https://github.com/cran/fields
Tip revision: 370c1161f275c7f191e9326102bfd61cae62d545 authored by Douglas Nychka on 11 May 2017, 17:23:03 UTC
version 8.15
version 8.15
Tip revision: 370c116
mKrig.Rd
%# fields is a package for analysis of spatial data written for
%# the R software environment .
%# Copyright (C) 2016
%# University Corporation for Atmospheric Research (UCAR)
%# Contact: Douglas Nychka, nychka@ucar.edu,
%# National Center for Atmospheric Research, PO Box 3000, Boulder, CO 80307-3000
%#
%# This program is free software; you can redistribute it and/or modify
%# it under the terms of the GNU General Public License as published by
%# the Free Software Foundation; either version 2 of the License, or
%# (at your option) any later version.
%# This program is distributed in the hope that it will be useful,
%# but WITHOUT ANY WARRANTY; without even the implied warranty of
%# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
%# GNU General Public License for more details.
%#
%# You should have received a copy of the GNU General Public License
%# along with the R software environment if not, write to the Free Software
%# Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
%# or see http://www.r-project.org/Licenses/GPL-2
\name{mKrig}
\alias{mKrig}
\alias{predict.mKrig}
\alias{mKrig.coef}
\alias{mKrig.trace}
\alias{print.mKrig}
\alias{summary.mKrig}
\alias{mKrigCheckXY}
\title{"micro Krig" Spatial process estimate of a curve or surface,
"kriging" with a known covariance function. }
\description{
This is a simple version of the Krig function that is
optimized for large data sets, sparse linear algebra, and a clear exposition of the
computations. Lambda, the smoothing parameter must be fixed.
This function is called higher level functions for maximum likelihood estimates of
covariance paramters.
}
\usage{
mKrig(x, y, weights = rep(1, nrow(x)), Z = NULL,
cov.function = "stationary.cov", cov.args = NULL,
lambda = 0, m = 2, chol.args = NULL, find.trA = TRUE,
NtrA = 20, iseed = 123, llambda = NULL, na.rm = FALSE,
collapseFixedEffect = TRUE,
...)
\method{predict}{mKrig}( object, xnew=NULL,ynew=NULL, grid.list = NULL,
derivative=0,
Z=NULL,drop.Z=FALSE,just.fixed=FALSE,collapseFixedEffect = TRUE, ...)
\method{summary}{mKrig}(object, ...)
\method{print}{mKrig}( x, digits=4,... )
mKrig.coef(object, y, collapseFixedEffect=TRUE)
mKrig.trace( object, iseed, NtrA)
mKrigCheckXY(x, y, weights, Z, na.rm)
}
%- maybe also 'usage' for other objects documented here.
\arguments{
\item{collapseFixedEffect}{ If replicated fields are given to mKrig (
\code{y} has multiple columns) there is the choice of estimating the
fixed effect coefficients (\code{d} in thereturned object) separately
for each replicate or pooling across replicates and deriving a single
estimate. If collapseFixedEffect is TRUE (default) the estimates are
pooled. }
\item{chol.args}{A list of optional arguments (pivot, nnzR) that will
be used with the call to the cholesky decomposition. Pivoting is done
by default to make use of sparse matrices when they are
generated. This argument is useful in some cases for sparse covariance
functions to reset the memory parameter nnzR. (See example below.)}
\item{cov.args}{A list of optional arguments that will be used in
calls to the covariance function.}
\item{cov.function}{The name, a text string of the covariance function.}
\item{derivative}{If zero the surface will be evaluated. If not zero
the matrix of partial derivatives will be computed.}
\item{digits}{Number of significant digits used in printed output.}
\item{drop.Z}{If true the fixed part will only be evaluated at the
polynomial part of the fixed model. The contribution from the other
covariates will be omitted.}
\item{find.trA}{
If TRUE will estimate the effective degrees of freedom using
a simple Monte Carlo method. This will add to the computational
burden by approximately NtrA solutions of the linear system but
the cholesky decomposition is reused.}
\item{grid.list}{A grid.list to evaluate the surface in place of specifying
arbitrary locations.}
\item{iseed}{Random seed ( using \code{set.seed(iseed)}) used to generate
iid normals for Monte Carlo estimate of the trace.}
\item{just.fixed}{If TRUE only the predictions for the fixed part of
the model will be evaluted.}
\item{lambda}{ Smoothing parameter or equivalently the ratio between
the nugget and process varainces.}
\item{llambda}{If not \code{NULL} then \code{lambda = exp( llambda)}}
\item{m}{ The degree of the polynomial used in teh fixed part is (m-1)}
\item{na.rm}{If TRUE NAs in y are omitted along with corresonding rows of x.}
\item{NtrA}{Number of Monte Carlo samples for the trace. But if NtrA is
greater than or equal to the number of observations the trace
is computed exactly.}
