%\VignetteEngine{knitr::knitr}
%\VignetteIndexEntry{Various GLGM examples}

\documentclass[12pt]{article}
\usepackage[margin=1in]{geometry}
\usepackage{caption}
\usepackage{subcaption}
\providecommand{\subfloat}[2][need a sub-caption]{\subcaptionbox{#1}{#2}}
\title{Various GLGM examples}
\author{Patrick Brown}

\begin{document}

\maketitle

<<knitr, include=FALSE>>=
require('knitr')
# very basic output that doesnt require extra latex packages

knit_hooks$set(source  =function(x, options) {
  paste0(c('\\begin{verbatim}', x, '\\end{verbatim}', ''),
      collapse = '\n')
})
knit_hooks$set(plot = function (x, options) 
    {
      paste0(knitr::hook_plot_tex(x, options), "\n")
   })
knit_hooks$set(chunk = function(x, options) {x})

hook_output <- function(x, options) {
  if (knitr:::output_asis(x, options)) return(x)
  paste0('\\begin{verbatim}\n', x, '\\end{verbatim}\n')
}

knit_hooks$set(output = hook_output)
knit_hooks$set(message = hook_output)
knit_hooks$set(warning = hook_output)

knitr::opts_chunk$set(out.width='0.48\\textwidth', 
  fig.align='default', fig.height=4, fig.width=6,
  tidy = FALSE, margins = 1)

knitr::knit_hooks$set(
  margins = function (before, options, envir) {
    if (!before) 
        return()
    graphics::par(mar = c(1.5 + 0.9 * options$margins, 1.5 + 
        0.9 * options$margins, 0.2, 0.2), mgp = c(1.45, 0.45, 
        0), cex = 1.25)
})
@    

This vignette is a bunch of examples, its primary purpose is to test the glgm function.

\section*{The data}

<<packages>>=
library("geostatsp")
data('swissRain')
swissRain = unwrap(swissRain)
swissAltitude = unwrap(swissAltitude)
swissBorder = unwrap(swissBorder)
swissRain$lograin = log(swissRain$rain)

swissAltitudeCrop = mask(swissAltitude,swissBorder)
@



<<inla, include=FALSE>>=
if(requireNamespace("INLA", quietly=TRUE) ) {
  INLA::inla.setOption(num.threads=2)
  # not all versions of INLA support blas.num.threads
  try(INLA::inla.setOption(blas.num.threads=2), silent=TRUE)
} else {
  print("INLA not installed, examples won't be run")
}
@




    

number of cells... smaller is faster but less interesting
<<cells>>=
if(!exists('fact')) fact = 1
fact
(Ncell = round(25*fact))
@



model with standard formula


<<swissINla>>=
swissFit =  geostatsp::glgm(
    formula = lograin~ CHE_alt,
    data = swissRain, 
    grid = Ncell,
    buffer = 10*1000,
    covariates=swissAltitudeCrop, 
    family="gaussian", 
    prior = list(
      sd=c(1,0.5), 
      sdObs = 1,
      range=c(500000, 0.5)),
    control.inla = list(strategy='gaussian')
  )
@



parameters
<<swissParams>>=
if(length(swissFit$parameters)) {
    knitr::kable(swissFit$parameters$summary[,c(1,3,5)], digits=3)
} else {
    print("INLA was not run, install the INLA package to see results")
}
@

Exceedance probabilities
<<swissExc>>=
if(length(swissFit$parameters)) {
  swissExc = excProb(
    x=swissFit,  random=TRUE,
    threshold=0)
} 
@

<<swissWithFormula, fig.cap = 'Swiss rain as in help file', fig.subcap = c('random','fitted', 'sd','range')>>=
if(length(swissFit$parameters)) {
  plot(swissExc, breaks = c(0, 0.2, 0.8, 0.95, 1.00001), 
    col=c('green','yellow','orange','red'))	
  
  plot(swissBorder, add=TRUE)		
  
  
  swissExcP = excProb(
    swissFit$inla$marginals.predict, 3, 
      template=swissFit$raster)
  plot(swissExcP, breaks = c(0, 0.2, 0.8, 0.95, 1.00001), 
    col=c('green','yellow','orange','red'))	
  plot(swissBorder, add=TRUE)

