geoR1 INTRODUCCION

8/13/2019 geoR1 INTRODUCCION

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geoR : Package for Geostatistical Data Analysis An illustrative session

Paulo J. Ribeiro Jr. & Peter J. DiggleLast update: 26 De! 2""#

http://www.leg.ufpr.br/geoR/geoRdoc/geoRintro.html

Cross-validation Getting started Exploratory Tools Parameter Estimation

Spatial Interpolation Bayesian analysis Simulation citing geoR

The packagegeoR provides unctions or geostatistical data analysis using theso t!are R "This document illustrates some #but not all $ $ o the capa%ilities o the package"

The o%&ective is to amiliarise the reader !ith thegeoR 's commands or dataanalysis and sho! some o the graphical outputs !hich can %e produced"The commands used here are &ust illustrative( providing %asic examples o the package handling")e did not attempt to per orm a de initive analysis o this data"

In !hat ollo!s the R commands are sho!n intypewriter fonts like this "

Typically( de ault arguments are used or the unction calls and the user isencouraged to inspect other arguments o the unctions" *or example( to see allthe arguments or the unctionvariog type+

args(variog)

The commands sho!n in this page are also availa%le in the ilegeoRintro"R "

)e re er to the geoR documentation or more details on the unctions included inthe packagegeoR "

%. 'AR'()G A * (+) A)D L+AD()G DA'A

, ter starting an R session( loadgeoR !ith the command+
http://www.leg.ufpr.br/geoR/geoRdoc/geoRintro.htmlhttp://www.leg.ufpr.br/geoR/geoRdoc/geoRintro.html#crosshttp://www.leg.ufpr.br/geoR/geoRdoc/geoRintro.html#startinghttp://www.leg.ufpr.br/geoR/geoRdoc/geoRintro.html#exploratoryhttp://www.leg.ufpr.br/geoR/geoRdoc/geoRintro.html#estimationhttp://www.leg.ufpr.br/geoR/geoRdoc/geoRintro.html#interpolationhttp://www.leg.ufpr.br/geoR/geoRdoc/geoRintro.html#bayesianhttp://www.leg.ufpr.br/geoR/geoRdoc/geoRintro.html#simulationhttp://www.leg.ufpr.br/geoR/geoRdoc/geoRintro.html#citinghttp://www.leg.ufpr.br/geoR/geoRdoc/geoRdoc/geoRintro.Rhttp://www.leg.ufpr.br/geoR/geoR.html#helphttp://www.leg.ufpr.br/geoR/geoR.html#helphttp://www.leg.ufpr.br/geoR/geoR.html#helphttp://www.leg.ufpr.br/geoR/geoR.html#helphttp://www.leg.ufpr.br/geoR/geoRdoc/geoRintro.htmlhttp://www.leg.ufpr.br/geoR/geoRdoc/geoRintro.html#crosshttp://www.leg.ufpr.br/geoR/geoRdoc/geoRintro.html#startinghttp://www.leg.ufpr.br/geoR/geoRdoc/geoRintro.html#exploratoryhttp://www.leg.ufpr.br/geoR/geoRdoc/geoRintro.html#estimationhttp://www.leg.ufpr.br/geoR/geoRdoc/geoRintro.html#interpolationhttp://www.leg.ufpr.br/geoR/geoRdoc/geoRintro.html#bayesianhttp://www.leg.ufpr.br/geoR/geoRdoc/geoRintro.html#simulationhttp://www.leg.ufpr.br/geoR/geoRdoc/geoRintro.html#citinghttp://www.leg.ufpr.br/geoR/geoRdoc/geoRdoc/geoRintro.Rhttp://www.leg.ufpr.br/geoR/geoR.html#help


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,&- , -*in . /0'1 0 . & 2&*a3 .21'2 /44 .22& 42 2

+%ata.s$**ary 5in. &st 6$. 5e%ian 5ean 'r% 6$. 5a3.

