Exponential, Matern, Radial Basis {fields} | R Documentation

## Covariance functions

### Description

Functional form of covariance function assuming the argument is a distance between locations.

### Usage

    Exponential(d, range = 1, alpha = 1/range, phi = 1)
    Matern (d , scale = 1, range = 1,alpha=1/range,
         smoothness = 0.5, nu= smoothness, phi=scale)
    Matern.cor.to.range(d, nu, cor.target=.5, guess=NULL,...)
    RadialBasis(d,M,dimension)

### Arguments

`d` | Vector of distances or for `Matern.cor.to.range` just a single distance.
`range` | Range parameter default is one. Note that the scale can also be specified through the "theta" scaling argument used in fields covariance functions)
`alpha` | 1/range
`scale` | Same as phi
`phi` | Marginal variance.
`smoothness` | Smoothness parameter in Matern. Controls the number of derivatives in the process. Default is 1/2 corresponding to an exponential covariance.
`nu` | Same as smoothness
`M` | Interpreted as a spline M is the order of the derivatives in the penalty.
`dimension` | Dimension of function
`cor.target` | Correlation used to match the range parameter. Default is .5.
`guess` | An optional starting guess for solution. This should not be needed.
`...` | Additional arguments to pass to the bisection search function.

### Details

Exponential:

phi* exp( -d/range)

Matern:

phi*con*(d^nu) * besselK(d , nu )

Matern covariance function transcribed from Stein's book page 31 nu==smoothness, alpha == 1/range

GeoR parameters map to kappa==smoothness and phi == range check for negative distances

`con` is a constant that normalizes the expression to be 1.0 when phi=1.0 and d=0.

Matern.cor.to.range: This function is useful to find Matern covariance parameters that are comparable for different smoothness parameters. Given a distance `d`, smoothness `nu`, target correlation `cor.target` and range `theta`, this function determines numerically the value of theta so that

`Matern( d, range=theta, nu=nu) == cor.target`

See the example for how this might be used.

Radial basis functions: C.m,d r^(2m-d) d- odd C.m,d r^(2m-d)ln(r) d-even where C.m.d is a constant based on spline theory and r is the radial distance between points. See `radbas.constant`.

### Value

For the covariance functions: a vector of covariances.
For Matern.cor.to.range: the value of the range parameter.

### Author(s)

Doug Nychka

### References

Stein, M.L. (1999) Statistical Interpolation of Spatial Data: Some Theory for Kriging. Springer, New York.

### See Also

stationary.cov, stationary.image.cov, Wendland,stationary.taper.cov rad.cov

### Examples

    # a Matern correlation function
     d<- seq( 0,10,,200)
     y<- Matern( d, range=1.5, smoothness=1.0)
     plot( d,y, type="l")

    # Several Materns of different smoothness with a similar correlation
    # range

    # find ranges for nu = .5, 1.0 and 2.0
    # where the correlation drops to .1 at a distance of 10 units.

     r1<- Matern.cor.to.range( 10, nu=.5, cor.target=.1)
     r2<- Matern.cor.to.range( 10, nu=1.0, cor.target=.1)
     r3<- Matern.cor.to.range( 10, nu=2.0, cor.target=.1)

    # note that these equivalent ranges
    # with respect to this correlation length are quite different
    # due the different smoothness parameters.

     d<- seq( 0, 15,,200)
     y<- cbind(  Matern( d, range=r1, nu=.5),
                 Matern( d, range=r2, nu=1.0),
                 Matern( d, range=r3, nu=2.0))

     matplot( d, y, type="l", lty=1, lwd=2)
     xline( 10)
     yline( .1)

[Package fields version 6.01 Index]