\item{object}{Object returned by mKrig. (Same as "x"
in the print function.)}
\item{weights}{Precision ( 1/variance) of each observation}
\item{x}{Matrix of unique spatial locations (or in print or surface
the returned mKrig object.)}
\item{xnew}{Locations for predictions.}
\item{y}{ Vector or matrix of observations at spatial locations,
missing values are not allowed! Or in mKrig.coef a new
vector of observations. If y is a matrix the columns are
assumed to be independent replicates of the spatial field. I.e.
observation vectors generated
from the same covariance and measurment error model but
independent from each other.
}
\item{ynew}{New observation vector. \code{mKrig} will reuse matrix
decompositions and find the new fit to these data.}
\item{Z}{ Linear covariates to be included in fixed part of the
model that are distinct from the default low order
polynomial in \code{x}. (NOTE the order of the polynomial
determined by \code{m})}
\item{\dots}{ In \code{mKrig} and \code{predict} additional arguments
that will be passed to the covariance function.}
}
\details{
This function is an abridged version of Krig. The m stand for micro
and this function focuses on the computations in Krig.engine.fixed
done for a fixed lambda parameter, for unique spatial locations and
for data without missing values.
These restrictions simplify the code for reading. Note that also
little checking is done and the spatial locations are not transformed
before the estimation. Because most of the operations are linear
algebra this code has been written to handle multiple data
sets. Specifically if the spatial model is the same except for
different observed values (the y's), one can pass \code{y} as a matrix
and the computations are done efficiently for each set. Note that
this is not a multivariate spatial model just an efficient computation
over several data vectors without explicit looping.A big difference in
the computations is that an exact expression for thetrace of the
smoothing matrix is (trace A(lambda)) is computationally expensive and
a Monte Carlo approximation is supplied instead.
See \code{predictSE.mKrig} for prediction standard errors and
\code{sim.mKrig.approx} to quantify the uncertainty in the estimated function using conditional
simulation.
\code{predict.mKrig} will evaluate the derivatives of the estimated
function if derivatives are supported in the covariance function. For
example the wendland.cov function supports derivatives.
\code{print.mKrig} is a simple summary function for the object.
\code{mKrig.coef} finds the "d" and "c" coefficients represent the
solution using the previous cholesky decomposition for a new data
vector. This is used in computing the prediction standard error in
predictSE.mKrig and can also be used to evalute the estimate
efficiently at new vectors of observations provided the locations and
covariance remain fixed.
Sparse matrix methods are handled through overloading the usual linear
algebra functions with sparse versions. But to take advantage of some
additional options in the sparse methods the list argument chol.args
is a device for changing some default values. The most important of
these is \code{nnzR}, the number of nonzero elements anticipated in
the Cholesky factorization of the postive definite linear system used
to solve for the basis coefficients. The sparse of this system is
essentially the same as the covariance matrix evalauted at the
observed locations. As an example of resetting \code{nzR} to 450000
one would use the following argument for chol.args in mKrig:
\code{ chol.args=list(pivot=TRUE,memory=list(nnzR= 450000))}
\code{mKrig.trace} This is an internal function called by \code{mKrig}
to estimate the effective degrees of freedom. The Kriging surface
estimate at the data locations is a linear function of the data and
can be represented as A(lambda)y. The trace of A is one useful
measure of the effective degrees of freedom used in the surface
representation. In particular this figures into the GCV estimate of
the smoothing parameter. It is computationally intensive to find the
trace explicitly but there is a simple Monte Carlo estimate that is
often very useful. If E is a vector of iid N(0,1) random variables
then the trace of A is the expected value of t(E)AE. Note that AE is
simply predicting a surface at the data location using the synthetic
observation vector E. This is done for \code{NtrA} independent N(0,1)
vectors and the mean and standard deviation are reported in the
\code{mKrig} summary. Typically as the number of observations is
increased this estimate becomse more accurate. If NtrA is as large as
the number of observations (\code{np}) then the algorithm switches to
finding the trace exactly based on applying A to \code{np} unit
vectors.
