log_lik: Natural logarithm of profile likelihood by the fast computing...

In FastGaSP: Fast and Exact Computation of Gaussian Stochastic Process

Description

This function computes the natural logarithm of the profile likelihood for the range and nugget parameter (if there is one) after plugging the closed form maximum likelihood estimator for the variance parameter.

Usage

log_lik(param, object)

Arguments

param	a vector of parameters. The first parameter is the natural logarithm of the inverse range parameter in the kernel function. If the data contain noise, the second parameter is the logarithm of the nugget-variance ratio parameter.
object	an object of class fgasp.

Value

The numerical value of natural logarithm of the profile likelihood.

Author(s)

Mengyang Gu [aut, cre]

Maintainer: Mengyang Gu <[e-mail]>

References

Hartikainen, J. and Sarkka, S. (2010). Kalman filtering and smoothing solutions to temporal gaussian process regression models, Machine Learning for Signal Processing (MLSP), 2010 IEEE International Workshop, 379-384.

M. Gu, Y. Xu (2017), Nonseparable Gaussian stochastic process: a unified view and computational strategy, arXiv:1711.11501.

M. Gu, X. Wang and J.O. Berger (2018), Robust Gaussian Stochastic Process Emulation, Annals of Statistics, 46, 3038-3066.

Examples

library(FastGaSP)
#--------------------------------------------------------------------------
# Example 1: comparing the fast and slow algorithms to compute the likelihood 
# of the Gaussian stochastic process for data with noises
#--------------------------------------------------------------------------

y_R<-function(x){
  sin(2*pi*x)
}

###let's test for 1000 observations
set.seed(1)
num_obs=1000
input=runif(num_obs)
output=y_R(input)+rnorm(num_obs,mean=0,sd=0.1)

##constucting the fgasp.model
fgasp.model=fgasp(input, output)

##range and noise-variance ratio (nugget) parameters 
param=c( log(1),log(.02))
## the log lik
log_lik(param,fgasp.model)
##time cost to compute the likelihood
time_cost=system.time(log_lik(param,fgasp.model))
time_cost[1]

##now I am comparing to the one with straightforward inversion

matern_5_2_kernel<-function(d,beta){  
  res=(1+sqrt(5)*beta*d + 5*beta^2*d^2/3 )*exp(-sqrt(5)*beta*d)
  res
}

##A function for computing the likelihood by the GaSP in a straightforward way
log_lik_GaSP_slow<-function(param,have_noise=TRUE,input,output){
  n=length(output)
  beta=exp(param[1])
  eta=0
  if(have_noise){
    eta=exp(param[2])
  }
  R00=abs(outer(input,input,'-'))
  R=matern_5_2_kernel(R00,beta=beta)
  R_tilde=R+eta*diag(n)
  #use Cholesky and one backsolver and one forward solver so it is more stable
  L=t(chol(R_tilde))
  output_t_R.inv= t(backsolve(t(L),forwardsolve(L,output )))
  S_2=output_t_R.inv%*%output
  
  -sum(log(diag(L)))-n/2*log(S_2)
}



##range and noise-variance ratio (nugget) parameters 
param=c( log(1),log(.02))
## the log lik
log_lik(param,fgasp.model)
log_lik_GaSP_slow(param,have_noise=TRUE,input=input,output=output)

##time cost to compute the likelihood
##More number of inputs mean larger differences
time_cost=system.time(log_lik(param,fgasp.model))
time_cost

time_cost_slow=system.time(log_lik_GaSP_slow(param,have_noise=TRUE,input=input,output=output))
time_cost_slow


#--------------------------------------------------------------------------
# Example 2: comparing the fast and slow algorithms to compute the likelihood 
# of the Gaussian stochastic process for data without a noise
#--------------------------------------------------------------------------
## Here is a function in the Sobolev Space with order 3
y_R<-function(x){
  j_seq=seq(1,200,1)
  record_y_R=0
  for(i_j in 1:200){
    record_y_R=record_y_R+2*j_seq[i_j]^{-2*3}*sin(j_seq[i_j])*cos(pi*(j_seq[i_j]-0.5)*x)

  }
  record_y_R
}


##generate some data without noise
num_obs=50
input=seq(0,1,1/(num_obs-1))

output=y_R(input)


##constucting the fgasp.model
fgasp.model=fgasp(input, output,have_noise=FALSE)

##range and noise-variance ratio (nugget) parameters 
param=c( log(1))
## the log lik
log_lik(param,fgasp.model)
log_lik_GaSP_slow(param,have_noise=FALSE,input=input,output=output)

FastGaSP documentation built on Sept. 5, 2021, 5:36 p.m.