&.&01 . ' &.& / .2' &.0& .101

Elementscovariate, bor%ers and or $nits.* !ill %e alsosummari/ed i present in thegeo%ata o%&ect"

2. Plotting data locations and alues

The unctionplot.geo%ata sho!s a 0 x 0 display !ith data locations#top plots$ and dataversus coordinates #%ottom plots$" *or an o%&ecto the class"geo%ata" the plot is produced %y the command+

plot(s& )


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1otice that the top-right plot is produced using the packagescatterplot3d " I this package is not installed a histogramo the data !ill replace this plot"

The unctionpoints.geo%ata produces a plot sho!ing the datalocations" ,lternatively( points indicating the data locations can %eadded to a current plot"There are options to speci y point si/es( patterns and colors( !hichcan %e set to %e proportional to the data values or speci ied.uantiles"Some examples o graphical outputs are illustrated %y thecommands and corresponding plots as sho!n %elo!")e start saving the current graphical parameters"

par.ori 7 par(no.rea%only = TR89)par(*frow = c( , ))points(s& , 3lab = ":oor% ;", ylab = ":oor%


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#. */pirical ariogra/s

Empirical variograms are calculated using the unctionvariog "There are options or theclassical or modulus estimator"Results can %e returned as variogram clouds( %inned or smoothedvariograms"

clo$%& 7 variog(s& , option = "clo$%", *a3.%ist=&)clo$% 7 variog(s& , option = "clo$%", esti*ator.type ="*o%$l$s", *a3.%ist=&)bin& 7 variog(s& , $vec=se ( ,&,l=&&))bin 7 variog(s& , $vec=se ( ,&,l=&&), esti*ator.type=

"*o%$l$s") par(*frow=c( , ))plot(clo$%&, *ain = "classical esti*ator")plot(clo$% , *ain = "*o%$l$s esti*ator")plot(bin&, *ain = "classical esti*ator")plot(bin , *ain = "*o%$l$s esti*ator")par(par.ori)


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*urthermore( the points o the variogram clouds can %e grouped intoclasses o distances #2%ins2$ and displayed !ith a %ox-plot or each %in"bin& 7 variog(s& ,$vec = se ( ,&,l=&&), bin.clo$% = T)bin 7 variog(s& ,$vec = se ( ,&,l=&&), esti*ator.type ="*o%$l$s", bin.clo$% = T)

par(*frow = c(&, ))plot(bin&, bin.clo$% = T, *ain = "classical esti*ator")plot(bin , bin.clo$% = T, *ain = "*o%$l$s esti*ator")

par(par.ori)


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Theoretical and empirical variograms can %e plotted and visuallycompared" *or example( the igure %elo! sho!s the theoreticalvariogram model used to simulate the datas& and t!o estimatedvariograms"bin& 7 variog(s& , $vec = se ( ,&,l=&&))plot(bin&)lines.vario*o%el(cov.*o%el = "e3p", cov.pars = c(&, .'),n$gget = , *a3.%ist = &, lw% = ')s*ooth 7 variog(s& , option = "s*ooth", *a3.%ist = &,n.points = & , kernel = "nor*al", ban% = . )lines(s*ooth, type ="l", lty = )legen%( .4, .', c("e*pirical", "e3ponential *o%el","s*oothe%"), lty = c(&,&, ), lw% = c(&,',&))

3irectional variograms can also %e computed %y theunctionvariog using the arguments%irection and tolerance " *or


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example( to compute a variogram or the direction 45 degrees !iththe de ault tolerance angle #00"6 degrees$ the command !ould %e+vario0 7 variog(s& , *a3.%ist = &, %irection=pi>')plot(vario0 )title(*ain = e3pression(paste("%irectional, angle = ", 0 ?%egree)))

and the plot is sho!n on the le t panel o the igure %elo!"

*or a .uick computation in our directions !e can use theunctionvariog4 and the corresponding plot is sho!n on the right

panel o the next igure"

vario.4 7 variog4(s& , *a3.%ist = &)plot(vario.4, lw%= )

#. PARA3*'*R * '(3A'(+)

The model parameters can %e estimated+o by "eye" + trying di erent models over empirical variograms #using

the unctionlines.vario*o%el $(o by least squares fit of empirical variograms + !ith options or

ordinary #78S$ and !eighted #)8S$ least s.uares #using theunctionvariofit $(

o by likelihood based methods + !ith options or maximum likelihood#98$ and restricted maximum likelihood #RE98$ #using the

unctionlikfit $(


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o Bayesian methods are also implemented and !ill %e presented inSection 6 #using the unctionkrige.bayes $"

