# Exponential: Covariance functions In fields: Tools for Spatial Data

## Description

Functional form of covariance function assuming the argument is a distance between locations. As they are defined here, they are in fact correlation functions. To set the marginal variance (sill) parameter, use the `rho` argument in `mKrig` or `Krig`. To set the nugget variance, use te `sigma2` argument in `mKrig` or `Krig`.

## Usage

 ```1 2 3 4 5``` ```Exponential(d, range = 1, alpha = 1/range, phi=1.0) Matern(d , range = 1,alpha=1/range, smoothness = 0.5, nu= smoothness, phi=1.0) Matern.cor.to.range(d, nu, cor.target=.5, guess=NULL,...) RadialBasis(d,M,dimension, derivative = 0) ```

## 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 `phi` This parameter option is added to be compatible with older versions of fields and refers to the marginal variance of the process. e.g. `phi* exp( -d/theta)` is the exponential covariance for points separated by distance and range theta. Throughout fields this parameter is equivalent to rho and it recommended that rho be used. If one is simulating random fields. See the help on `sim.rf` for more details. `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. `derivative` If greater than zero finds the first derivative of this function. `...` Additional arguments to pass to the bisection search function.

## Details

Exponential:

exp( -d/range)

Matern:

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 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.

 ```1 2 3``` ``` 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` for the computation of the constant. NOTE: Earlier versions of fields used ln(r^2) instead of ln(r) and so differ by a factor of 2.

## Value

For the covariance functions: a vector of covariances.

For Matern.cor.to.range: the value of the range parameter.

Doug Nychka

## References

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

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27``` ```# 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) ```