%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                                        %
%   Robust analysis of geostatistical data --- a vignette for the R package ``georob''   %
%                                                                                        %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% 2016-05-27 A. Papritz

% 2016-07-26 AP Version f\"{u}r georob_0.2-4.tar.gz
% 2016-08-22 AP Version f\"{u}r georob_0.3-0.tar.gz
% 2016-09-09 AP Anpassungen zur Reduktion der Rechenzeit
% 2016-11-14 AP Korrektur Fehler in Rotationsmatrix f\"{u}r geom. anisotrope Variogramme
% 2017-08-11 AP Korrektur von Tippfehlern
% 2017-10-20 AP Korrektur von Fehler bei Plot von Profillikelihood
% 2018-11-16 AP Kleine Erg\"{a}nzung Abschnitt \"{u}ber Berechnung von robusten Startwerten
% 2019-04-07 AP Anpassung fuer neue Argumente von fit.variogram.model
% 2020-02-05 AP Anpassung konditionale Verwendung von Objekten aus suggested packages

% cd ~/R.user.home/georob/vignettes/
% R CMD Sweave georob_vignette.Rnw;  R CMD Stangle georob_vignette.Rnw; pdflatex georob_vignette.tex; open georob_vignette.pdf;
% bibtex georob_vignette; pdflatex georob_vignette.tex;  open georob_vignette.pdf;

% sketches how model parameters are estimated robustly 

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\setlength{\parindent}{0mm}

\author{Andreas Papritz} \title{Tutorial and Manual for Geostatistical
Analyses with the R package \pkg{georob}}

%\VignetteIndexEntry{Tutorial and Manual for georob}

\begin{document}

<<preliminaries,echo=FALSE,results=hide,eval=TRUE>>=
options(
  SweaveHooks=list(
    fig=function(){
      par(
        mar=c(3.1, 3.1, 2.1, 2.1), mgp = c(1.5, 0.25, 0), tcl = -0.25
      )
    }
  ),
  width=80, str=strOptions(strict.width="cut"),
  digits=5
)
grDevices::pdf.options( useDingbats = FALSE )
t.t <- (1:9)
t.tick.locations <- c(
  t.t*0.0001, t.t*0.001, t.t*0.01, t.t*0.1, t.t, t.t*10, t.t*100, t.t*1000,
  t.t*10000, t.t*100000, t.t*1000000
)

t.tl <- c(1, 2, NA, NA, 5, NA, NA, NA, NA )
t.tick.labels <- as.character(
  c( t.tl*0.0001, t.tl*0.001, t.tl*0.01, t.tl*0.1, t.tl, t.tl*10, t.tl*100, t.tl*1000,
    t.tl*10000,  t.tl*100000,  t.tl*1000000
  )
)
# my.ncores <- parallel::detectCores()     # <<- Anpassen!!
my.ncores <- 1     # <<- Anpassen!!
@

\sloppy

\maketitle

\tableofcontents

\clearpage

%% 
 % Summary
 %%

\section{Summary}\label{sec:summary}

\pkg{georob} is a package for model-based Gaussian and robust analyses
of geostatistical data.  The software of the package performs two main
tasks:

\begin{itemize}
  \item It fits a linear model with spatially correlated errors to
  geostatistical data that are possibly contaminated by outliers.  The
  coefficients of the linear model (so-called external-drift) and the
  parameters of the variogram model are estimated by robust or
  Gaussian (restricted) maximum likelihood ([RE]ML).
  
  \item It computes from a fitted model object customary and robust
  external drift point and block Kriging predictions.
\end{itemize}

\citet{Kuensch-etal-2011} and \citet{Kuensch-etal-in-prep} explain the theoretical
foundations of the robust approach, and \citet{Diggle-Ribeiro-2007} is
a good reference for model-based Gaussian geostatistical analyses.
%

\smallskip

This document provides a practical introduction to model-based
Gaussian and robust analyses of geostatistical data.  It contains a
short summary of the modelling approach, illustrates the use of the
software with two examples and explains in some depth selected aspects
of (i) (robust) parameter estimation (ii) computing predictions by
(robust) Kriging and (iii) model building.



%% 
 % Introduction
 %%

% material taken from
% tools::Rd2latex("georob-package.Rd", "georob-package.tex", outputEncoding="utf-8")

\section{Introduction} \label{sec:introduction}

This section presents briefly

\begin{itemize}
  \item the modelling assumptions and model parametrization,
    
  \item  sketches how model parameters are estimated robustly and how
  robust Kriging predictions are computed, and 
  
  \item summarizes the main functionality of the package.
\end{itemize}

Further information on selected aspects can be found in
sections~\ref{sec:est-details} and \ref{sec:krig-details}.


\renewcommand{\\}{}

 
%%%%%%%%%%%%%%%%%%%%%%%%%%
% Model
  
\subsection{Model}\label{sec:intro-model}

We use the following model for the data \eqn{y_{i}=y(\mbox{\boldmath$s$\unboldmath}_{i})}{}:
%
\begin{equation}\label{eq:model}
  Y(\mbox{\boldmath$s$\unboldmath}_{i}) = 
  Z(\mbox{\boldmath$s$\unboldmath}_{i}) + \varepsilon_i =
  \mbox{\boldmath $x$\unboldmath}(\mbox{\boldmath$s$\unboldmath}_{i})^\mathrm{T}
  \mbox{\boldmath$\beta$\unboldmath} +
  B(\mbox{\boldmath$s$\unboldmath}_{i}) +
  \varepsilon_i,
\end{equation}
%
where \eqn{\mbox{\boldmath$s$\unboldmath}_{i}}{} denotes a data
location, \eqn{Z(\mbox{\boldmath$s$\unboldmath}_{i})=\mbox{\boldmath
$x$\unboldmath}(\mbox{\boldmath$s$\unboldmath}_{i})^\mathrm{T}
\mbox{\boldmath$\beta$\unboldmath} +
B(\mbox{\boldmath$s$\unboldmath}_{i})}{} is the so-called signal,
\eqn{\mbox{\boldmath
$x$\unboldmath}(\mbox{\boldmath$s$\unboldmath}_{i})^\mathrm{T}
\mbox{\boldmath$\beta$\unboldmath}}{} is the external drift,
\eqn{\{B(\mbox{\boldmath$s$\unboldmath})\}}{} is an unobserved
stationary or intrinsic Gaussian random field with zero mean, and
\eqn{\varepsilon_i}{} is an \emph{i.i.d} error from a possibly
long-tailed distribution with scale parameter \eqn{\tau}{}
(\eqn{\tau^2}{} is usually called nugget effect).  In vector form the
model is written as
%
\begin{equation}\label{eq:model-vector}
  \mbox{\boldmath $Y = X \beta + B + \varepsilon$\unboldmath},
\end{equation}
%
where \eqn{\mbox{\boldmath $X$\unboldmath}}{} is the model matrix with
the rows \eqn{\mbox{\boldmath
$x$\unboldmath}(\mbox{\boldmath$s$\unboldmath}_i)^\mathrm{T}}{}.

The (generalized) covariance matrix of the vector of  
spatial Gaussian random effects 
\eqn{\mbox{\boldmath$B$\unboldmath}}{} 
%
is denoted by 
%
\begin{equation}\label{eq:covariance-matrix-signal}
  \mathrm{E}[
  \mbox{\boldmath$B$\unboldmath}\,
  \mbox{\boldmath$B$\unboldmath}^\mathrm{T}] =
  \mbox{\boldmath$\Gamma$\unboldmath}_\theta =
  \sigma_{\mathrm{n}}^2\mbox{\boldmath$I$\unboldmath} +
  \sigma^2\mbox{\boldmath$V$\unboldmath}_\alpha = 
  \sigma^2_Z \, \mbox{\boldmath$V$\unboldmath}_{\alpha,\xi} = 
  \sigma^2_Z \, ((1-\xi) \, \mbox{\boldmath$I$\unboldmath} + 
  \xi\, \mbox{\boldmath$V$\unboldmath}_\alpha ),
\end{equation}
%
where 
\eqn{\sigma_{\mathrm{n}}^2}{} 
is the variance of seemingly uncorrelated micro-scale variation in 
\eqn{B(\mbox{\boldmath$s$\unboldmath})}{} 
that cannot be resolved with the chosen sampling design,  
\eqn{\sigma^2}{} is the variance of the captured auto-correlated variation in 
\eqn{B(\mbox{\boldmath$s$\unboldmath})}{},  
\eqn{\sigma_Z^2=\sigma_{\mathrm{n}}^2+\sigma^2}{} 
is the signal variance, and 
\eqn{\xi=\sigma^2/\sigma_Z^2}{}. 
To estimate both 
\eqn{\sigma_{\mathrm{n}}^2}{} and \eqn{\tau^2}{} (and not only their sum), one needs
replicated measurements for some of the 
\eqn{\mbox{\boldmath$s$\unboldmath}_i}{}.

We define \eqn{\mbox{\boldmath$V$\unboldmath}_{\alpha}}{} to be the
matrix with elements
%
\begin{equation}\label{eq:valpha}
  (\mbox{\boldmath$V$\unboldmath}_{\alpha})_{ij} =
  \gamma_0 - \gamma(|\mbox{\boldmath$A$\unboldmath}\;(
  \mbox{\boldmath$s$\unboldmath}_i-\mbox{\boldmath$s$\unboldmath}_j)|),
\end{equation}
%
where the constant \eqn{\gamma_0}{} is chosen large enough so that
\eqn{\mbox{\boldmath$V$\unboldmath}_{\alpha}}{} is positive definite,
\eqn{\gamma(\cdot)}{} is a valid stationary or intrinsic variogram,
and \eqn{\mbox{\boldmath$A$\unboldmath}
% =
% \mbox{\boldmath$A$\unboldmath}(\alpha, f_1, f_2; \omega, \phi,
% \zeta)
}{} is a matrix that is used to model geometrically anisotropic
auto-correlation (see section~\ref{sec:est-details-anisotropy}).

\smallskip

Two remarks are in order:

\begin{enumerate}


\item Clearly, the (generalized) covariance matrix of the observations 
\eqn{\mbox{\boldmath $Y$\unboldmath}}{} is given by
%
\begin{equation}\label{eq:covariance-matrix-response}
  \mathrm{Cov}[\mbox{\boldmath $Y$\unboldmath},\mbox{\boldmath $Y$\unboldmath}^\mathrm{T}]
        = \tau^2 \mbox{\boldmath$I$\unboldmath} + \mbox{\boldmath$\Gamma$\unboldmath}_\theta.      
\end{equation}
%
\item Depending on the context, the term ``variogram parameters''
denotes sometimes all parameters of a geometrically anisotropic
variogram model, but in places only the parameters of an isotropic
variogram model, i.e. \eqn{\sigma^2, \ldots, \alpha, \ldots}{} and
\eqn{f_1, \ldots, \zeta}{} are denoted by the term ``anisotropy
parameters''.  In the sequel
\eqn{\mbox{\boldmath$\theta$\unboldmath}}{} is used to denote all
variogram and anisotropy parameters except the nugget effect
\eqn{\tau^2}{}.

\end{enumerate}



%%%%%%%%%%%%%%%%%%%%%%%%%%
%  Estimation

\subsection{Estimation}\label{sec:intro-estimation}
The unobserved spatial random effects 
\eqn{\mbox{\boldmath$B$\unboldmath}}{} at the data locations 
\eqn{\mbox{\boldmath$s$\unboldmath}_i}{}
and the model parameters 
\eqn{\mbox{\boldmath$\beta$\unboldmath}}{}, $\tau^2$ and
\eqn{\mbox{\boldmath$\theta$\unboldmath}^\mathrm{T} = 
      (\sigma^2, \sigma_{\mathrm{n}}^2, \alpha, \ldots,  f_{1}, f_{2}, 
        \omega, \phi, \zeta)
    }{} 
are unknown and are estimated in \pkg{georob} either by Gaussian or 
robust restricted maximum likelihood (REML) or 
Gaussian maximum likelihood (ML). Here \var{\ldots} 
denote further parameters of the variogram such as the smoothness parameter
of the Whittle-Mat\'ern model. 

\smallskip

In brief, the robust REML method is based on the insight that for
given \eqn{\mbox{\boldmath$\theta$\unboldmath}}{} and \eqn{\tau^2}{} the
Kriging predictions (= BLUP) of
\eqn{\mbox{\boldmath$B$\unboldmath}}{} and the generalized least
squares (GLS = ML) estimates of
\eqn{\mbox{\boldmath$\beta$\unboldmath}}{} can be obtained
simultaneously by maximizing 
%
\begin{displaymath}
      - \sum_i 
        \left(
          \frac{
            y_i - 
            \mbox{\boldmath $x$\unboldmath}(\mbox{\boldmath$s$\unboldmath}_i)^\mathrm{T}
            \mbox{\boldmath$\beta$\unboldmath} -
            B(\mbox{\boldmath$s$\unboldmath}_i)
          }{\tau}
        \right)^2 - 
        \mbox{\boldmath$B$\unboldmath}^{\mathrm{T}} 
        \mbox{\boldmath$\Gamma$\unboldmath}^{-1}_\theta
        \mbox{\boldmath$B$\unboldmath}
\end{displaymath}
%
with respect to
\eqn{\mbox{\boldmath$B$\unboldmath}}{} and
\eqn{\mbox{\boldmath$\beta$\unboldmath}}{}, e.g.
% \Cite{Harville (1977)}.
\citet{Harville-1977}.

Hence, the BLUP of \eqn{\mbox{\boldmath$B$\unboldmath}}{},
ML estimates of \eqn{\mbox{\boldmath$\beta$\unboldmath}}{},  
\eqn{\mbox{\boldmath$\theta$\unboldmath}}{} and \eqn{\tau^2}{} 
are obtained by maximizing 
%
\begin{equation}\label{eq:ml-objective-function}
      - \log(\det(
        \tau^2 \mbox{\boldmath$I$\unboldmath} + 
        \mbox{\boldmath$\Gamma$\unboldmath}_\theta        
      )) -
      \sum_i 
        \left(
          \frac{
            y_i - 
            \mbox{\boldmath $x$\unboldmath}(\mbox{\boldmath$s$\unboldmath}_i)^\mathrm{T}
            \mbox{\boldmath$\beta$\unboldmath} -
            B(\mbox{\boldmath$s$\unboldmath}_i)
          }{\tau}
        \right)^2 - 
        \mbox{\boldmath$B$\unboldmath}^{\mathrm{T}} 
        \mbox{\boldmath$\Gamma$\unboldmath}^{-1}_\theta
        \mbox{\boldmath$B$\unboldmath}
\end{equation}
%
jointly with respect to \eqn{\mbox{\boldmath$B$\unboldmath}}{},
\eqn{\mbox{\boldmath$\beta$\unboldmath}}{},
\eqn{\mbox{\boldmath$\theta$\unboldmath}}{} and  \eqn{\tau^2}{}  or by solving the
respective estimating equations.

\smallskip

The estimating equations can then by robustified by 
\begin{itemize}

\item replacing the standardized errors, say
\eqn{\varepsilon_{i}/\tau}{}, by a bounded or re-descending
\eqn{\psi}{}-function, \eqn{\psi_c(\varepsilon_{i}/\tau)}{}, of them
\citep[e.g.][chap.~2]{Maronna-etal-2006} and by
\item
introducing suitable bias correction terms for Fisher consistency at
the Gaussian model,

\end{itemize}

see \citet{Kuensch-etal-2011} for details. 
The robustified estimating equations
are solved numerically by a combination of iterated re-weighted least squares
(IRWLS) to estimate \eqn{\mbox{\boldmath$B$\unboldmath}}{} and
\eqn{\mbox{\boldmath$\beta$\unboldmath}}{} for given 
\eqn{\mbox{\boldmath$\theta$\unboldmath}}{} and  \eqn{\tau^2}{} 
and non-linear root finding by the function 
\code{nleqslv()} of the \R{} package \pkg{nleqslv}
to get \eqn{\mbox{\boldmath$\theta$\unboldmath}}{} and  \eqn{\tau^2}{}.
The robustness of the procedure is controlled by the tuning parameter \eqn{c}{}
of the \eqn{\psi_c}{}-function. For \eqn{c \ge 1000}{} the algorithm computes
Gaussian (RE)ML estimates and customary plug-in Kriging predictions. 
Instead of solving the Gaussian (RE)ML estimating equations, our software then
maximizes the Gaussian (restricted) log-likelihood using  \code{nlminb()} or
\code{optim()}.

\smallskip

\pkg{georob} uses variogram models implemented in the \R{} package
\pkg{RandomFields} (see \code{RMmodel()}).  For most variogram
parameters, closed-form expressions of \eqn{\partial \gamma/ \partial
\theta_i}{} are used in the computations.  However, for the parameter
\eqn{\nu}{} of the models \code{"RMbessel"}, \code{"RMmatern"} and
\code{"RMwhittle"} \eqn{\partial \gamma/ \partial \nu}{} is evaluated
numerically by the function \code{numericDeriv()}, and this results in
an increase in computing time when \eqn{\nu}{} is estimated.


%%%%%%%%%%%%%%%%%%%%%%%%%%
%  Prediction

\subsection{Prediction}\label{sec:intro-prediction}
 
Robust plug-in external drift point Kriging predictions 
can be computed for an unsampled location 
\eqn{\mbox{\boldmath$s$\unboldmath}_0}{}
from the covariates
\eqn{\mbox{\boldmath$x$\unboldmath}(\mbox{\boldmath$s$\unboldmath}_0)}{},
the estimated parameters 
\eqn{\widehat{\mbox{\boldmath$\beta$\unboldmath}}}{},
\eqn{\widehat{\mbox{\boldmath$\theta$\unboldmath}}}{}
and the predicted random effects
\eqn{\widehat{\mbox{\boldmath$B$\unboldmath}}}{}
by
%
\begin{equation}\label{eq:robust-edk}
      \widehat{Y}(\mbox{\boldmath$s$\unboldmath}_0) = \widehat{Z}(\mbox{\boldmath$s$\unboldmath}_0) = 
      \mbox{\boldmath$x$\unboldmath}(\mbox{\boldmath$s$\unboldmath}_0)^\mathrm{T}
      \widehat{\mbox{\boldmath$\beta$\unboldmath}} + 
      \mbox{\boldmath$\gamma$\unboldmath}^\mathrm{T}_{\widehat{\theta}}(\mbox{\boldmath$s$\unboldmath}_0)
      \mbox{\boldmath$\Gamma$\unboldmath}^{-1}_{\widehat{\theta}}
      \widehat{\mbox{\boldmath$B$\unboldmath}},
\end{equation}
%
where
\eqn{\mbox{\boldmath$\Gamma$\unboldmath}_{\widehat{\theta}}}{}
is the estimated (generalized) covariance matrix of
\eqn{\mbox{\boldmath$B$\unboldmath}}{} and
\eqn{\mbox{\boldmath$\gamma$\unboldmath}_{\widehat{\theta}}(\mbox{\boldmath$s$\unboldmath}_0)}{}
is the vector with the estimated (generalized) covariances between 
\eqn{\mbox{\boldmath$B$\unboldmath}}{} and
\eqn{B(\mbox{\boldmath$s$\unboldmath}_0)}{}.
Kriging variances can be computed as well, based on approximated covariances of 
\eqn{\widehat{\mbox{\boldmath$B$\unboldmath}}}{} and 
\eqn{\widehat{\mbox{\boldmath$\beta$\unboldmath}}}{}
(see \citealp{Kuensch-etal-2011}, and appendices of
\citealp{Nussbaum-etal-2012,Nussbaum-etal-2014b}, for details).

The package \pkg{georob} provides in addition software for computing
robust external drift \emph{block} Kriging predictions.  The required
integrals of the (generalized) covariance function are computed by
functions of the \R{} package \pkg{constrainedKriging}
\citep{Hofer-Papritz-2011}.



%%%%%%%%%%%%%%%%%%%%%%%%%%
%  Functionality

\subsection{Functionality}\label{sec:intro-functionality}
For the time being, the functionality of \pkg{georob} is limited 
to robust geostatistical analyses of \emph{single} response variables. 
No software is currently available for robust multivariate geostatistical 
analyses.
\pkg{georob} offers functions for: 
\begin{enumerate}
  
  \item Robustly fitting a spatial linear model to data that are
  possibly contaminated by independent errors from a long-tailed
  distribution by robust REML (see functions \code{georob()} ---
  which also fits such models efficiently by Gaussian (RE)ML ---
  \code{profilelogLik()} and \code{control.georob()}).
  
  \item Extracting estimated model components (see
  \code{residuals.georob()},
  \code{rstandard.georob()}, 
  %   \code{rstudent.georob()},
  \code{ranef.georob()}).
  
  \item Robustly estimating sample variograms and for fitting
  variogram model functions to them (see
  \code{sample.variogram()} and
  \code{fit.variogram.model()}).
  
