"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 and a clear exposition of the computations. Lambda, the smoothing parameter must be fixed.

Usage

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mKrig(x, y, weights=rep(1, nrow(x)), cov.function="stationary.cov", 
                  cov.args = NULL, Z = NULL, lambda = 0, m = 2, 
                  chol.args = NULL, find.trA = TRUE, NtrA = 20, 
                  iseed = 123, llambda = NULL, ...)
  
## S3 method for class 'mKrig'
predict( object, xnew=NULL,ynew=NULL, grid.list = 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)

Arguments

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 mKrig.coef a new vector of observations. If y is a matrix the columns are assumed to be independent observations vectors generated from the same covariance and measurment error model.

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 x

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.

llambda

If not NULL then lambda = exp( llambda)

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 burden by approximately NtrA solutions of the linear system but the cholesky decomposition is reused.

grid.list

A grid.list to evaluate the surface in place of specifying arbitrary locations.

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 set.seed(iseed)) used to generate iid normals for Monte Carlo estimate of the trace.

...

In mKrig and predict additional arguments that will be passed to the covariance function.

object

Object returned by mKrig. (Same as "x" in the print function.)

xnew

Locations for predictions.

ynew

New observation vector. mKrig will reuse matrix decompositions and find the new fit to these data.

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.

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 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 predictSE.mKrig for prediction standard errors and sim.mKrig.approx to quantify the uncertainty in the estimated function using conditional simulation.

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 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 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.

Value

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 d coefficients vector. This is helpful to distguish between polynomial part and the extra covariates coefficients associated with Z.

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 np then the individual members of the Monte Carlo sample and sd(trA.info)/ sqrt(NtrA) is an estimate of the standard error. If NtrA is greater than or equal to np then these are the diagonal elements of A(lamdba).

GCV

Estimated value of the GCV function.

GCV.info

Monte Carlo sample of GCV functions

Author(s)

Doug Nychka, Reinhard Furrer, John Paige

References

http://cran.r-project.org/web/packages/fields/fields.pdf http://www.image.ucar.edu/~nychka/Fields/

See Also

Krig, surface.mKrig, Tps, fastTps, predictSurface, predictSE.mKrig, sim.mKrig.approx, mKrig.grid

Examples

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#
# 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 mKrig.MLE 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 mKrig.MLE 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

## Not run: 
 look2<- mKrig( x,y, cov.function="wendland.cov",k=2, theta=.9, lambda=.1)

## End(Not run)

# 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

## Not run:   
# 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
#
 look2<- mKrig( x,y, 
            cov.function="wendland.cov",k=2, theta=.1, lambda=1e2  )

## End(Not run)
# 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. 
## Not run: 
  mKrig( x,y, cov.function="wendland.cov",k=2, theta=dguess, 
            lambda=1e2) ->  look2

## End(Not run)

###############################################################################
# use precomputed distance matrix
#
## Not run:  
  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 )

## End(Not run)

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