# Maximizes likelihood for the process marginal variance (sigma) and nugget standard deviation (tau) parameters (e.g. lambda) over a many covariance models or covariance parameter values. These function are designed to explore the likelihood surface for different covariance parameters with the option of maximizing over tau and sigma. They used the `mKrig` base are designed for computational efficiency. ```r mKrigMLEGrid(x, y, weights = rep(1, nrow(x)), Z = NULL,ZCommon=NULL, mKrig.args = NULL, cov.function = "stationary.cov", cov.args = NULL, na.rm = TRUE, par.grid = NULL, reltol = 1e-06, REML = FALSE, GCV = FALSE, optim.args = NULL, cov.params.start = NULL, verbose = FALSE, iseed = NA) mKrigMLEJoint(x, y, weights = rep(1, nrow(x)), Z = NULL,ZCommon=NULL, mKrig.args = NULL, na.rm = TRUE, cov.function = "stationary.cov", cov.args = NULL, cov.params.start = NULL, optim.args = NULL, reltol = 1e-06, parTransform = NULL, REML = FALSE, GCV = FALSE, hessian = FALSE, iseed = 303, verbose = FALSE) profileCI(obj, parName, CIlevel = 0.95, REML = FALSE) mKrigJointTemp.fn(parameters, mKrig.args, cov.args, parTransform, parNames, REML = FALSE, GCV = FALSE, verbose = verbose, capture.env) ``` ## Arguments - `capture.env`: For the ML objective function the frame to save the results of the evaluation. This should be the environment of the function calling optim. - `CIlevel`: Confidence level. - `cov.function`: The name, a text string, of the covariance function. - `cov.args`: The arguments that would also be included in calls to the covariance function to specify the fixed part of the covariance model. This is the form of a list E.g.`cov.args= list( aRange = 3.5)` - `cov.params.start`: A list of initial starts for covariance parameters 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 (tau^2/ sigma^2) is most spatial applications and so if `lambda` is omitted then the component `lambda = .5` is added to this list. - `hessian`: If TRUE return the BFGS approximation to the hessian matrix at convergence. - `iseed`: Sets the random seed in finding the approximate Monte Carlo based GCV function and the effective degrees of freedom. This will not effect random number generation outside these functions. - `mKrig.args`: A list of additional parameters to supply to the `mKrig` function. E.g.`mKrig.args= list( m=1)`to set the regression function to be a constant function. `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). Use `mKrig.args= list( m = 0 )` to omit a fixed model and assume the observations have an expectation of zero. - `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 "aRange" 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 `...`. - `obj`: List returnerd from `mKrigMLEGrid` - `parName`: Name of parameter to find confidence interval. - `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. - `reltol`: Optim BFGS comvergence criterion. - `REML`: If TRUE use REML instead of the full log likelihood. - `GCV`: NOT IMPLEMENTED YET! A placeholder to implement optimization using an approximate cross-validation criterion. - `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` - `ZCommon`: Linear covariates to be included in fixed part of the model where a common parameter is estimated across all realizations. This option only makes sense for multiple realizations ( y is a matrix). ## 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 (tau) and the marginal variance of the process (sigma). It is useful to reparametrize as `sigma` and `lambda = tau^2/sigma`. The likelihood can be maximized analytically over sigma and the parameters in the fixed part of the model, this estimate of sigma 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. These functions may compute the effective degrees of freedomn ( see `mKrig.trace` ) using the random tace method and so need to generate some random normals. The `iseed` arguement can be used to set the seed for this with the default being the seed `303`. Note that the random number generation internal to these functions is coded so that it does not effect the random number stream outside these function calls. 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) } } ``` ## Returns ‘mKrigMLEGrid’ returns a list with the components: - **summary**: A matrix with each row giving the results of 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. ## References [https://github.com/dnychka/fieldsRPackage](https://github.com/dnychka/fieldsRPackage) ## Author(s) Douglas W. Nychka, John Paige ## See Also `mKrig` `Krig` `stationary.cov` `optim` ## Examples ```r ## Not run: #perform joint likelihood maximization over lambda and aRange. # NOTE: optim can get a bad answer with poor initial starts. data(ozone2) s<- ozone2$lon.lat z<- ozone2$y[16,] gridList<- list( aRange = seq( .4,1.0,length.out=20), lambda = 10**seq( -1.5,0,length.out=20) ) par.grid<- make.surface.grid( gridList) out<- mKrigMLEGrid( s,z, par.grid=par.grid, cov.args= list(smoothness=1.0, Covariance="Matern" ) ) outP<- as.surface( par.grid, out$summary[,"lnProfileLike.FULL"]) image.plot( outP$x, log10(outP$y),outP$z, xlab="aRange", ylab="log10 lambda") ## End(Not run) ## Not run: N<- 50 set.seed(123) x<- matrix(runif(2*N), N,2) aRange<- .2 Sigma<- Matern( rdist(x,x)/aRange , smoothness=1.0) Sigma.5<- chol( Sigma) tau<- .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) Y<- F.true + tau* matrix( rnorm(N*M), N,M) # find MLE for lambda with grid of ranges # and smoothness fixed in Matern par.grid<- list( aRange= seq( .1,.35,,8)) obj1b<- mKrigMLEGrid( x,Y, cov.args = list(Covariance="Matern", smoothness=1.0), cov.params.start=list( lambda = .5), par.grid = par.grid ) obj1b$summary # take a look # profile over aRange plot( par.grid$aRange, obj1b$summary[,"lnProfileLike.FULL"], type="b", log="x") ## End(Not run) ## Not run: # m=0 is a simple switch to indicate _no_ fixed spatial drift # (the default and highly recommended is linear drift, m=2). # However, m=0 results in MLEs that are less biased, being the correct model # -- in fact it nails it ! obj1a<- mKrigMLEJoint(x,Y, cov.args=list(Covariance="Matern", smoothness=1.0), cov.params.start=list(aRange =.5, lambda = .5), mKrig.args= list( m=0)) test.for.zero( obj1a$summary["tau"], tau, tol=.007) test.for.zero( obj1a$summary["aRange"], aRange, tol=.015) ## End(Not run) ########################################################################## # A bootstrap example # Here is a example of a more efficient (but less robust) bootstrap using # mKrigMLEJoint and tuned starting values ########################################################################## ## Not run: data( ozone2) obj<- spatialProcess( ozone2$lon.lat,ozone2$y[16,] ) ######### boot strap set.seed(123) M<- 25 # create M indepedent copies of the observation vector ySynthetic<- simSpatialData( obj, M) bootSummary<- NULL aRangeMLE<- obj$summary["aRange"] lambdaMLE<- obj$summary["lambda"] for( k in 1:M){ cat( k, " " ) # here the MLEs are found using the easy top level level wrapper # see mKrigMLEJoint for a more efficient strategy out <- mKrigMLEJoint(obj$x, ySynthetic[,k], weights = obj$weights, mKrig.args = obj$mKrig.args, cov.function = obj$cov.function.name, cov.args = obj$cov.args, cov.params.start = list( aRange = aRangeMLE, lambda = lambdaMLE) ) 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, aRange, 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( aRange = .18, smoothness = 1.1, lambda = .08), ) #look at lnLikelihood evaluations obj2$summary #compare to REML obj3<- mKrigMLEJoint(x,Y, cov.args=list(Covariance="Matern"), cov.params.start=list(aRange = .18, smoothness = 1.1, lambda = .08), , REML=TRUE) obj3$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(aRange=.2, lambda=.1), REML=TRUE) #look at lnLikelihood evaluations obj4$summary # nails it! ## End(Not run) ```

mKrigMLE function