}
\value{
\item{d}{Coefficients of the polynomial fixed part and if present
the covariates (Z).To determine which is which the logical vector
ind.drift also part of this object is TRUE for the polynomial
part. }
\item{c}{ Coefficients of the nonparametric part.}
\item{nt}{ Dimension of fixed part.}
\item{np}{ Dimension of c.}
\item{nZ}{Number of columns of Z covariate matrix (can be zero).}
\item{ind.drift}{Logical vector that indicates polynomial
coefficients in the \code{d} coefficients vector. This is helpful
to distguish between polynomial part and the extra covariates
coefficients associated with Z. }
\item{lambda.fixed}{The fixed lambda value}
\item{x}{Spatial locations used for fitting.}
\item{knots}{The same as x}
\item{cov.function.name}{Name of covariance function used.}
\item{args}{ A list with all the covariance arguments that were
specified in the call.}
\item{m}{Order of fixed part polynomial.}
\item{chol.args}{ A list with all the cholesky arguments that were
specified in the call.}
\item{call}{ A copy of the call to mKrig.}
\item{non.zero.entries}{ Number of nonzero entries in the covariance
matrix for the process at the observation locations.}
\item{shat.MLE}{MLE of sigma.}
\item{rho.MLE}{MLE or rho.}
\item{rhohat}{Estimate for rho adjusted for fixed model degrees of
freedom (ala REML).}
\item{lnProfileLike}{log Profile likelihood for lambda}
\item{lnDetCov}{Log determinant of the covariance matrix for the
observations having factored out rho.}
\item{Omega}{GLS covariance for the estimated parameters in the fixed
part of the model (d coefficients0.}
\item{qr.VT, Mc}{QR and cholesky matrix decompositions needed to
recompute the estimate for new observation vectors.}
\item{fitted.values, residuals}{Usual predictions from fit.}
\item{eff.df}{Estimate of effective degrees of freedom. Either the
mean of the Monte Carlo sample or the exact value. }
\item{trA.info}{If NtrA ids less than \code{np} then the individual
members of the Monte Carlo sample and \code{sd(trA.info)/ sqrt(NtrA)}
is an estimate of the standard error. If NtrA is greater than or equal
to \code{np} then these are the diagonal elements of A(lamdba).}
\item{GCV}{Estimated value of the GCV function.}
\item{GCV.info}{Monte Carlo sample of GCV functions}
}
\author{Doug Nychka, Reinhard Furrer, John Paige}
\seealso{ Krig, surface.mKrig, Tps, fastTps, predictSurface, predictSE.mKrig, sim.mKrig.approx,
\code{ \link{mKrig.grid}}}
\examples{
#
# Midwest ozone data 'day 16' stripped of missings
data( ozone2)
y<- ozone2$y[16,]
good<- !is.na( y)
y<-y[good]
x<- ozone2$lon.lat[good,]
# nearly interpolate using defaults (Exponential covariance range = 2.0)
# see also mKrigMLEGrid to choose lambda by maxmimum likelihood
out<- mKrig( x,y, theta = 2.0, lambda=.01)
out.p<- predictSurface( out)
surface( out.p)
#
# NOTE this should be identical to
# Krig( x,y, theta=2.0, lambda=.01)
##############################################################################
# an example using a "Z" covariate and the Matern family
# again see mKrigMLEGrid to choose parameters by MLE.
data(COmonthlyMet)
yCO<- CO.tmin.MAM.climate
good<- !is.na( yCO)
yCO<-yCO[good]
xCO<- CO.loc[good,]
Z<- CO.elev[good]
out<- mKrig( xCO,yCO, Z=Z, cov.function="stationary.cov", Covariance="Matern",
theta=4.0, smoothness=1.0, lambda=.1)
set.panel(2,1)
# quilt.plot with elevations
quilt.plot( xCO, predict(out))
# Smooth surface without elevation linear term included
surface( out)
set.panel()
