fields: mKrigMLE – R documentation

Become an expert in R — Interactive courses, Cheat Sheets, certificates and more!

mKrigMLE

Maximizes likelihood for the process marginal variance (rho) and nugget standard deviation (sigma) parameters (e.g. lambda) over a many covariance models or covariance parameter values.

Description

These function are designed to explore the likelihood surface for different covariance parameters with the option of maximizing over sigma and rho. They used the mKrig base are designed for computational efficiency.

Usage

mKrigMLEGrid(x, y, weights = rep(1, nrow(x)), Z = NULL, mKrig.args
                 = NULL, cov.fun = "stationary.cov", cov.args = NULL,
                 na.rm = TRUE, par.grid = NULL, relative.tolerance =
                 1e-04, REML = FALSE, optim.args = NULL,
                 cov.params.start = NULL, verbose = FALSE )

mKrigMLEJoint(x, y, weights = rep(1, nrow(x)), Z = NULL, mKrig.args = NULL, 
    na.rm = TRUE, cov.fun = "stationary.cov", cov.args = NULL, 
    cov.params.start = NULL, optim.args = NULL, abstol = 1e-04, 
    parTransform = NULL, find.trA = TRUE, REML = FALSE, verbose = FALSE) 


fastTpsMLE(x, y, weights = rep(1, nrow(x)), Z = NULL, ...,
                 par.grid=NULL, theta, lambda = NULL,
		 lambda.profile = TRUE,
                 verbose = FALSE, relative.tolerance = 1e-04)

mKrigJointTemp.fn(parameters, mKrig.args, cov.args, parTransform,
           parNames, REML = FALSE, capture.env)

Arguments

`abstol`	Absolute convergence tolerance used in optim.
`capture.env`	For the ML obective function the frame to save the results of the evaluation. This should be the environment of the function calling optim.
`cov.fun`	The name, a text string, of the covariance function.
`cov.args`	Additional arguments that would also be included in calls to the covariance function to specify the fixed part of the covariance model.
`find.trA`	This argument is used to determine if the the effective degrees of frreedom is found for given set of covariance parameters. It is more efficient not ot find this in the optimization. This is a logical and is passed to the mKrig function internally.
`cov.params.start`	A list of initial starts for covariance parameters over which the user wishes to perform likelihood maximization. The list contains the names of the parameters as well as the values. It usually makes sense to optimize over the important lambda parameter is most spatial applications and so if $`lambda`$ is omitted then the component $`lambda = .5` is added to this list.
`lambda`	The value of lambda to evaluate the likelihood.
`lambda.profile`	If TRUE will find the profile likelihood for `lambda`.
`mKrig.args`	A list of additional parameters to supply to the base `mKrig` function that are distinct from the covariance model. For example `mKrig.args= list( m=1 )` will set the polynomial to be just a constant term (degree = m -1 = 0).
`na.rm`	Remove NAs from data.
`optim.args`	Additional arguments that would also be included in calls to the optim function in joint likelihood maximization. If `NULL`, this will be set to use the "BFGS-" optimization method. See `optim` for more details. The default value is: `optim.args = list(method = "BFGS", control=list(fnscale = -1, ndeps = rep(log(1.1), length(cov.params.start)+1), abstol=1e-04, maxit=20))` Note that the first parameter is lambda and the others are the covariance parameters in the order they are given in `cov.params.start`. Also note that the optimization is performed on a transformed scale (based on the function `parTransform` ), and this should be taken into consideration when passing arguments to `optim`.
`parameters`	The parameter values for evaluate the likelihood.
`par.grid`	A list or data frame with components being parameters for different covariance models. A typical component is "theta" comprising a vector of scale parameters to try. If par.grid is "NULL" then the covariance model is fixed at values that are given in ....
`parNames`	Names of the parameters to optimize over.
`parTransform`	A function that maps the parameters to a scale for optimization or effects the inverse map from the transformed scale into the original values. See below for more details.
`relative.tolerance`	Tolerance used to declare convergence when maximizing likelihood over lambda.
`REML`	Currently using REML is not implemented.
`theta`	Range parameter for compact Wendland covariance. (see fastTps)
`verbose`	If `TRUE` print out interesting intermediate results.
`weights`	Precision ( 1/variance) of each observation
`x`	Matrix of unique spatial locations (or in print or surface the returned mKrig object.)
`y`	Vector or matrix of observations at spatial locations, missing values are not allowed! Or in mKrig.coef a new vector of observations. If y is a matrix the columns are assumed to be independent observations vectors generated from the same covariance and measurment error model.
`Z`	Linear covariates to be included in fixed part of the model that are distinct from the default low order polynomial in `x`
`...`	Other arguments assumed to covariance parameters for the covariance function.

