Auxil {GPM}

An auxiliary function used in calculating the negative log-likelehood and its gradient

Description

Calculates some auxiliary paramters to obtain the negative log-likelehood and its gradient.

Usage

Auxil(Omega, X, Y, CorrType, MinEig, Fn, n, dy)

Arguments

Omega

The vector storing all the hyperparameters of the correlation function. The length of Omega depends on the CorrType. See reference 1.

X

Matrix containing the training (aka design or input) data points. The rows and columns of X denote individual observation settings and input dimension, respectively.

Y

Matrix containing the output (aka response) data points. The rows and columns of Y denote individual observation responses and output dimension, respectively.

CorrType

The correlation function of the GP model. Choices include 'G' (default), 'PE', 'LBG', and 'LB'. See Fit and the references.

MinEig

The smallest eigen value that the correlation matrix is allowed to have, which in return determines the appraopriate nugget that should be added to the correlation matrix.

Fn

A matrix of 1's with nrow(X) rows and 1 column. See reference 1.

n

Number of observations, nrow(X).

dy

Number of responses, ncol(Y).

Details

Since Auxil is shared between NLogL and NLogL_G during optimization, ideally it should be run only once (e.g., via memoisation). Such an implementation is left for future editions.

Value

ALL A list containing the following components (based on CorrType, some other parameters are also stored in ALL):

• R The correlation matrix whose smallest eigen value is >= MinEig.

• L Cholesky decomposition of R.

• Raw_MinEig The smallest eigen value of R before adding Nug_opt.

• Nug_opt The added nugger to R.

• B

References

1. Bostanabad, R., Kearney, T., Tao, S., Apley, D. W. & Chen, W. (2018) Leveraging the nugget parameter for efficient Gaussian process modeling. Int J Numer Meth Eng, 114, 501-516.

2. Plumlee, M. & Apley, D. W. (2017) Lifted Brownian kriging models. Technometrics, 59, 165-177.

See Also

Fit to see how a GP model can be fitted to a training dataset.
Predict to use the fitted GP model for prediction.
Draw to plot the response via the fitted model.

Examples

# see the examples in the fitting function.