mKrig: "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

mKrig(x, y, weights = rep(1, nrow(x)), 
  lambda = 0, cov.function = "stationary.cov", 
    m = 2, chol.args=NULL,cov.args=NULL, ...)
## S3 method for class 'mKrig':
predict( object, xnew=NULL, derivative=0, ...)
## S3 method for class 'mKrig':
print( x, ... )

Arguments

Value

dCoefficients of the polynomial fixed part.cCoefficients of the nonparametric part.ntDimension of fixed part.npDimension of c.xSpatial locations used for fitting.cov.function.nameName of covariance function used.cov.argsA list with all the covariance arguments that were specified in the call.chol.argsA list with all the cholesky arguments that were specified in the call.callA copy of the call to mKrig.non.zero.entriesNumber of nonzero entries in the covariance matrix for the process at the observation locations.

Details

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 restriction simply the code for reading. Note that also little checking is done and the spatial locations are not transformed before the estimation.

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.

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))

Examples

Run this code

#
# 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 the taper scale is set to 1.5
# Default 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.args=list(theta = 1.5, k=2)
           ) -> out2

predict.surface( out2)-> out.p
surface( out.p)


# 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
  
mKrig( x,y, cov.function="wendland.cov",k=2, theta=.2, 
            lambda=.1)-> look2

# 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

mKrig( x,y, cov.function="wendland.cov",k=2, theta=.9, lambda=.1)-> look2

# 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
#
mKrig( x,y, 
            cov.function="wendland.cov",k=2, theta=.1, 
            lambda=1e2  )-> look2
# 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!

##################################################
# finding a good choice for theta 
##################################################
# Suppose the target is a spatial prediction using roughly 50 nearest neighbors
# (tapering covariances is effective for rouhgly 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=NULL, diag=FALSE)
# 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. 

mKrig( x,y, cov.function="wendland.cov",k=2, theta=dguess, 
            lambda=1e2) ->  look2

#
#    Using mKrig for evaluating  a solution on a big grid.
#    (Thanks to Jan Klennin for motivating this example.)

x<- RMprecip$x
y<- RMprecip$y

Tps( x,y)-> obj

# make up an 80X80 grid that has ranges of observations
# use same coordinate names as the x matrix

glist<- fields.x.to.grid(x, nx=80, ny=80) # this is a cute way to get a default grid that covers x

# convert grid list to actual x and y values ( try plot( Bigx, pch="."))
    make.surface.grid(glist)-> Bigx 

# include actual x locations along with grid. 
    Bigx<- rbind( x, Bigx)

# evaluate the surface on this set of points (exactly)

    predict(obj, x= Bigx)-> Bigy

# theta sets range for the compact covariance function 
# this will involve  less than 20 nearest neighbors tha have
# nonzero covariance

    theta<- c( 2.5*(glist$lon[2]-glist$lon[1]), 
                 2.5*(glist$lat[2]-glist$lat[1]))

# this is an interplotation of the values using a compact 
# but thin plate spline like covariance. 
    mKrig( Bigx,Bigy, cov.function="wendland.cov",k=4, theta=theta, 
                 lambda=0)->out2 
# the big evaluation this takes about 45 seconds on a Mac G4 latop
    predict.surface( out2, nx=400, ny=400)-> look

# the nice surface
surface( look)
    US( add=TRUE, col="white")