# Covariance functions

##### Exponential family, radial basis functions,cubic spline, compactly supported Wendland family and stationary covariances.

Given two sets of locations these functions compute the cross covariance matrix for some covariance families. In addition these functions can take advantage of spareness, implement more efficient multiplcation of the cross covariance by a vector or matrix and also return a marginal variance to be consistent with calls by the Krig function.

`stationary.cov`

and `Exp.cov`

have additional arguments for
precomputed distance matrices and for calculating only the upper triangle
and diagonal of the output covariance matrix to save time. Also, they
support using the `rdist`

function with `compact=TRUE`

or input
distance matrices in compact form, where only the upper triangle of the
distance matrix is used to save time.

Note: These functions have been been renamed from the previous fields functions
using 'Exp' in place of 'exp' to avoid conflict with the generic exponential
function (`exp(...)`

)in R.

- Keywords
- spatial

##### Usage

```
Exp.cov(x1, x2=NULL, theta = 1, p=1, distMat = NA,
C = NA, marginal = FALSE, onlyUpper=FALSE)
```Exp.simple.cov(x1, x2, theta =1, C=NA,marginal=FALSE)

Rad.cov(x1, x2, p = 1, m=NA, with.log = TRUE, with.constant = TRUE,
C=NA,marginal=FALSE, derivative=0)

cubic.cov(x1, x2, theta = 1, C=NA, marginal=FALSE)

Rad.simple.cov(x1, x2, p=1, with.log = TRUE, with.constant = TRUE,
C = NA, marginal=FALSE)

stationary.cov(x1, x2=NULL, Covariance = "Exponential", Distance = "rdist",
Dist.args = NULL, theta = 1, V = NULL, C = NA, marginal = FALSE,
derivative = 0, distMat = NA, onlyUpper = FALSE, ...)

stationary.taper.cov(x1, x2, Covariance="Exponential",
Taper="Wendland",
Dist.args=NULL, Taper.args=NULL,
theta=1.0,V=NULL, C=NA, marginal=FALSE,
spam.format=TRUE,verbose=FALSE,...)

wendland.cov(x1, x2, theta = 1, V=NULL, k = 2, C = NA,
marginal =FALSE,Dist.args = list(method = "euclidean"),
spam.format = TRUE, derivative = 0, verbose=FALSE)

##### Arguments

- x1
Matrix of first set of locations where each row gives the coordinates of a particular point.

- x2
Matrix of second set of locations where each row gives the coordinatesof a particular point. If this is missing x1 is used.

- theta
Range (or scale) parameter. This should be a scalar (use the V argument for other scaling options). Any distance calculated for a covariance function is divided by theta before the covariance function is evaluated.

- V
A matrix that describes the inverse linear transformation of the coordinates before distances are found. In R code this transformation is:

`x1 %*% t(solve(V))`

Default is NULL, that is the transformation is just dividing distance by the scalar value`theta`

. See Details below. If one has a vector of "theta's" that are the scaling for each coordinate then just express this as`V = diag(theta)`

in the call to this function.- C
A vector with the same length as the number of rows of x2. If specified the covariance matrix will be multiplied by this vector.

- marginal
If TRUE returns just the diagonal elements of the covariance matrix using the

`x1`

locations. In this case this is just 1.0. The marginal argument will trivial for this function is a required argument and capability for all covariance functions used with Krig.- p
Exponent in the exponential covariance family. p=1 gives an exponential and p=2 gives a Gaussian. Default is the exponential form. For the radial basis function this is the exponent applied to the distance between locations.

- m
For the radial basis function p = 2m-d, with d being the dimension of the locations, is the exponent applied to the distance between locations. (m is a common way of parametrizing this exponent.)

- with.constant
If TRUE includes complicated constant for radial basis functions. See the function

`radbad.constant`

for more details. The default is TRUE, include the constant. Without the usual constant the lambda used here will differ by a constant from spline estimators ( e.g. cubic smoothing splines) that use the constant. Also a negative value for the constant may be necessary to make the radial basis positive definite as opposed to negative definite.- with.log
If TRUE include a log term for even dimensions. This is needed to be a thin plate spline of integer order.

- Covariance
Character string that is the name of the covariance shape function for the distance between locations. Choices in fields are

`Exponential`

,`Matern`

- Distance
Character string that is the name of the distance function to use. Choices in fields are

`rdist`

,`rdist.earth`

- Taper
Character string that is the name of the taper function to use. Choices in fields are listed in help(taper).

- Dist.args
A list of optional arguments to pass to the Distance function.