  \item Model building by forward and backward selection of covariates
  for the external drift (see \code{waldtest.georob()},
  \code{step.georob()}, \code{add1.georob()}, \code{drop1.georob()},
  \code{extractAIC.georob()},  \code{logLik.georob()},
  \code{deviance.georob()}).  For a robust fit, the log-likelihood is
  not defined.  The function then computes the (restricted)
  log-likelihood of an equivalent Gaussian model with heteroscedastic
  nugget (see \code{deviance.georob()} for details).
  
  \item Assessing the goodness-of-fit and predictive power of the
  model by \var{K}-fold cross-validation (see
  \code{cv.georob()} and
  \code{validate.predictions()}).
  
  \item Computing robust external drift point and block Kriging
  predictions (see \code{predict.georob()},
  \code{control.predict.georob()}).
  
  \item Unbiased back-transformation of both point and block Kriging
  predictions of log-transformed data to the original scale of the
  measurements (see \code{lgnpp()}).
  
\end{enumerate}




\renewcommand{\\}{}

\clearpage

%% 
 % Analysis of meuse zinc data
 %%

\section{Model-based Gaussian analysis of \texttt{zinc}, data set \code{meuse}}\label{sec:meuse}

<<meuse-zinc-load,echo=FALSE,eval=TRUE>>=
if(file.exists("r_meuse_zinc_objects.RData")) load("r_meuse_zinc_objects.RData")
@

The package \pkg{sp} provides this data set.  According to the help
page, it ``gives locations and topsoil heavy metal concentrations,
along with a number of soil and landscape variable at the observation
locations, collected in a flood plain of the river Meuse, near the
village of Stein (NL)''. 
%
<<meuse-data,eval=TRUE>>=
data(meuse, package="sp")
levels(meuse$ffreq) <- paste("ffreq", levels(meuse$ffreq), sep="")
levels(meuse$soil) <- paste("soil", levels(meuse$soil), sep="")
str(meuse)
@
%
\citet{Bivand-etal-2013} use the data to illustrate geostatistical
analyses by the package \pkg{gstat} \citep{Pebesma-2004}.
%
We analyse here the data on \code{zinc} in the topsoil (Figure~\ref{fig:meuse}).

\subsection{Exploratory analysis}\label{sec:meuse-zinc-eda}

\bcf
% 
\begin{center}
  \includegraphics[width=0.45\textwidth]{0-overview-zinc-meuse.pdf}
\end{center}
% 
\ecf{Meuse data set: zinc concentration at 155 locations in floodplain
of river Meuse near Stein (NL) shown in Google Earth\texttrademark.}{meuse}
% 
Figure~\ref{fig:meuse} suggests that \code{zinc} concentration depends
on \code{dist}ance to the river.  We check graphically whether the
two factors \code{ffreq} (frequency of flooding) and \code{soil}
(type) also influence \code{zinc}:
% 
\bcf
%
<<meuse-zinc-eda-plot-1,fig=TRUE,height=4,width=6,eval=TRUE=>>=
library(lattice)
palette(trellis.par.get("superpose.symbol")$col)
plot(zinc~dist, meuse, pch=as.integer(ffreq), col=soil)
legend("topright", col=c(rep(1, nlevels(meuse$ffreq)), 1:nlevels(meuse$soil)),
  pch=c(1:nlevels(meuse$ffreq), rep(1, nlevels(meuse$soil))), bty="n",
  legend=c(levels(meuse$ffreq), levels(meuse$soil)))
@
% 
\ecf{Dependence of \code{zinc} concentration on \code{dist}ance to
river, frequency of flooding (\code{ffreq}) and \code{soil} type.}{meuse-eda-plot-1}
% 
\code{zinc} depends non-linearly on \code{dist} and seems in addition
to depend on \code{ffreq} (larger concentration at more often flooded
sites).  Furthermore, the scatter of \code{zinc} for given distance
increases with decreasing distance (= increasing \code{zinc}
concentration, heteroscedastic variation).  We use \code{log(zinc)} to
stabilize the variance:
% 
\bcf
%
<<meuse-zinc-eda-plot-2,fig=TRUE,height=4,width=6,eval=TRUE=>>=
xyplot(log(zinc)~dist | ffreq, meuse, groups=soil, panel=function(x, y, ...){
    panel.xyplot(x, y, ...)
    panel.loess(x, y, ...)
  }, auto.key=TRUE)
@
%
\ecf{Dependence of \code{zinc} on \code{dist}ance to river, frequency
of flooding (\code{ffreq}) and \code{soil} type.}{meuse-eda-plot-2}
%
The relation \code{log(zinc)\~{}dist} is still non-linear, hence we
transform \code{dist} by $\sqrt{\;\;}$:
% 
\bcf
%
<<meuse-zinc-eda-plot-3,fig=TRUE,height=4,width=6,eval=TRUE=>>=
xyplot(log(zinc)~sqrt(dist) | ffreq, meuse, groups=soil, panel=function(x, y, ...){
    panel.xyplot(x, y, ...)
    panel.loess(x, y, ...)
    panel.lmline(x, y, lty="dashed", ...)
  }, auto.key=TRUE)
@
%
\ecf{Dependence of \code{zinc} concentration on \code{dist}ance to
river, frequency of flooding (\code{ffreq}) and \code{soil} type.}{meuse-eda-plot-3}
%
which approximately linearizes the relation.

\smallskip

The slopes of the regression lines \code{log(zinc)\~{}sqrt(dist)} are
about the same for all levels of \code{ffreq}.  But the intercept of
\code{ffreq1} differs from the intercepts of the other levels.  Hence,
as an initial drift model we use
%
<<meuse-zinc-lm,eval=TRUE>>=
r.lm <- lm(log(zinc)~sqrt(dist)+ffreq, meuse)
summary(r.lm)
@
%

The residual diagnostic plots
%
\bcf
%
<<meuse-zinc-lm-resdiag,fig=TRUE,height=5,width=6,eval=TRUE=>>=
op <- par(mfrow=c(2, 2)); plot(r.lm); par(op)
@
%
\ecf{Residual diagnostic plots for linear drift model
\code{log(zinc)\~{}sqrt(dist)+ffreq}.}{meuse-zinc-lm-resdiag}
%
do not show violations of modelling assumptions.

\smallskip

Next, we compute the sample variogram of the residuals for the 4
directions N-S, NE-SW, E-W, SE-NW by the methods-of-moments estimator:
%
\bcf
%
<<meuse-zinc-lm-res-sv-1,fig=TRUE,width=6,height=4,results=hide,eval=TRUE>>=
library(georob)
plot(sample.variogram(residuals(r.lm), locations=meuse[, c("x","y")],
  lag.dist.def=100, max.lag=2000, xy.angle.def=c(0, 22.5, 67.5, 112.5, 157.5, 180),
  estimator="matheron"), type="l",
  main="sample variogram of residuals log(zinc)~sqrt(dist)+ffreq")
@
%
\ecf{Direction-dependent sample variogram of regression residuals of
\code{log(zinc)\~{}sqrt(dist)+ffreq}.}{meuse-zinc-lm-res-sv-1}
%
The residuals appear to be spatially dependent.  For the short lags
there is no clear dependence on direction, hence, we assume that
auto-correlation is isotropic.  

To complete the exploratory modelling exercise, we compute the
direction-indepdendent sample variogram and fit a spherical variogram
model by weighted non-linear least squares (using Cressie's weights)
%
\bcf
%
<<meuse-zinc-lm-res-sv-2,fig=TRUE,width=6,height=4,results=hide>>=
library(georob)
plot(r.sv <- sample.variogram(residuals(r.lm), locations=meuse[, c("x","y")],
  lag.dist.def=100, max.lag=2000, 
  estimator="matheron"), type="l",
  main="sample variogram of residuals log(zinc)~sqrt(dist)+ffreq")
lines(r.sv.spher <- fit.variogram.model(r.sv, variogram.mode="RMspheric",
  param=c(variance=0.1, nugget=0.05, scale=1000)))
@
<<meuse-zinc-lm-res-sv-2,fig=TRUE,width=6,height=4,results=hide,echo=FALSE,eval=TRUE>>=
library(georob)
plot(r.sv, type="l", main="sample variogram of residuals log(zinc)~sqrt(dist)+ffreq")
lines(r.sv.spher)
@
%
\ecf{Sample variogram of regression residuals of
\code{log(zinc)\~{}sqrt(dist)+ffreq} along with fitted spherical
variogram function.}{meuse-zinc-lm-res-sv-2}
%
and output the fitted variogram parameters
%
<<meuse-zinc-lm-res-spher,eval=TRUE>>=
summary(r.sv.spher)
@


\clearpage

%%%%%%%%%%%%%%%%%%%%%%%%%
% fitting model by robust REML

\subsection{Fitting  a spatial linear model by Gaussian (RE)ML}\label{sec:meuse-zinc-modelfit}

We fit the model that we developed in the exploratory analysis now by
Gaussian REML:
%
<<meuse-zinc-georob-reml,eval=TRUE>>=
r.georob.m0.spher.reml <- georob(log(zinc)~sqrt(dist)+ffreq, meuse, locations=~x+y,
  variogram.model="RMspheric", param=c(variance=0.1, nugget=0.05, scale=1000), 
  tuning.psi=1000)
@
%
<<meuse-zinc-summary-georob-reml,eval=TRUE>>=
summary(r.georob.m0.spher.reml)
@
%
The diagnostics at the begin of the \code{summary} output suggest that
maximization of the restricted log-likelihood by \code{nlminb()} was
successful.
%
Nevertheless, before we interprete the output, we compute the profile
log-likelihood for the range to see whether the maximization has found
the global maximum:
%
<<meuse-zinc-proflik-reml-scale-1a,eval=FALSE>>=
r.prfl.m0.spher.reml.scale <- profilelogLik(r.georob.m0.spher.reml,
  values=data.frame(scale=seq(500, 5000, by=50)))
@
%
<<meuse-zinc-proflik-reml-scale-1b,echo=FALSE>>=
r.prfl.m0.spher.reml.scale <- profilelogLik(r.georob.m0.spher.reml,
  values=data.frame(scale=seq(500, 5000, by=50)), ncores=my.ncores)
@
%
\bcf
%
<<meuse-zinc-proflik-reml-scale-2,fig=TRUE,width=6,height=4,results=hide,eval=TRUE>>=
plot(loglik~scale, r.prfl.m0.spher.reml.scale, type="l")
abline(v=coef(r.georob.m0.spher.reml, "variogram")["scale"], lty="dashed")
abline(h=r.georob.m0.spher.reml$loglik - 0.5*qchisq(0.95, 1), lty="dotted")
@
%
\ecf{Restricted profile log-likelihood for range parameter
(\code{scale}) of spherical variogram (vertical line: estimate of
\code{scale} returned by \code{georob()}; intersection of horizontal
line with profile defines a 95\% confidence region for \code{scale}
based on likelihood ratio test).
}{meuse-zinc-proflik-reml-scale-2}
%
Although the restricted log-likelihood is multimodal~---~which is
often observed for variogram models with compact support~---~we were
lucky to find the global maximum because the initial values of the
variogram parameters were close to the REML estimates.
%
Estimates of \code{scale} (range of variogram) and \code{variance}
(partial sill) are correlated, \code{nugget} and \code{scale}
less so:
%
\bcf
%
<<meuse-zinc-proflik-reml-scale-3,fig=TRUE,width=6,height=3,results=hide,eval=TRUE>>=
op <- par(mfrow=c(1,2), cex=0.66)
plot(variance~scale, r.prfl.m0.spher.reml.scale, ylim=c(0, max(variance)), type="l")
plot(nugget~scale, r.prfl.m0.spher.reml.scale, ylim=c(0, max(nugget)), type="l")
par(op)
@
%
\ecf{Estimates of \code{variance} (partial sill) and \code{nugget} as
a function of the estimate of the range (\code{scale}) of the
variogram.}{meuse-zinc-proflik-reml-scale-3}
%

\smallskip

We now study the \code{summary} output in detail: 
%
% The criteria controlling convergence of the maximization can be
% controlled by the arguments of \code{control.nlminb()}, see
% section~\ref{sec:control-georob}.  The maximized restricted
% log-likelihood is equal to -54.6.
%
The estimated variogram parameters are reported along with 95\%
confidence intervals that are computed based on the asymptotic normal
distribution of (RE)ML estimates from the observed Fisher information.

The dependence of \code{log(zinc)} on \code{sqrt(dist)} is highly
significant, as is the dependence on \code{ffreq}:
%
<<meuse-zinc-georob-reml-waldtest-1,eval=TRUE>>=
waldtest(r.georob.m0.spher.reml, .~.-ffreq)
@
%
We can test equality of all pairs of intercepts
 by functions of the package \pkg{multcomp}
%
<<meuse-zinc-georob-reml-multcomp,eval=TRUE>>=
if(requireNamespace("multcomp", quietly = TRUE)){
  summary(multcomp::glht(r.georob.m0.spher.reml, 
    linfct = multcomp::mcp(ffreq = c("ffreq1 - ffreq2 = 0", "ffreq1 - ffreq3 = 0",
    "ffreq2 - ffreq3 = 0"))))
} else {
  install.packages("multcomp")
  requireNamespace("multcomp", quietly = TRUE)
  summary(multcomp::glht(r.georob.m0.spher.reml, 
    linfct = multcomp::mcp(ffreq = c("ffreq1 - ffreq2 = 0", "ffreq1 - ffreq3 = 0",
    "ffreq2 - ffreq3 = 0"))))
}
# library(multcomp)
# summary(glht(r.georob.m0.spher.reml, 
#   linfct = mcp(ffreq = c("ffreq1 - ffreq2 = 0", "ffreq1 - ffreq3 = 0",
#   "ffreq2 - ffreq3 = 0"))))
@
%
As suspected only the intercept of \code{ffreq1} differs from the
others.
%
Adding the interaction
\code{sqrt(dist):ffreq} does not improve the model:
%
<<meuse-zinc-georob-reml-waldtest-2,eval=TRUE>>=
waldtest(r.georob.m0.spher.reml, .~.+sqrt(dist):ffreq)
@
%
nor does adding \code{soil}:
%
<<meuse-zinc-georob-reml-waldtest-3,eval=TRUE>>=
waldtest(r.georob.m0.spher.reml, .~.+soil)
@

\smallskip

Drift models may also be build by step-wise covariate selection based
on AIC, either keeping the variogram parameters fixed (default)
%
<<meuse-zinc-georob-reml-step-1,eval=FALSE>>=
step(r.georob.m0.spher.reml, scope=log(zinc)~ffreq*sqrt(dist)+soil)
@
%
\begin{Schunk}
\begin{Soutput}
Start:  AIC=106.91
log(zinc) ~ sqrt(dist) + ffreq

                   Df AIC Converged
<none>                107          
+ soil              2 107         1
+ ffreq:sqrt(dist)  2 110         1
- ffreq             2 166         1
- sqrt(dist)        1 178         1

Tuning constant:  1000 

Fixed effects coefficients:
(Intercept)   sqrt(dist)  ffreqffreq2  ffreqffreq3  
      7.094       -2.146       -0.526       -0.537  

Variogram:  RMspheric 
 variance(fixed)    snugget(fixed)     nugget(fixed)      scale(fixed)  
           0.123             0.000             0.056           872.400  
\end{Soutput}
\end{Schunk}
%
or re-estimating them afresh for each evaluated model
%
<<meuse-zinc-georob-reml-step-2,eval=FALSE>>=
step(r.georob.m0.spher.reml, scope=log(zinc)~ffreq*sqrt(dist)+soil, 
  fixed.add1.drop1=FALSE)
@
%
\begin{Schunk}
\begin{Soutput}
Start:  AIC=112.91
log(zinc) ~ sqrt(dist) + ffreq

                   Df AIC Converged
<none>                113         1
+ soil              2 113         1
+ ffreq:sqrt(dist)  2 116         1
- sqrt(dist)        1 144         1
- ffreq             2 158         1

Tuning constant:  1000 

Fixed effects coefficients:
(Intercept)   sqrt(dist)  ffreqffreq2  ffreqffreq3  
      7.094       -2.146       -0.526       -0.537  

Variogram:  RMspheric 
        variance    snugget(fixed)            nugget             scale  
           0.123             0.000             0.056           872.400  
\end{Soutput}
\end{Schunk}
%
which selects the same model.  Note that step-wise covariate
selection by \code{step.georob()}, \code{add1.georob()} and
\code{drop1.georob()} requires the \emph{non-restricted}
log-likelihood.  Before evaluating candidate models, the initial model
is therefore re-fitted by ML, which can be done by
%
<<meuse-zinc-georob-ml,eval=TRUE>>=
r.georob.m0.spher.ml <- update(r.georob.m0.spher.reml, 
  control=control.georob(ml.method="ML"))
@

<<meuse-zinc-georob-aic,eval=TRUE>>=
extractAIC(r.georob.m0.spher.reml, REML=TRUE)
extractAIC(r.georob.m0.spher.ml)
r.georob.m0.spher.ml
@

Models can be also compared by cross-validation
%
<<meuse-zinc-cv-1,eval=FALSE>>=
r.cv.m0.spher.reml <- cv(r.georob.m0.spher.reml, seed=3245, lgn=TRUE)
r.georob.m1.spher.reml <- update(r.georob.m0.spher.reml, .~.-ffreq)
r.cv.m1.spher.reml <- cv(r.georob.m1.spher.reml, seed=3245, lgn=TRUE)
@
%
<<meuse-zinc-cv-2,echo=FALSE>>=
r.cv.m0.spher.reml <- cv(r.georob.m0.spher.reml, seed=3245, lgn=TRUE, ncores=my.ncores)
r.georob.m1.spher.reml <- update(r.georob.m0.spher.reml, .~.-ffreq)
r.cv.m1.spher.reml <- cv(r.georob.m1.spher.reml, seed=3245, lgn=TRUE, ncores=my.ncores)
@
%
<<meuse-zinc-cv-summary,eval=TRUE>>=
summary(r.cv.m0.spher.reml)
summary(r.cv.m1.spher.reml)
@
%
Note that the argument \code{lng=TRUE} has the effect that the
cross-validation predictions of a log-transformed response are
transformed back to the original scale of the measurements.
%
%
\bcf
%
<<meuse-zinc-cv-plot,fig=TRUE,width=6,height=7.5,results=hide,eval=TRUE>>=
op <- par(mfrow=c(3,2))
plot(r.cv.m1.spher.reml, "sc")
plot(r.cv.m0.spher.reml, "sc", add=TRUE, col=2) 
abline(0, 1, lty="dotted")
legend("topleft", pch=1, col=1:2, bty="n", 
  legend=c("log(zinc)~sqrt(dist)", "log(zinc)~sqrt(dist)+ffreq"))
plot(r.cv.m1.spher.reml, "lgn.sc"); plot(r.cv.m0.spher.reml, "lgn.sc", add=TRUE, col=2) 
abline(0, 1, lty="dotted")
plot(r.cv.m1.spher.reml, "hist.pit")
plot(r.cv.m0.spher.reml, "hist.pit", col=2) 
plot(r.cv.m1.spher.reml, "ecdf.pit")
plot(r.cv.m0.spher.reml, "ecdf.pit", add=TRUE, col=2)
abline(0, 1, lty="dotted")
plot(r.cv.m1.spher.reml, "bs")
plot(r.cv.m0.spher.reml, add=TRUE, "bs", col=2) 
par(op)
@
%
\ecf{Diagnostic plots of cross-validation predictions of REML fits of
models \code{log(zinc)\~{}sqrt(dist)} (blue) and
\code{log(zinc)\~{}sqrt(dist)+ffreq} (magenta).}{meuse-zinc-cv-plot}

The simpler model gives less precise predictions (larger \code{rmse},
Brier score and therefore also larger continuous ranked probability
score \code{crps}), but it models prediction uncertainty better
\citep[PIT closer to uniform distribution, see section~\ref{sec:cv}
and][]{Gneiting-etal-2007}.  

