NuggetKriging::logLikelihood

Description

Get the Maximized Log-Likelihood of a NuggetKriging Model Object

Usage

  • Python

    # k = NuggetKriging(...)
    k.logLikelihood()
    
  • R

    k$logLikelihood()
    
  • Matlab/Octave

    % k = NuggetKriging(...)
    k.logLikelihood()
    

Details

See logLikelihoodFun.NuggetKriging for more details on the corresponding profile log-likelihood function.

Value

The value of the maximized profile log-likelihood \(\ell_{\texttt{prof}}(\widehat{\boldsymbol{\theta}},\,\widehat{\alpha})\) where \(\alpha:= \sigma^2 / (\sigma^2 + \nu^2)\) is the ratio of the variances \(\sigma^2\) for the GP and \(\sigma^2 + \nu^2\) for the GP \(+\) nugget. This is also the value \(\ell(\widehat{\boldsymbol{\theta}},\, \widehat{\alpha},\, \widehat{\sigma}^2,\, \widehat{\boldsymbol{\beta}})\) or \(\ell(\widehat{\boldsymbol{\theta}},\, \widehat{\sigma}^2,\, \widehat{\tau}^2, \, \widehat{\boldsymbol{\beta}})\) of the maximized log-likelihood.

Examples

f <- function(x) 1 - 1 / 2 * (sin(12 * x) / (1 + x) + 2 * cos(7 * x) * x^5 + 0.7)
set.seed(123)
X <- as.matrix(runif(10))
y <- f(X) + 0.1 * rnorm(nrow(X))

k <- NuggetKriging(y, X, kernel = "matern3_2", objective="LL")
print(k)

k$logLikelihood()

Results

* data: 10x[0.0455565,0.940467] -> 10x[0.149491,0.940566]
* trend constant (est.): 0.488124
* variance (est.): 0.0788813
* covariance:
  * kernel: matern3_2
  * range (est.): 0.275004
  * nugget (est.): 0.00347449
  * fit:
    * objective: LL
    * optim: BFGS
[1] 4.95114

Reference