Description Usage Arguments Details Value Author(s) References See Also Examples

Bayesian analysis of a computer model: obtaining the posterior distribution of the unknown parameters in the statistical model of Bayarri et al. (2007).

1 2 3 |

`object` |
An object of class |

`prior` |
The prior distribution assumed for the calibration parameters. This should be specified concatenating, |

`mcmcMultmle` |
A factor that is used to specify the prior for lambdaF and lambdaM (the precision parameters in the field and the bias, respectively). See details below. |

`prob.prop` |
The probability of proposing from the prior in the Metropolis-Hastings algorithm. See details below. |

`method` |
Method implemented in the Gibbs sampling. See details below. |

`n.iter` |
Number of total simulations. See details below. |

`nMH` |
Number of Metropolis-Hastings steps. See details below. |

`n.burnin` |
The number of iterations at the beginning of the MCMC that are thrown away. See details below. |

`n.thin` |
The thinning to be applied to the resulting MCMC sample. See details below. |

`verbose` |
A |

`...` |
Extra arguments to be passed to the function (still not implemented). |

The parameters in the statistical model which are treated as unknown are lambdaB (the precision of the bias, also called discrepancy term); lambdaF (the precision of the error) and the vector of calibration inputs. The function `bayesfit`

provides a sample from the posterior distribution of these parameters using a particular MCMC strategy. This function depends on two types of arguments detailed below: those defining the prior used, and those providing details for the MCMC sampling.

About the prior: the prior for lambdaB and lambdaF is the product of two independent exponential densities with the means being equal to the corresponding maximum likelihood estimates times the factor specified in `mcmcMultmle`

. Notice that the prior variance increases quadratically with this factor, and the prior becomes less informative as the factor increases.

The prior for each calibration parameter can be either a uniform distribution or a truncated normal and should be specified concatenating calls to `uniform`

and/or `normal`

. For example, in a problem with calibration parameters named "delta1" and "shift", a uniform(0,1) prior for "delta1" and for "shift" a normal density with mean 2 and standard deviation 1 truncated to the interval (0,3), the prior should be specified as

prior=c(uniform(var.name="delta1", lower=0, upper=1),

normal(var.name="shift", mean=2, sd=1, lower=0, upper=3))

About the MCMC. The algorithm implemented is based on a Gibbs sampling scheme. If `method=2`

then the emulator of the computer model and the bias are integrated out (analytically) and only the full conditionals for lambdaB, lambdaF and calibration parameters are sampled from. Else, if `method=1`

the computer model and the bias are part of the sampling scheme. The calibration parameters are sampled from their full conditional distribution using a Metropolis-Hastings algorithm with candidate samples proposed from a mixture of the prior specified and a uniform centered on the last sampled value. Here, the probability of a proposal coming from the prior is set by `prob.prop`

. The Metropolis-Hastings algorithm is run `nMH`

times before each sample is accepted. The default and preferred method is `method=2`

.

The MCMC is run a total of `n.iter`

iterations, of which the first `n.burnin`

are discarded. The remaining samples are thinned using the number specified in `n.thin`

.

`SAVE`

returns a copy of the `SAVE`

object used as argument to the function, but with the following slots filled (or replaced if they where not empty)

`method`

:the value given to

`method`

.`mcmcMultmle`

:the value given to

`mcmcMultmle`

.`n.iter`

:the value given to

`n.iter`

.`nMH`

:The value given to

`nMH`

.`mcmcsample`

:A named

`matrix`

with the simulated samples from the posterior distribution.`bayesfitcall`

:The

`call`

to`bayesfit`

.

Jesus Palomo, Rui Paulo and Gonzalo Garcia-Donato

Palomo J, Paulo R, Garcia-Donato G (2015). SAVE: An R Package for the Statistical Analysis of Computer Models. Journal of Statistical Software, 64(13), 1-23. Available from http://www.jstatsoft.org/v64/i13/

Bayarri MJ, Berger JO, Paulo R, Sacks J, Cafeo JA, Cavendish J, Lin CH, Tu J (2007). A Framework for Validation of Computer Models. Technometrics, 49, 138-154.

Craig P, Goldstein M, Seheult A, Smith J (1996). Bayes linear strategies for history matching of hydrocarbon reservoirs. In JM Bernardo, JO Berger, AP Dawid, D Heckerman, AFM Smith (eds.), Bayesian Statistics 5. Oxford University Press: London. (with discussion).

Higdon D, Kennedy MC, Cavendish J, Cafeo J, Ryne RD (2004). Combining field data and computer simulations for calibration and prediction. SIAM Journal on Scientific Computing, 26, 448-466.

Kennedy MC, O Hagan A (2001). Bayesian calibration of computer models (with discussion). Journal of the Royal Statistical Society B, 63, 425-464.

Roustant O., Ginsbourger D. and Deville Y. (2012). DiceKriging, DiceOptim: Two R Packages for the Analysis of Computer Experiments by Kriging-Based Metamodeling and Optimization. Journal of Statistical Software, 51(1), 1-55.

`plot`

,
`predictreality`

,
`validate`

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 | ```
## Not run:
library(SAVE)
#############
# load data
#############
data(spotweldfield,package='SAVE')
data(spotweldmodel,package='SAVE')
##############
# create the SAVE object which describes the problem and
# compute the corresponding mle estimates
##############
gfsw <- SAVE(response.name="diameter", controllable.names=c("current", "load", "thickness"),
calibration.names="tuning", field.data=spotweldfield,
model.data=spotweldmodel, mean.formula=~1,
bestguess=list(tuning=4.0))
##############
# obtain the posterior distribution of the unknown parameters
##############
gfsw <- bayesfit(object=gfsw, prior=c(uniform("tuning", upper=8, lower=0.8)),
n.iter=20000, n.burnin=100, n.thin=2)
# summary of the results
summary(gfsw)
## End(Not run)
``` |

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