- Taper.args
A list of optional arguments to pass to the Taper function.

`theta`

should always be the name for the range (or scale) paremeter.- spam.format
If TRUE returns matrix in sparse matrix format implemented in the spam package. If FALSE just returns a full matrix.

- k
The order of the Wendland covariance function. See help on Wendland.

- derivative
If nonzero evaluates the partials of the covariance function at locations x1. This must be used with the "C" option and is mainly called from within a predict function. The partial derivative is taken with respect to

`x1`

.- verbose
If TRUE prints out some useful information for debugging.

- distMat
If the distance matrix between

`x1`

and`x2`

has already been computed, it can be passed via this argument so it won't need to be recomputed.- onlyUpper
For internal use only, not meant to be set by the user. Automatically set to

`TRUE`

by`mKrigMLEJoint`

or`mKrigMLEGrid`

if`lambda.profile`

is set to`TRUE`

, but set to`FALSE`

for the final parameter fit so output is compatible with rest of`fields`

.If

`TRUE`

, only the upper triangle and diagonal of the covariance matrix is computed to save time (although if a non-compact distance matrix is used, the onlyUpper argument is set to FALSE). If`FALSE`

, the entire covariance matrix is computed. In general, it should only be set to`TRUE`

for`mKrigMLEJoint`

and`mKrigMLEGrid`

, and the default is set to`FALSE`

so it is compatible with all of`fields`

.- …
Any other arguments that will be passed to the covariance function. e.g.

`smoothness`

for the Matern.

##### Details

For purposes of illustration, the function
`Exp.cov.simple`

is provided in fields as a simple example and
implements the R code discussed below. List this function out as a
way to see the standard set of arguments that fields uses to define a
covariance function. It can also serve as a template for creating new
covariance functions for the `Krig`

and `mKrig`

functions. Also see the higher level function `stationary.cov`

to
mix and match different covariance shapes and distance functions.

A common scaling for stationary covariances: If `x1`

and
`x2`

are matrices where `nrow(x1)=m`

and `nrow(x2)=n`

then this function will return a mXn matrix where the (i,j) element
is the covariance between the locations `x1[i,]`

and
`x2[j,]`

. The exponential covariance function is computed as
exp( -(D.ij)) where D.ij is a distance between `x1[i,]`

and
`x2[j,]`

but having first been scaled by theta. Specifically if
`theta`

is a matrix to represent a linear transformation of the
coordinates, then let `u= x1%*% t(solve( theta))`

and ```
v=
x2%*% t(solve(theta))
```

. Form the mXn distance matrix with
elements:

`D[i,j] = sqrt( sum( ( u[i,] - v[j,])**2 ) )`

.

and the cross covariance matrix is found by `exp(-D)`

. The
tapered form (ignoring scaling parameters) is a matrix with i,j entry
`exp(-D[i,j])*T(D[i,j])`

. With T being a positive definite
tapering function that is also assumed to be zero beyond 1.

Note that if theta is a scalar then this defines an isotropic
covariance function and the functional form is essentially
`exp(-D/theta)`

.

Implementation: The function `r.dist`

is a useful FIELDS function
that finds the cross Euclidean distance matrix (D defined above) for
two sets of locations. Thus in compact R code we have

exp(-rdist(u, v))

Note that this function must also support two other kinds of calls:

If marginal is TRUE then just the diagonal elements are returned (in R
code `diag( exp(-rdist(u,u)) )`

).

If C is passed then the returned value is ```
exp(-rdist(u, v))
%*% C
```

.

Some details on particular covariance functions:

- Radial basis functions (
`Rad.cov`

: The functional form is Constant* rdist(u, v)**p for odd dimensions and Constant* rdist(u,v)**p * log( rdist(u,v) ) For an m th order thin plate spline in d dimensions p= 2*m-d and must be positive. The constant, depending on m and d, is coded in the fields function

`radbas.constant`

. This form is only a generalized covariance function -- it is only positive definite when restricted to linear subspace. See`Rad.simple.cov`

for a coding of the radial basis functions in R code.- Stationary covariance
`stationary.cov`

: Here the computation is to apply the function Covariance to the distances found by the Distance function. For example

`Exp.cov(x1,x2, theta=MyTheta)`

and

`stationary.cov( x1,x2, theta=MyTheta, Distance= "rdist", Covariance="Exponential")`

are the same. This also the same as

`stationary.cov( x1,x2, theta=MyTheta, Distance= "rdist", Covariance="Matern",smoothness=.5)`

.- Stationary tapered covariance
`stationary.taper.cov`

: The resulting cross covariance is the direct or Shure product of the tapering function and the covariance. In R code given location matrices,