  matplot(
    swissFit$parameters$sd$posterior[,'x'],
    swissFit$parameters$sd$posterior[,c('y','prior')],
    lty=1, col=c('black','red'), type='l',
    xlab='sd', ylab='dens', xlim = c(0,5))
  
  matplot(
    swissFit$parameters$range$posterior[,'x'],
    swissFit$parameters$range$posterior[,c('y','prior')],
    lty=1, col=c('black','red'), type='l',
    xlab='range', ylab='dens')
  } 
@

non-parametric elevation effect

<<swissNonParam, fig.cap = 'Swiss rain elevation rw2', fig.subcap = c('elevation effect','fitted')>>=

altSeq = exp(seq(
    log(100), log(5000),
    by = log(2)/5))
altMat = cbind(altSeq[-length(altSeq)], altSeq[-1], seq(1,length(altSeq)-1))

swissAltCut = classify(
  swissAltitudeCrop, 
  altMat
)
names(swissAltCut) = 'bqrnt'

  
  swissFitNp = geostatsp::glgm(
    formula = lograin ~ f(bqrnt, model = 'rw2', scale.model=TRUE, 
      values = 1:length(altSeq),
      prior = 'pc.prec', param = c(0.1, 0.01)),
    data=swissRain, 
    grid = Ncell, 
    covariates=swissAltCut,
    family="gaussian", buffer=20000,
    prior=list(
      sd=c(u = 0.5, alpha = 0.1), 
      range=c(50000,500000), 
      sdObs = c(u=1, alpha=0.4)), 
    control.inla=list(strategy='gaussian')
  )

if(length(swissFitNp$parameters)) {
  knitr::kable(swissFitNp$parameters$summary, digits=3)
  
  matplot(
    altSeq, 
    exp(swissFitNp$inla$summary.random$bqrnt[,
      c('0.025quant', '0.975quant', '0.5quant')]),
    log='xy',
    xlab ='elevation', ylab='rr',
  type='l',
    lty = 1,
    col=c('grey','grey','black')
    )
  
  swissExcP = excProb(swissFitNp$inla$marginals.predict, 
    3, template=swissFitNp$raster)
  plot(swissExcP, breaks = c(0, 0.2, 0.8, 0.95, 1.00001), 
    col=c('green','yellow','orange','red'))	
  plot(swissBorder, add=TRUE)
} 
@
  

intercept only, named response variable.  legacy priors
<<asHelpFile>>=
swissFit =  geostatsp::glgm("lograin", swissRain, Ncell, 
    covariates=swissAltitude, family="gaussian", buffer=20000,
    priorCI=list(sd=c(0.2, 2), range=c(50000,500000), sdObs = 2), 
    control.inla=list(strategy='gaussian')
)
if(length(swissFit$parameters))  
    knitr::kable(swissFit$parameters$summary[,c(1, 3:5, 8)], digits=4)
@




intercept only, add a covariate just to confuse glgm.

<<swissINterceptOnly, fig.cap = 'Swiss intercept only', fig.subcap = c('exc prob','range')>>=
  swissFit =  geostatsp::glgm(
    formula=lograin~1, 
    data=swissRain, 
    grid=Ncell, 
    covariates=swissAltitude, 
    family="gaussian", buffer=20000,
    priorCI=list(sd=c(0.2, 2), range=c(50000,500000)), 
    control.inla=list(strategy= 'gaussian'),
    control.family=list(hyper=list(prec=list(prior="loggamma", param=c(.1, .1))))
  )
  
  if(length(swissFit$parameters)) {

  knitr::kable(swissFit$parameters$summary[,c(1, 3:5, 8)], digits=3)
  
  swissExc = excProb(
    swissFit$inla$marginals.random$space, 0, 
    template=swissFit$raster)
  plot(swissExc, breaks = c(0, 0.2, 0.8, 0.95, 1.00001), 
    col=c('green','yellow','orange','red'))	
  plot(swissBorder, add=TRUE)		

    matplot(
    swissFit$parameters$range$posterior[,'x'],
    swissFit$parameters$range$posterior[,c('y','prior')],
    lty=1, col=c('black','red'), type='l',
    xlab='range', ylab='dens')
}
@




