predict {RobustCalibration}

Prediction for the robust calibration model

Description

Function to make prediction on Robust Calibration models after the rcalibration class has been constructed.

Usage

## S4 method for signature 'rcalibration'
predict(object, testing_input,X_testing=NULL,
                                 n_thinning=10,
                                 testing_output_weights=NULL,
                                 interval_est=NULL,interval_data=F,
                                 math_model=NULL,test_loc_index_emulator=NULL,...)

Arguments

object

an object of class rcalibration.

testing_input

a matrix containing the inputs where the predict is to perform prediction. To predict one observable input with multiple dimension, user should supply a row vector.

X_testing

a matrix of mean/trend for prediction.

n_thinning

number of points further thinning the MCMC posterior samples.

testing_output_weights

the weight of testing outputs.

interval_est

a vector for the the posterior credible interval. If interval_est is NULL, we do not compute the posterior credible interval. It can be specified as a vector of values ranging from zero to one. E.g. if interval_est=c(0.025, 0.975), the 95 posterior credible interval will be computed.

interval_data

a bool value to decide whether the experimental noise is included for computing the posterior credible interval.

math_model

a function for the math model to be calibrated.

test_loc_index_emulator

a vector of the location index from the ppgasp emulator to output. Only useful for vectorized output computer model emulated by the ppgasp emulator.

...

extra arguments to be passed to the function (not implemented yet).

Value

The returned value is a S4 CLass predictobj.rcalibration.

Author(s)

Mengyang Gu [aut, cre]

Maintainer: Mengyang Gu <[e-mail]>

References

A. O'Hagan and M. C. Kennedy (2001), Bayesian calibration of computer models, Journal of the Royal Statistical Society: Series B (Statistical Methodology, 63, 425-464.

Bayarri, Maria J and Berger, James O and Paulo, Rui and Sacks, Jerry and Cafeo, John A and Cavendish, James and Lin, Chin-Hsu and Tu, Jian (2007) A framework for validation of computer models. Technometrics. 49, 138–154.

M. Gu (2016), Robust Uncertainty Quantification and Scalable Computation for Computer Models with Massive Output, Ph.D. thesis., Duke University.

M. Gu and L. Wang (2017) Scaled Gaussian Stochastic Process for Computer Model Calibration and Prediction. arXiv preprint arXiv:1707.08215.

M. Gu (2018) Jointly Robust Prior for Gaussian Stochastic Process in Emulation, Calibration and Variable Selection . arXiv preprint arXiv:1804.09329.

Examples

#------------------------------------------------------------------------------
# Example: an example used in Susie Bayarri et. al. 2007 Technometrics paper
#------------------------------------------------------------------------------
    

##reality
test_funct_eg1<-function(x){
  3.5*exp(-1.7*x)+1.5
}


##math model
math_model_eg1<-function(x,theta){
  5*exp(-x*theta) 
}


## noise observations (sampled from reality + independent Gaussian noises)
## each has 3 replicates
input=c(rep(.110,3),rep(.432,3),rep(.754,3),rep(1.077,3),rep(1.399,3),rep(1.721,3),
        rep(2.043,3),rep(2.366,3),rep(2.688,3),rep(3.010,3))
output=c(4.730,4.720,4.234,3.177,2.966,3.653,1.970,2.267,2.084,2.079,2.409,2.371,1.908,1.665,1.685,
         1.773,1.603,1.922,1.370,1.661,1.757,1.868,1.505,1.638,1.390,1.275,1.679,1.461,1.157,1.530)


## calculating the average or the stack data
n_stack=length(output)/3
output_stack=rep(0,n_stack)
input_stack=rep(0,n_stack)
for(j in 1:n_stack){
  output_stack[j]=mean(output[ ((j-1)*3+1):(3*j)])
  input_stack[j]=mean(input[ ((j-1)*3+1):(3*j)])
}
output_stack=as.matrix(output_stack)
input_stack=as.matrix(input_stack)
## plot the output and stack output
#plot(input,output,pch=16,col='red')
#lines(input_stack,output_stack,pch=16,col='blue',type='p')