`x1`

and`x2`

and using Euclidean distance.`Covariance(rdist( x1, x2)/theta)*Taper( rdist( x1, x2)/Taper.args$theta)`

By convention, the

`Taper`

function is assumed to be identically zero outside the interval [0,1]. Some efficiency is introduced within the function to search for pairs of locations that are nonzero with respect to the Taper. This is done by the SPAM function`nearest.dist`

. This search may find more nonzero pairs than dimensioned internally and SPAM will try to increase the space. One can also reset the SPAM options to avoid these warnings. For spam.format TRUE the multiplication with the`C`

argument is done with the spam sparse multiplication routines through the "overloading" of the`%*%`

operator.

About the FORTRAN: The actual function `Exp.cov`

and
`Rad.cov`

call FORTRAN to
make the evaluation more efficient this is especially important when the
C argument is supplied. So unfortunately the actual production code in
Exp.cov is not as crisp as the R code sketched above. See
`Rad.simple.cov`

for a R coding of the radial basis functions.

##### Value

If the argument C is NULL the cross covariance matrix is returned. In general if nrow(x1)=m and nrow(x2)=n then the returned matrix will be mXn. Moreover, if x1 is equal to x2 then this is the covariance matrix for this set of locations.

If C is a vector of length n, then returned value is the multiplication of the cross covariance matrix with this vector.

##### See Also

Krig, rdist, rdist.earth, gauss.cov, Exp.image.cov, Exponential, Matern, Wendland.cov, mKrig

##### Examples

```
# NOT RUN {
# exponential covariance matrix ( marginal variance =1) for the ozone
#locations
out<- Exp.cov( ChicagoO3$x, theta=100)
# out is a 20X20 matrix
out2<- Exp.cov( ChicagoO3$x[6:20,],ChicagoO3$x[1:2,], theta=100)
# out2 is 15X2 matrix
# Kriging fit where the nugget variance is found by GCV
# Matern covariance shape with range of 100.
#
fit<- Krig( ChicagoO3$x, ChicagoO3$y, Covariance="Matern", theta=100,smoothness=2)
data( ozone2)
x<- ozone2$lon.lat
y<- ozone2$y[16,]
# Omit the NAs
good<- !is.na( y)
x<- x[good,]
y<- y[good]
# example of calling the taper version directly
# Note that default covariance is exponential and default taper is
# Wendland (k=2).
stationary.taper.cov( x[1:3,],x[1:10,] , theta=1.5, Taper.args= list(k=2,theta=2.0,
dimension=2) )-> temp
# temp is now a tapered 3X10 cross covariance matrix in sparse format.
is.spam( temp) # evaluates to TRUE
# should be identical to
# the direct matrix product
temp2<- Exp.cov( x[1:3,],x[1:10,], theta=1.5) * Wendland(rdist(x[1:3,],x[1:10,]),
theta= 2.0, k=2, dimension=2)
test.for.zero( as.matrix(temp), temp2)
# Testing that the "V" option works as advertized ...
x1<- x[1:20,]
x2<- x[1:10,]
V<- matrix( c(2,1,0,4), 2,2)
Vi<- solve( V)
u1<- t(Vi%*% t(x1))
u2<- t(Vi%*% t(x2))
look<- exp(-1*rdist(u1,u2))
look2<- stationary.cov( x1,x2, V= V)
test.for.zero( look, look2)
# Here is an example of how the cross covariance multiply works
# and lots of options on the arguments
Ctest<- rnorm(10)
temp<- stationary.cov( x,x[1:10,], C= Ctest,
Covariance= "Wendland",
k=2, dimension=2, theta=1.5 )
# do multiply explicitly
temp2<- stationary.cov( x,x[1:10,],
Covariance= "Wendland",
k=2, dimension=2, theta=1.5 )%*% Ctest
test.for.zero( temp, temp2)
# use the tapered stationary version
# cov.args is part of the argument list passed to stationary.taper.cov
# within Krig.
# This example needs the spam package.
#
# }
# NOT RUN {
Krig(x,y, cov.function = "stationary.taper.cov", theta=1.5,
cov.args= list(Taper.args= list(k=2, dimension=2,theta=2.0) )
) -> out2
# NOTE: Wendland is the default taper here.
# }
# NOT RUN {
# BTW this is very similar to
# }
# NOT RUN {
Krig(x,y, theta= 1.5)-> out
# }
# NOT RUN {
# }
```

*Documentation reproduced from package fields, version 10.0, License: GPL (>= 2)*