We finish modelling by plotting residual diagnostics of the model
\code{r.georob.m0.spher.reml} and comparing the estimated variogram
with the ML estimate and the model fitted previously to the sample
variogram of ordinary least squares (OLS) residuals:
%
\bcf%
<<meuse-zinc-georob-plot,fig=TRUE,width=6,height=5,results=hide,eval=TRUE>>=
op <- par(mfrow=c(2,2), cex=0.66)
plot(r.georob.m0.spher.reml, "ta"); abline(h=0, lty="dotted")
plot(r.georob.m0.spher.reml, "qq.res"); abline(0, 1, lty="dotted")
plot(r.georob.m0.spher.reml, "qq.ranef"); abline(0, 1, lty="dotted")
plot(r.georob.m0.spher.reml, lag.dist.def=100, max.lag=2000)
lines(r.georob.m0.spher.ml, col=2); lines(r.sv.spher, col=3)
par(op)
@
%
\ecf{Residual diagnostic plots of model
\code{log(zinc)\~{}sqrt(dist)+ffreq} and spherical variogram estimated
by REML (blue), ML (magenta) and fit to sample variogram
(darkgreen).}{meuse-zinc-georob-plot}
%
As expected REML estimates larger range and partial sill parameters.
The diagnostics plots do not reveal serious violations of modelling
assumptions.
%


\subsection{Computing Kriging predictions}\label{sec:lognormal-edk}

\subsubsection{Lognormal point Kriging}\label{sec:meuse-point-Kriging}

The data set \code{meuse.grid} (provided also by package \pkg{sp})
contains the covariates
%
<<meuse-zinc-meuse-grid,eval=TRUE>>=
data(meuse.grid)
levels(meuse.grid$ffreq) <- paste("ffreq", levels(meuse.grid$ffreq), sep="")
levels(meuse.grid$soil) <- paste("soil", levels(meuse.grid$soil), sep="")
coordinates(meuse.grid) <- ~x+y
gridded(meuse.grid) <- TRUE
@
%
for computing Kriging predictions of \code{log(zinc)} and transforming
them back to the original scale of the measurements by
%
<<meuse-zinc-meuse-point-Kriging,eval=TRUE>>=
r.pk <- predict(r.georob.m0.spher.reml, newdata=meuse.grid,
  control=control.predict.georob(extended.output=TRUE))
r.pk <- lgnpp(r.pk)
@
%
<<meuse-zinc-meuse-point-Kriging-lgn,eval=TRUE>>=
str(r.pk)
@
%
Note that 
\begin{itemize}
  \item \code{meuse.grid} was converted to a
  \code{SpatialPixelsDataFrame} object prior to Kriging.


  \item The argument
  \code{control=control.predict.georob(extended.output=TRUE)} of
  \code{predict.georob()} has the effect that all items required for
  the back-transformation are computed.
  
  \item The variables \code{lgn.{\itshape xxx}} in \code{r.pk} contain
  the back-transformed prediction items.
\end{itemize}

Finally, the function \code{spplot()} (package \pkg{sp}) allows to
easily plot prediction results:
%
\bcf[1]
%
<<meuse-zinc-point-Kriging-plot,fig=TRUE,width=9,height=3.5,results=hide,eval=TRUE>>=
brks <- c(25, 50, 75, 100, 150, 200, seq(500, 3500,by=500))
pred <- spplot(r.pk, zcol="lgn.pred", at=brks, main="prediction")
lwr <- spplot(r.pk, zcol="lgn.lower", at=brks, main="lower bound 95% PI")
upr <- spplot(r.pk, zcol="lgn.upper", at=brks, main="upper bound 95% PI")
plot(pred, position=c(0, 0, 1/3, 1), more=TRUE)
plot(lwr, position=c(1/3, 0, 2/3, 1), more=TRUE)
plot(upr, position=c(2/3, 0, 1, 1), more=FALSE)
@ 
%
\ecf{Lognormal point Kriging prediction of \code{zinc} (left:
prediction; middle and right: lower and upper bounds of point-wise 95\%
prediction intervals).}{meuse-zinc-point-Kriging-plot}
%

\subsubsection{Lognormal block Kriging}\label{sec:meuse-block-Kriging}

If \code{newdata} is a \code{SpatialPolygonsDataFrame} then
\code{predict.georob()} computes block Kriging predictions.  We
illustrate this here with the \code{SpatialPolygonsDataFrame}
\code{meuse.blocks} that is provided by the package
\pkg{constrainedKriging}:
%
<<meuse-zinc-block-Kriging,eval=TRUE>>=
data(meuse.blocks, package="constrainedKriging")
str(meuse.blocks, max=2)
@
%
\code{meuse.blocks} contains  \code{dist} as only covariate for the
blocks, therefore, we use the model \code{log(zinc)\~{}sqrt(dist)} for
computing the block Kriging predictions of \code{log(zinc)}:
%
<<meuse-zinc-meuse-block-Kriging-1a,eval=FALSE>>=
r.bk <- predict(r.georob.m1.spher.reml, newdata=meuse.blocks,
  control=control.predict.georob(extended.output=TRUE, pwidth=25, pheight=25, mmax=25))
@
%
<<meuse-zinc-meuse-block-Kriging-1b,echo=FALSE>>=
r.bk <- predict(r.georob.m1.spher.reml, newdata=meuse.blocks,
  control=control.predict.georob(extended.output=TRUE, pwidth=25, pheight=25))
@
%
and transform the predictions back by the approximately unbiased
back-transformation proposed by \citet{Cressie-2006} (see
section~\ref{eq:spatial-covariance-covariates}):
%
<<meuse-zinc-meuse-block-Kriging-2>>=
r.bk <- lgnpp(r.bk, newdata=meuse.grid)
@
%
Note the following points:
\begin{itemize}
  \item \code{pwidth} and \code{pheight} are the dimensions of the
  pixels used for efficiently computing ``block-block-'' and
  ``point-block- averages'' of the covariance function, see
  section~\ref{sec:block-Kriging}).
  
  \item \code{mmax} controls parallelization when computing Kriging
  predictions, see
  section~\ref{sec:Kriging-parallelized-computations}.

  \item \code{newdata} is used to pass point support covariates to
  \code{lgnpp()}, from which the spatial covariances of the
  covariates, needed for the back-transformation, are computed,
  \citep[see equation~\ref{eq:spatial-covariance-covariates}
  and][appendix~C]{Cressie-2006}.
\end{itemize}

We display the prediction results again by \code{spplot()}:
%
\bcf[1]
%
<<meuse-zinc-block-Kriging-plot,fig=TRUE,width=9,height=3.5,results=hide,eval=TRUE>>=
brks <- c(25, 50, 75, 100, 150, 200, seq(500, 3500,by=500))
pred <- spplot(r.bk, zcol="lgn.pred", at=brks, main="prediction")
lwr <- spplot(r.bk, zcol="lgn.lower", at=brks, main="lower bound 95% PI")
upr <- spplot(r.bk, zcol="lgn.upper", at=brks, main="upper bound 95% PI")
plot(pred, position=c(0, 0, 1/3, 1), more=TRUE)
plot(lwr, position=c(1/3, 0, 2/3, 1), more=TRUE)
plot(upr, position=c(2/3, 0, 1, 1), more=FALSE)
@ 
%
\ecf{Lognormal block Kriging prediction of \code{zinc} computed under
the assumption of permanence of log-normality (left: prediction; middle
and right: lower and upper bounds of point-wise 95\% prediction
intervals).}{meuse-zinc-block-Kriging-plot}
%

The assumption that both point values and block means follow
log-normal laws~---~which strictly cannot hold~---~does not much
impair the efficiency of the back-transformation as long as the blocks
are small \citep{Cressie-2006,Hofer-etal-2013}.  However, for
larger blocks, one should use the optimal predictor obtained by
\emph{averaging back-transformed point predictions}.  \code{lgnpp()}
allows to compute this as well.  We illustrate this here by predicting
the spatial mean over two larger square blocks:
%
\bcf[0.4]
<<meuse-zinc-block-Kriging-blocks,fig=TRUE,width=4,height=5,results=hide,eval=TRUE>>=
## define blocks
tmp <- data.frame(x=c(179100, 179900), y=c(330200, 331000))
blks <- SpatialPolygons(sapply(1:nrow(tmp), function(i, x){  
  Polygons(list(Polygon(t(x[,i] + 400*t(cbind(c(-1, 1, 1, -1, -1), c(-1, -1, 1, 1, -1)))),
  hole=FALSE)), ID=paste("block", i, sep=""))}, x=t(tmp)))
## compute spatial mean of sqrt(dist) for blocks
ind <- over(as(meuse.grid, "SpatialPoints"), blks)
tmp <- tapply(sqrt(meuse.grid$dist), ind, mean)
names(tmp) <- paste("block", 1:length(tmp), sep="")
## create SpatialPolygonsDataFrame
blks <- SpatialPolygonsDataFrame(blks, data=data.frame(dist=tmp^2))
## and plot
plot(as(meuse.grid, "SpatialPoints"), axes=TRUE)
plot(geometry(blks), add=TRUE, col=2)
@
%
\ecf{Point prediction grid and position of 2 target blocks for computing
optimal log-normal block predictor.}{meuse-zinc-block-Kriging-blocks}
%

We first compute block-Kriging predictions of \code{log(zinc)} and
back-transform them as before under the permanence of log-normality
assumption:
%
<<meuse-zinc-block-Kriging-3>>=
r.blks <- predict(r.georob.m1.spher.reml, newdata=blks, 
  control=control.predict.georob(extended.output=TRUE, pwidth=800, pheight=800))
r.blks <- lgnpp(r.blks, newdata=meuse.grid)
@
%
Note that we set \code{pwidth} and \code{pheight} equal to the
dimension of the blocks.  This is best for rectangular blocks
because the blocks are then represented by a single pixel, which
reduces computations.

Next we use \code{lgnpp()} for computing the optimal log-normal block
predictions.  As we need the full covariance matrix of the point
prediction errors for this we must re-compute the points predictions
of \code{log(zinc)} with the additional \code{control} argument
\code{full.covmat=TRUE}:
%
<<<meuse-zinc-block-Kriging-4a>>=
t.pk <- predict(r.georob.m0.spher.reml, newdata=as.data.frame(meuse.grid),
  control=control.predict.georob(extended.output=TRUE, full.covmat=TRUE))
str(t.pk)
@
%
\begin{Schunk}
\begin{Soutput}
List of 5
 $ pred           :'data.frame':	3103 obs. of  10 variables:
  ..$ x              : num [1:3103] 181180 181140 181180 181220 181100 ...
  ..$ y              : num [1:3103] 333740 333700 333700 333700 333660 ...
  ..$ pred           : num [1:3103] 7.05 7.06 6.8 6.57 7.07 ...
  ..$ se             : num [1:3103] 0.277 0.248 0.252 0.259 0.212 ...
  ..$ lower          : num [1:3103] 6.51 6.57 6.3 6.06 6.65 ...
  ..$ upper          : num [1:3103] 7.59 7.54 7.29 7.07 7.48 ...
  ..$ trend          : num [1:3103] 7.09 7.09 6.85 6.64 7.09 ...
  ..$ var.pred       : num [1:3103] 0.0779 0.0899 0.0822 0.0757 0.1027 ...
  ..$ cov.pred.target: num [1:3103] 0.0681 0.0818 0.0768 0.0717 0.0963 ...
  ..$ var.target     : num [1:3103] 0.135 0.135 0.135 0.135 0.135 ...
  ..- attr(*, "variogram.model")= chr "RMspheric"
  ..- attr(*, "param")= Named num [1:4] 0.1349 0 0.0551 876.5812
  .. ..- attr(*, "names")= chr [1:4] "variance" "snugget" "nugget" "scale"
  ..- attr(*, "psi.func")= chr "logistic"
  ..- attr(*, "tuning.psi")= num 1000
  ..- attr(*, "type")= chr "signal"
 $ mse.pred       : num [1:3103, 1:3103] 0.0766 0.0565 0.0606 0.0583 0.0373 ...
 $ var.pred       : num [1:3103, 1:3103] 0.0779 0.0833 0.0794 0.0748 0.0878 ...
 $ cov.pred.target: num [1:3103, 1:3103] 0.0681 0.0736 0.0714 0.0684 0.0781 ...
 $ var.target     : num [1:3103, 1:3103] 0.135 0.122 0.126 0.122 0.109 ...
\end{Soutput}
\end{Schunk}
The predictions are now stored in the list component \code{pred}, and
list components \code{mse.pred}, \code{var.pred}, \code{var.target}
and \code{cov.pred.target} store the covariance matrices of prediction
errors, predictions, prediction targets and the covariances between
predictions and targets.

Then we back-transform the predictions and
average them separately for the two blocks:
%
<<meuse-zinc-block-Kriging-5>>=
## index defining to which block the points predictions belong
ind <- over(geometry(meuse.grid), geometry(blks))
ind <- tapply(1:nrow(meuse.grid), factor(ind), function(x) x)
## select point predictions in block and predict block average
tmp <- t(sapply(ind, function(i, x){
  x$pred <- x$pred[i,]
  x$mse.pred <- x$mse.pred[i,i]
  x$var.pred <- x$var.pred[i,i]
  x$cov.pred.target <- x$cov.pred.target[i,i]
  x$var.target <- x$var.target[i,i]
  res <- lgnpp(x, is.block=TRUE)
  res
  }, x=t.pk))
colnames(tmp) <- c("opt.pred", "opt.se")
r.blks <- cbind( r.blks, tmp)
@
%
<<meuse-zinc-block-Kriging-6,eval=TRUE>>=
r.blks@data[, c("lgn.pred", "opt.pred", "lgn.se", "opt.se")]
@
%
Both the predictions and the prediction standard errors differ
somewhat for \code{block1}.  Based on the \citeauthor{Cressie-2006}'s
simulation study, we prefer the optimal prediction results.


<<meuse-zinc-cleanup-1,echo=FALSE,eval=TRUE>>=
palette("default")
@

<<meuse-zinc-cleanup-2,echo=FALSE>>=
# save(list=ls(pattern="^r\\."), file="r_meuse_zinc_objects.RData", version = 2)
save(list=c("r.sv", "r.sv.spher", "r.prfl.m0.spher.reml.scale", "r.cv.m0.spher.reml", 
    "r.cv.m1.spher.reml", "r.bk", "r.blks"), file="r_meuse_zinc_objects.RData", version = 2)
rm(list=ls(pattern="^r\\."))
@


\clearpage


%% 
 % Analysis of coalash data
 %%

\section{Robust analysis of \code{coalash} data}\label{sec:coalash}

<<ash-load,eval=TRUE,echo=FALSE>>=
if(file.exists("r_coalash_objects.RData")) load("r_coalash_objects.RData")
@
% 
This data set reports measurements of ash content [\% mass] in a
coal seam in Pennsylvania \citep{Gomez-Hazen-1970}.  A subset of the
data was analysed by \citet[section~2.2]{Cressie-1993a}
to illustrate techniques for robust exploratory analysis of
geostatistical data and for robust estimation of the sample variogram.

%%%%%%%%%%%%%%%%%%%%%%%%%
% exploratory analysis

\subsection{Exploratory analysis}\label{sec:coalash-eda}

The subset of the data analyzed by Cressie is available from the R package
\pkg{gstat} \citep{Pebesma-2004} as data frame \code{coalash}:
%
<<ash-data,eval=TRUE>>=
if(requireNamespace("gstat", quietly = TRUE)){
  data(coalash, package="gstat")
  summary(coalash)
} else {
  install.packages("gstat")
  requireNamespace("gstat", quietly = TRUE)
  data(coalash, package="gstat")
  summary(coalash)
}
# data(coalash, package="gstat")
# summary(coalash)
@
%
The coordinates are given as column and row numbers,  the spacing
between columns and rows is 2\,500~ft.
%
We display the spatial distribution of ash content by a ``bubble plot''
of the centred data:
%
\bcf[0.55]
%
<<ash-centred-bubbleplot1,fig=TRUE,height=6,width=4,eval=TRUE>>=
plot(y~x, coalash, cex=sqrt(abs(coalash - median(coalash))), 
  col=c("blue", NA, "red")[sign(coalash - median(coalash))+2], asp=1,
  main="coalash - median(coalash)", ylab="northing", xlab="easting")
points(y~x, coalash, subset=c(15, 50, 63, 73, 88, 111), pch=4); grid()
legend("topleft", pch=1, col=c("blue", "red"), legend=c("< 0", "> 0"), bty="n")
@
%
\ecf{``Bubble plot'' of coalash data centred by median (area of
symbols $\propto$ moduli of centred data; $\times$: observation identified by
\citeauthor{Cressie-1993a} as outlier).}{coalash-bubble-centred-data}
%
%
Ash content tends to decrease from west to east and is about
constant along the north-south direction:
%

\clearpage

\bcf
%
<<ash-coords-scatterplot,fig=TRUE,width=6,height=3,eval=TRUE>>=
op <- par(mfrow=c(1,2))
with(coalash, scatter.smooth(x, coalash, main="coalash ~ x"))
with(coalash, scatter.smooth(y, coalash, main="coalash ~ y"))
par(op)
@
%
\ecf{Coalash data plotted vs easting and
northing.}{coalash-trend}
%
%
There is a distinct outlier at position (5,6), other observations do
not clearly stand out from the marginal distribution:
%
\bcf[0.4]
%
<<ash-centerd-qqnorm,fig=TRUE,height=4,width=4,eval=TRUE>>=
qqnorm(coalash$coalash)
@
%
\ecf{Normal QQ plot of coalash data.}{coalash-normalqq}
%
%
\citeauthor{Cressie-1993a} identified the observations at locations
(7,3), (8,6), (6,8), (3,13) and (5,19) as \emph{local outliers}
(marked by $\times$ in Figure~\ref{fig:coalash-bubble-centred-data})
and the observations of row 2 as ``pocket of non-stationarity''.  One
could add to this list the observation at location (12,23), which is
clearly larger than the values at adjacent locations.

\bigskip

% exploratory model fit by lmrob

To further explore the data we fit a linear function of the
coordinates as drift to the data by a robust MM-estimator:
%
<<ash-lmrob,eval=TRUE>>=
library(robustbase)
r.lmrob <- lmrob(coalash~x+y, coalash)
summary(r.lmrob)
@
%
and check the fitted model by residual diagnostic plots:
%
\bcf
%
<<ash-diag-lmrob,fig=TRUE,width=6,height=5,eval=TRUE>>=
op <- par(mfrow=c(2,2))
plot(r.lmrob, which=c(1:2, 4:5))
par(op)
@
%
\ecf{Residual diagnostic plots for robust fit of the linear drift
model for coalash data.}{coalash-resdiag-lmrob}

Observations (5,6), (8,6) and (6,8) (Indices: 50, 111, 73) are
labelled as outliers.  Their
robustness weights,
%
\[
w_{i}= \frac{
\psi_{c}\big(\widehat{r}_{i}/\textrm{se}(\widehat{r}_{i})\big)}{\widehat{r}_{i}/\textrm{se}(\widehat{r}_{i})},
\]
%
---~$\widehat{r}_{i}$ are the regression residuals and
$\textrm{se}(\widehat{r}_{i})$ denote their standard errors~---~are
equal to $\approx$~0, 0.16 and 0.26, respectively.

\bigskip

% sample variogram of lmrob residuals

Next, we compute the isotropic sample variogram of the \code{lmrob}
residuals by various (robust) estimators \citep[e.g.][]{Lark-2000a}:
%
\bcf
%
<<ash-sv-res-lmrob-iso,fig=TRUE,width=6,height=4,results=hide,eval=TRUE>>=
library(georob)
plot(sample.variogram(residuals(r.lmrob), locations=coalash[, c("x","y")],
  lag.dist.def=1, max.lag=10, estimator="matheron"), pch=1, col="black",
  main="sample variogram of residuals coalash~x+y")
plot(sample.variogram(residuals(r.lmrob), locations=coalash[, c("x","y")],
  lag.dist.def=1, estimator="qn"), pch=2, col="blue", add=TRUE)
plot(sample.variogram(residuals(r.lmrob), locations=coalash[, c("x","y")],
  lag.dist.def=1, estimator="ch"), pch=3, col="cyan", add=TRUE)
plot(sample.variogram(residuals(r.lmrob), locations=coalash[, c("x","y")],
  lag.dist.def=1, estimator="mad"), pch=4, col="orange", add=TRUE)
legend("bottomright", pch=1:4, col=c("black", "blue", "cyan", "orange"),
  legend=paste(c("method-of-moments", "Qn", "Cressie-Hawkins", "MAD"), 
  "estimator"), bty="n")
@
%
\ecf{(Non-)robustly estimated isotropic sample variogram of robust
regression residuals of coalash data.}{coalash-isotropic-samplevariogram-lmrob}
%
%
Spatial dependence of the residuals is very weak, the outliers clearly
distort the method-of-moment estimate, but the various robust estimates
hardly differ.

\smallskip

To check whether drift removal accounted for directional effects we
compute the sample variogram separately for the N-S and W-E directions
(only Qn-estimator):
%
\bcf
%
<<ash-sv-res-lmrob-aniso,fig=TRUE,width=6,height=4,results=hide,eval=TRUE>>=
r.sv <- sample.variogram(residuals(r.lmrob), locations=coalash[, c("x","y")],
  lag.dist.def=1, max.lag=10, xy.angle.def=c(-0.1, 0.1, 89.9, 90.1),
  estimator="qn")
plot(gamma~lag.dist, r.sv, subset=lag.x < 1.e-6, xlim=c(0, 10), ylim=c(0, 1.4),
  pch=1, col="blue", 
  main="directional sample variogram of residuals (Qn-estimator)")
points(gamma~lag.dist, r.sv, subset=lag.y < 1.e-6, pch=3, col="orange")
legend("bottomright", pch=c(1, 3), col=c("blue", "orange"), 
  legend=c("N-S direction", "W-E direction"), bty="n")
@
%
\ecf{Direction-dependent sample variogram of robust regression
residuals of coalash data
($\mathrm{Q_{n}}$-estimate).}{coalash-anisotropic-samplevariogram-lmrob}
%
There is no indication that residual auto-correlation depends on
direction.