The ollo!ing commands sho! ho! to add a line !ith a variogram modelto a variogram plot"plot(variog(s& , *a3.%ist=&))lines.vario*o%el(cov.*o%el="e3p", cov.pars=c(&,.'), n$g= ,*a3.%ist=&)lines.vario*o%el(cov.*o%el="*at", cov.pars=c(.1/,. ), n$g= .&,kappa=&,*a3.%ist=&, lty= )lines.vario*o%el(cov.*o%el="sph", cov.pars=c(.1,.1),n$g= .&,*a3.%ist=&, lw%= )

In the parameter estimation unctionsvariofit and likfit the nuggete ect parameter can either %e estimated or set to a ixed value" The sameapplies or smoothness( anisotropy and trans ormation parameters" 7ptions

or taking trends into account are also included" Trends can %e speci ied as polynomial unctions o the coordinates and or linear unctions o givencovariates"

The commands %elo! sho!s models itted %y di erent methods !ithoptions or ixed or estimated nugget parameter" *eatures not illustratedhere include estimation o trends( anisotropy( smoothness and Box-Coxtrans ormation parameter"

@ itting *o%els with n$gget fi3e% to Bero*l 7 likfit(s& , ini = c(&, ./), fi3.n$gget = T)re*l 7 likfit(s& , ini = c(&, ./), fi3.n$gget = T, *etho% ="R5C")ols 7 variofit(bin&, ini = c(&, ./), fi3.n$gget = T,weights="e $al")wls 7 variofit(bin&, ini = c(&, ./), fi3.n$gget = T)


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@ itting *o%els with a fi3e% val$e for the n$gget*l.fn 7 likfit(s& , ini = c(&, ./), fi3.n$gget = T, n$gget =

.&/)re*l.fn 7 likfit(s& , ini = c(&, ./), fi3.n$gget = T, n$gget =

.&/, *etho% = "R5C")ols.fn 7 variofit(bin&,ini = c(&, ./), fi3.n$gget = T, n$gget =

.&/, weights="e $al")wls.fn 7 variofit(bin&, ini = c(&, ./), fi3.n$gget = T, n$gget =

.&/)

@ itting *o%els esti*ate% n$gget*l.n 7 likfit(s& , ini = c(&, ./), n$g = ./)re*l.n 7 likfit(s& , ini = c(&, ./), n$g = ./, *etho% = "R5C")ols.n 7 variofit(bin&, ini = c(&, ./), n$gget= ./,weights="e $al")wls.n 7 variofit(bin&, ini = c(&, ./), n$gget= ./)

@ Dow, plotting fitte% *o%els against e*pirical variogra*par(*frow = c(&,'))plot(bin&, *ain = e3pression(paste("fi3e% ", ta$E == )))

lines(*l, *a3.%ist = &)lines(re*l, lw% = , *a3.%ist = &)lines(ols, lty = , *a3.%ist = &)lines(wls, lty = , lw% = , *a3.%ist = &)legen%( ./, .',legen%=c("5C","R95C","!CF","GCF"),lty=c(&,&, , ),lw%=c(&, ,&, ),ce3= . )

plot(bin&, *ain = e3pression(paste("fi3e% ", ta$E == .&/)))lines(*l.fn, *a3.%ist = &)lines(re*l.fn, lw% = , *a3.%ist = &)lines(ols.fn, lty = , *a3.%ist = &)lines(wls.fn, lty = , lw% = , *a3.%ist = &)legen%( ./, .', legen%=c("5C","R95C","!CF","GCF"), lty=c(&,&, , ),lw%=c(&, ,&, ), ce3= . )

plot(bin&, *ain = e3pression(paste("esti*ate% ", ta$E )))lines(*l.n, *a3.%ist = &)lines(re*l.n, lw% = , *a3.%ist = &)lines(ols.n, lty = , *a3.%ist = &)lines(wls.n, lty = , lw% = , *a3.%ist = &)legen%( ./, .', legen%=c("5C","R95C","!CF","GCF"), lty=c(&,&, , ),lw%=c(&, ,&, ), ce3= . )

par(par.ori)


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Summary methods have %een !ritten to summari/e the resulting o%&ects"*or example( or the model !ith estimated nugget itted %y maximumlikelihood( typing+