## fit model using S-GaSP for the discrepancy
## one can change S and S_0 for the number of posterior and burn-in samples
## Normallly you may need a larger number of posterior sample
## you can set S=50000 and S_0=5000
## one may also change the sd of the proposal distribution using sd_proposal
model_sgasp=rcalibration(design=input_stack, observations=output_stack, p_theta=1,simul_type=1,
                         math_model=math_model_eg1,theta_range=matrix(c(0,10),1,2),
                         S=10000,S_0=2000,discrepancy_type='S-GaSP')

# one can  fit the GaSP model for discrepancy function by discrepancy_type='GaSP'
# one can  fit a model without the discrepancy function by discrepancy_type='no-discrepancy'

## posterior of the calibration parameter
#plot(model_sgasp@post_sample[,1],type='l',xlab='num',ylab=expression(theta))   
show(model_sgasp)

##

## test data set
testing_input=as.matrix(seq(0,6,0.02))

##perform prediction
prediction_sgasp=predict(model_sgasp,testing_input,math_model=math_model_eg1,
                         interval_est=c(0.025,0.975),interval_data=TRUE,
                         n_thinning =20 )

##real test output
testing_output=test_funct_eg1(testing_input)

##the prediction by S-GaSP
min_val=min(prediction_sgasp@mean,prediction_sgasp@interval,output,testing_output)
max_val=max(prediction_sgasp@mean,prediction_sgasp@interval,output,testing_output)

plot(testing_input,prediction_sgasp@mean,type='l',col='blue',xlab='x',ylab='y',
     ylim=c(min_val,max_val) )
lines(testing_input,prediction_sgasp@interval[,1],col='blue',lty=2)
lines(testing_input,prediction_sgasp@interval[,2],col='blue',lty=2)

lines(input,output,type='p')
lines(testing_input,prediction_sgasp@math_model_mean,col='blue',lty=3)

lines(testing_input,testing_output,type='l')

legend("topright", legend=c("reality", "predictive mean","95 percent posterior credible interval",
                            "predictive mean of the math model"),
                            col=c("black", "blue","blue","blue"), lty=c(1,1,2,3),cex=.6)


## MSE if the math model and discrepancy are used for prediction
mean((testing_output-prediction_sgasp@mean)^2)

## MSE if the math model is used for prediction 
mean((testing_output-prediction_sgasp@math_model_mean)^2)



##################################
#the example with a mean structure
##################################

##now let's fit  model with mean
model_sgasp_with_mean=rcalibration(design=input_stack, observations=output_stack,
                                   p_theta=1,X=matrix(1,dim(input_stack)[1],1),
                                   have_trend=TRUE,simul_type=1,
                                   math_model=math_model_eg1,
                                   theta_range=matrix(c(0,10),1,2),
                                   S=10000,S_0=2000,
                                   discrepancy_type='S-GaSP')

#posterior
#plot(model_sgasp_with_mean@post_sample[,1],type='l',xlab='num',ylab=expression(theta))   
show(model_sgasp_with_mean)

## test data set
testing_input=as.matrix(seq(0,6,0.02))


prediction_sgasp_with_mean=predict(model_sgasp_with_mean,testing_input, X_testing=matrix(1,dim
(testing_input)[1],1),
math_model=math_model_eg1,n_thinning = 50,
interval_est=c(0.025,0.975),interval_data=TRUE)


