predict.georob: Predict Method for Robustly Fitted Spatial Linear Models

Description

Robust and customary external drift Kriging prediction based on a spatial linear models fitted by georob. The predict method for the class georob computes fitted values, point and block Kriging predictions as well as model terms for display by termplot.

Usage

# S3 method for georob
predict(object, newdata, type =  c("signal", "response", "trend", "terms"), 
    terms = NULL, se.fit = TRUE, signif = 0.95, locations, 
    control = control.predict.georob(), verbose = 0, ...)
    
control.predict.georob(full.covmat = FALSE, extended.output = FALSE, 
    mmax = 10000, ncores = pcmp[["max.ncores"]], pwidth = NULL, pheight = NULL, 
    napp = 1, pcmp = control.pcmp())

Arguments

object

an object of class "georob", see georobObject.

newdata

an optional data frame, SpatialPointsDataFrame, SpatialPixelsDataFrame, SpatialGridDataFrame, SpatialPolygonsDataFrame or an (optional) object of class SpatialPoints, SpatialPixels or SpatialGrid, in which to look for variables with which to compute fitted values or Kriging predictions, see Details. If newdata is a SpatialPolygonsDataFrame then block Kriging predictions are computed, otherwise point Kriging predictions.

type

character keyword defining what target quantity should be predicted (computed). Possible values are

"signal": the “signal” $Z(\mbox{\boldmath$s$\unboldmath}) = \mbox{\boldmath$x$\unboldmath}(\mbox{\boldmath$s$\unboldmath})^\mathrm{T} \mbox{\boldmath$\beta$\unboldmath} + B(\mbox{\boldmath$s$\unboldmath})$ of the process (default),
"response": the observations $Y(\mbox{\boldmath$s$\unboldmath}) = Z(\mbox{\boldmath$s$\unboldmath}) + \varepsilon(\mbox{\boldmath$s$\unboldmath}),$
"trend": the external drift $\mbox{\boldmath$x$\unboldmath}(\mbox{\boldmath$s$\unboldmath})^\mathrm{T} \mbox{\boldmath$\beta$\unboldmath},$
"terms": the model terms.

terms

If type = "terms", which terms (default is all terms).

se.fit

logical, only used if type is equal to "terms", see predict.lm.

signif

positive numeric equal to the tolerance or confidence level for computing respective intervals. If NULL no intervals are computed.

locations

an optional one-sided formula specifying what variables of newdata are the coordinates of the prediction points (default: object[["locations.objects"]]$locations).

control

a list with the arguments full.covmat, extended.output, mmax, ncores, pwidth, pheight, napp and pcmp or a function such as control.predict.georob that generates such a list.

full.covmat

logical controlling whether the full covariance matrix of the prediction errors is returned (TRUE) or only the vector with its diagonal elements (FALSE, default), see Value for an explanation of the effect of full.covmat.

extended.output

logical controlling whether the covariance matrices of the Kriging predictions and of the data should be computed, see Details (default FALSE).

mmax

integer equal to the maximum number (default 10000) of prediction items, computed in a sub-task, see Details.

ncores

positive integer controlling how many cores are used for parallelized computations, see Details.

pwidth, pheight, napp

numeric scalars, used to tune numeric integration of semi-variances for block Kriging, see preCKrige.

pcmp

a list of arguments passed to pmm or a function such as control.pcmp that generates such a list (see control.pcmp for allowed arguments).

verbose

positive integer controlling logging of diagnostic messages to the console. verbose = 0 (default) largely suppresses such messages.

...

arguments passed to control.predict.georob.

Value

If type is equal to "terms" then a vector, a matrix, or a list with prediction results along with bounds and standard errors, see predict.lm. Otherwise, the structure and contents of the output generated by predict.georob are determined by the class of newdata and the logical flags full.covmat and extended.output:

If full.covmat is FALSE then the result is an object of a "similar" class as newdata (data frame, SpatialPointsDataFrame, SpatialPixelsDataFrame SpatialGridDataFrame, SpatialPolygonsDataFrame).

The data frame or the data slot of the Spatial...DataFrame objects have the following components:

the coordinates of the prediction points (only present if newdata is a data frame).
pred: the Kriging predictions (or fitted values).
se: the root mean squared prediction errors (Kriging standard errors).
lower, upper: the limits of tolerance/confidence intervals,
trend, var.pred, cov.pred.target, var.target: only present if extended.output is TRUE, see Details.

If full.covmat is TRUE then predict.georob returns a list with the following components:

pred: a data frame or a Spatial...DataFrame object as described above for full.covmat = FALSE.
mse.pred: the full covariance matrix of the prediction errors, $Y(\mbox{\boldmath$s$\unboldmath})-\widehat{Y}(\mbox{\boldmath$s$\unboldmath})$ or $Z(\mbox{\boldmath$s$\unboldmath})-\widehat{Z}(\mbox{\boldmath$s$\unboldmath})$ see Details.
var.pred: the full covariance matrix of the Kriging predictions, see Details.
cov.pred.target: the full covariance matrix of the predictions and the prediction targets, see Details.
var.target: the full covariance matrix of the prediction targets, see Details.

