For a given covariance function estimates the nugget (sigma^2) and process variance (rho) and the range parameter (theta) by restricted maximum likelihood and then computes the spatial model with these estimated parameters. Other parameters of the covariance are kept fixed and need to be specified.
1 2 3 4 5 6 7 8 9 10 11  spatialProcess(x, y, weights = rep(1, nrow(x)), Z = NULL,
mKrig.args = list(m = 2),
cov.function = "stationary.cov", cov.args = list(Covariance = "Matern",
smoothness = 1), theta.start = NULL, lambda.start = 0.5,
theta.range = NULL,
abstol = 1e04, na.rm = TRUE, verbose = FALSE, ...)
## S3 method for class 'spatialProcess'
print(x, digits = 4, ...)
## S3 method for class 'spatialProcess'
plot(x, digits = 4, which = 1:4, ...)

x 
Observation locations 
y 
Observation values 
weights 
Weights for the error term (nugget) in units of reciprocal variance. 
Z 
A matrix of extra covariates for the fixed part of spatial model. E.g. elevation for fitting climate data over space. 
mKrig.args 
Arguments passed to the mKrig function. 
cov.function 
A character string giving the name of the covariance function for the spatial component. 
cov.args 
A list specifying parameters and other components of the covariance function. 
theta.start 
Starting value for MLE fitting of the scale (aka range) parameter. If omitted the starting value is taken from a grid search ove theta. 
lambda.start 
Starting value for MLE fitting of the lambda parameter. Note lambda
is the ratio of the nugget variance to the process variance. In code variables this is

theta.range 
A range for the ML search to estimate theta. Default is based on quantiles of the location pairwise distances. 
na.rm 
If TRUE NAs are removed from the data. 
verbose 
If TRUE print out intermediate information for debugging. 
... 
Any other arguments that will be passed to the 
abstol 
The absolute tolerance bound used to judge convergence. This is applied to the difference in log likelihood values. 
digits 
Number of significant digits in printed summary 
which 
The vector 1:4 or any subset of 1:4, giving the plots to draw. See the description ofthese plots below. 
This function makes many choices for the user in terms of defaults and it is important to be aware of these. The spatial model is
Y.k= P(x.k) + Z(x.k)%*%d2 + g(x.k) + e.k
where ".k" means subscripted by k, Y.k is the dependent variable
observed at location x.k. P is a low degree polynomial (default is a
linear function in the spatial coordinates) and Z is a matrix of covariates (optional) that enter as a linear model the fixed part. g is a mean zero,
Gaussian stochastic process with a marginal variance of rho and a
scale (or range) parameter, theta. The measurement errors, e.k, are
assumed to be uncorrelated, normally distributed with mean zero and
standard deviation sigma. If weights are supplied then the variance of e is assumed to be sigma^2/ weights
.
Perhaps the most important aspect of this function is that the range (theta), nugget (sigma**2) and process variance (rho) parameters for the covariance are estimated by restricted maximum likelihood and this is the model that is then used for spatial prediction. Geostatistics usaually refers to sigma**2 + rho as the "sill" and often these parameters are estimated by variogram fitting rather than maximum likelihood. To be consistent with spline models and to focus on the key part of model we reparametrize as lambda= sigma**2/ rho and rho. Thinking about h as the spatial signal and e as the noise lambda can be interpreted as the noise to signal variance ratio in this spatial context.(See the comparision with fitting the geoR model in the examples section.)
The likelihood and the cross valdiation function can be concentrated to only depend on lambda and theta and so in reported the optimiztation of these two criterion we focus on this form of the parameters. Once lambda and theta are found, the MLE for rho has a closed form and of course then sigma is then determined from lambda and rho.