*l.n

!ill produce the output+likfit# esti*ate% *o%el para*eters# beta ta$s sig*as phi

. 00 . . /& .&1

likfit# *a3i*ise% log likelihoo% = 1'./020

!hilst a more detailed summary is o%tained !ith+ s$**ary(*l.n)

F$**ary of the para*eter esti*ation

9sti*ation *etho%# *a3i*$* likelihoo%

Para*eters of the *ean co*ponent (tren%)# beta

. 00

Para*eters of the spatial co*ponent# correlation f$nction# e3ponential (esti*ate%) variance para*eter sig*as (partial sill) =

. /& (esti*ate%) cor. fct. para*eter phi (range para*eter) =

.&1 anisotropy para*eters# (fi3e%) anisotropy angle = ( %egrees ) (fi3e%) anisotropy ratio = &


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Para*eter of the error co*ponent# (esti*ate%) n$gget =

Transfor*ation para*eter# (fi3e%) Io3 :o3 para*eter = & (no transfor*ation)

5a3i*ise% Cikelihoo%# log.C n.para*s AJ: IJ:

1'./020 4. & /.&'2& &1/.//21

:all#likfit(geo%ata = s& , ini.cov.pars = c(&, ./), n$gget = ./)

T!o kinds o variogramenvelopes computed %y simulation are illustratedin the igure %elo!"The plot on the le t-hand side sho!s an envelope %ased on permutations o

the data values across the locations( i"e" envelopes %uilt under theassumption o no spatial correlation"The envelopes sho!n on the right-hand side are %ased on simulations roma given set o model parameters( in this example the parameter estimates

rom theWLS variogram it" This envelope sho!s the varia%ility o theempirical variogram"

env.*c 7 variog.*c.env(s& , obK.var=bin&)env.*o%el 7 variog.*o%el.env(s& , obK.var=bin&, *o%el=wls)

par(*frow=c(&, ))plot(bin&, envelope=env.*c)plot(bin&, envelope=env.*o%el)par(par.ori)


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Pro ile likelihoods #:-3 and 0-3$ are computed %y the unctionproflik ";ere !e sho! the pro ile likelihoods or the covariance parameters o themodel !ithout nugget e ect previously itted %ylikfit "

4AR)()G: R5))()G ' * )*,' 7+33A)D 7A) 8* '(3*97+) 53()G

prof 7 proflik(*l, geo%ata = s& , sill.val = se ( .41, , l=&&),range.val = se ( .&, ./ , l=&&), $ni.only = ACF9)

par(*frow=c(&,'))plot(prof, nlevels=&0)par(par.ori)

2. 7R+ 9 AL(DA'(+)

The unction3vali% per orms cross-validation either using theleaving-one-out strategy or using a di erent set o locations provided %y the user"*or the irst strategy( data points are removed one %y one and predicted %ykriging using the remaining data" The commands %elo! illustrates cross-validation or the models itted %y maximum likelihood and !eighted leasts.uares" In the irst t!o calls the model parameters remains the same orthe prediction at each location" In the next t!o calls the model parametersare re-estimated each time a point is removed rom the data-set" Graphicalresults are sho!n or the cross-validation results !here the leaving-one-out strategy com%ined !ith the !ls estimates or the parameters !as used"3v.*l 7 3vali%(s& , *o%el=*l)


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3v.wls 7 3vali%(s& , *o%el=wls)

4AR)()G: R5))()G ' * )*,' 7+33A)D 7A) 8* '(3*97+) 53()G3vR.*l 7 3vali%(s& , *o%el=*l, reest=TR89)3vR.wls 7 3vali%(s& , *o%el=wls, reest=TR89, variog.obK=bin&)

par(*fcol = c(/, ), *ar=c(',',./,./), *gp=c(&./,. , ))plot(3v.wls)par(par.ori)


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3. PA'(AL ()'*RP+LA'(+)


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Conventional geostatistical spatial interpolation #kriging $ can %e per ormed !ith options or+

o Simple kriging o Ordinary kriging

o Trend universal! kriging o #ternal trend kriging

There are additional options or Box-Cox trans ormation #and %acktrans ormation o the results$ and anisotropic models"Simulations can %e dra!n rom the resulting predictive distri%utions ire.uested"

,s a irst example consider the prediction at our locations la%eled$% &% '%( and indicated in the igure %elo!"

