# Covariance functions

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##### 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$theta) By convention, the Taper function is assumed to be identically zero outside the interval [0,1]. Some efficiency is introduced within the function to search for pairs of locations that are nonzero with respect to the Taper. This is done by the SPAM function nearest.dist. This search may find more nonzero pairs than dimensioned internally and SPAM will try to increase the space. One can also reset the SPAM options to avoid these warnings. For spam.format TRUE the multiplication with the C argument is done with the spam sparse multiplication routines through the "overloading" of the %*% operator. About the FORTRAN: The actual function Exp.cov and Rad.cov call FORTRAN to make the evaluation more efficient this is especially important when the C argument is supplied. So unfortunately the actual production code in Exp.cov is not as crisp as the R code sketched above. See Rad.simple.cov for a R coding of the radial basis functions. ##### Value If the argument C is NULL the cross covariance matrix is returned. In general if nrow(x1)=m and nrow(x2)=n then the returned matrix will be mXn. Moreover, if x1 is equal to x2 then this is the covariance matrix for this set of locations. If C is a vector of length n, then returned value is the multiplication of the cross covariance matrix with this vector. ##### See Also Krig, rdist, rdist.earth, gauss.cov, Exp.image.cov, Exponential, Matern, Wendland.cov, mKrig ##### Aliases • Exp.cov • Exp.simple.cov • Rad.cov • Rad.simple.cov • stationary.cov • stationary.taper.cov • wendland.cov • cubic.cov ##### Examples # NOT RUN { # exponential covariance matrix ( marginal variance =1) for the ozone #locations out<- Exp.cov( ChicagoO3$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)

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