Often the lambda parameter is difficult to interpret when covariates and a linear function of the coordinates is included and also when the range becomes large relative to the size of the spatial domain. For this reason it is convenient to report the effective degrees of freedom (also referred to trA in R code and the output summaries) associated with the predicted surface or curve. This measure has a one to one relationship with lamdba and is easier to interpret. For example an eff degrees of freedom that is very small suggests that the surface is rwell represented by a low ordoer polynomial. Degrees of freedom close to the number of locations indicates a surface that is close to interpolating the observations and suggests a small or zero value for the nugget variance.
The default covariance model is assumed to follow a Matern
with smoothness set to 1.0. This is implementd using the
stationary.cov
covariance that can take a argument for
the form of the covariance, a sill and range parameters and
possibily additional parameter might comtrol the shape.
See the example below how to switch to another model. (Note that the exponential is also part of the Matern family with smoothness set to .5. )
The parameter estimation is done by MLESpatialProcess
and the returned list from this function is added to the Krig
output object that is returned by this function. The estimate
is a version of maximum likelihood where the observations are
transfromed to remove the fixed linear part of the model. If
the user just wants to fix the range parameter theta then
Krig
can be used.
NOTE: The defaults for the optim
function used in MLESpatialProcess are:
1 2 3 4 5 
There is always a hazard in providing a simple to use method that makes many default choices for the spatial model. As in any analysis be aware of these choices and try alternative models and parameter values to assess the robustness of your conclusions. Also examine the residuals to check the adequacy of the fit. See the examples below for some help in how to do this easily in fields. Also see quilt.plot to get an quick plot to discern spatial paterns.
plot method provides a panel of 4 diagnositic plots of the fit.
Use set.panel(2,2)
to see all 4 at once. The third plot gives the likelihood and
GCV functions as a function of lambda evaluated at the global MLE for theta.
This is based on the gird evaluations in the component MLEInfo$MLEProfileLambda.
The fourth
plot is a profile likelihood trace for theta having maximized over lambda and is based on the component MLEInfo$MLEGrid.
print method gives a summary of the fit.
An object of classes mKrig
and SpatialProcess
. The main difference
from mKrig is an extra component, MLEInfo
that has the results of the grid
evaluation over theta (maximizing lamdba), joint maximization over theta and lambda,
and a grid evaluation over lambda with theta fixed at its MLE.
Doug Nychka
Tps, MLESpatialProcess, mKrigMLEGrid, mKrigMLEJoint, plot.Krig, predict.mKrig, predictSE.mKrig
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157  data( ozone2)
# x is a two column matrix where each row is a location in lon/lat
# coordinates
x< ozone2$lon.lat
# y is a vector of ozone measurements at day 16 a the locations.
y< ozone2$y[16,]
obj< spatialProcess( x, y)
# summary of model
summary( obj)
# diagnostic plots
set.panel(2,2)
plot(obj)
# plot 1 data vs. predicted values
# plot 2 residuals vs. predicted
# plot 3 criteria to select the smoothing
# parameter lambda = sigma^2 / rho
# the x axis has transformed lambda
# in terms of effective degrees of freedom
# to make it easier to interpret
# Note that here the GCV function is minimized
# while the REML is maximzed.
# plot 4 the log profile likelihood used to
# determine theta.
#
# predictions on a grid
surface( obj)
#(see also predictSurface for more control on evaluation grid
# and plotting)
#
## Not run:
# working with covariates and filling in missing station data
# using an ensemble method
# see the example under help(sim.spatialProcess) to see how to
# handle a conditional simulation on a grid of predictions with
# covariates.
data(COmonthlyMet)
fit1E< spatialProcess(CO.loc,CO.tmin.MAM.climate, Z=CO.elev,
theta.range= c(.25, 2.0) )
set.panel( 2,2)
plot( fit1E)
# conditional simulation at missing data
notThere< is.na(CO.tmin.MAM.climate )
xp < CO.loc[notThere,]
Zp < CO.elev[notThere]
infill< sim.spatialProcess( fit1E, xp=xp,
Z= Zp, M= 10)
