kernel {GPBayes}

A wraper to build different kinds of correlation matrices with distance as arguments

Description

This function wraps existing built-in routines to construct a covariance matrix based on data type, covariance type, and distance type with distances as inputs. The constructed covariance matrix can be directly used for GaSP fitting and and prediction for spatial data, spatio-temporal data, and computer experiments.

Usage

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

Arguments

d

a matrix or a list of distances

range

a vector of range parameters, which could be a scalar.

tail

a vector of tail decay parameters, which could be a scalar.

nu

a vector of smoothness parameters, which could be a scalar.

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; 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 correlation matrix

Author(s)

Pulong Ma [e-mail]

See Also

CH, matern, ikernel, GPBayes-package, GaSP

Examples

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