##plot for the S-GaSP 
##for this example, with a mean structure, it fits much better
min_val=min(prediction_sgasp_with_mean@mean,output,testing_output,
prediction_sgasp_with_mean@interval[,1])
max_val=max(prediction_sgasp_with_mean@mean,output,testing_output,
prediction_sgasp_with_mean@interval[,2])

plot(testing_input,prediction_sgasp_with_mean@mean,type='l',col='blue',xlab='x',
     ylab='y',ylim=c(min_val,max_val) )
#lines(testing_input,prediction_sgasp_with_mean@interval[,1],col='blue',lty=2)
#lines(testing_input,prediction_sgasp_with_mean@interval[,2],col='blue',lty=2)

lines(input,output,type='p')
lines(testing_input,prediction_sgasp_with_mean@math_model_mean,col='blue',lty=3)
lines(testing_input,prediction_sgasp_with_mean@interval[,1],col='blue',lty=2)
lines(testing_input,prediction_sgasp_with_mean@interval[,2],col='blue',lty=2)

lines(testing_input,testing_output,type='l')

legend("topright", legend=c("reality", "predictive mean", "predictive mean of the math model"),
       col=c("black", "blue","blue"), lty=c(1,1,3),cex=.6)


## MSE if the math model and discrepancy are used for prediction
mean((testing_output-prediction_sgasp_with_mean@mean)^2)

## MSE if the math model is used for prediction 
mean((testing_output-prediction_sgasp_with_mean@math_model_mean)^2)


## Not run: 
  #-------------------------------------------------------------
  #the example with the emulator
  #-------------------------------------------------------------
  
  n_design=80
  
  design_simul=matrix(runif(n_design*2),n_design,2)
  #library(lhs)
  #design_simul=maximinLHS(n=n_design,k=2)
  
  design_simul[,1]=6*design_simul[,1]   ##the first one is the observed input x
  design_simul[,2]=10*design_simul[,2]   ##the second one is the calibration parameter \theta
  
  output_simul=math_model_eg1(design_simul[,1],design_simul[,2])
  
  
  ##this is a little slow compared with the previous model
  
  model_sgasp_with_mean_emulator=rcalibration(design=input_stack, observations=output_stack,
                                              p_theta=1,simul_type=0, 
                                              have_trend=T,X=matrix(1,dim(input_stack)[1],1),
                                              input_simul=design_simul, output_simul=output_simul,
                                              theta_range=matrix(c(0,10),1,2),
                                              S=10000,S_0=2000,discrepancy_type='S-GaSP')
  
  ##now the output is a list
  show(model_sgasp_with_mean_emulator)
  
  ##here is the plot
  plot(model_sgasp_with_mean_emulator@post_sample[,4],type='l',xlab='num',ylab=expression(theta))   
  plot(model_sgasp_with_mean_emulator@post_value,type='l',xlab='num',ylab='posterior value')   
  
  
  prediction_sgasp_with_mean_emulator=predict(model_sgasp_with_mean_emulator,testing_input,
                                              X_testing=matrix(1,dim(testing_input)[1],1),
                                              interval_est=c(0.025,0.975),
                                              interval_data=TRUE)
  
  ##for this example, with a mean structure, it fits much better
  min_val=min(prediction_sgasp_with_mean_emulator@mean,output,testing_output,
              prediction_sgasp_with_mean_emulator@math_model_mean)
  max_val=max(prediction_sgasp_with_mean_emulator@mean,output,testing_output,
              prediction_sgasp_with_mean_emulator@math_model_mean)
  
  plot(testing_input,prediction_sgasp_with_mean_emulator@mean,type='l',col='blue',xlab='x',
       ylab='y',ylim=c(min_val,max_val) )
  #lines(testing_input,prediction_sgasp_with_mean@interval[,1],col='blue',lty=2)
  #lines(testing_input,prediction_sgasp_with_mean@interval[,2],col='blue',lty=2)
  
  lines(input,output,type='p')
  lines(testing_input,prediction_sgasp_with_mean_emulator@math_model_mean,col='blue',lty=3)
  
  lines(testing_input,testing_output,type='l')
  
  legend("topright", legend=c("reality", "predictive mean", "predictive mean of the math model"),
         col=c("black", "blue","blue"), lty=c(1,1,3),cex=.6)
  
  
  ## MSE if the math model and discrepancy are used for prediction
  mean((testing_output-prediction_sgasp_with_mean_emulator@mean)^2)
  
  ## MSE if the math model is used for prediction 
  mean((testing_output-prediction_sgasp_with_mean_emulator@math_model_mean)^2)


## End(Not run)