deriv_kernel: A wraper to construct the derivative of correlation matrix...

In GPBayes: Tools for Gaussian Process Modeling in Uncertainty Quantification

A wraper to construct the derivative of correlation matrix with respect to correlation parameters

Description

This function wraps existing built-in routines to construct the derivative of correlation matrix with respect to correlation parameters.

Usage

deriv_kernel(d, range, tail, nu, covmodel)

Arguments

d	a matrix or a list of distances returned from distance.
range	a vector of range parameters
tail	a vector of tail decay parameters
nu	a vector of smoothness parameters
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.

Value

a list of matrices

Author(s)

Pulong Ma [e-mail]

See Also

CH, matern, kernel, GPBayes-package, GaSP

Examples

input = seq(0,1,length=10)
d = distance(input,input,type="isotropic",dtype="Euclidean")
dR = deriv_kernel(d,range=0.5,tail=0.2,nu=2.5,
         covmodel=list(family="CH",form="isotropic"))

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