%%%%%%%%%%%%%%%%%%%%%%%%%
% fitting model by robust REML

\subsection{Fitting  a spatial linear model robust REML}\label{sec:coalash-modelfit}

Based on the results of the exploratory analysis, we fit a model with a
linear drift in the coordinates and an isotropic exponential variogram
to the data by robust REML. By default, \code{georob()} uses a
scaled and shifted ``logistic'' $\psi_{c}$-function 
\[ 
  \psi_{c}(x) = \tanh(x/c)
\]
with a tuning constant $c=2$ for robust REML, but the
popular Huber function and a redescending $\psi$-function based on the
$t$-distribution are also implemented (see section~\ref{sec:control-georob}):
%
<<ash-georob-robust-1,eval=TRUE>>=
r.georob.m0.exp.c2 <- georob(coalash~x+y, coalash, locations=~x+y, 
  variogram.model="RMexp", param=c(variance=0.1, nugget=0.9, scale=1))
@
%
<<ash-summary-georob-robust-1,eval=TRUE>>=
summary(r.georob.m0.exp.c2)
@
%
The diagnostics at the begin of the \code{summary} output report that
\code{nleqslv()} found the roots of the estimating equations of the
variogram parameters (reported numbers are function values evaluated
at the root).  The criteria controlling convergence of the
root-finding algorithm can be controlled by the arguments of
\code{control.nleqslv()}, see section~\ref{sec:control-georob}.
%

\smallskip

The drift coefficients confirm that there is no clear change of ash
content along the N-S direction.  A quadratic drift
function does not fit the data any better:
%
<<ash-waldtest-quadratic-robust,eval=TRUE>>=
waldtest(update(r.georob.m0.exp.c2, .~.+I(x^2)+I(y^2)+I(x*y)), r.georob.m0.exp.c2)
@
%
We simplify the drift model by dropping $y$:
%
<<ash-georob-robust-2,eval=TRUE>>=
r.georob.m1.exp.c2 <- update(r.georob.m0.exp.c2, .~.-y)
@
%
<<ash-summary-georob-robust-2,eval=TRUE>>=
r.georob.m1.exp.c2
@
%
and plot the robust REML estimate of the variogram along with a robust
estimate of the sample variogram of the robust REML regression residuals:
%
\bcf
%
<<ash-georob-robust-3,fig=TRUE,width=6,height=4,eval=TRUE>>=
plot(r.georob.m1.exp.c2, lag.dist.def=1, max.lag=10, estimator="qn", col="blue")
@
%
\ecf{Robust REML estimate of exponential
variogram).}{coalasl-robust-reml-variogram}
%
As already seen before, residual auto-correlation is weak.
%
% residual diagnostics

\smallskip

Next, we check the fit of the model by residual diagnostic plots:
%
\bcf[0.9]
%
<<ash-diag-georob-robust-2-1,fig=TRUE,width=6.64,height=6.64,eval=TRUE>>=
op <- par(mfrow=c(2,2))
plot(r.georob.m1.exp.c2, what="ta")
plot(r.georob.m1.exp.c2, what="sl")
plot(r.georob.m1.exp.c2, what="qq.res"); abline(0, 1, lty="dotted")
plot(r.georob.m1.exp.c2, what="qq.ranef"); abline(0, 1, lty="dotted")
par(op)
@
%
\ecf{Residual diagnostic plots for robust REML fit of coalash
data.}{coalash-resdiag-robust-reml}
%
Note that the \code{plot} method for class \code{georob} displays for
\code{what = "ta"} or \code{what = "sl"} the regression residuals $y_{i}
-\mb{x}(\mb{s}_{i}) \widehat{\mb{\beta}}$, for \code{what = "qq.res"}
the standardized errors $\widehat{\varepsilon}_{i} /
\widehat{\mathrm{se}}(\widehat{\varepsilon}_{i}) = \big(y_{i}
-\mb{x}(\mb{s}_{i}) \widehat{\mb{\beta}} - \widehat{B}(\mb{s}_{i})
\big) / \widehat{\mathrm{se}}(\widehat{\varepsilon}_{i})$ and for
\code{what = "qq.ranef"} the standardized random effects $
\widehat{B}(\mb{s}_{i})/\widehat{\mathrm{se}}(
\widehat{B}(\mb{s}_{i}))$.

\smallskip

The robustness weights of the outliers identified so far are equal to
%
<<ash-rw-georob-robust-2-1,eval=TRUE>>=
round(cbind(coalash[, c("x", "y")], 
  rweights=r.georob.m1.exp.c2[["rweights"]])[c(15, 50, 63, 73, 88, 111, 192),],
  2)
@
%
In addition, the observations
%
<<ash-rw-georob-robust-2-1,eval=TRUE>>=
sel <- r.georob.m1.exp.c2[["rweights"]] <= 0.8 & 
  !1:nrow(coalash) %in% c(15, 50, 63, 73, 88, 111, 192)
round(cbind(coalash[, c("x", "y")], 
  rweights=r.georob.m1.exp.c2[["rweights"]])[sel,],
  2)
@
%
have weights $\leq 0.8$.  All these outliers are marked in
Figure~\ref{fig:coalash-bubble-residuals-robust-reml} by $\times$:
%
\bcf[0.55]
%
<<ash-diag-georob-robust-2-2,fig=TRUE,height=6,width=4,eval=TRUE>>=
plot(y~x, coalash, cex=sqrt(abs(residuals(r.georob.m1.exp.c2))), 
  col=c("blue", NA, "red")[sign(residuals(r.georob.m1.exp.c2))+2], asp=1,
  main="estimated errors robust REML", xlab="northing", ylab="easting")
points(y~x, coalash, subset=r.georob.m1.exp.c2[["rweights"]]<=0.8, pch=4); grid()
legend("topleft", pch=1, col=c("blue", "red"), legend=c("< 0", "> 0"), bty="n")
@
%
\ecf{``Bubble plot'' of independent errors $\widehat{\varepsilon}$ estimated by
robust REML (area of symbols $\propto$ moduli of residuals; $\times$: observation
with robustness weights $w_{i} \leq 0.8$).}{coalash-bubble-residuals-robust-reml}
%
Comparison with Figure~\ref{fig:coalash-bubble-centred-data} reveals
that the 5 additional mild outliers were visible in this plot as well,
but were not identified by Cressie's exploratory analysis.

% comparison Gaussian vs robust REML fit

\smallskip

For comparison, we fit the same model by Gaussian REML by setting the
tuning constant of the $\psi_c$-function $c \geq 1000$:
%
<<ash-georob-gaussian-1,eval=TRUE>>=
r.georob.m1.exp.c1000 <- update(r.georob.m1.exp.c2, tuning.psi=1000)
@
<<ash-georob-summary-gaussian-1,eval=TRUE>>=
summary(r.georob.m1.exp.c1000)
@
% 
and compare the Gaussian and robust REML estimates of the variogram:
%
\bcf
%
<<ash-georob-gaussian-2,fig=TRUE,width=6,height=4,eval=TRUE>>=
plot(r.georob.m1.exp.c1000, lag.dist.def=1, max.lag=10, estimator="matheron")
plot(r.georob.m1.exp.c2, lag.dist.def=1, max.lag=10, estimator="qn", add = TRUE, 
  col="blue")
plot(update(r.georob.m1.exp.c2, subset=-50), lag.dist.def=1, max.lag=10, estimator="qn", 
  add = TRUE, col="orange")
legend("bottomright", lt=1, col=c("black","blue", "orange"), 
  legend =c("Gaussian REML", "robust REML", "Gaussian REML without outlier (5,6)" ), bty="n")
@
%
\ecf{Gaussian and robust REML estimate of exponential
variogram.}{coalash-gaussian-robust-reml-variogram}
%
% Note that by default for Gaussian REML, \code{georob()} maximizes a
% restricted profile loglikelihood that only depends on
% \eqn{\xi=\sigma^2/\sigma_Z^2}{}, \eqn{\eta = \sigma_Z^2 /
% \tau^2} and $\alpha$ with respect to these parameters, and the
% variance of the signal
% \eqn{\sigma_Z^2=\sigma_{\mathrm{n}}^2+\sigma^2}{} is estimated by an
% explicit expression that depends on these parameters
% \citep[e.g.][p.~113]{Diggle-Ribeiro-2007}, see help page of
% \code{georob()}.
%
The outliers inflate mostly the nugget effect (by $\approx$ 20~\%) and less
so the signal variance and range parameter (by $\approx$ 10~\%).  When
we eliminate the severest outlier at (5,6) then the Gaussian and robust REML
estimates of the variogram hardly differ. 

\smallskip

% \bcf[0.9]
% %
% <<ash-georob-diag-gaussian-robust-1,fig=TRUE,width=6.64,height=3.32,eval=TRUE>>=
% op <- par(mfrow=c(1,2), cex=5/6)
% plot(r.georob.m1.exp.c1000, what="qq.res"); abline(0, 1, lty="dotted")
% plot(r.georob.m1.exp.c1000, what="qq.ranef"); abline(0, 1, lty="dotted")
% @
% %
% \ecf{Normal QQ plots of standardized errors and random effects
% (Gaussian REML).}{coalash-resdiag-1-robust-reml}

Gaussian REML masks the estimated independent errors
($\widehat{\varepsilon}_{i}$) somewhat at the cost of inflated
estimates of random effects ($\widehat{B}(\mb{s}_{i})$):


\bcf[0.8]
%
<<ash-georob-diag-gaussian-robust-2,fig=TRUE,width=6.64,height=3.32,eval=TRUE>>=
op <- par(mfrow=c(1,2), cex=5/6)
plot(residuals(r.georob.m1.exp.c2), residuals(r.georob.m1.exp.c1000), 
  asp = 1, main=expression(paste("Gaussian vs robust ", widehat(epsilon))),
  xlab=expression(paste("robust ", widehat(epsilon))), 
  ylab=expression(paste("Gaussian ", widehat(epsilon))))
abline(0, 1, lty="dotted")
plot(ranef(r.georob.m1.exp.c2), ranef(r.georob.m1.exp.c1000), 
  asp = 1, main=expression(paste("Gaussian vs robust ", italic(widehat(B)))),
  xlab=expression(paste("robust ", italic(widehat(B)))), 
  ylab=expression(paste("Gaussian ", italic(widehat(B)))))
abline(0, 1, lty="dotted")
@
%
\ecf{Comparison of estimated errors $\widehat{\varepsilon}_{i}$ and
random effects $\widehat{B}(\mb{s}_{i})$ for Gaussian and robust REML
fit of coalash data.}{coalash-resdiag-2-robust-reml}

\smallskip

We compare the Gaussian and robust REML fit by 10-fold cross-validation:
%
\begin{Schunk}
\begin{Sinput}
> r.cv.georob.m1.exp.c2 <- cv(r.georob.m1.exp.c2, seed=1)
> r.cv.georob.m1.exp.c1000 <- cv(r.georob.m1.exp.c1000, seed=1)
\end{Sinput}
\begin{Soutput}
Warning message:
In cv.georob(r.georob.m1.exp.c2, seed = 1, ) :
  lack of covergence for  1 cross-validation sets
\end{Soutput}
\end{Schunk}
%
The robustfied estimating equations could not be solved for one
cross-valiation subset.  This may happen if the initial guesses of the
variogram parameters are too far away from the root (see
section~\ref{sec:est-details-initial-values}).  Sometimes it helps
then to suppress the computation of robust guesses of the variogram
parameters and to use the robust parameter estimates computed from the
whole data set as initial values (see argument \code{initial.param} of
\code{control.georob()} and
section~\ref{sec:est-details-initial-values}):

<<ash-cv-georob-gaussian-robust-1,eval=FALSE>>=
r.cv.georob.m1.exp.c2 <- cv(r.georob.m1.exp.c2, seed=1,
  control=control.georob(initial.param=FALSE))
r.cv.georob.m1.exp.c1000 <- cv(r.georob.m1.exp.c1000, seed=1)
@
% %
<<ash-cv-georob-gaussian-robust-2,echo=FALSE>>=
r.cv.georob.m1.exp.c2 <- cv(r.georob.m1.exp.c2, seed=1, 
  control=control.georob(initial.param=FALSE), ncores=my.ncores)
r.cv.georob.m1.exp.c1000 <- cv(r.georob.m1.exp.c1000, seed=1, ncores=my.ncores)
@
%
By default, \code{cv.georob()} partitions the data set into 10
geographically compact subsets of adjacent locations (see argument
\code{method} of \code{cv.georob()} and section~\ref{sec:cv}):
%
\bcf[0.55]
%
<<ash-cv-georob-subsets,fig=TRUE,height=6,width=4,eval=TRUE>>=
plot(y~x, r.cv.georob.m1.exp.c2$pred, asp=1, col=subset, pch=as.integer(subset))
@
%
\ecf{Default method for defining cross-validation subsets by
\code{kmeans()} using argument \code{method="block"} of
\code{cv.georob()}.}{coalash-cv-devault-partition}

\smallskip

We compute now summary statistics of the (standardized)
cross-validation prediction errors for the two model fits (see
section~\ref{sec:cv}):
%
<<ash-summary-cv-georob-gaussian-robust,eval=TRUE>>=
summary(r.cv.georob.m1.exp.c1000, se=TRUE)
summary(r.cv.georob.m1.exp.c2, se=TRUE)
@
%
The statistics of cross-validation errors are marginally better for
the robust fit.

\smallskip

For robust REML, the standard errors of the cross-validation errors
are likely too small.  This is due to the fact that for the time
being, the Kriging variance of the \emph{response} $Y$ is approximated
by adding the estimated nugget $\widehat{\tau}^2$ to the Kriging
variance of the signal $Z$.  This approximation likely underestimates
the mean squared prediction error of the response if the errors come
from a long-tailed distribution.  Hence, the summary statistics of the
\emph{standardized prediction errors} (\code{msse, medsse}) should
be interpreted with caution.  The same is true for the
continuous-ranked probability score (\code{crps}), which is computed
under the assumption that the predictive distribution of $Y$ is
Gaussian even if the errors come from a long-tailed distribution.
This cannot strictly hold.

\smallskip

For illustration, we nevertheless show here some diagnostics plots of
criteria of the cross-validation prediction errors that further depend
on the predictive distributions of the response:
%
\bcf[0.9]
%
<<ash-diag-cv-georob-gaussian-robust,fig=TRUE,width=6.64,height=9.96,eval=TRUE>>=
op <- par(mfrow=c(3, 2))
plot(r.cv.georob.m1.exp.c2, type="ta", col="blue")
plot(r.cv.georob.m1.exp.c1000, type="ta", col="orange", add=TRUE)
abline(h=0, lty="dotted")
legend("topleft", pch=1, col=c("orange", "blue"), legend=c("Gaussian", "robust"), bty="n")
plot(r.cv.georob.m1.exp.c2, type="qq", col="blue")
plot(r.cv.georob.m1.exp.c1000, type="qq", col="orange", add=TRUE)
abline(0, 1, lty="dotted")
legend("topleft", lty=1, col=c("orange", "blue"), legend=c("Gaussian", "robust"), bty="n")
plot(r.cv.georob.m1.exp.c2, type="ecdf.pit", col="blue", do.points=FALSE)
plot(r.cv.georob.m1.exp.c1000, type="ecdf.pit", col="orange", add=TRUE, do.points=FALSE)
abline(0, 1, lty="dotted")
legend("topleft", lty=1, col=c("orange", "blue"), legend=c("Gaussian", "robust"), bty="n")
plot(r.cv.georob.m1.exp.c2, type="bs", col="blue")
plot(r.cv.georob.m1.exp.c1000, type="bs", col="orange", add=TRUE)
legend("topright", lty=1, col=c("orange", "blue"), legend=c("Gaussian", "robust"), bty="n")
plot(r.cv.georob.m1.exp.c1000, type="mc", main="Gaussian REML")
plot(r.cv.georob.m1.exp.c2, type="mc", main="robust REML")
par(op)
@
%
\ecf{Diagnostic plots of the standardized cross-validation errors, the
probability integral transform (PIT), the Brier score and of the
predictive distributions for Gaussian and robust
REML.}{coalash-diag-cv-gaussian-robust}
%

In general, the diagnostics are slightly better for robust than for
Gaussian REML: The empirical distribution of the probability integral
transform (PIT) is closer to a uniform distribution, the Brier score
(BS) is smaller and the average predictive distribution $\bar{F}$ is
closer to the empirical distribution of the data $\widehat{G}$
\citep[see section~\ref{sec:cv} and][for more details about the
interpretation of these plots]{Gneiting-etal-2007}.


%%%%%%%%%%%%%%%%%%%%%%%%%
% Kriging coalash data

\subsection{Computing robust Kriging predictions}\label{sec:coalash-Kriging}

% point Kriging

\subsubsection{Point Kriging}\label{sec:coalash-point-Kriging}

For point Kriging  we must first generate a fine-meshed
grid of predictions points (optionally with the covariates for the
drift) that is passed as argument \code{newdata} to
\code{predict.georob()}.  \code{newdata} can be an customary
\code{data.frame}, an object of class \code{SpatialPointsDataFrame},
\code{SpatialPixelsDataFrame} or \code{SpatialGridDataFrame} or
\code{SpatialPoints}, \code{SpatialPixels} or \code{SpatialGrid}, all
provided by the package \pkg{sp}.  If \code{newdata} is a
\code{SpatialPoints}, \code{SpatialPixels} or a
\code{SpatialGrid} object then the drift model may only use the coordinates
as covariates (universal Kriging), as we do it here.
%
<<ash-Kriging-grid,eval=TRUE>>=
coalash.grid <- expand.grid(x=seq(-1, 17, by=0.2),
  y=seq( -1, 24, by=0.2))
coordinates( coalash.grid) <- ~x+y  # convert to SpatialPoints
gridded( coalash.grid) <- TRUE      # convert to SpatialPixels
fullgrid( coalash.grid) <- TRUE     # convert to SpatialGrid
str(coalash.grid, max=2)
@
%

Computing (robust) plug-in Kriging predictions is then straightforward:
%
<<ash-pKriging-1,eval=FALSE>>=
r.pk.m1.exp.c2 <- predict(r.georob.m1.exp.c2, newdata=coalash.grid)
r.pk.m1.exp.c1000 <- predict(r.georob.m1.exp.c1000, newdata=coalash.grid)
@
% %
<<ash-pKriging-2,echo=FALSE>>=
r.pk.m1.exp.c2 <- predict(r.georob.m1.exp.c2, newdata=coalash.grid, control=control.predict.georob(ncores=my.ncores))
r.pk.m1.exp.c1000 <- predict(r.georob.m1.exp.c1000, newdata=coalash.grid, control=control.predict.georob(ncores=my.ncores))
@
%
and plotting the results by the function \code{spplot()} of the
package \pkg{sp} as well:
% 
% \clearpage
% 
%
\bcf[1]
%
<<ash-pKriging-plot-robust-gaussian-1,fig=TRUE,width=9.96,height=9.96,eval=TRUE>>=
pred.rob <- spplot(r.pk.m1.exp.c2, "pred", at=seq(8, 12, by=0.25), 
  main="robust Kriging prediction", scales=list(draw=TRUE))
pred.gauss <- spplot(r.pk.m1.exp.c1000, "pred", at=seq(8, 12, by=0.25), 
  main="Gaussian Kriging prediction", scales=list(draw=TRUE))
se.rob <- spplot(r.pk.m1.exp.c2, "se", at=seq(0.35, 0.65, by=0.025), 
  main="standard error robust Kriging", scales=list(draw=TRUE))
se.gauss <- spplot(r.pk.m1.exp.c1000, "se", at=seq(0.35, 0.65, by=0.025), 
  main="standard error Gaussian Kriging", scales=list(draw=TRUE))
plot(pred.rob, pos=c(0, 0.5, 0.5, 1), more=TRUE)
plot(pred.gauss, pos=c(0.5, 0.5, 1, 1), more=TRUE)
plot(se.rob, pos=c(0, 0, 0.5, 0.5), more=TRUE)
plot(se.gauss, pos=c(0.5, 0, 1, 0.5), more=FALSE)
@
%
\ecf{Robust (left) and Gaussian (right) point Kriging predictions (top) and
Kriging standard errors (bottom).}{coalash-pKriging-robust-gaussian}
%
By default, \code{predict.georob()} predicts the \emph{signal}
$Z(\mb{s})$, but predictions of the \emph{response} $Y(\mb{s})$ or
estimates of \emph{model terms} and \emph{trend} (drift) can be
computed as well (see argument \code{type} of \code{predict.georob()}
and section~\ref{sec:prediction-targets}).  Apart from Kriging
predictions and standard errors, bounds (\code{lower}, \code{upper})
of point-wise 95\%-prediction intervals (plug-in) are computed for
assumed Gaussian predictive distributions.

