mKrig {fields} | R Documentation |
This is a simple version of the Krig function that is optimized for large data sets and a clear exposition of the computations. Lambda, the smoothing parameter must be fixed.
mKrig(x, y, weights = rep(1, nrow(x)), Z=NULL, lambda = 0, cov.function = "stationary.cov", m = 2, chol.args=NULL,cov.args=NULL, find.trA = TRUE, NtrA = 20, iseed = 123, ...) ## S3 method for class 'mKrig' predict( object, xnew=NULL,ynew=NULL, derivative=0, Z=NULL, drop.Z=FALSE,just.fixed=FALSE, ...) ## S3 method for class 'mKrig' summary(object, ...) ## S3 method for class 'mKrig' print( x, digits=4,... ) mKrig.coef(object, y) mKrig.trace( object, iseed, NtrA)
x |
Matrix of unique spatial locations (or in print or surface the returned mKrig object.) |
y |
Vector or matrix of observations at spatial locations, missing values
are not allowed! Or in |
weights |
Precision ( 1/variance) of each observation |
Z |
Linear covariates to be included in fixed part of the model that are distinct from the default low order polynomial in |
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. |
lambda |
Smoothing parameter or equivalently the ratio between the nugget and process varainces. |
cov.function |
The name, a text string of the covariance function. |
m |
The degree of the polynomial used in teh fixed part is (m-1) |
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.) |
cov.args |
A list of optional arguments that will be used in calls to the covariance function. |
find.trA |
If TRUE will estimate the effective degrees of freedom using a simple Monte Carlo
method. This will add to the computational by approximately |
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. |
iseed |
Random seed ( using |
... |
In |
object |
Object returned by mKrig. (Same as "x" in the print function.) |
xnew |
Locations for predictions. |
ynew |
New observation vector. |
derivative |
If zero the surface will be evaluated. If not zero the matrix of partial derivatives will be computed. |
just.fixed |
If TRUE only the predictions for the fixed part of the model will be evaluted. |
digits |
Number of significant digits used in printed output. |
This function is an abridged version of Krig that 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 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.
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.
print.mKrig
is a simple summary function for the object.
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
predict.se.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 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 nzR
to 450000 one would use the following
argument for chol.args in mKrig:
chol.args=list(pivot=TRUE,memory=list(nnzR= 450000))
mKrig.trace
This is an internal function called by 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 NtrA
independent N(0,1)
vectors and the mean and standard deviation are reported in the
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 (np
) then the algorithm switches to
finding the trace exactly based on applying A to np
unit
vectors.
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. |
c |
Coefficients of the nonparametric part. |
nt |
Dimension of fixed part. |
np |
Dimension of c. |
nZ |
Number of columns of Z covariate matrix (can be zero). |
ind.drift |
Logical vector that indicates polynomial coefficients in the
|
lambda.fixed |
The fixed lambda value |
x |
Spatial locations used for fitting. |
knots |
The same as x |
cov.function.name |
Name of covariance function used. |
args |
A list with all the covariance arguments that were specified in the call. |
m |
Order of fixed part polynomial. |
chol.args |
A list with all the cholesky arguments that were specified in the call. |
call |
A copy of the call to mKrig. |
non.zero.entries |
Number of nonzero entries in the covariance matrix for the process at the observation locations. |
shat.MLE |
MLE of sigma. |
rho.MLE |
MLE or rho. |
rhohat |
Estimate for rho adjusted for fixed model degrees of freedom (ala REML). |
lnProfileLike |
log Profile likelihood for lambda |
lnDetCov |
Log determinant of the covariance matrix for the observations having factored out rho. |
Omega |
GLS covariance for the estimated parameters in the fixed part of the model (d coefficients0. |
qr.VT, Mc |
QR and cholesky matrix decompositions needed to recompute the estimate for new observation vectors. |
fitted.values, residuals |
Usual predictions from fit. |
eff.df |
Estimate of effective degrees of freedom. Either the mean of the Monte Carlo sample or the exact value. |
trA.info |
If NtrA ids less than |
GCV |
Estimated value of the GCV function. |
GCV.info |
Monte Carlo sample of GCV functions |
Doug Nychka, Reinhard Furrer
Krig, surface.mKrig, Tps, fastTps, mKrig.grid
# # 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) mKrig( x,y, theta = 2.0, lambda=.01)-> out # # NOTE this should be identical to # Krig( x,y, theta=2.0, lambda=.01) # 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. 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 predict.surface( out2)-> out.p surface( out.p) # 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) # an example using a "Z" covariate and the Matern family data(COmonthlyMet) y<- CO.tmin.MAM.climate good<- !is.na( y) y<-y[good] x<- CO.loc[good,] Z<- CO.elev[good] out<- mKrig( x,y, 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( x, predict(out)) # quilt.plot without elevation linear term included quilt.plot( x, predict(out, drop.Z=TRUE)) set.panel() # here is a series of examples with a bigger problem # 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 predict.surface(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 ## Not run: mKrig( x,y, cov.function="wendland.cov",k=2, theta=.9, lambda=.1)-> look2 ## End(Not run) # The warning resets the memory allocation for the covariance matirx according the # 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)) mKrig( x,y, cov.function="wendland.cov",k=2, theta=.9, lambda=1e-2)-> look2 # as a check notice that # print( look2) # report 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 # 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 # ## Not run: mKrig( x,y, cov.function="wendland.cov",k=2, theta=.1, lambda=1e2 )-> look2 ## End(Not run) # to avoid the warning mKrig( x,y, cov.function="wendland.cov",k=2, theta=.1, lambda=1e2, chol.args=list(pivot=TRUE,memory=list(nnzR= 450000)) )-> look2 # success! # 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) mKrig( x,Y,cov.function="wendland.cov",k=2, theta=.1, lambda=1e2 )-> look3 # 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. ## Not run: mKrig( x,y, cov.function="wendland.cov",k=2, theta=dguess, lambda=1e2) -> look2 ## End(Not run)