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`

.

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)
``` |

`d` |
Vector of distances or for |

`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. |

`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. |

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.

Radial basis functions:

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.

For the covariance functions: a vector of covariances.

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

Doug Nychka

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

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

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)
``` |

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