covmodelCMB: Computes values of covariance functions

In rcosmo: Cosmic Microwave Background Data Analysis

Description

This function computes the covariances given the separation distance of locations. Options for different covariance functions on spheres are available. The function extends the function cov.spatial for additional covariance models on spheres.

Usage

covmodelCMB(
  obj,
  cov.model = "matern",
  cov.pars = stop("no cov.pars argument provided"),
  kappa = 0.5
)

Arguments

obj	Vector of distances between pairs of spatial locations.
cov.model	A type of the correlation function. Available choices are: "matern", "exponential","spherical", "powered.exponential", "cauchy", "gencauchy", "pure.nugget", "askey", "c2wendland", "c4wendland", "sinepower", "multiquadric". Default is "matern"
cov.pars	A vector with two covariance parameters. The first parameter corresponds to the variance sigma^2. The second parameter corresponds to the range phi of the correlation function.
kappa	A smoothness parameter of the correlation function.

Details

The function returns the value of the covariance C(h) at the distance h. The covariance function has the form

C(h) = sigma^2 * rho(h/phi).

The parameters of the covariance are positive numbers sigma^2, phi and kappa.

Expressions for the correlation functions which are not included in the package geoR:

askey

rho(h/phi) = (1 - h/phi)^{kappa}, if h < phi;

0, otherwise.

c2wendland

rho(h/phi) = (1 + kappa * h/phi) * (1 - h/phi)^{kappa}, if h < phi;

0, otherwise.

c4wendland

rho(h/phi) = (1 + kappa * h/phi + (kappa^2 - 1) * (h/phi)^2 / 3) * (1 - h/phi)^{kappa}, if h < phi;

0, otherwise.

sinepower

rho(h/phi) = 1 - (sin(h/(2 phi))) ^{kappa}

multiquadric

C(h) = (1 - phi) ^{(2 * kappa)} / (1 + phi^2 - 2 * phi * cos(h))^{kappa}, 0<phi<1

Additional information can be found in the section Details in cov.spatial.

Value

Values of a covariance function for the given distances.

References

geoR package, cov.spatial

T. Gneiting. Strictly and non-strictly positive definite functions on spheres. Bernoulli 19 (2013), no. 4, 1327-1349.

Examples

## Compute Askey variogram at x = pi/4

1 - covmodelCMB(pi/4, cov.pars = c(1, pi), kappa = 3, cov.model = "askey" )

## Plot of the Askey covariance function

v1.f <- function(x, ...) {covmodelCMB(x, ...)}
curve( v1.f(x, cov.pars = c(1, 0.03), kappa = 3, cov.model = "askey"),
from = 0, to = 0.1, xlab = "distance", ylab = expression(cov(h)), lty = 2,
main = "covariance")

rcosmo documentation built on Dec. 11, 2021, 9:29 a.m.