An introduction to the package geoR

geoR : Package for Geostatistical Data Analysis
An illustrative session

Paulo J. Ribeiro Jr. & Peter J. Diggle
Last update: 26/Dez/2003

Cross-validation Getting started Exploratory Tools Parameter Estimation
Spatial Interpolation Bayesian analysis Simulation citing geoR

The package geoR provides functions for geostatistical data analysis using the software R.
This document illustrates some (but not all !) of the capabilities of the package.

The objective is to familiarise the reader with the geoR's commands for data analysis and show some of the graphical outputs which can be produced.
The commands used here are just illustrative, providing basic examples of the package handling.
We did not attempt to perform a definitive analysis of this data.

In what follows the R commands are shown in

typewriter fonts like this.

Typically, default arguments are used for the function calls and the user is encouraged to inspect other arguments of the functions. For example, to see all the arguments for the function variog type:


The commands shown in this page are also available in the file geoRintro.R.

We refer to the geoR documentation for more details on the functions included in the package geoR.

  2. After starting an R session, load geoR with the command:


    If the installation directory for the package is the default location for R packages, type:

    library(geoR, lib.loc="PATH_TO_geoR")

    where "PATH_TO_geoR" is the path to the directory where geoR was installed. If the package is loaded correctly the following message will be displayed:

    geoR: functions for geostatistical data analysis
    geoR version 1.3-17 is now loaded

    Typically, data are stored as an object (a list) of the class "geodata".
    An object of this class contains at least the coordinates of data locations and the data values.

    Click for information on how to read data from an ASCII (text) file.
    We refer to the documentation for the functions as.geodata and read.geodata for more information on how to import/convert data and on the definitions for the class "geodata".

    For the examples included in this document we use the data set s100 included in the geoR distribution.
    To load this data type:


    1. A quick summary for the geodata object
    2. A quick summary of the data can be obtained typing


      which will return a summary of the coordinates and data values like this

                 [,1]       [,2]
      min 0.005638006 0.01091027
      max 0.983920544 0.99124979
         Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
      -1.1680  0.2730  1.1050  0.9307  1.6100  2.8680

      Elements covariate, borders and or units.m will be also summarized if present in the geodata object.

    3. Plotting data locations and values
    4. The function plot.geodata shows a 2 x 2 display with data locations (top plots) and data versus coordinates (bottom plots). For an object of the class "geodata" the plot is produced by the command:


      Notice that the top-right plot is produced using the package scatterplot3d. If this package is not installed a histogram of the data will replace this plot.

      The function points.geodata produces a plot showing the data locations. Alternatively, points indicating the data locations can be added to a current plot.
      There are options to specify point sizes, patterns and colors, which can be set to be proportional to the data values or specified quantiles.
      Some examples of graphical outputs are illustrated by the commands and corresponding plots as shown below.
      We start saving the current graphical parameters.

      par.ori <- par(no.readonly = TRUE)
      par(mfrow = c(2,2))
      points(s100, xlab = "Coord X", ylab = "Coord Y")	
      points(s100, xlab = "Coord X", ylab = "Coord Y", pt.divide = "rank.prop")
      points(s100, xlab = "Coord X", ylab = "Coord Y", cex.max = 1.7, 
             col = gray(seq(1, 0.1, l=100)), pt.divide = "equal")
      points(s100, pt.divide = "quintile", xlab = "Coord X", ylab = "Coord Y")

    5. Empirical variograms
    6. Empirical variograms are calculated using the function variog. There are options for the classical or modulus estimator.
      Results can be returned as variogram clouds, binned or smoothed variograms.

      cloud1 <- variog(s100, option = "cloud", max.dist=1)
      cloud2 <- variog(s100, option = "cloud", estimator.type = "modulus", max.dist=1)
      bin1 <- variog(s100, uvec=seq(0,1,l=11))
      bin2  <- variog(s100, uvec=seq(0,1,l=11), estimator.type= "modulus")
      plot(cloud1, main = "classical estimator")
      plot(cloud2, main = "modulus estimator")
      plot(bin1, main = "classical estimator")
      plot(bin2, main = "modulus estimator")

      Furthermore, the points of the variogram clouds can be grouped into classes of distances ("bins") and displayed with a box-plot for each bin.

      bin1 <- variog(s100,uvec = seq(0,1,l=11), = T)
      bin2 <- variog(s100,uvec = seq(0,1,l=11), estimator.type = "modulus", = T)
      par(mfrow = c(1,2))
      plot(bin1, = T, main = "classical estimator")
      plot(bin2, = T, main = "modulus estimator")