plot(s& +coor%s, 3li*=c( ,&. ), yli*=c( ,&. ), 3lab=":oor% ;",ylab=":oor%


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The output is a list including the predicted values #kc4+pre%ict $ and thekriging variances #kc4+krige.var $"

Consider no! a second example" The goal is to per orm prediction on agrid covering the area and to display the results" ,gain( !e use ordinarykriging" The commands are+@ %efining the gri%pre%.gri% 7 e3pan%.gri%(se ( ,&, l=/&), se ( ,&, l=/&))@ kriging calc$lationskc 7 krige.conv(s& , loc = pre%.gri%, krige = krige.control(obK.*= *l))@ %isplaying pre%icte% val$esi*age(kc, loc = pre%.gri%, col=gray(se (&, .&,l=' )), 3lab=":oor%;", ylab=":oor%


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,s an example consider a model !ithout nugget and including uncertaintyin the mean( sill and range parameters" Prediction at the our locationsindicated a%ove is per ormed %y typing a command like+

WARNING: RUNNING THE NEXT COMMAN CAN !E TIME"CON#UMING

bsp4 7 krige.bayes(s& , loc = loci, prior =prior.control(phi.%iscrete = se ( ,/,l=& &), phi.prior="rec"),o$tp$t=o$tp$t.control(n.post=/ ))

;istograms sho!ing posterior distri%ution or the model parameters can %e plotted %y typing+

par(*frow=c(&,'), *ar=c(',',./,./), *gp=c( ,&, ))hist(bsp4+posterior+sa*ple+beta, *ain="", 3lab=e3pression(beta),prob=T)hist(bsp4+posterior+sa*ple+sig*as , *ain="",3lab=e3pression(sig*aE ), prob=T)hist(bsp4+posterior+sa*ple+phi, *ain="", 3lab=e3pression(phi),prob=T)par(par.ori)


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lines(bsp4, *a3.%ist = &. , s$** = "*o%e", post="par",lw% = , lty= )legen%( . /, .4, legen% = c("variogra* posterior *ean", "variogra*posterior *e%ian", "para*eters posterior *o%e"), lty = c(&, , ),lw% = c(&,&, ), ce3 = .1)

The next igure sho!s predictive distri%utions at the our selectedlocations"3ashed lines sho! Gaussian distri%utions !ith mean and variance given %y results o ordinary kriging o%tained in Section >"The ull lines correspond to the Bayesian prediction" The plot sho!sresults o density estimation using samples rom the predictivedistri%utions"

par(*frow=c( , ), *ar=c(',',./,./), *gp=c(&./,. , )) for(i in )L kp3 7 se (kc4+pre% i- '?s rt(kc4+krige.var i-),kc4+pre% i- M'?s rt(kc4+krige.var i-), l=& )

kpy 7 %nor*(kp3, *ean=kc4+pre% i-, s%=s rt(kc4+krige.var i-))bp 7 %ensity(bsp4+pre%ic+si* i,-)r3 7 range(c(kp3, bp+3))ry 7 range(c(kpy, bp+y))plot(cbin%(r3, ry), type="n", 3lab=paste("Cocation", i),

ylab="%ensity", 3li*=c( 4, 4), yli*=c( ,&.&))lines(kp3, kpy, lty= )lines(bp)

Npar(par.ori)


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Consider no!( under the same model assumptions( o%taining simulationsrom the predictive distri%utions on a grid o points covering the area" The

commands to de ine the grid and per orm Bayesian prediction are+

pre%.gri% 7 e3pan%.gri%(se ( ,&, l='&), se ( ,&, l='&))

4AR)()G: R5))()G ' * )*,' 7+33A)D 7A) 8* '(3*97+) 53()Gbsp 7 krige.bayes(s& , loc = pre%.gri%, prior =

prior.control(phi.%iscrete = se ( ,/,l=/&)),o$tp$t=o$tp$t.control(n.pre%ictive= ))

9aps !ith the summaries and simulations o the predictive distri%utioncan %e plotted as ollo!s"

par(*frow=c( , ))


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i*age(bsp, loc = pre%.gri%, *ain = "pre%icte%",col=gray(se (&, .&,l=' )))i*age(bsp, val ="variance", loc = pre%.gri%,

*ain = "pre%iction variance", col=gray(se (&, .&,l=' )))i*age(bsp, val = "si*$lation", n$*ber.col = &, loc = pre%.gri%,