\smallskip

The Kriging predictions visibly differ around the outlier at (5,6) and
the Kriging standard errors are larger for Gaussian Kriging because
the outliers inflated the non-robustly estimated sill of the variogram
(Figure~\ref{fig:coalash-gaussian-robust-reml-variogram}).
%
To see the effects of outliers more clearly, we plot the relative
difference of  Gaussian and robust Kriging predictions and the ratio
of the respective Kriging variances:
%
\bcf[1]
%
<<ash-pKriging-plot-robust-gaussian-2,results=hide,fig=TRUE,width=9.96,height=4.98,eval=TRUE>>=
library(lattice)
# rel. difference of predictions
r.pk.m1.exp.c2$reldiff.pred <- (r.pk.m1.exp.c1000$pred -
  r.pk.m1.exp.c2$pred) / r.pk.m1.exp.c2$pred * 100
reldiff.pred <- spplot(r.pk.m1.exp.c2, "reldiff.pred", at=-1:7, 
  main="Gaussian - robust Kriging predictions", scales=list(draw=TRUE))
# ratio Kriging variances
r.pk.m1.exp.c2$ratio.msep <- r.pk.m1.exp.c1000$se^2 / 
  r.pk.m1.exp.c2$se^2 * 100
ratio.msep <- spplot(r.pk.m1.exp.c2, "ratio.msep",  at=105:115, 
  main="ratio of Gaussian to robust Kriging variances",scales=list(draw=TRUE))
plot(reldiff.pred, pos=c(0, 0, 0.5, 1), more=TRUE)
#  add bubble plot of centred data colored by "robustness" weights
rw <- cut(r.georob.m1.exp.c2$rweights, seq(0.2, 1, by = 0.2))
trellis.focus("panel", 1, 1)
panel.points(coalash$x, coalash$y, lwd=2, 
  cex=sqrt(abs(coalash$coalash - median((coalash$coalash)))), 
  col=colorRampPalette(c("yellow", "orange", grey(0.4)))(4)[as.numeric(rw)])
panel.text(rep(17, nlevels(rw)+1), 0:nlevels(rw), pos=2, cex=0.8, 
  labels=c(rev(levels(rw)), "rob. weights"), 
  col=c(rev(colorRampPalette(c("yellow", "orange", grey(0.4)))(4)), "white"))
trellis.unfocus()

plot(ratio.msep, pos=c(0.5, 0, 1, 1), more=FALSE)
@
%
\ecf{Relative differences of Gaussian and robust point Kriging predictions
(\%, left) and ratio of Gaussian to robust point Kriging variances (\%,
right).  The area of the ``bubbles'' in the left panel is proportional
to the moduli of centred ash content and their color codes the
``robustness'' weights of robust
REML.}{coalash-diff-Kriging-gaussian-robusts}
%
The Kriging predictions differ by more than $\pm$ 1~\% around
observations with robustness weights $\leq 0.6$, and the efficiency of
the Gaussian relative to robust Kriging varies between 106--114~\%.

% block Kriging

\subsubsection{Block Kriging}\label{sec:coalash-block-Kriging}

First, we must define the blocks (and the covariates) for which
predictions should be computed.  The package \pkg{georob} computes
block Kriging predictions for \code{newdata} objects of class
\code{SpatialPolygonsDataFrame}s provided by \pkg{sp}:
%
\bcf[0.4]
<<ash-Kriging-polygons,fig=TRUE,height=6,width=4,eval=TRUE>>=
tmp <- expand.grid(x = seq(2.5, 16.5, by=4), y=seq(2, 22, by=4))
rownames(tmp) <- paste("block", rownames(tmp), sep="")
# create SpatialPolygonsDataFrame
coalash.polygons <- sapply(1:nrow(tmp), function(i, x){  
  Polygons(list(Polygon(
        t(x[,i] + t(cbind(c(-2, 2, 2, -2, -2), c(-2, -2, 2, 2, -2)))),
        hole=FALSE)), ID=paste("block", i, sep=""))},
  x=t(tmp))
coalash.polygons <- SpatialPolygonsDataFrame(SpatialPolygons(coalash.polygons), 
  data = tmp)
summary(coalash.polygons)
plot(coalash.polygons, col="grey", axes=TRUE); points(y~x, coalash)
@
%
\ecf{Geometry of blocks for which Kriging predictions are
computed (dots are data locations).}{coalash-block-geometry}
%
Note that coordinates of the  block centres are the covariates for the
drift model of the block means.
%
Block-block- and point-block-means of the semi-variance are computed by
\code{predict.georob()} efficiently by functions of the R packages
\pkg{constrainedKriging} and \pkg{spatialCovariance}
\citep[see section~\protect\ref{sec:predict-georob} and][]{Hofer-Papritz-2011,Clifford-2005}:
%
<<ash-bKriging-1>>=
r.bk.m1.exp.c2 <- predict(r.georob.m1.exp.c2, newdata=coalash.polygons,
  control=control.predict.georob(pwidth=4, pheight=4, full.covmat=TRUE))
r.bk.m1.exp.c1000 <- predict(r.georob.m1.exp.c1000, newdata=coalash.polygons,
  control=control.predict.georob(pwidth=4, pheight=4, full.covmat=TRUE))
@
%
Since the blocks are squares of constant size, choosing a square
``integration pixel'' of the same dimension (arguments \code{pwidth},
\code{pheight} of \code{control.predict.georob()}) allows to compute
the required mean semi-variances most efficiently \citep[see][for
details]{Hofer-Papritz-2011}.  The third argument
\code{full.covmat=TRUE} of \code{control.predict.georob()} has the
effect that the full covariance matrix of the predictions errors of
the block means is computed (and stored as list component
\code{mse.pred} in \code{r.bk.m1.exp.c2}):
%
<<ash-str-bKriging,eval=TRUE>>=
str(r.bk.m1.exp.c2, max=2)
@
%
We can compute from these covariances the Kriging variances of the
spatial means over groups of block, in particular of the spatial mean
over the whole domain:
%
<<ash-gKriging,eval=TRUE>>=
c(pred=mean(r.bk.m1.exp.c2$pred$pred),
  se=sqrt(sum(r.bk.m1.exp.c2$mse.pred))/24)
@
%
which is of course the same as predicting the mean over the whole
domain by block Kriging:
%
<<ash-bKriging-2,eval=TRUE>>=
coalash.domain <- rbind(c(0.5,0), c(16.5,0), c(16.5,24), c(0.5,24), c(0.5,0))
coalash.domain <- SpatialPolygonsDataFrame(
  SpatialPolygons(list(Polygons(list(Polygon(coalash.domain)), ID= "domain"))), 
  data=data.frame(x=8.5,y=12,row.names="domain"))
slot(predict(r.georob.m1.exp.c2, newdata=coalash.domain,
  control=control.predict.georob(pwidth=16, pheight=24)), "data")
@

We conclude this section by graphs of the robust and Gaussian block
Kriging predictions:
%
\bcf[1]
%
<<ash-bKriging-plot-robust-gaussian-1,fig=TRUE,width=9.96,height=9.96,eval=TRUE>>=
pred.rob <- spplot(r.bk.m1.exp.c2$pred, "pred", at=seq(8, 11, by=0.25), 
  main="robust Kriging prediction", scales=list(draw=TRUE))
pred.gauss <- spplot(r.bk.m1.exp.c1000$pred, "pred", at=seq(8, 11, by=0.25), 
  main="Gaussian Kriging prediction", scales=list(draw=TRUE))
se.rob <- spplot(r.bk.m1.exp.c2$pred, "se", at=seq(0.15, 0.45, by=0.025), 
  main="standard error robust Kriging", scales=list(draw=TRUE))
se.gauss <- spplot(r.bk.m1.exp.c1000$pred, "se", at=seq(0.15, 0.45, by=0.025), 
  main="standard error Gaussian Kriging", scales=list(draw=TRUE))
plot(pred.rob, pos=c(0, 0.5, 0.5, 1), more=TRUE)
plot(pred.gauss, pos=c(0.5, 0.5, 1, 1), more=TRUE)
plot(se.rob, pos=c(0, 0, 0.5, 0.5), more=TRUE)
plot(se.gauss, pos=c(0.5, 0, 1, 0.5), more=FALSE)
@
%
\ecf{Robust (left) and Gaussian (right) block Kriging predictions (top) and
Kriging standard errors (bottom).}{coalash-Kriging-robust-gaussian}
%
The outlier at (5,6) also affects the the prediction of the block
means and robust block Kriging is again more efficient than Gaussian
block Kriging.

<<ash-results-save-1,echo=FALSE>>=
# save(list=ls(pattern="^r\\."), file="r_coalash_objects.RData", version = 2)
save(list=c(
    "r.cv.georob.m1.exp.c2", "r.cv.georob.m1.exp.c1000", 
    "r.pk.m1.exp.c2", "r.pk.m1.exp.c1000", "r.bk.m1.exp.c2", "r.bk.m1.exp.c1000"
  ), file="r_coalash_objects.RData", version = 2)
rm(list=ls(pattern="^r\\."))
@

\clearpage


% \input

%% 
 % Details about parameter estimation
 %%

% material taken from
% tools::Rd2latex("georob.Rd", "../../../vignettes/georob.tex", outputEncoding="utf-8")
% tools::Rd2latex("control.georob.Rd", "../../../vignettes/control.georob.tex", outputEncoding="utf-8")

\section{Details about parameter estimation}\label{sec:est-details}

\subsection{Implemented variogram models}\label{sec:est-details-models}

Currently, estimation of the parameters of the following models is
implemented (see argument \code{variogram.model} of \code{georob()}):

\code{"RMaskey"}, \code{"RMbessel"}, \code{"RMcauchy"},
\code{"RMcircular"}, \code{"RMcubic"}, \code{"RMdagum"}, \\{}
\code{"RMdampedcos"}, \code{"RMdewijsian"}, \code{"RMexp"} (default),
\code{"RMfbm"}, \code{"RMgauss"}, \\{} \code{"RMgencauchy"},
\code{"RMgenfbm"}, \code{"RMgengneiting"}, \code{"RMgneiting"},
\code{"RMlgd"}, \\{} \code{"RMmatern"}, \code{"RMpenta"}, \code{"RMqexp"},
\code{"RMspheric"}, \code{"RMstable"}, \code{"RMwave"}, \\{}
\code{"RMwhittle"}. 

\smallskip

Some of these models have in addition to \code{variance},
\code{snugget}, \code{nugget} and \code{scale} further parameters.
Initial values of these parameters (\code{param}) and fitting flags
(\code{fit.param}) must be passed to \code{georob()} by the same names
as used by the functions \code{RM...()} of the package
\pkg{RandomFields} (see \code{RMmodel())}.  Use the
function \code{param.names()} to list additional parameters of a
given variogram.model.

\smallskip

The arguments \code{fit.param} and \code{fit.aniso} are used to
control what variogram and anisotropy parameters are estimated and
which are kept at the constant initial values.  The functions
\code{default.fit.param()} and \code{default.fit.aniso()} set
reasonable default values for these arguments.  Note, as an aside,
that the function \code{default.aniso()} sets (default) values of the
anisotropy parameters for an isotropic variogram.


\subsection{Estimating parameters of power function variogram}\label{sec:est-details-intrisic-models}
 
The intrinsic variogram model \code{RMfbm} is over-parametrized when
both the \code{variance} (plus possibly \code{snugget}) and the
\code{scale} are estimated.  Therefore, to estimate the parameters of this
model, \code{scale} must be kept fixed at an arbitrary value by using
\code{fit.param = default.fit.param(scale = FALSE)}.


\subsection{Estimating parameters of geometrically anisotropic variograms}\label{sec:est-details-anisotropy}

Subsection~\ref{sec:intro-model} describes how covariances are
modelled in general.  Here we describe in detail how geometrically
anisotropic variogram are parametrized:

\smallskip

The matrix \eqn{\mbox{\boldmath$A$\unboldmath} =
\mbox{\boldmath$A$\unboldmath}(\alpha, f_1, f_2; \omega, \phi, \zeta)
}{} (see equation \ref{eq:valpha}) maps an arbitrary point on an
ellipsoidal surface with constant (generalized) covariance in \eqn{
\mathrm{I}\!\mathrm{R}^3}{}, centred on the origin, and having lengths
of semi-principal axes, \eqn{\mbox{\boldmath$p$\unboldmath}_1}{},
\eqn{\mbox{\boldmath$p$\unboldmath}_2}{},
\eqn{\mbox{\boldmath$p$\unboldmath}_3}{}, equal to
\eqn{|\mbox{\boldmath$p$\unboldmath}_1|=\alpha}{},
\eqn{|\mbox{\boldmath$p$\unboldmath}_2|=f_1\,\alpha}{} and
\eqn{|\mbox{\boldmath$p$\unboldmath}_3|=f_2\,\alpha}{}, \eqn{0 < f_2
\leq f_1 \leq 1}{}, respectively, onto the surface of the unit ball
centred on the origin.
%
The orientation of the ellipsoid is defined by the three angles
\eqn{\omega}{}, \eqn{\phi}{} and \eqn{\zeta}{}: 
\begin{description}
  
  \item[\eqn{\omega}{}] is the azimuth of \eqn{\mbox{\boldmath$p$\unboldmath}_1}{}
  (= angle between north and the projection  
  of
  \eqn{\mbox{\boldmath$p$\unboldmath}_1}{}
  onto the \eqn{x}{}-\eqn{y}{}-plane, 
  measured from north to south positive clockwise in degrees),
  
  
  \item[\eqn{\phi}{}] is 90 degrees minus the altitude of 
  \eqn{\mbox{\boldmath$p$\unboldmath}_1}{}
  (= angle between the zenith and
  \eqn{\mbox{\boldmath$p$\unboldmath}_1}{},
  measured from zenith to nadir positive clockwise in degrees), and
  
  
  \item[\eqn{\zeta}{}] is the angle between 
  \eqn{\mbox{\boldmath$p$\unboldmath}_2}{}
  and the direction of the line, say \eqn{y^\prime}{}, 
  defined by the intersection between the
  \eqn{x}{}-\eqn{y}{}-plane and the plane orthogonal to 
  \eqn{\mbox{\boldmath$p$\unboldmath}_1}{} running through the origin
  (\eqn{\zeta}{} is measured from \eqn{y^\prime}{} positive counter-clockwise in degrees).
  
\end{description}


The transformation matrix is given by
%
\begin{equation}\label{eq:coordinate-transformation-matrix}
  \mbox{\boldmath$A$\unboldmath}=
  \left(\begin{array}{ccc} 1/\alpha & 0 & 0\\
  0 & 1/(f_1\,\alpha) & 0\\
  0 & 0 & 1/(f_2\,\alpha) \\
  \end{array}\right) ( \mbox{\boldmath$C$\unboldmath}_1,
  \mbox{\boldmath$C$\unboldmath}_2, \mbox{\boldmath$C$\unboldmath}_3)
\end{equation}
%
where
%
\begin{eqnarray}\label{eq:coordinate-rotation-matrix}
  \mbox{\boldmath$C$\unboldmath}_1^\mathrm{T} & = & ( \sin\omega \sin\phi, -\cos\omega \cos\zeta - \sin\omega \cos\phi \sin\zeta, \cos\omega \sin\zeta - \sin\omega \cos\phi \cos\zeta ) \nonumber \\
  \mbox{\boldmath$C$\unboldmath}_2^\mathrm{T} & = & ( \cos\omega \sin\phi, \sin\omega \cos\zeta - \cos\omega \cos\phi \sin\zeta, -\sin\omega \sin\zeta - \cos\omega \cos\phi\cos\zeta )\nonumber \\
  \mbox{\boldmath$C$\unboldmath}_3^\mathrm{T} & = & (\cos\phi, \sin\phi \sin\zeta, \sin\phi \cos\zeta )
\end{eqnarray}
%
To model geometrically anisotropic variograms in 
\eqn{ \mathrm{I}\!\mathrm{R}^2}{} 
one has to set \eqn{\phi=90}{} and \eqn{f_2 = 1}{},
and for \eqn{f_1 = f_2 = 1}{} 
one obtains the model for isotropic auto-correlation 
with range parameter \eqn{\alpha}{}. 
Note that for isotropic auto-correlation the software processes data for 
which \eqn{d}{} may exceed 3.


\smallskip

Some additional remarks might be helpful:

\begin{itemize}
  
  \item The first semi-principal axis points into the direction with 
  the farthest reaching auto-correlation, which is described by the range
  parameter \code{scale} (\eqn{\alpha}{}).
  
  \item The ranges in the direction of the second and third
  semi-principal axes are given by \eqn{f_1\alpha}{} and \eqn{f_2
  \alpha}{}, with \eqn{0 < f_2 \leq f_1 \leq 1}{}.
  
  \item The default values for \code{aniso} (\eqn{f_1=1}{}, \eqn{f_2=1}{})
  define an isotropic variogram model.
  
  \item Valid ranges for the angles characterizing the orientation of
  the semi-variance ellipsoid are (in degrees): \eqn{\omega}{} [0, 180],
  \eqn{\phi}{} [0, 180], \eqn{\zeta}{} [-90, 90].  
  
\end{itemize}


\subsection{Estimating variance of micro-scale variation}\label{sec:est-details-microscale}

Simultaneous estimation of the variance of the micro-scale variation
(\code{snugget}, \eqn{\sigma_\mathrm{n}^2}{}), which appears seemingly
as spatially uncorrelated with a given sampling design, and of the
variance (\code{nugget}, \eqn{\tau^2}{}) of the independent errors
requires that for some locations
\eqn{\mbox{\boldmath$s$\unboldmath}_i}{} replicated observations are
available.  Locations less or equal than \code{zero.dist} apart are
thereby considered as being coincident (see
\code{control.georob()}).

%
\subsection{Estimating variance parameters by Gaussian (RE)ML}\label{sec:est-details-gaussian-reml}

Unlike robust REML, where robustified estimating equations are solved
for the variance parameters \code{nugget} (\eqn{\tau^2}{}),
\code{variance} (\eqn{\sigma^2}{}), and possibly \code{snugget}
(\eqn{\sigma_{\mathrm{n}}^2}{}), for Gaussian (RE)ML the
variances can be re-parametrized to

\begin{itemize}

\item the signal variance 
\eqn{\sigma_Z^2 = \sigma^2 + \sigma_{\mathrm{n}}^2}{},

\item the inverse relative nugget 
\eqn{\eta = \sigma_Z^2 / \tau^2}{} and

\item the relative auto-correlated signal variance
\eqn{\xi = \sigma^2/\sigma_Z^2}{}.

\end{itemize}


\code{georob()} maximizes then a (restricted) \emph{profile
log-likelihood} that depends only on \eqn{\eta}{}, \eqn{\xi}{},
\eqn{\alpha}{}, \ldots, and \eqn{\sigma_Z^2}{} is estimated by an
explicit expression that depends on these parameters
\citep[e.g.][p.~113]{Diggle-Ribeiro-2007}.  This is usually more
efficient than maximizing the (restricted) log-likelihood with respect
to the original variance parameters \eqn{\tau^2}{},
\eqn{\sigma_{\mathrm{n}}^2}{} and \eqn{\sigma^2}{}.  \code{georob()}
chooses the parametrization automatically, but the user can control it
by the argument \code{reparam} of the function
\code{control.georob()}.



\subsection{Constraining estimates of variogram parameters}\label{sec:est-details-constraining-parameters}

Parameters of variogram models can vary only within certain bounds
(see \code{param.bounds()} and \code{RMmodel()} for allowed ranges).
\code{georob()} uses three mechanisms to constrain parameter estimates
to permissible ranges:

\begin{enumerate}
  
  \item \emph{Parameter transformations}: By default, all variance
  (\code{variance}, \code{snugget}, \code{nugget}), the range
  \code{scale} and the anisotropy parameters \code{f1} and \code{f2}
  are log-transformed before solving the estimating equations or
  maximizing the restricted log-likelihood and this warrants that the
  estimates are always positive (see \code{control.georob()} and
  section~\ref{sec:parameter-transformations} for detailed
  explanations how to control parameter transformations).
  
  \item \emph{Checking permissible ranges}: The additional parameters
  of the variogram models such as the smoothness parameter \eqn{\nu}{}
  of the Whittle-Mat\'ern model are forced to stay in the permissible
  ranges by signalling an error to \code{nleqslv()}, \code{nlminb()} or
  \code{optim()} if the current trial values are invalid.  These
  functions then graciously update the trial values of the parameters
  and carry their task on.  However, it is clear that such a procedure
  likely gets stuck at a point on the boundary of the parameter space
  and is therefore just a workaround for avoiding runtime errors due
  to invalid parameter values.
  
  \item \emph{Exploiting the functionality of
  {\upshape\code{nlminb()}} and {\upshape\code{optim()}}}: If a
  spatial model is fitted non-robustly, then the arguments
  \code{lower}, \code{upper} (and \code{method} of \code{optim()}) can
  be used to constrain the parameters (see \code{control.optim()} how
  to pass them to \code{optim()}).  For \code{optim()} one has to use
  the arguments \code{method = "L-BFGS-B"}, \code{lower = \var{l}},
  \code{upper = \var{u}}, where \var{l} and \var{u} are numeric
  vectors with the lower and upper bounds of the \emph{transformed}
  parameters in the order as they appear in\\{} \code{c(variance,
  snugget, nugget, scale, ...)[fit.param], aniso[fit.aniso])},\\{}
  where \code{...} are additional parameters of isotropic variogram
  models (use \\{} \code{param.names(variogram.model)} to display the
  names and the order of the additional parameters for
  \code{variogram.model}).
  