      Theoretical and empirical variograms can be plotted and visually compared. For example, the figure below shows the theoretical variogram model used to simulate the data s100 and two estimated variograms.

      bin1 <- variog(s100, uvec = seq(0,1,l=11))
      lines.variomodel(cov.model = "exp", = c(1,0.3), nugget = 0, max.dist = 1,  lwd = 3)
      smooth <- variog(s100, option = "smooth", max.dist = 1, n.points = 100, kernel = "normal", band = 0.2)
      lines(smooth, type ="l", lty = 2)
      legend(0.4, 0.3, c("empirical", "exponential model", "smoothed"), lty = c(1,1,2), lwd = c(1,3,1))

      Directional variograms can also be computed by the function variog using the arguments direction and tolerance. For example, to compute a variogram for the direction 60 degrees with the default tolerance angle (22.5 degrees) the command would be:

      vario60 <- variog(s100, max.dist = 1, direction=pi/3) 
      title(main = expression(paste("directional, angle = ", 60 * degree)))	    

      and the plot is shown on the left panel of the figure below.

      For a quick computation in four directions we can use the function variog4 and the corresponding plot is shown on the right panel of the next figure.

      vario.4 <- variog4(s100, max.dist = 1)
      plot(vario.4, lwd=2)

  5. The model parameters can be estimated:

    The following commands show how to add a line with a variogram model to a variogram plot.

    plot(variog(s100, max.dist=1))
    lines.variomodel(cov.model="exp",,.3), nug=0, max.dist=1)
    lines.variomodel(cov.model="mat",,.2), nug=0.1, kappa=1,max.dist=1, lty=2)
    lines.variomodel(cov.model="sph",,.8), nug=0.1,max.dist=1, lwd=2)

    In the parameter estimation functions variofit and likfitthe nugget effect parameter can either be estimated or set to a fixed value. The same applies for smoothness, anisotropy and transformation parameters. Options for taking trends into account are also included. Trends can be specified as polynomial functions of the coordinates and/or linear functions of given covariates.

    The commands below shows models fitted by different methods with options for fixed or estimated nugget parameter. Features not illustrated here include estimation of trends, anisotropy, smoothness and Box-Cox transformation parameter.

    # Fitting models with nugget fixed to zero
    ml <- likfit(s100, ini = c(1,0.5), fix.nugget = T)
    reml <- likfit(s100, ini = c(1,0.5), fix.nugget = T, method = "RML")
    ols <- variofit(bin1, ini = c(1,0.5), fix.nugget = T, weights="equal")
    wls <- variofit(bin1, ini = c(1,0.5), fix.nugget = T)
    # Fitting models with a fixed value for the nugget
    ml.fn <- likfit(s100, ini = c(1,0.5), fix.nugget = T, nugget = 0.15)
    reml.fn <- likfit(s100, ini = c(1,0.5), fix.nugget = T, nugget = 0.15, method = "RML")
    ols.fn <- variofit(bin1,ini = c(1,0.5), fix.nugget = T, nugget = 0.15, weights="equal")
    wls.fn <- variofit(bin1, ini = c(1,0.5), fix.nugget = T, nugget = 0.15)
    # Fitting models estimated nugget
    ml.n <- likfit(s100, ini = c(1,0.5), nug = 0.5)
    reml.n <- likfit(s100, ini = c(1,0.5), nug = 0.5, method = "RML")
    ols.n <- variofit(bin1, ini = c(1,0.5), nugget=0.5, weights="equal")
    wls.n <- variofit(bin1, ini = c(1,0.5), nugget=0.5)
    # Now, plotting fitted models against empirical variogram
    par(mfrow = c(1,3))
    plot(bin1, main = expression(paste("fixed ", tau^2 == 0)))
    lines(ml, max.dist = 1)
    lines(reml, lwd = 2, max.dist = 1)
    lines(ols, lty = 2, max.dist = 1)
    lines(wls, lty = 2, lwd = 2, max.dist = 1)
    legend(0.5, 0.3, legend=c("ML","REML","OLS","WLS"),lty=c(1,1,2,2),lwd=c(1,2,1,2), cex=0.7)
    plot(bin1, main = expression(paste("fixed ", tau^2 == 0.15)))
    lines(ml.fn, max.dist = 1)
    lines(reml.fn, lwd = 2, max.dist = 1)
    lines(ols.fn, lty = 2, max.dist = 1)
    lines(wls.fn, lty = 2, lwd = 2, max.dist = 1)
    legend(0.5, 0.3, legend=c("ML","REML","OLS","WLS"), lty=c(1,1,2,2), lwd=c(1,2,1,2), cex=0.7)
    plot(bin1, main = expression(paste("estimated  ", tau^2)))
    lines(ml.n, max.dist = 1)
    lines(reml.n, lwd = 2, max.dist = 1)
    lines(ols.n, lty = 2, max.dist = 1)
    lines(wls.n, lty =2, lwd = 2, max.dist = 1)
    legend(0.5, 0.3, legend=c("ML","REML","OLS","WLS"), lty=c(1,1,2,2), lwd=c(1,2,1,2), cex=0.7)

