Kriging steps

Kriging models can be used in different steps depending on the goal.

  • Trend estimation If only the trend parameters \(\beta_k\) are unknown, these can be estimated by Generalized Least Squares. This step separates the observed response \(y_i\) into a trend and component \(\widehat{\mu}(\mathbf{x}_i)\) a non-trend component. The non-trend component involves a smooth GP component \(\widehat{\zeta}(\mathbf{x}_i)\) and, optionally, a nugget or noise component \(\widehat{\varepsilon}(\mathbf{x}_i)\) or \(\widehat{\varepsilon}_i\).

  • Fit Find estimates of the parameters, including the covariance parameters. Several methods are implemented, all relying on the optimization of a function of the covariance parameters called the objective. This objective can relates to frequentist estimation methods: Maximum-Likelihood (ML) and Leave-One-Out Cross-Validation. It can also be a Bayesian Marginal Posterior Density, in relation with specific priors, in which case the estimate will be a Maximum A Posteriori (MAP). Mind that in libKriging only point estimates will be given for the correlation parameters.

  • Update Update a model object by processing \(n^\star\) new observations. Once this step is achieved, the predictions will be based on the full set of \(n + n^\star\) observations. The covariance parameters can optionally be updated by using the new observations when computing the fitting objective.

  • Predict Given \(n^\star\) “new” inputs \(\mathbf{x}^\star_i\) forming the rows of a matrix \(\mathbf{X}^\star\), compute the Gaussian distribution of \(\mathbf{y}^\star\) conditional on \(\mathbf{y}\). As long as the covariance parameters are regarded as known, the conditional distribution is Gaussian, and is characterized by its expectation vector and its covariance matrix. These are often called the Kriging mean and the Kriging covariance.

  • Simulate Given \(n^\star\) “new” inputs \(\mathbf{x}^\star_i\) forming the rows of a matrix \(\mathbf{X}^\star\), draw a sample of \(n_{\texttt{sim}}\) vectors \(\mathbf{y}^\star\) \(k=1\), \(\dots\), \(n_{\texttt{sim}}\) from the distribution of \(y(\mathbf{x})\) conditional on the observations.

By “Kriging” one often means the prediction step. The fit step is generally the most costly one in terms of computation because the fit objective has to be evaluated repeatedly (say dozens of times) and each evaluation involves \(O(n^3)\) elementary operations.