covariates are in data
<<covInData, fig.cap = 'covaraites in data', fig.subcap = c('predict.mean','range')>>=
newdat = swissRain
newdat$elev = extract(swissAltitude, swissRain, ID=FALSE)
swissLandType = unwrap(swissLandType)
  swissFit =  geostatsp::glgm(lograin~ elev + land,
    newdat, Ncell, 
    covariates=list(land=swissLandType),
    family="gaussian", buffer=40000,
    priorCI=list(sd=c(0.2, 2), range=c(50000,500000)), 
    control.inla = list(strategy='gaussian'),
    control.family=list(hyper=list(prec=list(prior="loggamma", 
          param=c(.1, .1))))
  )

  if(length(swissFit$parameters)) {
  knitr::kable(swissFit$parameters$summary, digits=3)
  
  plot(swissFit$raster[['predict.mean']])
   plot(swissBorder, add=TRUE)    

    matplot(
    swissFit$parameters$range$posterior[,'x'],
    swissFit$parameters$range$posterior[,c('y','prior')],
    lty=1, col=c('black','red'), type='l',
    xlab='range', ylab='dens')
}
@

formula, named list elements
<<swissFitNamedList>>=
swissFit =  geostatsp::glgm(lograin~ elev,
    swissRain, Ncell, 
    covariates=list(elev=swissAltitude), 
    family="gaussian", buffer=20000,
    priorCI=list(sd=c(0.2, 2), range=c(50000,500000)), 
    control.mode=list(theta=c(1.9,0.15,2.6),restart=TRUE),
    control.inla = list(strategy='gaussian'),
    control.family=list(hyper=list(prec=list(prior="loggamma", 
          param=c(.1, .1))))
  )
if(length(swissFit$parameters))
    swissFit$parameters$summary[,c(1,3,5)]
@

categorical covariates
<<swissFitCategorical, fig.cap = 'categorical covariates', fig.subcap = c('map','range')>>=
swissFit =  geostatsp::glgm(
    formula = lograin ~ elev + factor(land),
    data = swissRain, grid = Ncell, 
    covariates=list(elev=swissAltitude,land=swissLandType), 
    family="gaussian", buffer=20000,
    prior=list(sd=c(0.2, 0.5), range=c(100000,0.5)), 
    control.inla=list(strategy='gaussian'),
    control.family=list(hyper=list(
      prec=list(prior="loggamma", 
          param=c(.1, .1))))
  )
if(length(swissFit$parameters)) {

  knitr::kable(swissFit$parameters$summary[,c(1,3,5)], digits=3)
  
  
  plot(swissFit$raster[['predict.mean']])
   plot(swissBorder, add=TRUE)    

    matplot(
    swissFit$parameters$range$posterior[,'x'],
    swissFit$parameters$range$posterior[,c('y','prior')],
    lty=1, col=c('black','red'), type='l',
    xlab='range', ylab='dens')
} 
@

put some missing values in covaritates
also dont put factor() in formula
<<someMissing>>=
temp = values(swissAltitude)
temp[seq(10000,12000)] = NA
values(swissAltitude) = temp
  
swissFitMissing =  geostatsp::glgm(rain ~ elev + land,swissRain,  Ncell, 
    covariates=list(elev=swissAltitude,land=swissLandType), 
    family="gaussian", buffer=20000,
    prior=list(sd=c(0.2, 0.5), range=c(100000,0.5)), 
    control.inla = list(strategy='gaussian'),
    control.family=list(hyper=list(prec=list(prior="loggamma", 
          param=c(.1, .1))))
)
if(length(swissFitMissing$parameters))
      knitr::kable(swissFitMissing$parameters$summary[,1:5], digits=3)
@

covariates in data, factors

<<covInDataLevels>>=
newdat = swissRain
newdat$landOrig = extract(swissLandType, swissRain, ID=FALSE)
newdat$landRel = relevel(newdat$landOrig, 'Mixed forests')

swissFit =  geostatsp::glgm(
    formula = lograin~ elev + landOrig,
    data=newdat, 
    covariates=list(elev = swissAltitude),
    grid=geostatsp::squareRaster(swissRain,Ncell), 
    family="gaussian", buffer=0,
    prior=list(sd=c(0.2, 0.5), range=c(100000,0.5)), 
    control.inla = list(strategy='gaussian'),
    control.family=list(hyper=list(prec=list(prior="loggamma", 
          param=c(.1, .1))))
)

swissFitR =  geostatsp::glgm(
    formula = lograin~ elev + landRel,
    data=newdat, 
    grid=geostatsp::squareRaster(swissRain,Ncell), 
    covariates=list(elev = swissAltitude, landRel = swissLandType),
    family="gaussian", buffer=0,
    prior=list(sd=c(0.2, 0.5), range=c(100000,0.5)), 
    control.inla = list(strategy='gaussian'),
    control.family=list(hyper=list(prec=list(prior="loggamma", 
          param=c(.1, .1))))
  )
@

