R: Maximizes likelihood for the process marginal variance (rho)...

mKrigMLE {fields} | R Documentation

## 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, lambda = NULL, lambda.profile = TRUE,
        relative.tolerance = 1e-04,
        REML = FALSE, 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, lambda.start = 0.5, cov.params.start
                     = NULL, optim.args = NULL, abstol = 1e-04,
                     parTransform = NULL, 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.
`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.
`lambda` | If `lambda.profile=FALSE` the values of lambda to evaluate the likelihood if "TRUE" the starting values for the optimization. If lambda is NA then the optimum value from previous search is used as the starting value. If lambda is NA and it is the first value the starting value defaults to 1.0.
`lambda.start` | The initial guess for lambda in the joint log-likelihood maximization process
`lambda.profile` | If `TRUE` maximize likelihood over 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 to pass to the mKrig 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 the 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. It is this expression that is then maximized numerically over lambda when `lambda.profile = TRUE`.

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 giving the results for evaluating the likelihood for each covariance model.
`par.grid` | The par.grid argument used.
`cov.args.MLE` | The list of covariance arguments (except for lambda) that have the largest likelihood over the list covariance models. NOTE: To fit the surface at the largest likelihood among those tried `do.call( "mKrig", c(obj$mKrig.args, obj$cov.args.MLE,list(lambda=obj$lambda.opt)) )` where `obj` is the list returned by this function.
`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.
`par.MLE` | The maximum likelihood estimates.
`parTransform` | The transformation of the parameters used in the optimziation.

### Author(s)

Douglas W. Nychka, John Paige

### References

http://cran.r-project.org/web/packages/fields/fields.pdf http://www.image.ucar.edu/~nychka/Fields/

### See Also

`mKrig` `Krig` `stationary.cov` `optim`

### Examples

    #perform joint likelihood maximization over lambda and theta.
    #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.start=.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(cbind(obj$lnLike.eval[,1:2])), obj$lnLike.eval[,5],
                 xlim=c(-1.2,-.40), ylim=c( -1,1), zlim=c( -625, -610))
                 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)%*% matrix( rnorm(N*M), N,M)
      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")

      ## Not run:
    # m=0 is a simple switch to indicate _no_ fixed spatial drift
    # (the default and highly recommended  is linear drift, m=2).
    # this results in MLEs that are less biased -- in fact it nails it !
      obj1a<- mKrigMLEJoint(x,Y,
                        cov.args=list(Covariance="Matern", smoothness=1.0),
                        cov.params.start=list(theta=.5), lambda.start=.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)

    ## Not run:
    #perform joint likelihood maximization over lambda, theta, and smoothness.
    #optim can get a bad answer with poor initial guesses.
    obj2<- mKrigMLEJoint(x,Y,
                          cov.args=list(Covariance="Matern", smoothness=1),
                          cov.params.start=list(theta=.2),
                           lambda.start=.1)

    #look at lnLikelihood  evaluations
    obj2$summary
    #compare to REML
    obj3<- mKrigMLEJoint(x,Y,
                          cov.args=list(Covariance="Matern", smoothness=1),
                          cov.params.start=list(theta=.2),
                           lambda.start=.1, REML=TRUE)

    ## End(Not run)
    ## Not run:
    #look at lnLikelihood  evaluations
    obj3$summary
    # check convergence of MLE to true fit with no fixed part
    #
    obj4<- mKrigMLEJoint(x,Y,
                          mKrig.args= list( m=0),
                          cov.args=list(Covariance="Matern", smoothness=1),
                          cov.params.start=list(theta=.2),
                           lambda.start=.1, REML=TRUE)
    #look at lnLikelihood  evaluations
    obj4$summary
    # nails it!

    ## End(Not run)

[Package fields version 9.6.1 Index]