#########################################################################
# Interpolate using tapered version of the exponential,
# the taper scale is set to 1.5 default taper covariance is the Wendland.
# Tapering will done at a scale of 1.5 relative to the scaling
# done through the theta passed to the covariance function.
data( ozone2)
y<- ozone2$y[16,]
good<- !is.na( y)
y<-y[good]
x<- ozone2$lon.lat[good,]
mKrig( x,y,cov.function="stationary.taper.cov",
theta = 2.0, lambda=.01,
Taper="Wendland", Taper.args=list(theta = 1.5, k=2, dimension=2)
) -> out2
# Try out GCV on a grid of lambda's.
# For this small data set
# one should really just use Krig or Tps but this is an example of
# approximate GCV that will work for much larger data sets using sparse
# covariances and the Monte Carlo trace estimate
#
# a grid of lambdas:
lgrid<- 10**seq(-1,1,,15)
GCV<- matrix( NA, 15,20)
trA<- matrix( NA, 15,20)
GCV.est<- rep( NA, 15)
eff.df<- rep( NA, 15)
logPL<- rep( NA, 15)
# loop over lambda's
for( k in 1:15){
out<- mKrig( x,y,cov.function="stationary.taper.cov",
theta = 2.0, lambda=lgrid[k],
Taper="Wendland", Taper.args=list(theta = 1.5, k=2, dimension=2) )
GCV[k,]<- out$GCV.info
trA[k,]<- out$trA.info
eff.df[k]<- out$eff.df
GCV.est[k]<- out$GCV
logPL[k]<- out$lnProfileLike
}
#
# plot the results different curves are for individual estimates
# the two lines are whether one averages first the traces or the GCV criterion.
#
par( mar=c(5,4,4,6))
matplot( trA, GCV, type="l", col=1, lty=2,
xlab="effective degrees of freedom", ylab="GCV")
lines( eff.df, GCV.est, lwd=2, col=2)
lines( eff.df, rowMeans(GCV), lwd=2)
# add exact GCV computed by Krig
out0<- Krig( x,y,cov.function="stationary.taper.cov",
theta = 2.0,
Taper="Wendland", Taper.args=list(theta = 1.5, k=2, dimension=2),
spam.format=FALSE)
lines( out0$gcv.grid[,2:3], lwd=4, col="darkgreen")
# add profile likelihood
utemp<- par()$usr
utemp[3:4] <- range( -logPL)
par( usr=utemp)
lines( eff.df, -logPL, lwd=2, col="blue", lty=2)
axis( 4)
mtext( side=4,line=3, "-ln profile likelihood", col="blue")
title( "GCV ( green = exact) and -ln profile likelihood", cex=2)
#########################################################################
# here is a series of examples with bigger datasets
# using a compactly supported covariance directly
set.seed( 334)
N<- 1000
x<- matrix( 2*(runif(2*N)-.5),ncol=2)
y<- sin( 1.8*pi*x[,1])*sin( 2.5*pi*x[,2]) + rnorm( 1000)*.1
look2<-mKrig( x,y, cov.function="wendland.cov",k=2, theta=.2,
lambda=.1)
# take a look at fitted surface
predictSurface(look2)-> out.p
surface( out.p)