<<covInDataLevelsResults1>>=
levels(newdat$landOrig)
levels(newdat$landRel)
@

<<covInDataLevelsResults2>>=
if(length(swissFit$parameters)) {
    levels(swissFit$inla$.args$data$landOrig)
    levels(swissFitR$inla$.args$data$landRel)
  }
@
<<covInDataLevelsResults3>>=
if(length(swissFit$parameters)) {
    knitr::kable(swissFit$parameters$summary[,c(1,3,5)], digits=3)
    knitr::kable(swissFitR$parameters$summary[,c(1,3,5)], digits=3)
}
@

covariates are in data, interactions
<<interactionsData>>=
newdat = swissRain
newdat$elev = extract(swissAltitude, swissRain, ID=FALSE)
@

<<interactionsRun>>=
swissFit =  geostatsp::glgm(
    formula = lograin~ elev : land,
    data=newdat, 
    grid=geostatsp::squareRaster(swissRain,Ncell), 
    covariates=list(land=swissLandType),
    family="gaussian", buffer=0,
    prior=list(sd=c(0.2, 0.5), range=c(100000,0.5)), 
    control.inla = list(strategy='gaussian'),
    control.family=list(hyper=list(prec=list(prior="loggamma", 
          param=c(.1, .1))))
)
if(length(swissFit$parameters)) {
  knitr::kable(swissFit$parameters$summary[,c(1,3,5)], digits=3)
}
@

<<interactions, fig.cap = 'interactions', fig.subcap = c('map','range')>>=
if(length(swissFit$parameters)) {
   plot(swissFit$raster[['predict.mean']])
    plot(swissBorder, add=TRUE)    

    matplot(
    swissFit$parameters$range$posterior[,'x'],
    swissFit$parameters$range$posterior[,c('y','prior')],
    lty=1, col=c('black','red'), type='l',
    xlab='range', ylab='dens')
}
@


categorical tests

<<categoricalData>>=
  data('loaloa')
  loaloa = unwrap(loaloa)
  ltLoa = unwrap(ltLoa)
  elevationLoa = unwrap(elevationLoa)
  eviLoa = unwrap(eviLoa)

  rcl = rbind(
    # wedlands and mixed forests to forest
    c(5,2),c(11,2),
# savannas to woody savannas
    c(9,8),
    # croplands and urban changed to crop/natural mosaid
    c(12,14),c(13,14))
ltLoaR = classify(ltLoa, rcl)
levels(ltLoaR) = levels(ltLoa)
    
elevationLoa = elevationLoa - 750
elevLow = min(elevationLoa, 0)
elevHigh = max(elevationLoa, 0)

eviLoa2 = (eviLoa - 1e7)/1e6

covList = list(elLow = elevLow, elHigh = elevHigh, 
    land = ltLoaR, evi=eviLoa2)
@

<<longTestsLoa>>=
 loaFit = geostatsp::glgm(
    y ~ 1 + land + evi + elHigh + elLow + 
      f(villageID, prior = 'pc.prec', param = c(log(2), 0.5),
       model="iid"),
    loaloa,
    Ncell, 
    covariates=covList, 
    family="binomial", Ntrials = loaloa$N,
    shape=2, buffer=25000,
    prior = list(
      sd=log(2), 
      range = 100*1000),
    control.inla = list(strategy='gaussian')
    )
@

<<longTestsTable>>=
if(length(loaFit$parameters)) {
  knitr::kable(loaFit$par$summary[,c(1,3,5)], digits=3)
} 
@