*ain = "a si*$lation fro*Onthe pre%ictive %istrib$tion",col=gray(se (&, .&,l=' )))i*age(bsp, val = "si*$lation", n$*ber.col = ,loc = pre%.gri%,

*ain = "another si*$lation fro* On the pre%ictive%istrib$tion", col=gray(se (&, .&,l=' )))par(par.ori)

)ote: *urther examples or the unctionkrige.bayes are given in theilee#amples)krige)bayes)*
http://www.leg.ufpr.br/geoR/geoRdoc/examples.krige.bayes.Rhttp://www.leg.ufpr.br/geoR/geoRdoc/examples.krige.bayes.R


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$. i/ulation of Gaussian Rando/ ;ields

The unctiongrf generates simulations o Gaussian random ields onregular or irregular sets o locationsSome o its unctionality is illustrated %y the next commands"

si*& 7 grf(& , cov.pars=c(&, . /))points.geo%ata(si*&, *ain="si*$late% locations an% val$es")plot(si*&, *a3.%ist=&, *ain="tr$e an% e*pirical variogra*s")

si* 7 grf(44&, gri%="reg", cov.pars=c(&, . /))i*age(si* , *ain="a s*all ish si*$lation", col=gray(se (&, .&,

l=' )))


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si*' 7 grf(4 4 &, gri%="reg", cov.pars=c(& , . ), *et="circ")i*age(si*', *ain="si*$lation on a fine gri%", col=gray(se (&, .&,l=' )))

17TE+ !e recommend the packageRan%o* iel%s or a morecomprehensive implementation or simulation o Gaussian Random *ields"
http://cran.r-project.org/src/contrib/PACKAGES.html#RandomFieldshttp://cran.r-project.org/src/contrib/PACKAGES.html#RandomFields


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%. 7iting geoR

The unctioncite.geoR() sho!s in ormation on ho! to cite geoR in pu%lications" cite.geoR()

To cite geoR in p$blications, $se

RJI9JR! r., P. . Q JSSC9, P. . ( &) geoR# A package for geostatistical analysis. R D9GF, ol &, Do , &/ &1. JFFD &0 2'0'&.

Please cite geoR when $sing it for %ata analysisU

A IibTe; entry for CaTe; $sers is

VArticleL, title = LLgeoRN# a package for geostatisticalanalysisN, a$thor = LRibeiro r., P. . an% iggle, P. .N, Ko$rnal = LR D9GFN, year = L &N, vol$*e = L&N, n$*ber = L N, pages = L&/ &1N, issn = L&0 2 '0'&N, $rl = Lhttp#>>cran.R proKect.org>%oc>RnewsN N

Site maintained %y+Paulo ?" Ri%eiro ?r" #Paulo"Ri%eiro@est"u pr"%r $

'utorials on t0e usage of t0e package geoR%. *


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=" i/ulating fro/ t0e generalised linear geostatistical /odel "

#. (llustrating geostatistical calculations step by step

:" *it0 co ariates + example on ho! to construct a covariateon the prediction points to %e used in the kriging calculations"

6" =riging in presence of co ariates: a note + comments on the usageo the arguments trend)l and trend)d "

?. Generic tutorials

1. A traditional geostatistical data analysis + example o the %asicsteps or a simple geostatistical analysis"

0" Some commands or a standard geostatistical analysis: illustratethe package syntax or exploratory analysis #including variograms$( parameter estimation and kriging prediction"

3. An introduction to geoR + !e% page !ith the main unctionality othe package geoR "

>" 7o//ands file !ith contents sho!n during a pratical sessionduring a short course on geostatistics"

6" A basic e


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0" (nference and prediction for t0e bi ariate Gaussian co//onco/ponent /odel +

. +t0er data9sets Bin addition to t0e ones already included in t0epackageC

Data this is a directory !ith data-sets used in some o the tutorials" Typically there are t!o iles corresponding to each data set+

o iles !ith extension .dat are the data iles"o iles !ith extension .t


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statistics and co/putational /et0ods " 8ecture notes in statistics( vol":H=( p" >=-D4( Springer"

S1 88( T"( RIBEIR7 ?r( P"?" RJ3I1( ;" 1 ne2 conceptual model forregional spatio-temporal dynamics% the substrate-tracking metapopulation

model) #su%mitted$ *EJE1( 8"( RIBEIR7 ?r( P"?"( 3e S9E3T( *" 3IGG8E( P"?"

