# a 2-d example
# fitting a surface to ozone
# measurements. Exponential covariance, range parameter is 20 (in miles)
fit <- Krig(ozone$x, ozone$y, theta=20)
summary( fit) # summary of fit
set.panel( 2,2)
plot(fit) # four diagnostic plots of fit
set.panel()
surface( fit, type="C") # look at the surface
# predict at data
predict( fit)
# predict using 7.5 effective degrees of freedom:
predict( fit, df=7.5)
# predict on a grid ( grid chosen here by defaults)
out<- predict.surface( fit)
surface( out, type="C") # option "C" our favorite
# predict at arbitrary points (10,-10) and (20, 15)
xnew<- rbind( c( 10, -10), c( 20, 15))
predict( fit, xnew)
# standard errors of prediction based on covariance model.
predict.se( fit, xnew)
# surface of standard errors on a default grid
predict.surface.se( fit)-> out.p # this takes some time!
surface( out.p, type="C")
points( fit$x)
# Using anohter stationary covariance.
# smoothness is the shape parameter for the Matern.
fit <- Krig(ozone$x, ozone$y, Covariance="Matern", theta=10, smoothness=1.0)
summary( fit)
#
# Roll your own: creating very simple user defined Gaussian covariance
#
test.cov <- function(x1,x2,theta,marginal=FALSE,C=NA){
# return marginal variance
if( marginal) { return(rep( 1, nrow( x1)))}
# find cross covariance matrix
temp<- exp(-(rdist(x1,x2)/theta)**2)
if( is.na(C[1])){
return( temp)}
else{
return( temp%*%C)}
}
#
# use this and put in quadratic polynomial fixed function
fit.flame<- Krig(flame$x, flame$y, cov.function="test.cov", m=3, theta=.5)
#
# note how range parameter is passed to Krig.
# BTW: GCV indicates an interpolating model (nugget variance is zero)
#
# take a look ...
surface(fit.flame, type="I")
#
# Thin plate spline fit to ozone data using the radial
# basis function as a generalized covariance function
#
# p=2 is the power in the radial basis function (with a log term added for
# even dimensions)
# If m is the degree of derivative in penalty then p=2m-d
# where d is the dimension of x. p must be greater than 0.
# In the example below p = 2*2 - 2 = 2
#
out<- Krig( ozone$x, ozone$y,cov.function="Rad.cov",
m=2,p=2,scale.type="range")
# See also the Fields function Tps
# out should be identical to Tps( ozone$x, ozone$y)
#
# A Knot example
data(ozone2)
y16<- ozone2$y[16,]
# there are some missing values -- remove them
good<- !is.na( y16)
y<- y16[good]
x<- ozone2$lon.lat[ good,]
#
# the knots can be arbitrary but just for fun find them with a space
# filling design. Here we select 50 from the full set of 147 points
#
xknots<- cover.design( x, 50, num.nn= 75)$design # select 50 knot points
out<- Krig( x, y, knots=xknots, cov.function="Exp.cov", theta=300)
summary( out)
# note that that trA found by GCV is around 17 so 50>17 knots may be a
# reasonable approximation to the full estimator.
#
# the plot
surface( out, type="C")
US( add=TRUE)
points( x, col=2)
points( xknots, cex=2, pch="O")
# A correlation model example
# fit krig surface using a mean and sd function to standardize
# first get stats from 1987 summer Midwest O3 data set
# Compare the function Tps to the call to Krig given above
# fit tps surfaces to the mean and sd points.
# (a shortcut is being taken here just using the lon/lat coordinates)
data(ozone2)
stats.o3<- stats( ozone2$y)
mean.o3<- Tps( ozone2$lon.lat, c( stats.o3[2,]))
sd.o3<- Tps( ozone2$lon.lat, c( stats.o3[3,]))
#
# Now use these to fit particular day ( day 16)
# and use great circle distance
#NOTE: there are some missing values for day 16.
fit<- Krig( ozone2$lon.lat, y16,
theta=350, mean.obj=mean.o3, sd.obj=sd.o3,
Covariance="Matern", Distance="rdist.earth",
smoothness=1.0,
na.rm=TRUE) #
# the finale
surface( fit, type="I")
US( add=TRUE)
points( fit$x)
title("Estimated ozone surface")
#
#
# explore some different values for the range and lambda using REML
theta <- seq( 300,400,,10)
PLL<- matrix( NA, 10,80)
# the loop
for( k in 1:10){
# call to Krig with different ranges
# also turn off warnings for GCV search
# to avoid lots of messages. (not recommended in general!)
PLL[k,]<- Krig( ozone2$lon.lat[good,], y16[good],
cov.function="stationary.cov",
theta=theta[k], mean.obj=mean.o3, sd.obj=sd.o3,
Covariance="Matern",smoothness=.5,
Distance="rdist.earth", nstep.cv=80,
give.warnings=FALSE)$gcv.grid[,7]
#
# gcv.grid is the grid search output from
# the optimization for estimating different estimates for lambda including
# REML
# default grid is equally spaced in eff.df scale ( and should the same across theta)
# here
}
# see the 2 column of $gcv.grid to get the effective degress of freedom.
cat( "all done", fill=TRUE)
contour( theta, 1:80, PLL)Run the code above in your browser using DataLab