# 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

 ```1 2``` ```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:

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}

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

 ``` 1 2 3 4 5 6 7 8 9 10``` ```## 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 Feb. 14, 2020, 5:06 p.m.