    Summary methods have been written to summarize the resulting objects.
    For example, for the model with estimated nugget fitted by maximum likelihood, typing:


    will produce the output:
    likfit: estimated model parameters:
       beta   tausq sigmasq     phi 
     0.7766  0.0000  0.7517  0.1827 
    likfit: maximised log-likelihood = -83.5696
    whilst a more detailed summary is obtained with:
    > summary(ml.n)
    Summary of the parameter estimation
    Estimation method: maximum likelihood 
    Parameters of the mean component (trend):
    Parameters of the spatial component:
       correlation function: exponential
          (estimated) variance parameter sigmasq (partial sill) =  0.7517
          (estimated) cor. fct. parameter phi (range parameter)  =  0.1827
       anisotropy parameters:
          (fixed) anisotropy angle = 0  ( 0 degrees )
          (fixed) anisotropy ratio = 1
    Parameter of the error component:
          (estimated) nugget =  0
    Transformation parameter:
          (fixed) Box-Cox parameter = 1 (no transformation)
    Maximised Likelihood:
       log.L n.params      AIC      BIC 
    -83.5696   4.0000 175.1391 185.5598 
    likfit(geodata = s100, = c(1, 0.5), nugget = 0.5)

    Two kinds of variogram envelopes computed by simulation are illustrated in the figure below.
    The plot on the left-hand side shows an envelope based on permutations of the data values across the locations, i.e. envelopes built under the assumption of no spatial correlation.
    The envelopes shown on the right-hand side are based on simulations from a given set of model parameters, in this example the parameter estimates from the WLS variogram fit. This envelope shows the variability of the empirical variogram. <-, obj.var=bin1)
    env.model <- variog.model.env(s100, obj.var=bin1, model=wls)
    plot(bin1, envelope=env.model)

    Profile likelihoods (1-D and 2-D) are computed by the function proflik. Here we show the profile likelihoods for the covariance parameters of the model without nugget effect previously fitted by likfit.


    prof <- proflik(ml, geodata = s100, sill.val = seq(0.48, 2, l=11),
    		range.val = seq(0.1, 0.52, l=11), uni.only = FALSE)
    plot(prof, nlevels=16)

  7. The function xvalid performs cross-validation either using the leaving-one-out strategy or using a different set of locations provided by the user. For the first strategy, data points are removed one by one and predicted by kriging using the remaining data. The commands below illustrates cross-validation for the models fitted by maximum likelihood and weighted least squares. In the first two calls the model parameters remains the same for the prediction at each location. In the next two calls the model parameters are re-estimated each time a point is removed from the data-set. Graphical results are shown for the cross-validation results where the leaving-one-out strategy combined with the wls estimates for the parameters was used. <- xvalid(s100, model=ml)
    xv.wls <- xvalid(s100, model=wls)
    xvR.wls <- xvalid(s100, model=wls, reest=TRUE, variog.obj=bin1)
    par(mfcol = c(5,2), mar=c(3,3,.5,.5), mgp=c(1.5,.7,0))

  9. Conventional geostatistical spatial interpolation (kriging) can be performed with options for:

    There are additional options for Box-Cox transformation (and back transformation of the results) and anisotropic models.
    Simulations can be drawn from the resulting predictive distributions if requested.

    As a first example consider the prediction at four locations labeled 1, 2, 3, 4 and indicated in the figure below.

    plot(s100$coords, xlim=c(0,1.2), ylim=c(0,1.2), xlab="Coord X", ylab="Coord Y")
    loci <- matrix(c(0.2, 0.6, 0.2, 1.1, 0.2, 0.3, 1.0, 1.1), ncol=2)
    text(loci, as.character(1:4), col="red")
    polygon(x=c(0,1,1,0), y=c(0,0,1,1), lty=2)

    The command to perform ordinary kriging using the parameters estimated by weighted least squares with nugget fixed to zero would be:

    	    kc4 <- krige.conv(s100, locations = loci, krige = krige.control(obj.m = wls))

    The output is a list including the predicted values (kc4$predict) and the kriging variances (kc4$krige.var).