  
\end{enumerate}


%
\subsection{Computing robust initial estimates of parameters for robust REML}\label{sec:est-details-initial-values}

To solve the robustified estimating equations for
\eqn{\mbox{\boldmath$B$\unboldmath}}{} and
\eqn{\mbox{\boldmath$\beta$\unboldmath}}{} the following initial
estimates are used:

\begin{itemize}

  \item \eqn{ \widehat{\mbox{\boldmath$B$\unboldmath}}=
  \mbox{\boldmath$0$\unboldmath},}{} if this turns out to be
  infeasible, initial values can be passed to \code{georob()} by the
  argument \code{bhat} of \code{control.georob()}.
  
  \item \eqn{\widehat{\mbox{\boldmath$\beta$\unboldmath}}}{} is either
  estimated robustly by the function \code{lmrob()}, \code{rq()} or
  non-robustly by \code{lm()} (see argument \code{initial.fixef} of
  \code{control.georob()}).
  

\end{itemize}


Finding the roots of the robustified estimating equations of the
variogram and anisotropy parameters is more sensitive to a good choice
of initial values than maximizing the Gaussian (restricted)
log-likelihood with respect to the same parameters.  If the initial
values for \code{param} and \code{aniso} are not sufficiently close to
the roots of the system of nonlinear equations, then \code{nleqslv()}
may fail to find them.  Setting \code{initial.param = TRUE} (see
\code{control.georob()}) allows one to find initial values that are
often sufficiently close to the roots so that \code{nleqslv()}
converges.  This is achieved by:

\begin{enumerate}
  
  \item Initial values of the regression parameters are computed by
  \code{lmrob()} irrespective of the choice for \code{initial.fixef}
  (see \code{control.georob()}).
  
  \item Observations with ``robustness weights'' of the \code{lmrob}
  fit, satisfying\\{}
  \eqn{\psi_c(\widehat{\varepsilon}_i/\widehat{\tau})/(\widehat{\varepsilon}_i/\widehat{\tau})
  \leq \mbox{\code{min.rweight}}}{}, are discarded (see
  \code{control.georob()}).
  
  \item The model is fit to the pruned data set by Gaussian REML using 
  \code{optim()} or \code{nlminb()}.
  
  \item The resulting estimates of the variogram parameters
  (\code{param}, \code{aniso}) are used as initial estimates for the
  subsequent robust fit of the model by \code{nleqslv()}.
  
  
\end{enumerate}


Note that for step 3 above, initial values of \code{param} (and
possibly \code{aniso}) must be provided to \code{georob()}.



%
% \subsection{Approximation of covariances of fixed and random effects and residuals}\label{sec:est-details-covariances}
% 
% The robustified estimating equations of robust REML depend on the
% covariances of \eqn{\widehat{\mbox{\boldmath$B$\unboldmath}}}{}.
% These covariances (and the covariances of
% \eqn{\mbox{\boldmath$B$\unboldmath}-\widehat{\mbox{\boldmath$B$\unboldmath}}}{},
% \eqn{\widehat{\mbox{\boldmath$\beta$\unboldmath}}}{},
% \eqn{\widehat{\mbox{\boldmath$\varepsilon$\unboldmath}}}{},
% \eqn{\widehat{\mbox{\boldmath$\varepsilon$\unboldmath}} +
% \widehat{\mbox{\boldmath$B$\unboldmath}}}{}) are approximated by
% expressions that in turn depend on the variances of
% \eqn{\varepsilon}{}, \eqn{\psi_c(\varepsilon/\tau)}{} and the
% expectation of \eqn{\psi_c'(\varepsilon/\tau)\ (= \partial / \partial
% \varepsilon\, \psi_c(\varepsilon/\tau))}{}.  The arguments
% \code{error.family.estimation}, \code{error.family.cov.effects} and
% \code{error.family.cov.residuals} of \code{control.georob()} control
% what parametric distribution for \eqn{\varepsilon}{} is used to
% compute the variances of \eqn{\varepsilon}{},
% \eqn{\psi_c(\varepsilon/\tau)}{} and the expectation of
% \eqn{\psi_c'(\varepsilon/\tau)}{} when
% 
% \begin{itemize}
%   
%   \item solving the estimating equations (\code{error.family.estimation}),
%   
%   \item computing the covariances of
%   \eqn{\widehat{\mbox{\boldmath$\beta$\unboldmath}}}{},
%   \eqn{\widehat{\mbox{\boldmath$B$\unboldmath}}}{} and
%   \eqn{\mbox{\boldmath$B$\unboldmath}-\widehat{\mbox{\boldmath$B$\unboldmath}}}{}
%   (\code{error.family.cov.effects}) and
%   
%   \item computing the covariances of
%   \eqn{\widehat{\mbox{\boldmath$\varepsilon$\unboldmath}}=\mbox{\boldmath$Y$\unboldmath}
%   - \mbox{\boldmath$X$\unboldmath}
%   \widehat{\mbox{\boldmath$\beta$\unboldmath}} -
%   \widehat{\mbox{\boldmath$B$\unboldmath}}}{} and \eqn{\widehat{\mbox{\boldmath$\varepsilon$\unboldmath}} +
%   \widehat{\mbox{\boldmath$B$\unboldmath}}
%   =\mbox{\boldmath$Y$\unboldmath} - \mbox{\boldmath$X$\unboldmath}
%   \widehat{\mbox{\boldmath$\beta$\unboldmath}}}{} (\code{error.family.cov.residuals}).
%   
% \end{itemize}
% 
% Possible options are: \code{"gaussian"} or \code{"long.tailed"}.  In
% the latter case, the PDF of \eqn{\varepsilon}{} is assumed to be
% proportional to \eqn{1/\tau \, \exp(-\rho_c(\varepsilon/\tau))}{}, where
% \eqn{\psi_c(x)=\rho_c'(x)}{}.


\subsection{Estimating parameters of ``nested'' variogram models}\label{sec:nested-variograms}

As a further option, \code{georob()} allows to estimate parameters of
so-called ``nested'' variogram models.  For this one assumes that the
signal process $Z(\mb{s})$ is the sum of several independent
auto-correlated Gaussian processes $B_{k}(\mb{s})$
%
\begin{equation}
  B(\mb{s}) = \sum_{k=1}^{K} B_{k}(\mb{s}), 
\end{equation}
%
each characterized by a parametric variogram function with parameters
($\sigma^2_{Z,k}$, $\mb{\theta}_{k}$), see equation
\ref{eq:covariance-matrix-signal}.
%
Initial values for nested variogram models are passed to
\code{georob()} by the argument \code{variogram.object}, which must be
a list of length $K$.  The $k$th component of \code{variogram.object}
is itself a list with the mandatory components \code{variogram.model}
and \code{param} and the optional components \code{fit.param},
\code{aniso} and \code{fit.aniso}.  Note that sensible defaults are
used if the optional components are missing.  Note further that
\code{nugget} and \code{snugget} may be specified for all $k$ model
strctures but these variances are summed-up and assigned to the first
model structure ($k=1$) and all \code{nugget} and \code{snugget} are
set to zero for $k > 1$.


\subsection{Controlling \code{georob()} by the function \code{control.georob()}}\label{sec:control-georob}

% tools::Rd2latex("control.georob.Rd", "../../../vignettes/control.georob.tex", outputEncoding="utf-8")
% tools::Rd2latex("pmm.Rd", "../../../vignettes/pmm.tex", outputEncoding="utf-8")

All control and tuning parameters except the robustness tuning
constant $c$ of the $\psi_{c}$-function (argument \code{tuning.psi} of
\code{georob()}) are set by the arguments of the function
\code{control.georob()}, which generates a list that is passed by the
\code{control} argument to \code{georob()}.
% 
This section describes in some detail how to control \code{georob()}
by the various arguments of \code{control.georob()}.


\subsubsection{Gaussian (RE)ML estimation}\label{sec:gaussian-reml}

Gaussian (RE)ML estimates are computed provided \code{tuning.psi}
$\geq 1000$.  
%
Use the argument \code{ml.method} to select either \code{"ML"} or
\code{"REML"} (default) and the argument \code{reparam} to
control whether the re-parametrized (default  \code{TRUE}) or the original
variogram parameters are estimated (see section~\ref{sec:est-details-gaussian-reml}).
%
The function used to maximize the (restricted) log-likelihood is
chosen by the argument \code{maximizer} (default \code{"nlminb"}).
%
Use the argument \code{nlminb} along with the function
\code{control.nlminb()} (or the argument \code{optim} along with
\code{control.optim()} to pass arguments to \code{nlminb()}
(or \code{optim()}), in particular the argument \code{rel.tol}, which
controls convergence.
%
The argument \code{hessian} controls whether confidence intervals of
variogram parameters are computed from the observed Fisher information
based on the asymptotic normal distribution of (RE)ML estimates
(default \code{TRUE}, see \code{summary.georob()}).


\subsubsection{Robust REML estimation}\label{sec:robust-reml}

The argument \code{psi.func} selects the $\psi_{c}$-function for robust
REML. Apart from a shifted and scaled logistic CDF
%
\[ 
  \psi_{c}(x) = \tanh(x/c)
\]
%
(\code{"logistic"}, default), the Huber (\code{"huber"}) or a
re-descending $\psi_c$-function based on the $t$-distribution
(\code{"t.dist"})
%
\[ 
  \psi_{c}(x) = \frac{c^2\,x}{c^2+x^2}
\]
can be used.

The argument \code{initial.fixef} chooses the function for computing
(robust) initial estimates of \mb{\beta}.  Possible choices are
\code{"lmrob"} (default) \code{"rq"} and \code{"lm"}.  
%
Use the argument \code{lmrob} along with the function
\code{lmrob.control()} (or the argument \code{rq} along with the
function \code{control.rq()} to pass arguments to \code{lmrob()} (or
\code{rq()}).
%
The argument \code{initial.param} controls whether robust initial
estimates of the variogram parameters are computed (default
\code{TRUE}, see section~\ref{sec:est-details-initial-values}).

For given variogram parameters, the estimating equations for
\mb{\beta} and \mb{B} are solved by iterated re-weighted least squares
(IRWLS).  The argument \code{irwls.maxiter} sets the maximum number of
iterations (default 50), and the argument \code{irwls.ftol} controls
convergence: Convergence is assumed if the largest absolute function
value at the current root is less than \code{irwls.ftol} (default
$10^{-5}$).
%
The current estimates $\widehat{\mb{\beta}}$ and $\widehat{\mb{B}}$
are then plugged into the estimating equations for \mb{\theta} and
$\tau^2$ that are solved in turn by \code{nleqslv()}.  Use the
argument \code{nleqslv} along with the function
\code{control.nleqslv()} to pass arguments to \code{nleqslv()}, in
particular \code{ftol}, which controls convergence for root finding.


\subsubsection{Approximation of covariances of fixed and random effects and residuals}\label{sec-covariances-fixed-random-efects}

The robustified estimating equations of robust REML depend on the
covariances of \eqn{\widehat{\mbox{\boldmath$B$\unboldmath}}}{}.
These covariances (and the covariances of
\eqn{\mbox{\boldmath$B$\unboldmath}-\widehat{\mbox{\boldmath$B$\unboldmath}}}{},
\eqn{\widehat{\mbox{\boldmath$\beta$\unboldmath}}}{},
\eqn{\widehat{\mbox{\boldmath$\varepsilon$\unboldmath}}}{},
\eqn{\widehat{\mbox{\boldmath$\varepsilon$\unboldmath}} +
\widehat{\mbox{\boldmath$B$\unboldmath}}}{}) are approximated by
expressions that in turn depend on the variances of
\eqn{\varepsilon}{}, \eqn{\psi_c(\varepsilon/\tau)}{} and the
expectation of \eqn{\psi_c'(\varepsilon/\tau)\ (= \partial / \partial
\varepsilon\, \psi_c(\varepsilon/\tau))}{}.  The arguments
\code{error.family.estimation}, \code{error.family.cov.effects} and
\code{error.family.cov.residuals} of \code{control.georob()} control
what parametric distribution for \eqn{\varepsilon}{} is used to
compute the variances of \eqn{\varepsilon}{},
\eqn{\psi_c(\varepsilon/\tau)}{} and the expectation of
\eqn{\psi_c'(\varepsilon/\tau)}{} when

\begin{itemize}
  
  \item solving the estimating equations (\code{error.family.estimation}),
  
  \item computing the covariances of
  \eqn{\widehat{\mbox{\boldmath$\beta$\unboldmath}}}{},
  \eqn{\widehat{\mbox{\boldmath$B$\unboldmath}}}{} and
  \eqn{\mbox{\boldmath$B$\unboldmath}-\widehat{\mbox{\boldmath$B$\unboldmath}}}{}
  (\code{error.family.cov.effects}) and
  
  \item computing the covariances of
  \eqn{\widehat{\mbox{\boldmath$\varepsilon$\unboldmath}}=\mbox{\boldmath$Y$\unboldmath}
  - \mbox{\boldmath$X$\unboldmath}
  \widehat{\mbox{\boldmath$\beta$\unboldmath}} -
  \widehat{\mbox{\boldmath$B$\unboldmath}}}{} and \eqn{\widehat{\mbox{\boldmath$\varepsilon$\unboldmath}} +
  \widehat{\mbox{\boldmath$B$\unboldmath}}
  =\mbox{\boldmath$Y$\unboldmath} - \mbox{\boldmath$X$\unboldmath}
  \widehat{\mbox{\boldmath$\beta$\unboldmath}}}{} (\code{error.family.cov.residuals}).
  
\end{itemize}

Possible options are: \code{"gaussian"} or \code{"long.tailed"}.  In
the latter case, the PDF of \eqn{\varepsilon}{} is assumed to be
proportional to \eqn{1/\tau \, \exp(-\rho_c(\varepsilon/\tau))}{}, where
\eqn{\psi_c(x)=\rho_{c}'(x)}{}.

\smallskip

% \subsubsection{Controlling computation of (co-)variances}

The logical arguments \code{cov.{\itshape xxx}} and
\code{full.cov.{\itshape xxx}} of \code{control.georob()} control what
(co-)variances should be computed by \code{georob()}.  \code{full.cov.{\itshape xxx}}
controls whether the full covariance matrix (\texttt{TRUE}) or only the
variances (\texttt{FALSE}) are computed.
%
\begin{tabbing}
  \code{delta.bhat.betahat} \quad \= (co-)variances of \quad \= default \texttt{cov.\itshape xxx}\quad \= default  \texttt{full.cov.\itshape xxx}\kill
  \texttt{\itshape xxx} \> (co-)variances of \>  default \texttt{cov.\itshape xxx} \> default  \texttt{full.cov.\itshape xxx} \\
  \texttt{bhat} \> $\widehat{\mb{B}}$ \>  \texttt{TRUE} \> \texttt{FALSE} \\
  \texttt{betahat} \> $\widehat{\mb{\beta}}$ \> \texttt{TRUE} \> -- \\
  \texttt{delta.bhat} \> $\mb{B} - \widehat{\mb{B}}$ \> \texttt{TRUE} \> \texttt{TRUE} \\
  \texttt{delta.bhat.betahat} \> $\mb{B} - \widehat{\mb{B}}$ and $\widehat{\mb{\beta}}$ \> \texttt{TRUE} \> -- \\
  \texttt{ehat} \> $\widehat{\mb{\varepsilon}}$ \> \texttt{TRUE} \> \texttt{FALSE} \\
  \texttt{ehat.p.bhat} \> $\widehat{\mb{\varepsilon}} + \widehat{\mb{B}}$ \> \texttt{FALSE} \> \texttt{FALSE} \\
\end{tabbing}


\subsubsection{Transformations of variogram parameters for (RE)ML estimation}\label{sec:parameter-transformations}

The arguments \code{param.tf}, \code{fwd.tf}, \code{deriv.fwd.tf},
\code{bwd.tf} of \code{control.georob()} define the transformations of
the variogram parameters for RE(ML) estimation.  Implemented are
currently \code{"log"}, \code{"logit1"}, \code{"logit2"},
\code{"logit3"} (various variants of logit-transformation, see code of
function \code{fwd.transf}) and \code{"identity"} (= no)
transformations.  These are the possible values that the many
arguments of the function \code{param.transf()} accept (as quoted
character strings) and these are the names of the list components
returned by \code{fwd.transf()}, \code{dfwd.transf()} and
\code{bwd.transf()}.  Additional transformations can be implemented by:

\begin{enumerate}
  \item Extending the function definitions by arguments like
  
  \code{fwd.tf = fwd.transf(my.fun = function(x) your transformation)},\\{}
  \code{deriv.fwd.tf = dfwd.transf(my.fun = function(x) your derivative)},\\{}
  \code{bwd.tf = bwd.transf(my.fun = function(x) your back-transformation)},
  
  \item Assigning to a given argument of \code{param.transf} the name of
  the new function, e.g.\\{} \code{variance = "my.fun"}.  
\end{enumerate}

Note that the values given for the arguments of \code{param.transf()}
must match a name of the functions returned by \code{fwd.transf()},
\code{dfwd.transf()} and \code{bwd.transf()}.


\subsubsection{Miscellaneous arguments of \code{control.georob()}}\label{sec:georob-control-miscellaneous}

\code{control.georob()} has the following additional arguments:

\begin{ldescription}

  \item[\code{bhat}] initial values for the spatial random effects
  \eqn{\widehat{\mbox{\boldmath$B$\unboldmath}}}{}, with
  \eqn{\widehat{\mbox{\boldmath$B$\unboldmath}}=\mbox{\boldmath$0$\unboldmath}}{}
  if \code{bhat} is equal to \code{NULL} (default).
  
  \item[\code{force.gradient}] logical controlling whether the estimating
  equations or the gradient of the Gaussian restricted log-likelihood are
  evaluated even if all variogram parameters are fixed (default
  \code{FALSE}).
  
  \item[\code{min.condnum}] positive numeric.  Minimum acceptable ratio of smallest to
  largest singular value of the model matrix
  \eqn{\mbox{\boldmath$X$\unboldmath}}{} (default \code{1.e-12}).
  
  \item[\code{zero.dist}] positive numeric equal to the maximum distance, separating two
  sampling locations that are still considered as being coincident.
  
\end{ldescription}

Note that \code{georob()} can fit models with rank-deficient model
matrices.  It uses then the Moore-Penrose inverse of the matrix
$\mb{X} \mb{V}_{\alpha,\xi}^{-1}\mb{X}$ to compute the projection
matrices $\mb{A}_{\alpha}$ and $\mb{P}_{\alpha}$ 
\citep[see][]{Kuensch-etal-in-prep}.


\subsection{Parallelized computations}\label{sec:parallization}

Parallelized computations shorten computing time for large data sets
(\eqn{n>1000}{}).  \code{georob()} and other functions of the package
therefore use on non-windows OS the function \code{mclapply()}
(forking, package \pkg{parallel}) and on windows OS the functions
\code{parLapply} (socket cluster, package \pkg{parallel}) as well as
\code{sfLapply()} (socket cluster, package \pkg{snowfall}) for
parallelized computations.  The following tasks may be executed in
parallel:

\begin{enumerate}
  
  \item \label{item:parallel-other} Simultaneously fitting multiple
  models by functions \code{cv.georob()},
  \code{profilelogLik()} (section~\ref{sec:model-assessment}),
  \code{add1.georob()}, \code{drop1.georob()} and
  \code{step.georob()} (section~\ref{sec:model-building});
 
  \item \label{item:parallel-predict} computing Kriging predictions by
  \code{predict.georob()} (section~\ref{sec:predict-georob});
  
  \item \label{item:parallel-pmm} matrix multiplication by function \code{pmm()};
    
  \item \label{item:parallel-gcr} computing the (generalized)
  covariance matrix $\mb{\Gamma}_{\theta}$ of the data by the
  (non-exported) function \code{f.aux.gcr()}.
  
\end{enumerate}

For tasks \ref{item:parallel-other} and \ref{item:parallel-predict}
the functions have an argument \code{ncores} to control how many cores
should be used for parallel computations.
%
For tasks \ref{item:parallel-pmm} and \ref{item:parallel-gcr} one can
use the argument \code{pcmp} of \code{control.georob()} along with the
function \code{pcmp.control()} to control parallelized computations.
The function \code{pcmp.control()} has the following arguments:

\begin{ldescription}
\item[\code{pmm.ncores}] number (integer, default 1) of cores used for parallelized
matrix multiplication.

\item[\code{gcr.ncores}] number (integer, default 1) of cores used for parallelized
computation of (generalized) covariance matrix.

\item[\code{max.ncores}] allowed maximum number of cores (integer,
default all cores of a machine) used for parallelized computations.

\item[\code{f}] number (integer, default 2) of tasks assigned to each core in
  parallelized computations.

\item[\code{sfstop}] logical controlling whether the SNOW socket
cluster is stopped after each parallelized computations on windows OS
(default \code{FALSE}).