#0550$ Bayesian methodology to stochastic capture 5one determination3conditioning on transmissivity measurements " )ater Resources Research(=D # $+ ::4>-::H>"

RIBEIR7 ?R"( P"?" and 3IGG8E( P"?" #055:$ geoR+ , package orgeostatistical analysis" *-6 WS ol :( 1o 0" ISS1 :45 -=4=:"do!nload+http+ cran"r-pro&ect"org doc Rne!s

RIBEIR7 ?R"( P"?" and BR7)1( P"E" #055:$ Some !ords on the R pro&ect"The 4SB1 Bulletin ol :( 1o D #9arch$"do!nload+ *isba)pdf

C;RISTE1SE1( 7"*"( 3IGG8E( P"?" ,13 RIBEIR7 ?R( P"?"#055:$" 1nalysing positive-valued spatial data3 the transformed ,aussianmodel) 4n 9onestie/( P"( ,llard( 3" and *roidevaux #eds$(Geo*) ((( 9Geostatistics for en iron/ental applications " Kuantitative Geology andGeostatistics( Llu!er Series(%%( 0DH--0 D"abstract

3IGG8E( P"?" ,13 RIBEIR7 ?R( P"?" #0555$"3odel9basedgeostatistics " Caxam%u+ ,ssociaMNo Brasileira de EstatOstica" #:>SI1,PE - SimpQsio 1acional de Pro%a%ilidade e EstatOstica$"

preface

3IGG8E( P"?" ,13 RIBEIR7 ?R( P"?" #055:$" Bayesian in erence inGaussian model-%ased geostatistics",eographical and nvironmental

7odelling #to appear$abstract

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RIBEIR7 ?R( P"?" ,13 3IGG8E( P"?" #: $" geoS3 1 geostatisticallibrary for S-/L8S) Technical report ST- -5 ( 3ept o 9aths and Stats(8ancaster


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determined %y the scientist himsel " *or the latter the locations are given %y astochastic point process" In general it is GeostGeostatistical data versus point process data+ analysis o second-order characteristicsatistical data versus point process data+ analysis o second-order characteristics not possi%le to extend agiven marked point process to a random ield %ecause o the interactions amongthe locations and the marks o the point process" ;o!ever( such an extension is possi%le in the so called random ield model !hich is there ore o particularinterest in data analysis as a re erence model" Second-order characteristicsdescri%e the association %et!een the random varia%les U#sF:$ and U#sF0$located at the locations sF: and sF0 " Kuantities like pair correlation( markcorrelation and mark variogram unctions are use ul in order to assess thesecond-order characteristics o marked point processes( !hilecovariance correlation unctions and the variogram are commonly used or therandom ields" The goal o this paper is to analy/e the practical implications oall the a%ove mentioned characteristics using examples rom ecology and( ingeneral( rom environmental science ields" Comparisons %et!een statistics in thegeostatistical and the point process context are developed"

:ey2ords3 geostatistics% model based inference% Bayesian inference% spatialinterpolation)

#:$ 8ancaster


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Abstract

This paper considers data /F:( ldots(/Fn assumed to stem rom a reali/ation oa spatial process U and collected at sites sF:( ldots(sFn " The random ieldand the marked point process are t!o kinds o spatial processes" The ormer isde ined in every point o the area o interest and the sample positions can %edetermined %y the scientist himsel " *or the latter the locations are given %y astochastic point process" In general it is not possi%le to extend a given marked point process to a random ield %ecause o the interactions among the locationsand the marks o the point process" ;o!ever( such an extension is possi%le in theso called random ield model !hich is there ore o particular interest in dataanalysis as a re erence model" Second-order characteristics descri%e theassociation %et!een the random varia%les U#sF:$ and U#sF0$ located at thelocations sF: and sF0 " Kuantities like pair correlation( mark correlation andmark variogram unctions are use ul in order to assess the second-ordercharacteristics o marked point processes( !hile covariance correlation unctionsand the variogram are commonly used or the random ields" The goal o this paper is to analy/e the practical implications o all the a%ove mentionedcharacteristics using examples rom ecology and( in general( rom environmentalscience ields" Comparisons %et!een statistics in the geostatistical and the point process context are developed"

