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MLESpatialProcess

Estimates key covariance parameters for a spatial process.


Description

Maximizes the likelihood to determine the nugget variance (sigma^2), the sill ( rho) and the range (theta) for a spatial process.

Usage

MLESpatialProcess( x, y, weights = rep(1, nrow(x)), Z = NULL,
                            mKrig.args = NULL,
                          cov.function = "stationary.cov", 
                              cov.args = list(Covariance = "Matern",
                                            smoothness = 1),  
			     gridTheta = NULL,
                                 gridN = 15,
                            optim.args = NULL,
                                 na.rm = TRUE,
                               verbose = FALSE,
                               abstol  = 1e-4,
                                  REML = FALSE,
                      cov.params.start = NULL,
                               ...)

Arguments

x

A matrix of spatial locations with rows indexing location and columns the dimension (e.g. longitude/latitude)

y

The spatial observations. This can be a matrix of reoplicated spatial data.

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

mKrig.args

A list containing other objects to pass to mKrig.

gridN

Number of points to use in grid search over theta.

cov.function

The name of the covariance function (See help on Krig for details. )

gridTheta

A grid of range parameters to search over. The default is to use a range based on the quantiles of pairwise distances of the spatial locations.

cov.args

A list with arguments for the covariance functions. These are usually parameters and other options such as the type of distance function.

optim.args

Additional arguments passed to the optim function for likelihood maximization. The default value is: optim.args = list(method = "BFGS", control = list(fnscale = -1, parscale = c(0.5, 0.5), ndeps = c(0.05,0.05)))

na.rm

If TRUE remove missing values in y and corresponding locations in x.

verbose

If TRUE print out intermediate information for debugging.

abstol

Absolute tolerance used to judeg convergence in optim.

REML

If TRUE use maximize the restricted Likelihood instead of the concentrated likelihood.(Preliminary experience suggests this does not make much difference.)

cov.params.start

A list with each component being the name of a covariance paramater to optimize over and the value being the starting value. This is the same format used by optim. If NULL no additional parameters are optimized and their values are fixed in the cov.args list.

...

Additional arguments to pass to the mKrig function.

Details

MLESpatialProcess is designed to be a simple and easy to use function for maximizing the likelihood for a Gaussian spatial process. For other fixed, covariance parameters, the likelihood is maximized over the nugget and sill parameters using the mKrig function. lambda and theta are optimized using the mKrigMLEJoint function on a log scale. This is a wrapper for two lower level functions to make spatialProcess readable. We recommend using mKrigMLEJoint and mKrigMLEGrid for direct optimzation and grid searches over parameters and consulting those two commented functions for an exact description of how the computations are being done. At the very basic level the likelihood function is evaluated by calling the mKrig function with the covariance parameters being adjusted to test different values. So the lowest level likelihood evaluate and what one might use at the highest level are the same workhorse function, mKrig.

Note the likelihood can be maximized analytically over the parameters of the fixed part of the spatial model and with the nugget (sigma) and sill (rho) reduced to the single parameter lambda= sigma^2/rho. The likelihood is maximized numerically over lambda and theta if there are additional covariance parameters ( such as smoothness for the Matern) these need to be fixed and so the MLE is found for the covariance conditional on these additional parameter values. From a practical point of view it is often difficult to estimate just these three from a moderate spatial data set and the user is encourage to try different combinations of fixing covariance parameters with ML for the remaining ones.

MLESpatialProcess.fast is an older fields function also using the optim function to maximize the likelihood computed from the mKrig function. It will eventually be removed from later versions of fields but perhaps is still useful as a cross check on these newer functions

Value

MLESpatialProcess: A list (with sublists!) that includes components:

summary A vector with the optimized parameters (aka MLEs) along with log likelihood at the maximum and information from optim on convergence. Also included are the effecive degrees of freedom of the model at the MLEs and the GCV function value.

MLEJoint A list with components: summary, lnLike.eval, optimResults, pars.MLE, parTransform. summary gives the results from the joint optimization over all parameters. lnLike.eval is a table that has captured all the likelihood evaluations as optim has searched for the maxmimum. Note that if the method has converged many of these parameter value choices will be close the final converged value but often still useful to investigate the likelihood surface around the maximum. optimResults is the list returned by the optim function.

pars.MLE are the parameters specifically optimized over. par.Transform the function used to transform the parameters for a more suitable optimization. The default is a log transformation for lambda and theta.