<<longTestsLoaPLot, fig.cap='categorical', fig.subcap = c('predict','range')>>=
if(length(loaFit$parameters)) {
  plot(loaFit$raster[['predict.exp']])

matplot(
      loaFit$parameters$range$posterior[,'x'],
      loaFit$parameters$range$posterior[,c('y','prior')],
    lty=1, col=c('black','red'), type='l',
    xlab='range', ylab='dens')
}
@


prior for observation standard deviation
<<LongTestsSwiss>>=  
swissFit =  geostatsp::glgm( formula="lograin",data=swissRain, grid=Ncell,
    covariates=swissAltitude, family="gaussian", buffer=20000,
    prior=list(sd=0.5, range=200000, sdObs=1), 
    control.inla = list(strategy='gaussian')
)
@

\section*{no data checks}

a model with little data, posterior should be same as prior

<<noData, fig.height=3, fig.width=4, fig.cap = 'no data, pc priors', fig.subcap = c('beta','sd','range','scale')>>=  
  
data2 = vect(cbind(c(1,0), c(0,1)),
    atts=data.frame(y=c(0,0), offset=c(-50,-50), x=c(-1,1)),
    crs = '+proj=merc')

  
resNoData = res = geostatsp::glgm(
  data=data2, grid=Ncell, 
    formula=y~1 + x+offset(offset), 
    prior = list(sd=0.5, range=0.1),
    family="poisson",
    buffer=0.5, 
    control.fixed=list(
      mean.intercept=0, prec.intercept=1,
      mean=0,prec=4),
    control.mode = list(theta = c(0.651, 1.61), restart=TRUE),
    control.inla = list(strategy='gaussian')
  )
  
if(length(res$parameters)) {
# beta
  plot(res$inla$marginals.fixed[['x']], col='blue', type='l',
    xlab='beta',lwd=3)
  xseq = res$inla$marginals.fixed[['x']][,'x']
  lines(xseq, dnorm(xseq, 0, 1/2),col='red',lty=2,lwd=3)
  legend("topright", col=c("blue","red"),lty=1,legend=c("prior","post'r"))
    
# sd
    matplot(
      res$parameters$sd$posterior[,'x'],
      res$parameters$sd$posterior[,c('y','prior')], 
      xlim = c(0, 4),
      type='l', col=c('red','blue'),xlab='sd',lwd=3, ylab='dens')
    legend("topright", col=c("blue","red"),lty=1,legend=c("prior","post'r"))
  
  
# range
    matplot(
      res$parameters$range$posterior[,'x'],
      res$parameters$range$posterior[,c('y','prior')], 
      xlim = c(0, 1.5),
      type='l', col=c('red','blue'),xlab='range',lwd=3, ylab='dens')
    legend("topright", col=c("blue","red"),lty=1,legend=c("prior","post'r"))

        matplot(
      res$parameters$scale$posterior[,'x'],
      res$parameters$scale$posterior[,c('y','prior')], 
      xlim = c(0, 2/res$parameters$summary['range','0.025quant']),
#      ylim = c(0, 10^(-3)), xlim = c(0,1000),
      type='l', col=c('red','blue'),xlab='scale',lwd=3, ylab='dens')
    legend("topright", col=c("red","blue"),lty=1,legend=c("post'r","prior"))
}
@

<<noDataQuantile, fig.height=3, fig.width=4, fig.cap = 'no data quantile priors', fig.subcap = c('intercept','sd','range','scale')>>=  


  resQuantile = res = geostatsp::glgm(
    data=data2, 
    grid=25, 
    formula=y~1 + x+offset(offset), 
    prior = list(
      sd=c(lower=0.2, upper=2), 
      range=c(lower=0.02, upper=0.5)),
    family="poisson", buffer=1, 
    control.fixed=list(
      mean.intercept=0, prec.intercept=1,
      mean=0,prec=4),
    control.inla = list(strategy='gaussian')
  )
  
if(length(res$parameters)) {
# beta
  plot(res$inla$marginals.fixed[['x']], col='blue', type='l',
    xlab='beta',lwd=3)
  xseq = res$inla$marginals.fixed[['x']][,'x']
  lines(xseq, dnorm(xseq, 0, 1/2),col='red',lty=2,lwd=3)
  legend("topright", col=c("blue","red"),lty=1,legend=c("prior","post'r"))
    
# sd
    matplot(
      res$parameters$sd$posterior[,'x'],
      res$parameters$sd$posterior[,c('y','prior')], 
      xlim = c(0, 4),
      type='l', col=c('red','blue'),xlab='sd',lwd=3, ylab='dens')
    legend("topright", col=c("blue","red"),lty=1,legend=c("prior","post'r"))
  