Details

The observational model follows the same as that described in the Krig function and thus the two primary covariance parameters for a stationary model are the nugget standard deviation (sigma) and the marginal variance of the process (rho). It is useful to reparametrize as rho and lambda = sigma^2/rho. The likelihood can be maximized analytically over rho and the parameters in the fixed part of the model, this estimate of rho can be substituted back into the likelihood to give a expression that is just a function of lambda and the remaining covariance parameters. This operation is called concentrating the likelhood by maximizing over a subset of parameters

For these kind of computations there has to be some device to identify parameters that are fixed and those that are optimized. For mKrigMLEGrid and mKrigMLEJoint the list cov.args should have the fixed parameters. For example this is how to fix a lambda value in the model. The list cov.params.start should be list with all parameters to optimize. The values for each component are use as the starting values. This is how the optim function works.

Note that fastTpsMLE is a convenient variant of this more general version to use directly with fastTps, and mKrigMLEJoint is similar to mKrigMLEGrid, except it uses the optim function to optimize over the specified covariance parameters and lambda jointly rather than optimizing on a grid. Unlike mKrigMLEJoint, it returns an mKrig object.

For mKrigMLEJoint the default transformation of the parameters is set up for a log/exp transformation:

parTransform <- function(ptemp, inv = FALSE) {
            if (!inv) {
                log(ptemp)
            }
            else {
                exp(ptemp)
            }
        }

Value

mKrigMLEGrid returns a list with the components:

`summary`	A matrix with eah row giving the results for evaluating the likelihood for each covariance model.
`par.grid`	The par.grid argument used. A matrix where rows are the combination of parameters considered.
`call`	The calling arguments to this function.

mKrigMLEJoint returns a list with components:

`summary`	A vector giving the MLEs and the log likelihood at the maximum
`lnLike.eval`	A table containing information on all likelihood evaluations performed in the maximization process.
`optimResults`	The list returned from the optim function. Note that the parameters may be transformed values.
`par.MLE`	The maximum likelihood estimates.
`parTransform`	The transformation of the parameters used in the optimziation.

Author(s)

Douglas W. Nychka, John Paige

References

https://github.com/NCAR/Fields

Examples

## Not run: 
#perform joint likelihood maximization over lambda and theta. 
# NOTE: optim can get a bad answer with poor initial starts.
data(ozone2)
x<- ozone2$lon.lat
y<- ozone2$y[16,]
obj<- mKrigMLEJoint(x,y, 
                      cov.args=list(Covariance="Matern", smoothness=1.0), 
                      cov.params.start=list(theta=.2, lambda = .1) 
                      )
#                      
# check  lnLikeihood evaluations that were culled from optim
# these are in obj$lnLike.eval
# funny ranges are set to avoid  very low likelihood values

quilt.plot( log10(obj$lnLike.eval[,c("theta","lambda") ] ), obj$lnLike.eval[,"lnProfileLike.FULL"])
points( log10(obj$pars.MLE[1]), log10(obj$pars.MLE[2]), pch=16, col="grey" )