\item[\code{allow.recursive}] logical controlling whether nested parallelized
computation should be allowed (default \code{TRUE}).

\end{ldescription}

Generating child processes requires itself resources and increasing
the number of cores for tasks \ref{item:parallel-pmm} and
\ref{item:parallel-gcr} does not always result in reduced computing
time.  A sensible default for the number of cores for tasks
\ref{item:parallel-pmm} and \ref{item:parallel-gcr} is likely
2.  \code{max.ncores} controls how many child processes
are generated in total.  This can be used to prevent that child
processes generate themselves children which may result in too many
child processes.

\smallskip

Note, however, that very substantial reductions in computing time
results when one uses the \strong{OpenBLAS} library instead of the
reference BLAS library that ships with R, see
\url{http://www.openblas.net/} and R FAQ for your OS. With OpenBLAS no
gains are obtained by using more than one core for matrix
multiplication, and one should therefore use the default argument
\code{pmm.ncores = 1} for \code{control.pcmp()}.
%

\clearpage
%



%% 
 % Details Kriging
 %%

% material taken from
% tools::Rd2latex("predict.georob.Rd", "../../../vignettes/predict.georob.tex", outputEncoding="utf-8")
% tools::Rd2latex("lgnpp.Rd", "../../../vignettes/lgnpp.tex", outputEncoding="utf-8")
% tools::Rd2latex("validate.predictions.Rd", "../../../vignettes/validate.predictions.tex", outputEncoding="utf-8")

\section{Details about Kriging}\label{sec:krig-details}


\subsection{Functionality of \texttt{predict.georob()}}\label{sec:predict-georob}

The \code{predict} method of class \code{georob} computes customary or
robust external drift point or block plug-in Kriging predictions (see
equation \ref{eq:robust-edk}).  
%
Data about the prediction targets are passed as argument
\code{newdata} to \code{predict.georob()}, where \code{newdata} can be
either an ordinary \strong{data frame}, a \strong{SpatialPoints-},
\strong{SpatialPixels-}, \strong{SpatialGrid} or a
\strong{SpatialPolygonsDataFrame}, a \strong{SpatialPoints},
\strong{SpatialPixels} or a \strong{SpatialGrid} object, all the
latter provided by the package \pkg{sp}.  If \code{newdata} is a
\code{SpatialPoints}, \code{SpatialPixels} or a \code{SpatialGrid}
object then the drift model may only use the coordinates as covariates
(universal Kriging).  If \code{newdata} is a
\code{SpatialPolygonsDataFrame} then block Kriging predictions are
computed, otherwise point Kriging predictions.


\subsubsection{Prediction targets}\label{sec:prediction-targets}

% \smallskip
% 
The argument \code{type} controls what quantities
\code{predict.georob()} computes (given here for a target point
\mb{s}, but the same quantities are also computed for a block):
%
\begin{itemize}
  
  \item \code{"signal"}: the ``signal''
  \eqn{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),
  
  \item \code{"response"}: the observations
  \eqn{Y(\mbox{\boldmath$s$\unboldmath}) =
  Z(\mbox{\boldmath$s$\unboldmath}) +
  \varepsilon(\mbox{\boldmath$s$\unboldmath}),}{}
  
  \item \code{"trend"}: the external drift
  \eqn{\mbox{\boldmath$x$\unboldmath}(\mbox{\boldmath$s$\unboldmath})^\mathrm{T}
  \mbox{\boldmath$\beta$\unboldmath},}{}
  
  \item \code{"terms"}: the model terms.  The argument \code{terms}
  can then be used to select the terms (default
  all terms) and \code{se.fit} controls whether standard errors of the
  terms are computed (default \code{TRUE}).
  
\end{itemize}


\subsubsection{Further control}\label{sec:control-predict-misc}

Use the argument \code{control} along with the function
\code{control.predict.georob()} to further control what
\code{predict.georob()} actually does:


\paragraph{Covariance matrices} 
The argument \code{full.covmat} of \code{control.predict.georob()}
controls whether the full covariance matrices of prediction errors,
fitted trend, etc.\ are computed (\code{TRUE}) or only the vector with
their variances (\code{FALSE}, default).
%

\paragraph{Computing auxiliary items for log-normal Kriging}
Use the argument \code{extended.output = TRUE} of
\code{control.predict.georob()} to compute all quantities required for
the (approximately) unbiased back-transformation of Kriging
predictions of log-transformed data to the original scale of the
measurements by \code{lgnpp()} (see
section~\ref{sec:krig-details-lognormal}).  In more detail, the
following items are computed:

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

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



\paragraph{Contents and structure of returned object}

Depending on the values passed to \code{type}, \code{full.covmat} and
\code{extended.output}, \code{predict.georob()} returns the following object:

\begin{enumerate}
  
  \item If \code{type} is equal to \code{"terms"} then a vector, a
  matrix, or a list with prediction results along with bounds and
  standard errors, see \code{predict.lm()}.
  
  \item If \code{type} is not equal to \code{"terms"} and
  \code{full.covmat} is \code{FALSE} then the result is an object of
  the same class as \code{newdata} (data frame,
  \code{{SpatialPointsDataFrame}},
  \code{{SpatialPixelsDataFrame}}
  \code{{SpatialGridDataFrame}}, \\{}
  \code{{SpatialPolygonsDataFrame}}).
  The data frame or the \code{data} slot of the
  \code{Spatial{\itshape xxx\,}DataFrame} objects have the following components:
  
  \begin{itemize}
    
    \item the coordinates of the prediction points (only present if
    \code{newdata} is a data frame).
    
    \item \code{pred}: the Kriging predictions (or fitted values).
    
    \item \code{se}: the root mean squared prediction errors (Kriging
    standard errors).
    
    \item \code{lower}, \code{upper}: the limits of
    tolerance/confidence intervals (the confidence level is set by the
    argument \code{signif}) of \code{predict.georob()}),
    
    \item \code{trend}, \code{var.pred}, \code{cov.pred.target},
    \code{var.target}: only present if \code{extended.output} is \code{TRUE},
    see above.
    
  \end{itemize}
  
  \item If \code{type} is not equal to \code{"terms"} and
  \code{full.covmat} is \code{TRUE} then \code{predict.georob()}
  returns a list with the following components:
  
  \begin{itemize}
    
    \item \code{pred}: a data frame or a \code{Spatial{\itshape xxx\,}DataFrame}
    object as described above for\\{} \code{full.covmat = FALSE}.
    
    \item \code{mse.pred}: the full covariance matrix of the
    prediction errors,
    \eqn{Y(\mbox{\boldmath$s$\unboldmath})-\widehat{Y}(\mbox{\boldmath$s$\unboldmath})}{}
    or
    \eqn{Z(\mbox{\boldmath$s$\unboldmath})-\widehat{Z}(\mbox{\boldmath$s$\unboldmath})}{}
    see above.
    
    \item \code{var.pred}: the full covariance matrix of the Kriging
    predictions, see above.
    
    \item \code{cov.pred.target}: the full covariance matrix of the
    predictions and the prediction targets, see above.
    
    \item \code{var.target}: the full covariance matrix of the
    prediction targets, see above.
        
  \end{itemize}
  
\end{enumerate}

\clearpage

\bcf[1]

\begin{center}
  \includegraphics[width=0.8\textwidth]{0-block-discretization}
\end{center}

\ecf{Approximation of four blocks $B_{1}$, \ldots, $B_{4}$ by
a group of 16 pixels $PX_{1}$ \ldots, $PX_{16}$.  The original
geometry of blocks with the grid of the pixels is shown on the
left and the approximation on the
right.}{block-discretization}


\subsubsection{Block Kriging}\label{sec:block-Kriging}

\code{georob.predict()} uses functions of the package
\pkg{constrainedKriging} \citep{Hofer-Papritz-2011} for efficiently
computing the required integrals of the covariance function.
%
The target blocks are approximated for this by sets of
\emph{rectangular pixels} arranged on a regular lattice
(Figure~\ref{fig:block-discretization}).  The integrals can then be
computed very efficiently because the PDF of the distance between two
points, that are uniformly distributed in two rectangles on a regular
lattice, can be computed by closed-form expressions
\citep{Clifford-2005}.
%
Also the PDF of the distance between a fixed point and a point
uniformly distributed in a rectangle is available in closed form
\citep{Hofer-Papritz-2011}.
%
Hence, for a two-dimensional study region, only 1-dimensional
integrals of the covariance function must be evaluated numerically
instead of 2- and 4-dimensional integrals \citep[see][for
details]{Hofer-Papritz-2011}.

\smallskip

The arguments \code{pwidth} and \code{pheight} of
\code{control.predict.georob()} define the dimension of the pixel used
for the approximation of the blocks, and \code{napp} defines how many
repetitions of the approximation by randomly placed grid of pixels
should be averaged (see help page of the function \code{preCKrige} of
package \pkg{constrainedKriging} for more details).

\smallskip

For the time being \pkg{constrainedKriging} does not integrate the
following (generalized) covariances : \code{RMaskey}, \code{RMdagum},
\code{RMdewijsian}, \code{RMfbm} and \code{RMgenfbm}.  Hence, these
models cannot be used for block Kriging.


\subsubsection{Parallelized computations}\label{sec:Kriging-parallelized-computations}

\code{predict.georob()} computes
its results in parallel.  The parallelization is controlled by the
arguments \code{mmax} and \code{ncores} of
\code{control.predict.georob()} (and \code{pcmp} along with the
function \code{control.pcmp()}, see section~\ref{sec:parallization}):
% 
% \begin{ldescription}
%   
%   \item[\code{mmax}] integer equal to the maximum number (default
%   \code{10000}) of prediction items, computed in a sub-task, see
%   below.
%   
%   \item[\code{ncores}] positive integer controlling how many cores are
%   used for parallelized computations, see below.
%   
% \end{ldescription}
% 
If there are \eqn{m}{} items to compute, the task is split into
\code{ceiling(m/mmax)} sub-tasks that are then distributed to
\code{ncores} CPUs.  Evidently, \code{ncores = 1} suppresses parallel
execution.  By default, the function uses all available CPUs as
returned by \code{{detectCores()}}.  Note that if \code{full.covmat} is
\code{TRUE} \code{mmax} must exceed \eqn{m}{} (and parallel execution
is not possible).

\smallskip






\subsection{Lognormal Kriging}\label{sec:krig-details-lognormal}

The function \code{lgnpp()} back-transforms point or block Kriging
predictions of a log-transformed response variable computed by
\code{predict.georob()}.  Alternatively, the function averages
log-normal point Kriging predictions for a block and approximates the
mean squared prediction error of the block mean.
%
The items required for the back-transformation are computed by
\code{predict.georob()} if the argument \code{control =
control.georob.predict(extended.output = TRUE)} is used, see
section~\ref{sec:predict-georob}.

% In more
% detail, the following items are then computed:
% 
% \begin{itemize}
%   
%   
%   \item \code{trend}: the fitted values,
%   \eqn{\mbox{\boldmath$x$\unboldmath}(\mbox{\boldmath$s$\unboldmath})\mathrm{^T}\widehat{\mbox{\boldmath$\beta$\unboldmath}}}{},
%   
%   \item \code{var.pred}: the variances of the Kriging predictions, 
%   \eqn{\mathrm{Var}_{\hat{\theta}}[\widehat{Y}(\mbox{\boldmath$s$\unboldmath})]}{} or 
%   \eqn{\mathrm{Var}_{\hat{\theta}}[\widehat{Z}(\mbox{\boldmath$s$\unboldmath})]}{},
%   
%   \item \code{cov.pred.target}: the covariances between the predictions and the
%   prediction targets,\\{}
%   \eqn{\mathrm{Cov}_{\hat{\theta}}[\widehat{Y}(\mbox{\boldmath$s$\unboldmath}),Y(\mbox{\boldmath$s$\unboldmath})]}{} or 
%   \eqn{\mathrm{Cov}_{\hat{\theta}}[\widehat{Z}(\mbox{\boldmath$s$\unboldmath}),Z(\mbox{\boldmath$s$\unboldmath})]}{},
%   
%   
%   \item \code{var.target}: the variances of the prediction targets
%   \eqn{\mathrm{Var}_{\hat{\theta}}[Y(\mbox{\boldmath$s$\unboldmath})]}{} or 
%   \eqn{\mathrm{Var}_{\hat{\theta}}[Z(\mbox{\boldmath$s$\unboldmath})]}{}.
%   
%   
% \end{itemize}
% 
% 
% Note that the component \code{var.pred} is also  present if 
% \code{type} is equal to \code{"trend"}, irrespective of the choice for \code{extended.output}.  
% This component contains then the variances of the fitted values.


\smallskip

\code{lgnpp()} then performs the following three tasks:

%
\subsubsection{Back-transformation of point Kriging predictions of a log-transformed response}\label{sec:krig-details-lognormal-point}

The usual, marginally unbiased back-transformation for log-normal point
Kriging is used:

\begin{equation}\label{eq:point-lk-predictor}
  \widehat{U}(\mbox{\boldmath$s$\unboldmath}) = \exp( \widehat{Z}(\mbox{\boldmath$s$\unboldmath}) + 
  1/2 (  \mathrm{Var}_{\hat{\theta}}[ Z(\mbox{\boldmath$s$\unboldmath})]
  - \mathrm{Var}_{\hat{\theta}}[\widehat{Z}(\mbox{\boldmath$s$\unboldmath})])), 
\end{equation}
%
\begin{eqnarray}\label{eq:point-lk-msep}
  \lefteqn{\mathrm{Cov}_{\hat{\theta}}[
  U(\mbox{\boldmath$s$\unboldmath}_i) - \widehat{U}(\mbox{\boldmath$s$\unboldmath}_i),
  U(\mbox{\boldmath$s$\unboldmath}_j) - \widehat{U}(\mbox{\boldmath$s$\unboldmath}_j) 
  ] = \mu_{\hat{\theta}}(\mbox{\boldmath$s$\unboldmath}_i) \mu_{\hat{\theta}}(\mbox{\boldmath$s$\unboldmath}_j)
  \big\{
  \exp(\mathrm{Cov}_{\hat{\theta}}[Z(\mbox{\boldmath$s$\unboldmath}_i),Z(\mbox{\boldmath$s$\unboldmath}_j)])} \hspace{2cm}\nonumber \\
   &&  \quad\quad 
   -2\exp(\mathrm{Cov}_{\hat{\theta}}[\widehat{Z}(\mbox{\boldmath$s$\unboldmath}_i),Z(\mbox{\boldmath$s$\unboldmath}_j)]) 
    +\exp(\mathrm{Cov}_{\hat{\theta}}[\widehat{Z}(\mbox{\boldmath$s$\unboldmath}_i),\widehat{Z}(\mbox{\boldmath$s$\unboldmath}_j)])
  \big\},
\end{eqnarray}
where \eqn{\widehat{Z}}{} and \eqn{\widehat{U}}{} denote the
log- and back-transformed predictions of the signal,
and
%
\begin{equation}\label{eq:point-lk-mean}
  \mu_{\hat{\theta}}(\mbox{\boldmath$s$\unboldmath}) \approx 
      \exp(\mbox{\boldmath$x$\unboldmath}(\mbox{\boldmath$s$\unboldmath})\mathrm{^T}\widehat{\mbox{\boldmath$\beta$\unboldmath}} 
      + 1/2 \mathrm{Var}_{\hat{\theta}}[Z(\mbox{\boldmath$s$\unboldmath})]).    
\end{equation}
%
The expressions for the required covariance terms can be found in the
Appendices of \citet{Nussbaum-etal-2012,Nussbaum-etal-2014b}.  
%
Instead of the signal \eqn{Z(\mbox{\boldmath$s$\unboldmath})}{},
predictions of the log-transformed response
\eqn{Y(\mbox{\boldmath$s$\unboldmath})}{} or the estimated trend
\eqn{\mbox{\boldmath$x$\unboldmath}(\mbox{\boldmath$s$\unboldmath})^\mathrm{T}\widehat{\mbox{\boldmath$\beta$\unboldmath}}}{}
of the log-transformed data can be back-transformed.  The above
transformations are used if the \code{object} passed as first argument
to \code{lgnpp()} contains point Kriging predictions and if the
argument \code{is.block = FALSE} and the argument \code{all.pred} is
missing.


%
\subsubsection{Back-transformation of block Kriging predictions of a log-transformed response}\label{sec:krig-details-lognormal-block-permanence}

Block Kriging predictions of a log-transformed response variable are
back-transformed by the approximately unbiased transformation proposed
by \citet[Appendix~C]{Cressie-2006}

\begin{equation}\label{eq:block-lk-predictor}
  \widehat{U}(A) = \exp( \widehat{Z}(A) + 1/2 \{
    \mathrm{Var}_{\hat{\theta}}[Z(\mbox{\boldmath$s$\unboldmath})] + \widehat{\mbox{\boldmath$\beta$\unboldmath}}\mathrm{^T}
    \mbox{\boldmath$M$\unboldmath}(A) \widehat{\mbox{\boldmath$\beta$\unboldmath}} -
    \mathrm{Var}_{\hat{\theta}}[\widehat{Z}(A)]  
    \}),
\end{equation}
%
\begin{eqnarray}\label{eq-msep-block-lk-predictor}
\mathrm{E}_{\hat{\theta}}[\{U(A) - \widehat{U}(A))^2] & = & \mu_{\hat{\theta}}(A)^2  \big\{
    \exp(\mathrm{Var}_{\hat{\theta}}[Z(A)]) \nonumber \\
    &&  - 2 \, \exp(\mathrm{Cov}_{\hat{\theta}}[\widehat{Z}(A),Z(A)]) + \exp(\mathrm{Var}_{\hat{\theta}}[\widehat{Z}(A)])
    \big\}
\end{eqnarray}

where \eqn{\widehat{Z}(A)}{} and \eqn{\widehat{U}(A)}{} are the log- and
back-transformed predictions of the block mean \eqn{U(A)}{}, respectively,
\eqn{\mbox{\boldmath$M$\unboldmath}(A)}{} is the spatial
covariance matrix of the covariates

\begin{equation}  \label{eq:spatial-covariance-covariates}
\mbox{\boldmath$M$\unboldmath}(A) = 1/|A| \int_A 
  ( \mbox{\boldmath$x$\unboldmath}(\mbox{\boldmath$s$\unboldmath}) - \mbox{\boldmath$x$\unboldmath}(A) )
  ( \mbox{\boldmath$x$\unboldmath}(\mbox{\boldmath$s$\unboldmath}) - \mbox{\boldmath$x$\unboldmath}(A) )\mathrm{^T} \,d\mbox{\boldmath$s$\unboldmath}
\end{equation}

within the block $A$, where 
\begin{equation}  \label{eq:spatial-mean-covariates}
\mbox{\boldmath$x$\unboldmath}(A) = 1/|A| \int_A \mbox{\boldmath$x$\unboldmath}(\mbox{\boldmath$s$\unboldmath}) \,d\mbox{\boldmath$s$\unboldmath}  
\end{equation}
  and 

\begin{equation}  \label{eq:spatial-mean}
  \mu_{\hat{\theta}}(A) \approx \exp(\mbox{\boldmath$x$\unboldmath}(A)\mathrm{^T}
    \widehat{\mbox{\boldmath$\beta$\unboldmath}} + 1/2 \mathrm{Var}_{\hat{\theta}}[Z(A)]). 
\end{equation}

This back-transformation is based on the assumption that both the
point data \eqn{U(\mbox{\boldmath$s$\unboldmath})}{} and the block
means \eqn{U(A)}{} follow log-normal laws, which strictly cannot hold.
But for small blocks the assumption works well as the bias and the
loss of efficiency caused by this assumption are small
\citep{Cressie-2006,Hofer-etal-2013}.

\smallskip

The above formulae are used by \code{lgnpp()} if \code{object}
contains block Kriging predictions in the form of a
\code{\LinkA{SpatialPolygonsDataFrame}{SpatialPolygonsDataFrame}}.  To
approximate \eqn{\mbox{\boldmath$M$\unboldmath}(A)}{M(A)}{}, one needs
the covariates on a fine grid for the whole study domain in which the
blocks lie.  The covariates are passed  to \code{lgnpp()} as argument \code{newdata}, where
\code{newdata} can be any spatial data frame accepted by
\code{predict.georob}.  For evaluating
\eqn{\mbox{\boldmath$M$\unboldmath}(A)}{} the geometry of the blocks
is taken from the \code{polygons} slot of the
\code{SpatialPolygonsDataFrame} passed as \code{object} to
\code{lgnpp()}.