MLEGrid A list with results from profiling the likelihood over theta, the range parameter. Here the component summary is a table giving the values tried for theta and the corresponding parameters that maximize the likelihood with this fixed value for theta. Also included in the table are some convergence information, effective degrees of freedom, and the GCV criterion.

MLEProfileLambda Results from profiling the likelihood over lambda. Format is the same as that described for MLEGrid.

call The call made to this function.

timingTable Some timing information for the joint optimzation, as the column labeled timeOptim, and the two profiling computations with timeGrid being for theta and timeProfile being for lambda.

MLESpatialProcess.fast has been depreciated and is included for backward compatibility.

Author(s)

Doug Nychka, John Paige

See Also

Examples

#
# Trying out on a simulated data set
# Generate observation locations (50 is really too small but this is
#  just to make this run quickly)

  n <- 50
 
  set.seed(124)
  x = matrix(runif(2*n), nrow=n)
#generate observations at the locations
  trueTheta = .1
  trueSigma = .01
  covMat = exp( -rdist(x,x) /trueTheta )
# As rcode:    y = t(covMat)%*% (rnorm(n))  +  trueSigma * rnorm( n)
  y  <-  t(covMat) %*% (rnorm(n))  +  trueSigma * rnorm( n)
# Use exponential covariance estimate constant function for mean
  out <-  MLESpatialProcess(x, y, 
                        smoothness = .5,
                        mKrig.args = list( m = 1), 
                        )
  out$summary
## Not run: 
#Now a (small) grid search over the smoothness
# add replicated data  fields ( i.e. independent copies drawn from same covariance model)
# to make this stable
  set.seed(124)
# Y = t(covMat)
  Y = t(covMat)%*% matrix(rnorm(n*200), n,200)  +
                  trueSigma * matrix((rnorm(n*200)), n,200)
#This may take a few seconds
testSmoothness = c(.5, 1.5, 2.0)
for( nu in testSmoothness){
  out = MLESpatialProcess(x, Y, cov.args = list(Covariance="Matern"),
                  smoothness = nu, cov.params.start= list( lambda=.5)) 
  print( out$MLEJoint$summary)
}


## End(Not run)

# example with a covariate  
## Not run: 
data(COmonthlyMet)
ind<- !is.na( CO.tmean.MAM.climate)
x<- CO.loc[ind,]
y<- CO.tmean.MAM.climate[ind]
elev<- CO.elev[ind]
obj2<-  MLESpatialProcess( x,y)
obj3<-  MLESpatialProcess( x,y, Z=elev)

# elevation makes a difference
obj2$MLEJoint$summary
obj3$MLEJoint$summary

  
## End(Not run)
 ## Not run: 
# fits for first 10 days from ozone data
data( ozone2) 	 
NDays<- 5
O3MLE<- NULL
for( day in 1: NDays){
	cat( day, " ")
	ind<- !is.na(ozone2$y[day,] )
	x<- ozone2$lon.lat[ind,]
	y<- ozone2$y[day,ind]
	print( length( y))
	 out<- MLESpatialProcess( x,y,
	            Distance="rdist.earth")$MLEJoint$summary
	O3MLE<- rbind( O3MLE, out)
}
# NOTE: names from summary:
#[1] "lnProfileLike.FULL" "lambda"             "theta"             
#[4] "sigmaMLE"           "rhoMLE"             "funEval"           
#[7] "gradEval"           "eff.df"             "GCV"  
plot( log(O3MLE[,"lambda"]), log(O3MLE[,"theta"]))

## End(Not run)

fields

Tools for Spatial Data

v11.6
GPL (>= 2)
Authors
Douglas Nychka [aut, cre], Reinhard Furrer [aut], John Paige [aut], Stephan Sain [aut], Florian Gerber [aut], Matthew Iverson [aut], University Corporation for Atmospheric Research [cph]
Initial release
2020-10-06

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