#
# interpretation is that these infilled values are all equally plausible
# given the observations and also given the estimated covariance model
#
# for extra credit one could now standardized the infilled values to have
# conditional mean and variance from the exact computations
# e.g. predict( fit1E, xp=CO.loc[!good,], Z= CO.elev[!good])
# and predictSE(fit1E, xp=CO.loc[!good,], Z= CO.elev[!good])
# with these standardization one would still preserve the correlations
# among the infilled values that is also important for considering them as a
# multivariate prediction.
# conditional simulation on a grid but not using the covariate of elevation
fit2< spatialProcess(CO.loc,CO.tmin.MAM.climate,
theta.range= c(.25, 2.0) )
# note larger range parameter
# create 2500 grids using handy function
gridList < fields.x.to.grid( fit2$x, nx=50,ny=50)
xGrid< make.surface.grid( gridList)
ensemble< sim.spatialProcess( fit2, xp=xGrid, M= 5)
# this is an "n^3" computation so increasing the grid size
# can slow things down for computation
image.plot( as.surface( xGrid, ensemble[1,]))
set.panel()
## End(Not run)
## Not run:
data( ozone2)
# x is a two column matrix where each row is a location in lon/lat
# coordinates
x< ozone2$lon.lat
# y is a vector of ozone measurements at day 16 a the locations.
y< ozone2$y[16,]
# a comparison to using an exponential and Wendland covariance function
# and great circle distance  just to make range easier to interpret.
obj < spatialProcess( x, y,
Distance = "rdist.earth")
obj2< spatialProcess( x, y,
cov.args = list(Covariance = "Exponential"),
Distance = "rdist.earth" )
obj3< spatialProcess( x, y,
cov.args = list(Covariance = "Wendland",
dimension = 2,
k = 2),
Distance = "rdist.earth")
# obj2 could be also be fit using the argument:
# cov.args = list(Covariance = "Matern", smoothness=.5)
#
# Note very different range parameters  BTW these are in miles
# but similar nugget variances.
obj$pars
obj2$pars
obj3$pars
# since the exponential is Matern with smoothness == .5 the first two
# fits can be compared in terms of their likelihoods
# the REML value is slightly higher for obj verses obj2 (598.4 > 596.7)
# these are the _negative_ log likelihoods so suggests a preference for the
# exponential model
#
# does it really matter in terms of spatial prediction?
set.panel( 3,1)
surface( obj)
US( add=TRUE)
title("Matern sm= 1.0")
surface( obj2)
US( add=TRUE)
title("Matern sm= .5")
surface( obj3)
US( add=TRUE)
title("Wendland k =2")
# prediction standard errors
# these take a while because prediction errors are based
# directly on the Kriging weight matrix
# see mKrig for an alternative.
set.panel( 2,1)
out.p< predictSurfaceSE( obj, nx=40,ny=40)
surface( out.p)
US( add=TRUE)
title("Matern sm= 1.0")
points( x, col="magenta")
#
out.p< predictSurfaceSE( obj, nx=40,ny=40)
surface( out.p)
US( add=TRUE)
points( x, col="magenta")
title("Matern sm= .5")
## End(Not run)
set.panel(1,1)
## Not run:
### comparison with GeoR
data(ozone2)
x< ozone2$lon.lat
y< ozone2$y[16,]
good<!is.na(y)
x1< x[good,]
y1< y[good]
obj< spatialProcess( x, y, mKrig.args= list(m=1), smoothness = .5 )
library( geoR)
ml.n < likfit(coords= x1, data=y1, ini = c(570, 3), nug = 50)
# compare to
stuffFields< obj$MLEInfo$MLEJoint$summary[c(1,3,4,5)]
stuffGeoR< c( ml.n$loglik, ml.n$phi, sqrt(ml.n$nugget),ml.n$sigmasq)
test.for.zero( stuffFields, stuffGeoR, tol=.005)
## End(Not run)

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