spatialProcess: Estimates a spatial process model.

Description

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.

Usage

spatialProcess(x, y, cov.function = "stationary.cov",
         cov.args = list(Covariance = "Matern", smoothness = 1),
         ngrid = 10, theta.grid = NULL, ...)

Arguments

Value

An object with classes Krig and spatialProcess. See help on these two functions for details.

Details

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= f(x.k) = P(x.k) + Z(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 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.

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 Z as the spatial signal and e as the noise lambda can be interpreted as the noise to signal variance ratio in this spatial context. 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.

There is always a danger 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.

Examples

Run this code

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)
#  
# 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")
set.panel(1,1)
# working with covariates and filling in missing station data
# using an ensemble method
#
data(COmonthlyMet)
  fit1E<- spatialProcess(CO.loc,CO.tmin.MAM.climate, Z=CO.elev   )
# conditional simulation at missing data
  good<- !is.na(CO.tmin.MAM.climate ) 
  infill<- sim.Krig( fit1E, xp=CO.loc[!good,],  Z= CO.elev[!good], M= 100)
#  
# 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.