Details

If newdata is an object of class SpatialPoints, SpatialPixels or SpatialGrid then the drift model may only use the coordinates as covariates (universal Kriging). Furthermore the names used for the coordinates in newdata must be the same as in data when creating object (argument locations of predict.georob should not be used). Note that the result returned by predict.georob is then an object of class SpatialPointsDataFrame, SpatialPixelsDataFrame or SpatialGridDataFrame.

The predict method for class georob uses the package parallel for parallelized computation of Kriging predictions. If there are $m$ items to predict, the task is split into ceiling(m/mmax) sub-tasks that are then distributed to ncores CPUs. Evidently, ncores = 1 suppresses parallel execution. By default, the function uses all available CPUs as returned by detectCores. Note that if full.covmat is TRUE mmax must exceed $m$ (and parallel execution is not possible).

The argument extended.output = TRUE is used to compute all quantities required for (approximately) unbiased back-transformation of Kriging predictions of log-transformed data to the original scale of the measurements by lgnpp. In more detail, the following items are computed:

trend: the fitted values, $\mbox{\boldmath$x$\unboldmath}(\mbox{\boldmath$s$\unboldmath})\mathrm{^T}\widehat{\mbox{\boldmath$\beta$\unboldmath}}$,
var.pred: the variances of the Kriging predictions, $\mathrm{Var}_{\hat{\theta}}[\widehat{Y}(\mbox{\boldmath$s$\unboldmath})]$ or $\mathrm{Var}_{\hat{\theta}}[\widehat{Z}(\mbox{\boldmath$s$\unboldmath})]$,
cov.pred.target: the covariances between the predictions and the prediction targets, $\mathrm{Cov}_{\hat{\theta}}[\widehat{Y}(\mbox{\boldmath$s$\unboldmath}),Y(\mbox{\boldmath$s$\unboldmath})]$ or $\mathrm{Cov}_{\hat{\theta}}[\widehat{Z}(\mbox{\boldmath$s$\unboldmath}),Z(\mbox{\boldmath$s$\unboldmath})]$,
var.target: the variances of the prediction targets $\mathrm{Var}_{\hat{\theta}}[Y(\mbox{\boldmath$s$\unboldmath})]$ or $\mathrm{Var}_{\hat{\theta}}[Z(\mbox{\boldmath$s$\unboldmath})]$.

Note that the component var.pred is also present if type is equal to "trend", irrespective of the choice for extended.output. This component contains then the variances of the fitted values.

References

Nussbaum, M., Papritz, A., Baltensweiler, A. and Walthert, L. (2012) Organic Carbon Stocks of Swiss Forest Soils, Institute of Terrestrial Ecosystems, ETH Zurich and Swiss Federal Institute for Forest, Snow and Landscape Research (WSL), pp. 51. http://dx.doi.org/10.3929/ethz-a-007555133

K<U+009F>nsch, H. R., Papritz, A., Schwierz, C. and Stahel, W. A. (2011) Robust estimation of the external drift and the variogram of spatial data. Proceedings of the ISI 58th World Statistics Congress of the International Statistical Institute. http://e-collection.library.ethz.ch/eserv/eth:7080/eth-7080-01.pdf

Examples

Run this code

# NOT RUN {
data(meuse)

data(meuse.grid)
coordinates(meuse.grid) <- ~x+y
meuse.grid.pixdf <- meuse.grid
gridded(meuse.grid.pixdf) <- TRUE
  
library(constrainedKriging)
data(meuse.blocks)

r.logzn.rob <- georob(log(zinc) ~ sqrt(dist), data = meuse, locations = ~ x + y,
    variogram.model = "RMexp", param = c(variance = 0.15, nugget = 0.05, scale = 200),
    tuning.psi = 1., control = control.georob(cov.bhat = TRUE, full.cov.bhat = TRUE))
        
## point predictions of log(Zn)
r.pred.points <- predict(r.logzn.rob, newdata = meuse.grid.pixdf, 
    control = control.predict.georob(extended.output = TRUE, full.covmat = TRUE))
str(r.pred.points$pred@data)

## back-transformation of point predictions
r.backtf.pred.points <- lgnpp(r.pred.points)
str(r.pred.points$pred@data)

spplot(r.backtf.pred.points[["pred"]], zcol = "lgn.pred", main = "Zn content")

## predicting mean Zn content for whole area
r.block <- lgnpp(r.pred.points, is.block = TRUE, all.pred = r.backtf.pred.points[["pred"]])
r.block

## block predictions of log(Zn)
r.pred.block <- predict(r.logzn.rob, newdata = meuse.blocks, 
    control = control.predict.georob(extended.output = TRUE,
        pwidth = 75, pheight = 75))
r.backtf.pred.block <- lgnpp(r.pred.block, newdata = meuse.grid)

spplot(r.backtf.pred.block, zcol = "lgn.pred", main = "block means Zn content")
# }
# NOT RUN {
# }

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