%
\subsubsection{Back-transformation and averaging of point Kriging predictions of a log-transformed response}\label{sec:krig-details-lognormal-block-averaging}

\code{lgnpp()} allows as a further option to back-transform and
\emph{average} point Kriging predictions passed as \code{object} to
the function \citep[optimal log-normal block Kriging,
see][]{Cressie-2006}.  One then assumes that the predictions in
\code{object} refer to points that lie in \emph{a single} block.
Hence, one uses the approximation

\begin{equation}  \label{eq:discrete-approx-spatial-mean-1}
\widehat{U}(A) \approx \frac{1}{K} \sum_{s_i \in A} \widehat{U}(\mbox{\boldmath$s$\unboldmath}_i)
\end{equation}

to predict the block mean \eqn{U(A)}{}, where \eqn{K}{} is the number
of points in \eqn{A}{}.  The mean squared error of prediction can be
approximated by

\begin{equation}  \label{eq:discrete-approx-msep-block-mean-1}
  \mathrm{E}_{\hat{\theta}}[\{U(A) - \widehat{U}(A)\}^2] \approx \frac{1}{K^2}
    \sum_{s_i \in A} \sum_{s_j \in A} 
    \mathrm{Cov}_{\hat{\theta}}[
    U(\mbox{\boldmath$s$\unboldmath}_i) - \widehat{U}(\mbox{\boldmath$s$\unboldmath}_i),
    U(\mbox{\boldmath$s$\unboldmath}_j) - \widehat{U}(\mbox{\boldmath$s$\unboldmath}_j) 
    ].
\end{equation}
In most instances, the evaluation of the above double sum is not feasible
because a large number of points is used to discretize the block \eqn{A}{}.
\code{lgnpp()} then uses the following approximations to compute the
mean squared error
\citep[see][Appendices]{Nussbaum-etal-2012,Nussbaum-etal-2014b}:

\begin{itemize}
  
  
  \item Point prediction results are passed as \code{object} to \code{lgnpp()}
  only for a \emph{random sample of points in \eqn{A}{}} (of size \eqn{k}{}),
  for which the evaluation of the above double sum is feasible.
  
  \item The prediction results for the \emph{complete set of points}
  within the block are passed as argument \code{all.pred} to
  \code{lgnpp}.  These results are used to compute \eqn{\widehat{U}(A)}{}.
  
  \item The mean squared error is then approximated by  
  
  \begin{eqnarray}\label{eq:discrete-approx-msep-block-mean-2}
    \lefteqn{\mathrm{E}_{\hat{\theta}}[\{U(A) - \widehat{U}(A)\}^2]  \approx
    \frac{1}{K^2} \sum_{s_i \in A} \mathrm{E}_{\hat{\theta}}[ \{ U(\mbox{\boldmath$s$\unboldmath}_i) - \widehat{U}(\mbox{\boldmath$s$\unboldmath}_i)\}^2]}  \nonumber\\
    && + \frac{K-1}{K k (k-1)} \sum_{s_i \in \mathrm{sample}}\sum_{s_j \in \mathrm{sample}, s_j \neq s_i}
    \mathrm{Cov}_{\hat{\theta}}[
    U(\mbox{\boldmath$s$\unboldmath}_i) - \widehat{U}(\mbox{\boldmath$s$\unboldmath}_i),
    U(\mbox{\boldmath$s$\unboldmath}_j) - \widehat{U}(\mbox{\boldmath$s$\unboldmath}_j) 
    ].
  \end{eqnarray}
  
  The first term of the RHS (and \eqn{\widehat{U}(A)}{}) can be
  computed from the point Kriging results contained in
  \code{all.pred}, and the double sum is evaluated from the full
  covariance matrices of the predictions and the respective targets,
  passed to \code{lgnpp()} as \code{object} (one has to use the
  arguments \code{control=control.predict.georob(full.covmat=TRUE)}
  for \code{predict.georob()} when computing the point Kriging
  predictions stored in \code{object}).
  
  \item If the prediction results are not available for the complete set
  of points in \eqn{A}{} then \code{all.pred} may be equal to \eqn{K}{}.  The
  block mean is then approximated by
  
  \begin{equation}  \label{eq:discrete-approx-spatial-mean-2}
  \widehat{U}(A) \approx \frac{1}{k} \sum_{s_i \in \mathrm{sample}}
  \widehat{U}(\mbox{\boldmath$s$\unboldmath}_i)
  \end{equation}
  
  and the first term of the RHS of the expression for the mean squared
  error by
  
  \begin{equation}  \label{eq:discrete-approx-msep-block-mean-3}
    \frac{1}{kK} \sum_{s_i \in \mathrm{sample}} \mathrm{E}_{\hat{\theta}}[ \{
  U(\mbox{\boldmath$s$\unboldmath}_i) -
  \widehat{U}(\mbox{\boldmath$s$\unboldmath}_i)\}^2].
  \end{equation}
  
  \item By drawing samples repeatedly and passing the related Kriging
  results as \code{object} to \code{lgnpp()}, one can reduce the error
  of the approximation of the mean squared error.
  
  
\end{itemize}

\clearpage


%% 
 % Details model building and assessment
 %%

% material taken from
% tools::Rd2latex("georobModelBuilding.Rd", "../../../vignettes/georobModelBuilding.tex", outputEncoding="utf-8")
% tools::Rd2latex("S3methods.georob.Rd", "../../../vignettes/S3methods.georob.tex", outputEncoding="utf-8")
% tools::Rd2latex("plot.georob.Rd", "../../../vignettes/plot.georob.tex", outputEncoding="utf-8")
% tools::Rd2latex("profilelogLik.Rd", "../../../vignettes/profilelogLik.tex", outputEncoding="utf-8")
% tools::Rd2latex("cv.georob.Rd", "../../../vignettes/cv.georob.tex", outputEncoding="utf-8")

\section{Building models and assessing fitted models}\label{sec:model-building-assessment}

%%%%%%%%%%%%%%%%%%%%%
% model building

\subsection{Model building} \label{sec:model-building}

% step, drop1, add1

\paragraph{Wald tests} The \code{waldtest} method for class
\texttt{georob} can be used to test hypotheses about the fixed effects
of a model.  Note that this function uses \emph{conditional} $F$- or
$\chi^2$-tests \citep[section~2.4.2]{Pinheiro-Bates-2000}, i.e.\ it fixes
the variogram parameters at the values of the more general model of
each comparison (see help page of function \code{waldtest()} of
package \pkg{lmtest} for details).

\smallskip

Besides \code{waldtest.georob()} the functions of the package
\pkg{multcomp} can be used to test general linear hypotheses about the
fixed effects of the model.


\paragraph{Log-likelihood and AIC}

The \code{deviance} method for class \code{georob} returns for
Gaussian (RE)ML fits the \emph{residual deviance}
%
\deqn{(\mbox{\boldmath $Y -X
\widehat{\beta}$})^{\mathrm{T}} (\widehat{\tau}^2
\mbox{\boldmath$I$\unboldmath} +
\mbox{\boldmath$\Gamma$\unboldmath}_{\widehat{\theta}})^{-1}
(\mbox{\boldmath $Y -X \widehat{\beta}$}),}{} 
%
and the \code{logLik} and \code{extractAIC} methods extract for class
\code{georob} the respective goodness-of-fit criteria, depending on
the argument \code{REML} either for {ML} (default) or {REML}.

\smallskip

For a robust REML fit the deviance is not defined because there is no
robustified log-likelihood.  \code{deviance.georob()} then computes
(with a warning) the residual deviance of an equivalent Gaussian model
with heteroscedastic nugget effect \eqn{\widehat{\tau}^2/\mbox{\boldmath
$w$\unboldmath}}{}, where \eqn{\mbox{\boldmath $w$\unboldmath}}{} are
the ``robustness weights'' \deqn{w_{i} =
\psi_c(\widehat{\varepsilon}_i/\widehat{\tau})/(\widehat{\varepsilon}_i/\widehat{\tau}).}{}
%
For robust REML, \code{logLik()} and \code{extractAIC()} return the
respective criteria also for the equivalent Gaussian model with
heteroscedastic nugget effect.


\paragraph{Stepwise selection of covariates}

The \code{add1} and \code{drop1} methods for class \code{georob}
compute all the single terms that can be added or dropped from the
model according to their \code{scope} argument.  By default, the
variogram parameters are kept fixed at the values fitted for
\code{object}.  Use the argument \code{fixed = FALSE} if the variogram
parameters should be re-estimate afresh for each evaluated term.  Then
either the variogram parameters in \code{object\$initial.objects}
(\code{use.fitted.param = FALSE}) or the fitted parameters of
\code{object} (\code{use.fitted.param = TRUE}) are used as initial
values.

\smallskip

The \code{step} method for class \code{georob}\footnote{%
The package \pkg{georob} provides a
generic \code{step} function and re-defines the function
\code{step()} of the package \pkg{stats} as \code{default} method.}
%
allows even finer control whether the variogram parameters are kept
fixed or re-estimated when evaluating single terms.  Two argument of
\code{step.georob()} control the behaviour:

\begin{ldescription}
  
  \item[\code{fixed.add1.drop1}] logical controlling whether the
  variogram parameters are \emph{not} adjusted when \code{add}ing or
  \code{drop}ping model terms by \code{add1.georob()} and
  \code{drop1.georob()} (default \code{TRUE}).
  
  \item[\code{fixed.step}] logical controlling whether the variogram
  parameters are \emph{not} adjusted after having called
  \code{add1.georob()} and \code{drop1.georob()} in a cycle of
  \code{step.georob()} (default \code{TRUE}).  For \code{fixed.step =
  FALSE} the parameters are estimated afresh for the new model that
  was chosen in the previous cycle.
  
\end{ldescription}

Of course, model building based on AIC is only sound for Gaussian
(RE)ML as there is no log-likelihood for robust REML. For robust REML
fits \code{add1.georob()}, \code{drop1.georob()} and
\code{step.georob()} use AIC values evaluated for an equivalent
Gaussian model with heteroscedastic nugget effect, see above, and it
is currently not known  whether this is a valid approach for model
building.

\smallskip

A last remark: Use the argument \code{ncores} to let
\code{add1.georob()} and \code{drop1.georob()} evaluate single
terms in parallel.


%%%%%%%%%%%%%%%%%%%%%
% model assessment

\subsection{Assessing fitted models}\label{sec:model-assessment}

\subsubsection{Model diagnostics}\label{sec:model-diagnostics}

The methods required for residual diagnostics (\code{fitted},
\code{residuals}, \code{rstandard}, \code{ranef}) work for objects of
class \code{georob} (either as \code{default} or specific
\code{georob} method).

\smallskip

Note that the are two kind of residuals: \code{residuals.georob()}
extracts either the estimated independent errors
\eqn{\widehat{\varepsilon}(\mbox{\boldmath$s$\unboldmath})}{} or the
sum of the latter quantities and the spatial random effects
\eqn{\widehat{B}(\mbox{\boldmath$s$\unboldmath})}{} (regression
residuals).  \code{rstandard.georob()} does the same but standardizes
the residuals to unit variance.  \code{ranef.georob()} extracts
\eqn{\widehat{B}(\mbox{\boldmath$s$\unboldmath})}{} with the option to
standardize them as well (by argument \code{standard}).

\smallskip

Diagnostics plots are created by the \code{plot} method for class
\code{georob}: Depending on the value of the argument \code{type} the
following plots are created:

\begin{itemize}

\item \code{"variogram"}: the estimated variogram (default),

\item \code{"covariance"}: the estimated covariance function,

\item \code{"correlation"}: the estimated correlation function,

\item \code{"scale-location"}: square root of absolute regression
residuals plotted against fitted values (Scale-Location plot),

\item \code{"ta"}: regression residuals plotted against fitted values
(Tukey-Anscombe plot),

\item \code{"qq.res"}: normal QQ plot of standardized errors
\eqn{\hat{\varepsilon}}{},

\item \code{"qq.ranef"}: normal QQ plot of standardized random
effects \eqn{\hat{B}}{}.

\end{itemize}


\subsubsection{Log-likelihood profiles}\label{sec:likelihood-profiles}

The function \code{profilelogLik()} computes for an array of fixed
values of variogram parameters the profile log-likelihood by
maximizing the (restricted) log-likelihood with respect to the
remaining variogram parameters, the fixed and random effects.  Of
course, the maximized profile log-likelihood values are meaningful
only for Gaussian (RE)ML fits (for robust fits the calculated values
refer to the equivalent Gaussian model with heteroscedastic nugget
effect, see above).  Use the argument \code{ncores} to fit multiple
models in parallel.  

\smallskip

\code{profilelogLik()} uses the function \code{{update}} to
re-estimated the model with partly fixed variogram parameters.
Therefore, any argument accepted by \code{{georob()}} (except
\code{data}) can be changed when fitting the model.  Some of them
(e.g. \code{verbose}) are explicit arguments of \code{profilelogLik},
but also the remaining ones can be passed by \code{...} to the
function.

\smallskip

\code{profilelogLik()} returns its results as a
data frame.  Customary graphics functions can be used for display of
log-likelihood profiles and dependence of estimated parameters on each
other.


%%%%%%%%%%%%%%%%%%%%%
% model cross-validation

\subsection{Cross-validation}\label{sec:cv}

\subsubsection{Computing cross-validation predictions}\label{sec:computing-cv-predictions}

The function \code{cv.georob()} assesses the goodness-of-fit of a
spatial linear model by \var{K}-fold cross-validation.  In more
detail, the model is fitted \var{K} times by robust (or Gaussian)
(RE)ML, excluding each time \var{1/K}th of the data.  The fitted
models are used to compute robust (or customary) external Kriging
predictions for the omitted observations.  If the response variable is
log-transformed then the Kriging predictions can be optionally
transformed back to the original scale of the measurements.  Use the
argument \code{lgn = TRUE} for this.

Practitioners in geostatistics commonly cross-validate a fitted model
without re-estimating the model parameters with the reduced data sets.
This is clearly an unsound practice
\citep[section~7.10]{Hastie-etal-2009}.  Therefore, the argument
\code{re.estimate} should always be set to \code{TRUE}.  The
alternative is provided only for historic reasons.  The argument
\code{return.fit} and \code{reduced.output} control whether results of
the model fit are returned by \code{cv.georob()}.

\smallskip

By default, \code{cv.georob()} fits the models in parallel to the
cross-validation sets.  Use the argument \code{ncores} to control
parallelized computations.

\paragraph{Defining the cross-validation subsets}

The argument \code{method} controls how the data set is partitioned
into the $K$ subsets: For \code{method = "block"} (default) the
function \code{kmeans()} is used to form geographically compact
subsets of data locations and for \code{method = "random"} simple
random sampling is used to form the subsets.  In analogy to the block
bootstrap in time series analysis \citep{Kuensch-1989}, the first
method should be preferred for model assessment, while the latter
might be more informative for assessing prediction precision.  Instead
of using \code{method} (along with the argument \code{seed}) to form
the subsets, the argument \code{sets} can be used to pass the
definition of a partition to \code{cv.georob()}.

Irrespective of the method used to define the subsets, coincident
sampling locations are assigned to the same subset, except when one
uses the argument \code{duplicate.in.same.set = FALSE}.

\paragraph{Further control}

When the external drift model contains factors it may happen that
observations are missing for some factor levels in some of the
subsets.  The argument \code{mfl.action} controls what then happens:
For \code{mfl.action = "stop"} \code{cv.georob()} stops with an error
message.  For \code{mfl.action = "offset"} the effect of the
respective factor (estimated with all the data) is treated as an
\code{offset} term and \code{cv.georob()} estimates only the remaining
terms of the external drift model.

\smallskip

\code{cv.georob()} uses the function \code{{update()}} to re-estimated
the model with the subsets.  Therefore, any argument accepted by
\code{georob()} except \code{data} can be changed when re-fitting the
model.  Some of them (e.g. \code{formula}, \code{subset}, etc.)  are
explicit arguments of \code{cv.georob()}, but also the remaining ones
can be passed by \code{...} to the function.

\smallskip

Sometimes, the estimated variograms differ considerably between the
cross-validation subsets.  Using common initial values for estimating
the model is then numerically inefficient.  Therefore, the arguments
\code{param} and \code{aniso} accept in addition to vectors of initial
variogram parameters matrices with $K$ rows of initial values.  The
$i$th row of the matrix then contains the initial variogram parameters
that are used to fit the model to the $i$th cross-validation subset.


\subsubsection{Criteria for assessing (cross-)validation prediction errors}\label{sec:validating-cv-predictions}

The function \code{validate.predictions()} computes the items required
to evaluate (and plot) the diagnostic criteria proposed by
\citet{Gneiting-etal-2007} for assessing the \emph{calibration} and
the \emph{sharpness} of probabilistic predictions of
\mbox{(cross-)}validation data.  To this aim,
\code{validate.predictions()} uses the assumption that the prediction
errors
\eqn{Y(\mbox{\boldmath$s$\unboldmath})-\widehat{Y}(\mbox{\boldmath$s$\unboldmath})}{}
follow normal distributions with zero mean and standard deviations
equal to the Kriging standard errors.  This assumption is an
approximation if the errors \eqn{\varepsilon}{\epsilon} come from a
long-tailed distribution.  Furthermore, for the time being, the
Kriging variance of the \emph{response} \eqn{Y}{} is approximated by
adding the estimated nugget \eqn{\widehat{\tau}^2}{} to the Kriging
variance of the signal \eqn{Z}{}.  This approximation likely
underestimates the mean squared prediction error of the response if
the errors come from a long-tailed distribution.  Hence, for robust
Kriging, the standard errors of the \mbox{(cross-)}validation errors
are likely too small. 

\smallskip

Notwithstanding these difficulties and imperfections,
\code{validate.predictions()} computes

\begin{itemize}
  
  
  \item the \emph{probability integral transform} (PIT),
  
  \begin{equation}\label{eq:cv-pit}
    \mathrm{PIT}_i = F_i(y_i),
  \end{equation}
  
  where \eqn{F_i(y_i)}{} denotes the (plug-in) predictive CDF evaluated
  at \eqn{y_i}{}, the value of the \eqn{i}{}th
  \mbox{(cross-)}validation datum,
  
  \item the \emph{average predictive CDF}  
  
  \begin{equation}\label{eq:cv-average-pred-dist}
    \bar{F}_n(y)=1/n \sum_{i=1}^n F_i(y),
  \end{equation}
  
  where \eqn{n}{} is the number of (cross-)validation observations and
  the \eqn{F_i}{} are evaluated at \eqn{N}{} quantiles equal to the set
  of distinct \eqn{y_i}{} (or a subset of size \eqn{N}{} of them),
  
  \item the \emph{Brier Score}  
  
  \begin{equation}\label{ea:cv-bs}
    \mathrm{BS}(y) = 1/n \sum_{i=1}^n \left(F_i(y) - I(y_i \leq y)
    \right)^2,
  \end{equation}
  
  where \eqn{I(x)}{} is the indicator function for the event \eqn{x}{},
  and the Brier score is again evaluated at the unique values of the
  (cross-)validation observations (or a subset of size \eqn{N}{} of
  them),
  
  \item the \emph{averaged continuous ranked probability score}, CRPS, a
  strictly proper scoring criterion to rank predictions, which is
  related to the Brier score by
  
  \begin{equation}\label{eq:cv-crps}
    \mathrm{CRPS} = \int_{-\infty}^\infty \mathrm{BS}(y) \,dy.
  \end{equation}
  
  
\end{itemize}


\citet{Gneiting-etal-2007} proposed the following plots to validate
probabilistic predictions:

\begin{itemize}


\item A histogram (or a plot of the empirical CDF) of the PIT values.
For ideal predictions, with observed coverages of prediction intervals
matching nominal coverages, the PIT values have an uniform
distribution.

\item Plots of \eqn{\bar{F}_n(y)}{} and of the empirical CDF of
the data, say \eqn{\widehat{G}_n(y)}{}, and of their
difference, \eqn{\bar{F}_n(y)-\widehat{G}_n(y)}{}
vs. \eqn{y}{}.  The forecasts are said to be \emph{marginally calibrated}
if \eqn{\bar{F}_n(y)}{} and \eqn{\widehat{G}_n(y)}{}
match.

\item A plot of \eqn{\mathrm{BS}(y)}{} vs.  \eqn{y}{}.  Probabilistic
predictions are said to be \emph{sharp} if the area under this curve,
which equals CRPS, is minimized.


\end{itemize}


The \code{plot()} method for class \code{cv.georob} allows to create
these plots, along with scatter-plots of observations and predictions,
Tukey-Anscombe and normal QQ plots of the standardized prediction
errors, and \code{summary.cv.georob()} computes the mean and
dispersion statistics of the (standardized) \mbox{(cross-)}validation
prediction errors.

% (by a call to
% \code{validate.prediction} with argument \code{statistic = "st"}, see
% \emph{Value}) and the averaged continuous ranked probability score
% (\code{crps}).  If present in the \code{cv.georob} object, the error
% statistics are also computed for the errors of the unbiasedly
% back-transformed predictions of a log-transformed response.  If \code{se}
% is \code{TRUE} then these statistics are evaluated separately for the
% \eqn{K}{} cross-validation subsets and the standard errors of the means of
% these statistics are returned as well.
% 
% The \code{print} method for class \code{cv.georob} returns the mean
% and dispersion statistics of the (standardized) prediction errors.
% 
% The method \code{rstudent} returns for class \code{cv.georob} the
% standardized prediction errors.


\bibliographystyle{natbib}\bibliography{georob}

\end{document}