  
# range
    matplot(
      res$parameters$range$posterior[,'x'],
      res$parameters$range$posterior[,c('y','prior')], 
      xlim = c(0, 1.2*res$parameters$summary['range','0.975quant']),
#      xlim = c(0, 1), ylim = c(0,5),
      type='l', col=c('red','blue'),xlab='range',lwd=3, ylab='dens')
    legend("topright", col=c("red","blue"),lty=1,legend=c("post'r","prior"))

# scale
    matplot(
      res$parameters$scale$posterior[,'x'],
      res$parameters$scale$posterior[,c('y','prior')], 
      xlim = c(0, 2/res$parameters$summary['range','0.025quant']),
#      ylim = c(0, 10^(-3)), xlim = c(0,1000),
      type='l', col=c('red','blue'),xlab='scale',lwd=3, ylab='dens')
    legend("topright", col=c("red","blue"),lty=1,legend=c("post'r","prior"))
}
@




No data, legacy priors

<<noDataLegacy, fig.height=3, fig.width=4, fig.cap='No data, legacy priors', fig.subcap = c('intercept','beta','sd','range')>>=  

resLegacy = res = geostatsp::glgm(data=data2, 
    grid=20, 
    formula=y~1 + x+offset(offset), 
    priorCI = list(
      sd=c(lower=0.3,upper=0.5), 
      range=c(lower=0.25, upper=0.4)),
    family="poisson",
    buffer=0.5, 
    control.fixed=list(
      mean.intercept=0, 
      prec.intercept=1,
      mean=0, prec=4),
    control.inla = list(strategy='gaussian'),
    control.mode=list(theta=c(2, 2),restart=TRUE)
  )
  
if(length(res$parameters)) {
# intercept
  plot(res$inla$marginals.fixed[['(Intercept)']], col='blue', type='l',
    xlab='intercept',lwd=3)
  xseq = res$inla$marginals.fixed[['(Intercept)']][,'x']
  lines(xseq, dnorm(xseq, 0, 1),col='red',lty=2,lwd=3)
  legend("topright", col=c("blue","red"),lty=1,legend=c("prior","post'r"))
  
# beta
  plot(res$inla$marginals.fixed[['x']], col='blue', type='l',
    xlab='beta',lwd=3)
  xseq = res$inla$marginals.fixed[['x']][,'x']
  lines(xseq, dnorm(xseq, 0, 1/2),col='red',lty=2,lwd=3)
  legend("topright", col=c("blue","red"),lty=1,legend=c("prior","post'r"))
    
# sd
    matplot(
      res$parameters$sd$posterior[,'x'],
      res$parameters$sd$posterior[,c('y','prior')], 
      type='l', col=c('red','blue'),xlab='sd',lwd=3, ylab='dens')
    legend("topright", col=c("blue","red"),lty=1,legend=c("prior","post'r"))
  
  
# range
    matplot(
      res$parameters$range$posterior[,'x'],
      res$parameters$range$posterior[,c('y','prior')], 
      type='l', col=c('red','blue'),xlab='range',lwd=3, ylab='dens')
    legend("topright", col=c("blue","red"),lty=1,legend=c("prior","post'r"))
}
@


specifying spatial formula

<<spaceFormula, fig.cap='spatial formula provided', fig.subcap = c('one','two')>>=

swissRain$group = 1+rbinom(length(swissRain), 1, 0.5)
theGrid = geostatsp::squareRaster(swissRain, Ncell, buffer=10*1000)

swissFit = geostatsp::glgm(
    formula = rain ~ 1,
    data=swissRain, 
    grid=theGrid, 
    family="gaussian",
    spaceFormula = ~ f(space, model='matern2d', 
      nrow = nrow(theGrid), ncol = ncol(theGrid),
      nu = 1, replicate = group),
      control.inla = list(strategy='gaussian'),
)

if(length(swissFit$parameters)) {
  swissFit$rasterTwo = setValues(
    rast(swissFit$raster, nlyrs=2),
    as.matrix(swissFit$inla$summary.random$space[
      ncell(theGrid)+values(swissFit$raster[['space']]), 
      c('mean','0.5quant')]))
  plot(swissFit$raster[['random.mean']])

  plot(swissFit$rasterTwo[['mean']])
}
@


\end{document}
