loglik: A wraper to compute the natural logarithm of the integrated...

In GPBayes: Tools for Gaussian Process Modeling in Uncertainty Quantification

A wraper to compute the natural logarithm of the integrated likelihood function

Description

This function wraps existing built-in routines to construct the natural logarithm of the integrated likelihood function. The constructed loglikelihood can be directly used for numerical optimization

Usage

loglik(par, output, H, d, covmodel, smooth, smoothness_est)

Arguments

par	a numerical vector, with which numerical optimization routine such as optim can be carried out directly. When the confluent Hypergeometric class is used, it is used to hold values for range, tail, nugget, and nu if the smoothness parameter is estimated. When the Matérn class or powered-exponential class is used, it is used to hold values for range, nugget, and nu if the smoothness parameter is estimated. The order of the parameter values in par cannot be changed. For tensor or ARD form correlation functions, range and tail becomes a vector.
output	a matrix of outputs
H	a matrix of regressors in the mean function of a GaSP model.
d	an R object holding the distances. It should be a distance matrix for constructing isotropic correlation matrix, or a list of distance matrices along each input dimension for constructing tensor or ARD types of correlation matrix.
covmodel	a list of two strings: family, form, where family indicates the family of covariance functions including the Confluent Hypergeometric class, the Matérn class, the Cauchy class, the powered-exponential class. form indicates the specific form of covariance structures including the isotropic form, tensor form, automatic relevance determination form. family CH The Confluent Hypergeometric correlation function is given by C(h) = \frac{\Gamma(\nu+\alpha)}{\Gamma(\nu)} \mathcal{U}\left(\alpha, 1-\nu, \left(\frac{h}{\beta}\right)^2\right), where \alpha is the tail decay parameter. \beta is the range parameter. \nu is the smoothness parameter. \mathcal{U}(\cdot) is the confluent hypergeometric function of the second kind. For details about this covariance, see Ma and Bhadra (2023; \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/01621459.2022.2027775")}). cauchy The generalized Cauchy covariance is given by C(h) = \left\{ 1 + \left( \frac{h}{\phi} \right)^{\nu} \right\}^{-\alpha/\nu}, where \phi is the range parameter. \alpha is the tail decay parameter. \nu is the smoothness parameter with default value at 2. matern The Matérn correlation function is given by C(h)=\frac{2^{1-\nu}}{\Gamma(\nu)} \left( \frac{h}{\phi} \right)^{\nu} \mathcal{K}_{\nu}\left( \frac{h}{\phi} \right), where \phi is the range parameter. \nu is the smoothness parameter. \mathcal{K}_{\nu}(\cdot) is the modified Bessel function of the second kind of order \nu. exp This is the Matérn correlation with \nu=0.5. This covariance should be specified as matern with smoothness parameter \nu=0.5. matern_3_2 This is the Matérn correlation with \nu=1.5. This covariance should be specified as matern with smoothness parameter \nu=1.5. matern_5_2 This is the Matérn correlation with \nu=2.5. This covariance should be specified as matern with smoothness parameter \nu=2.5. powexp The powered-exponential correlation function is given by C(h)=\exp\left\{-\left(\frac{h}{\phi}\right)^{\nu}\right\}, where \phi is the range parameter. \nu is the smoothness parameter. gauss The Gaussian correlation function is given by C(h)=\exp\left(-\frac{h^2}{\phi^2}\right), where \phi is the range parameter. form isotropic This indicates the isotropic form of covariance functions. That is, C(\mathbf{h}) = C^0(\|\mathbf{h}\|; \boldsymbol \theta), where \| \mathbf{h}\| denotes the Euclidean distance or the great circle distance for data on sphere. C^0(\cdot) denotes any isotropic covariance family specified in family. tensor This indicates the tensor product of correlation functions. That is, C(\mathbf{h}) = \prod_{i=1}^d C^0(|h_i|; \boldsymbol \theta_i), where d is the dimension of input space. h_i is the distance along the ith input dimension. This type of covariance structure has been often used in Gaussian process emulation for computer experiments. ARD This indicates the automatic relevance determination form. That is, C(\mathbf{h}) = C^0\left(\sqrt{\sum_{i=1}^d\frac{h_i^2}{\phi^2_i}}; \boldsymbol \theta \right), where \phi_i denotes the range parameter along the ith input dimension.
smooth	The smoothness parameter \nu in a correlation function.
smoothness_est	a logical value indicating whether the smoothness parameter is estimated.

Value

The natural logarithm of marginal or integrated likelihood

Author(s)

Pulong Ma [e-mail]

See Also

CH, matern, gp.optim, GPBayes-package, GaSP

GPBayes documentation built on May 29, 2024, 7:30 a.m.