# this works because the number of nonzero elements within distance theta
# are less than the default maximum allocated size of the
# sparse covariance matrix.
# see spam.options() for the default values
# The following will give a warning for theta=.9 because
# allocation for the covariance matirx storage is too small.
# Here theta controls the support of the covariance and so
# indirectly the number of nonzero elements in the sparse matrix
\dontrun{
look2<- mKrig( x,y, cov.function="wendland.cov",k=2, theta=.9, lambda=.1)
}
# The warning resets the memory allocation for the covariance matrix
# according the to values 'spam.options(nearestdistnnz=c(416052,400))'
# this is inefficient becuase the preliminary pass failed.
# the following call completes the computation in "one pass"
# without a warning and without having to reallocate more memory.
spam.options(nearestdistnnz=c(416052,400))
look2<- mKrig( x,y, cov.function="wendland.cov",k=2,
theta=.9, lambda=1e-2)
# as a check notice that
# print( look2)
# reports the number of nonzero elements consistent with the specifc allocation
# increase in spam.options
# new data set of 1500 locations
set.seed( 234)
N<- 1500
x<- matrix( 2*(runif(2*N)-.5),ncol=2)
y<- sin( 1.8*pi*x[,1])*sin( 2.5*pi*x[,2]) + rnorm( N)*.01
\dontrun{
# the following is an example of where the allocation (for nnzR)
# for the cholesky factor is too small. A warning is issued and
# the allocation is increased by 25% in this example
#
look2<- mKrig( x,y,
cov.function="wendland.cov",k=2, theta=.1, lambda=1e2 )
}
# to avoid the warning
look2<-mKrig( x,y,
cov.function="wendland.cov", k=2, theta=.1,
lambda=1e2, chol.args=list(pivot=TRUE, memory=list(nnzR= 450000)))
###############################################################################
# fiting multiple data sets
#
#\dontrun{
y1<- sin( 1.8*pi*x[,1])*sin( 2.5*pi*x[,2]) + rnorm( N)*.01
y2<- sin( 1.8*pi*x[,1])*sin( 2.5*pi*x[,2]) + rnorm( N)*.01
Y<- cbind(y1,y2)
look3<- mKrig( x,Y,cov.function="wendland.cov",k=2, theta=.1,
lambda=1e2 )
# note slight difference in summary because two data sets have been fit.
print( look3)
#}
##################################################################
# finding a good choice for theta as a taper
# Suppose the target is a spatial prediction using roughly 50 nearest neighbors
# (tapering covariances is effective for roughly 20 or more in the situation of
# interpolation) see Furrer, Genton and Nychka (2006).
# take a look at a random set of 100 points to get idea of scale
set.seed(223)
ind<- sample( 1:N,100)
hold<- rdist( x[ind,], x)
dd<- (apply( hold, 1, sort))[65,]
dguess<- max(dd)
# dguess is now a reasonable guess at finding cutoff distance for
# 50 or so neighbors
# full distance matrix excluding distances greater than dguess
# but omit the diagonal elements -- we know these are zero!
hold<- nearest.dist( x, delta= dguess,upper=TRUE)
# exploit spam format to get quick of number of nonzero elements in each row
hold2<- diff( hold@rowpointers)
# min( hold2) = 55 which we declare close enough
# now the following will use no less than 55 nearest neighbors
# due to the tapering.
\dontrun{
mKrig( x,y, cov.function="wendland.cov",k=2, theta=dguess,
lambda=1e2) -> look2
}
###############################################################################
# use precomputed distance matrix
#
\dontrun{
y1<- sin( 1.8*pi*x[,1])*sin( 2.5*pi*x[,2]) + rnorm( N)*.01
y2<- sin( 1.8*pi*x[,1])*sin( 2.5*pi*x[,2]) + rnorm( N)*.01
Y<- cbind(y1,y2)
#precompute distance matrix in compact form
distMat = rdist(x, compact=TRUE)
look3<- mKrig( x,Y,cov.function="stationary.cov", theta=.1,
lambda=1e2, distMat=distMat )
#precompute distance matrix in standard form
distMat = rdist(x)
look3<- mKrig( x,Y,cov.function="stationary.cov", theta=.1,
lambda=1e2, distMat=distMat )
}
}
\references{
%% ~put references to the literature/web site here ~
http://cran.r-project.org/web/packages/fields/fields.pdf
http://www.image.ucar.edu/~nychka/Fields/
}
% Add one or more standard keywords, see file 'KEYWORDS' in the
% R documentation directory.
\keyword{spatial }