# some synthetic data with replicates
  N<- 50
  set.seed(123)
  x<- matrix(runif(2*N), N,2)
  theta<- .2
  Sigma<-  Matern( rdist(x,x)/theta , smoothness=1.0)
  Sigma.5<- chol( Sigma)
  sigma<- .1
  #  250 independent spatial data sets but a common covariance function 
  #    -- there is little overhead in
  #        MLE across independent realizations and a good test of code validity.
  M<-250
  #F.true<- t( Sigma.5)
  F.true<- t( Sigma.5)%*% matrix( rnorm(N*M), N,M)
  Y<-  F.true +  sigma* matrix( rnorm(N*M), N,M)

# find MLE for lambda with grid of ranges 
# and smoothness fixed in Matern                     
 par.grid<- list( theta= seq( .1,.35,,8))
  obj1b<- mKrigMLEGrid( x,Y,
     cov.args = list(Covariance="Matern", smoothness=1.0), 
        par.grid = par.grid
                    )
  obj$summary # take a look
# profile over theta
  plot( par.grid$theta, obj1b$summary[,"lnProfileLike.FULL"],
    type="b", log="x")
 
## End(Not run)
 
  ## Not run: 
# m=0 is a simple switch to indicate _no_ fixed spatial drift
# (the default and highly recommended  is linear drift, m=2). 
# However, m=0 results in MLEs that are less biased, being the correct model
# -- in fact it nails it !
  obj1a<- mKrigMLEJoint(x,Y, 
                    cov.args=list(Covariance="Matern", smoothness=1.0), 
                    cov.params.start=list(theta =.5, lambda = .5),
                     mKrig.args= list( m=0))
 
 test.for.zero( obj1a$summary["sigmaMLE"], sigma, tol=.0075)
 test.for.zero( obj1a$summary["theta"], theta, tol=.05)

## End(Not run) 

##########################################################################
# A bootstrap example
# Here is a example of a more efficient (but less robust) bootstrap using 
# mKrigMLEJoint and tuned starting values
##########################################################################
## Not run: 
data( ozone2)
obj<- spatialProcess( ozone2$lon.lat,ozone2$y[16,] )

######### boot strap 
  set.seed(123)
  M<- 50
# create M indepedent copies of the observation vector
  ySynthetic<- simSpatialData( obj, M)
  
  bootSummary<- NULL

  for(  k in 1:M){
  cat( k, " " )
# here the MLEs are found using the easy top level level wrapper
# see mKrigMLEJoint for a more efficient strategy
  out <- mKrigMLEJoint(obj$x, ySynthetic[,k],
                 weights = obj$weights,
              mKrig.args = obj$mKrig.args,
                 cov.fun = obj$cov.function.name,
                cov.args = obj$cov.args, 
        cov.params.start = list( theta = obj$theta.MLE,
                                lambda = obj$lambda.MLE)
                      )
  newSummary<- out$summary
  bootSummary<- rbind( bootSummary, newSummary)
  }
   cat(  " ", fill=TRUE )
  obj$summary
  stats( bootSummary)
  

## End(Not run)
## Not run: 
#perform joint likelihood maximization over lambda, theta, and smoothness.  
#note: finding smoothness is not a robust optimiztion 
#      can get a bad answer with poor initial guesses.
obj2<- mKrigMLEJoint(x,Y, 
                      cov.args=list(Covariance="Matern"), 
                      cov.params.start=list( theta = .18,
                                        smoothness = 1.1,
                                            lambda =.08),
                       )

#look at lnLikelihood  evaluations
obj2$summary
#compare to REML
obj3<- mKrigMLEJoint(x,Y, 
                      cov.args=list(Covariance="Matern"), 
                      cov.params.start=list(theta = .18, 
                                       smoothness = 1.1,
                                           lambda = .08),
                       , REML=TRUE)
obj3$summary                      

## End(Not run)
## Not run: 
#look at lnLikelihood  evaluations

# check convergence of MLE to true fit with no fixed part
# 
obj4<- mKrigMLEJoint(x,Y, 
                      mKrig.args= list( m=0),
                      cov.args=list(Covariance="Matern", smoothness=1), 
                      cov.params.start=list(theta=.2, lambda=.1),
                       REML=TRUE)
#look at lnLikelihood  evaluations
obj4$summary
# nails it!

## 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