cor.to.par {GPBayes}

Find the correlation parameter given effective range

Description

This function finds the correlation parameter given effective range

Usage

cor.to.par(
  d,
  param,
  family = "CH",
  cor.target = 0.05,
  lower = NULL,
  upper = NULL,
  tol = .Machine$double.eps
)

Arguments

d

a numerical value containing the effective range

param

a list containing correlation parameters. The specification of param should depend on the covariance model. If the parameter value is NULL, this function will find its value given the effective range via root-finding function uniroot.

• For the Confluent Hypergeometric class, range is used to denote the range parameter \beta. tail is used to denote the tail decay parameter \alpha. nu is used to denote the smoothness parameter \nu.

• For the generalized Cauchy class, range is used to denote the range parameter \phi. tail is used to denote the tail decay parameter \alpha. nu is used to denote the smoothness parameter \nu.

• For the Matérn class, range is used to denote the range parameter \phi. nu is used to denote the smoothness parameter \nu. When \nu=0.5, the Matérn class corresponds to the exponential covariance.

• For the powered-exponential class, range is used to denote the range parameter \phi. nu is used to denote the smoothness parameter. When \nu=2, the powered-exponential class corresponds to the Gaussian covariance.

family

a string indicating the type of covariance structure. The following correlation functions are implemented:

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. Note that this parameterization of the CH covariance is different from the one in Ma and Bhadra (2023). 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.

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

The exponential correlation function is given by

C(h)=\exp(-h/\phi),

where \phi is the range parameter. This is the Matérn correlation with \nu=0.5.

matern_3_2

The Matérn correlation with \nu=1.5.

matern_5_2

The Matérn correlation with \nu=2.5.

cor.target

a numerical value. The default value is 0.05, which means that correlation parameters are searched such that the correlation is approximately 0.05.

lower

a numerical value. This sets the lower bound to find the correlation parameter via the R function uniroot.

upper

a numerical value. This sets the upper bound to find the correlation parameter via the R function uniroot.

tol

a numerical value. This sets the precision of the solution with default value specified as the machine precision .Machine$double.eps in R.

Value

a numerical value of correlation parameters

Author(s)

Pulong Ma [e-mail]

See Also

GPBayes-package, GaSP, kernel, ikernel

Examples

range = cor.to.par(1,param=list(tail=0.5,nu=2.5), family="CH")
tail = cor.to.par(1,param=list(range=0.5,nu=2.5), family="CH")
range = cor.to.par(1,param=list(nu=2.5),family="matern")