. See prob.theta for details


Boolean, with TRUE meaning to save a loadable file stage[123].<date>, containing the interim value of phi and the corresponding optimand to directory at each evalution of the optimizer. If FALSE, don't write the files


The directory to write files to; only matters if do.filewrite is TRUE


In function stage2(), Boolean with default FALSE meaning to carry out a full optimization, and TRUE meaning to restrict the scope to isotroic roughness matrices. See details section below


Boolean, with default TRUE meaning to print interim values of phi at each evaluation


In stage2(), a Boolean argument, with default FALSE meaning to use the real value for matrix V.temp, and TRUE meaning to use a standing that is the same size but contains fictitious values. The only time to set use.standin to TRUE is when debugging as it runs several orders of magnitude faster


Boolean, with default TRUE meaning to hold the value of rho constant at one (1)


In stage3(), the starting point of the optimization with default NULL meaning to use the maximum likelihood point of the apriori distribution (ie phi$theta.apriori$mean)


Hyperparameters. Used as initial values for the hyperparameters in the optimization routines


The three functions documented here carry out the multi-stage optimization detailed in KOH2001 (actually, KOH2001 only defined stage 1 and stage 2, which estimated the hyperparameters. What is here called “stage3()” estimates the true value of \(\theta\) given the hyperparameters).

stage1() carries out stage 1 of KOH2001 which is used to estimate \(\psi_1\) using optimization.

In function stage2(), setting argument isotropic to TRUE will force phi$omegastar_x to be a function of a length one scalar. The value of phi$omegastar_x used will depend on pdm.maker.psi2() (an internal function appearing in In stage2(), several kludges are made. The initial conditions are provided by argument phi. The relevant part of this is phi$psi2.

Function stage2() estimates \(\psi_2\) and \(\rho\) and \(\lambda\), using optimization. Note that \(\psi_2\) includes \(\sigma_2^2\) in addition to omegastar_X (in the toy case, \(\psi_2\) has three elements: the first two are the diagonal of omegastar_x and the third is \(\sigma_2^2\) but this information is encoded in, which changes from application to application).

Function stage3() attempts to find the maximum likelihood estimate of \(\theta\), given hyperparameters and observations, using optimization


  • 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

  • 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


  • stage1
  • stage2
  • stage3
stage1(D1=D1.toy,y=y.toy,H1=H1.toy, maxit=5,, phi=phi.toy)

##now try with a slightly bigger dataset:
##Examples below take a few minutes to run:

jj <- , D2.toy)
y.toy <- jj$y.toy
z.toy <- jj$z.toy
d.toy <- jj$d.toy

phi.toy.stage1 <- stage1(D1=D1.toy, y=y.toy, H1=H1.toy, maxit=10,, phi=phi.toy)

phi.toy.stage2 <- stage2(D1=D1.toy, D2=D2.toy, H1=H1.toy, H2=H2.toy,
 y=y.toy, z=z.toy, extractor=extractor.toy,, E.theta=E.theta.toy, Edash.theta=Edash.theta.toy,
maxit=3, phi=phi.toy.stage1)

stage3(D1=D1.toy, D2=D2.toy, H1=H1.toy, H2=H2.toy, d=d.toy, maxit=3, phi=phi.toy.stage2)

# Now try with the true values of the hyperparameters:
phi.true <- phi.true.toy(phi=phi.toy)

stage3(D1=D1.toy, D2=D2.toy, H1=H1.toy, H2=H2.toy, d=d.toy, maxit=3, phi=phi.true)

# }
Documentation reproduced from package calibrator, version 1.2-8, License: GPL-2

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