:ey2ords3 second-order characteristics% mark correlation function% geostatistics%marked point process)

#:$


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Bto be presented in t0e %?F ()AP* 2?92 July 2""" 7a


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!ith explicit stochastic models rarely declared and( conse.uently( little use madeo the likelihood-%ased methods o in erence !hich are central to modernstatistics"

3iggle( 9oyeed and Ta!n #: D$ used the phrasemodel-based geostatistics todescri%e an approach to geostatistical pro%lems %ased on the application o

ormal statistical methods under an explicitly assumed stochastic model" Thiscourse takes the same point o vie!"

The course is an applied statistical counterpart to 9ichael Stein's excellent %ook#Stein( : $( !hich gives a rigorous mathematical theory o kriging"

)e illustrate the methodology %y applying it to real data-sets" )e have !ritten ali%rary #geoF $ o 9PL5 unctions( and itsR counterpart #geoR $ to implementthe methods descri%ed in the course" This so t!are( together !ith the data-setsused in the course( is reely availa%le rom the !e%-address+http#>>www.*aths.lancs.ac.$k>W%iggle

*inally( !e thank ,BE #,ssociaMNo Brasileira de EstatOstica$ or the opportunityto present our ideas at the :>X SI1,PE and our colleagues in 8ancaster(


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*or iles in this ormat you !ill need to #i$ do!nload one o the iles %elo!V #ii$ uncompress it usingWin5ip #)indo!s$or gun5ip #8inux pages per sheeto compressed "ps ile !ith0 pages per sheet

o compressed "ps ile !ith: page per sheet

)otes for t0e geoR&geo# tutorial

The tutorial session !ill %e %ased on a Tecnical Report descri%ing the so t!are(availa%le incompressed postscript )ps)g5! or pdf ormat" Select one o the

options %elo! or do!nload":" pd ormatThis ile can %e visualised using the 1dobe 1crobat reader

o click to do!nload the ile in pd ormat

0" Compressed postscript ormat"

*or iles in this ormat you !ill need to #i$ do!nload one o the iles %elo!V #ii$ uncompress it usingWin5ip #)indo!s$or gun5ip #8inux


34/36

Bto be presented in t0e %?F ()AP* 2?92 July 2""" 7a


35/36

!ith explicit stochastic models rarely declared and( conse.uently( little use madeo the likelihood-%ased methods o in erence !hich are central to modernstatistics"

3iggle( 9oyeed and Ta!n #: D$ used the phrasemodel-based geostatistics todescri%e an approach to geostatistical pro%lems %ased on the application o

ormal statistical methods under an explicitly assumed stochastic model" Thiscourse takes the same point o vie!"

The course is an applied statistical counterpart to 9ichael Stein's excellent %ook#Stein( : $( !hich gives a rigorous mathematical theory o kriging"

)e illustrate the methodology %y applying it to real data-sets" )e have !ritten ali%rary #geoF $ o 9PL5 unctions( and itsR counterpart #geoR $ to implementthe methods descri%ed in the course" This so t!are( together !ith the data-setsused in the course( is reely availa%le rom the !e%-address+http#>>www.*aths.lancs.ac.$k>W%iggle

*inally( !e thank ,BE #,ssociaMNo Brasileira de EstatOstica$ or the opportunityto present our ideas at the :>X SI1,PE and our colleagues in 8ancaster(


36/36

*or iles in this ormat you !ill need to #i$ do!nload one o the iles %elo!V #ii$ uncompress it usingWin5ip #)indo!s$or gun5ip #8inux pages per sheeto compressed "ps ile !ith0 pages per sheet

o compressed "ps ile !ith: page per sheet

)otes for t0e geoR&geo# tutorial

The tutorial session !ill %e %ased on a Tecnical Report descri%ing the so t!are(availa%le incompressed postscript )ps)g5! or pdf ormat" Select one o the

options %elo! or do!nload":" pd ormatThis ile can %e visualised using the 1dobe 1crobat reader

o click to do!nload the ile in pd ormat

0" Compressed postscript ormat"

*or iles in this ormat you !ill need to #i$ do!nload one o the iles %elo!V #ii$ uncompress it usingWin5ip #)indo!s$or gun5ip #8inux

geoR1 INTRODUCCION

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