    Consider now a second example. The goal is to perform prediction on a grid covering the area and to display the results. Again, we use ordinary kriging. The commands are:

    # defining the grid
    pred.grid <-  expand.grid(seq(0,1, l=51), seq(0,1, l=51))
    # kriging calculations
    kc <- krige.conv(s100, loc = pred.grid, krige = krige.control(obj.m = ml))
    # displaying predicted values
    image(kc, loc = pred.grid, col=gray(seq(1,0.1,l=30)), xlab="Coord X", ylab="Coord Y")

  11. Bayesian analysis for Gaussian models is implemented by the function krige.bayes. It can be performed for different "degrees of uncertainty", meaning that model parameters can be treated as fixed or random.

    As an example consider a model without nugget and including uncertainty in the mean, sill and range parameters. Prediction at the four locations indicated above is performed by typing a command like:

    bsp4 <- krige.bayes(s100, loc = loci, prior = prior.control(phi.discrete = seq(0,5,l=101), phi.prior="rec"), output=output.control(

    Histograms showing posterior distribution for the model parameters can be plotted by typing:

    par(mfrow=c(1,3), mar=c(3,3,.5,.5), mgp=c(2,1,0))
    hist(bsp4$posterior$sample$beta, main="", xlab=expression(beta), prob=T)
    hist(bsp4$posterior$sample$sigmasq, main="", xlab=expression(sigma^2), prob=T)
    hist(bsp4$posterior$sample$phi, main="", xlab=expression(phi), prob=T)

    Using summaries of these posterior distributions (means, medians or modes) we can check the "estimated Bayesian variograms" against the empirical variogram, as shown in the next figure. Notice that it is also possible to compare these estimates with other fitted variograms such as the ones computed in Section 3.

    plot(bin1, ylim = c(0,1.5))
    lines(bsp4, max.dist = 1.2, summ = mean)
    lines(bsp4, max.dist = 1.2, summ = median, lty = 2)
    lines(bsp4, max.dist = 1.2, summ = "mode", post="par",lwd = 2, lty = 2)
    legend(0.25, 0.4, legend = c("variogram posterior mean", "variogram posterior median", "parameters posterior mode"), lty = c(1,2,2), lwd = c(1,1,2), cex = 0.8)

    The next figure shows predictive distributions at the four selected locations.
    Dashed lines show Gaussian distributions with mean and variance given by results of ordinary kriging obtained in Section 4.
    The full lines correspond to the Bayesian prediction. The plot shows results of density estimation using samples from the predictive distributions.

    par(mfrow=c(2,2), mar=c(3,3,.5,.5), mgp=c(1.5,.7,0))
       for(i in 1:4){
            kpx <- seq(kc4$pred[i] - 3*sqrt(kc4$krige.var[i]), kc4$pred[i] +3*sqrt(kc4$krige.var[i]), l=100)
    	kpy <- dnorm(kpx, mean=kc4$pred[i], sd=sqrt(kc4$krige.var[i]))
    	bp <- density(bsp4$predic$sim[i,])
    	rx <- range(c(kpx, bp$x))
    	ry <- range(c(kpy, bp$y))
    	plot(cbind(rx, ry), type="n", xlab=paste("Location", i), ylab="density", xlim=c(-4, 4), ylim=c(0,1.1))
    	lines(kpx, kpy, lty=2)

    Consider now, under the same model assumptions, obtaining simulations from the predictive distributions on a grid of points covering the area. The commands to define the grid and perform Bayesian prediction are:

    pred.grid <-  expand.grid(seq(0,1, l=31), seq(0,1, l=31))
    bsp <- krige.bayes(s100, loc = pred.grid, prior = prior.control(phi.discrete = seq(0,5,l=51)), 

    Maps with the summaries and simulations of the predictive distribution can be plotted as follows.

    image(bsp, loc = pred.grid, main = "predicted", col=gray(seq(1,0.1,l=30)))
    image(bsp, val ="variance", loc = pred.grid, 
          main = "prediction variance", col=gray(seq(1,0.1,l=30)))
    image(bsp, val = "simulation", number.col = 1, loc = pred.grid, 
          main = "a simulation from\nthe predictive distribution", col=gray(seq(1,0.1,l=30)))
    image(bsp, val = "simulation", number.col = 2,loc = pred.grid, 
          main = "another simulation from \n the predictive distribution", col=gray(seq(1,0.1,l=30)))

    Note: Further examples for the function krige.bayes are given in the file examples.krige.bayes.R

  12. Simulation of Gaussian Random Fields
  13. The function grf generates simulations of Gaussian random fields on regular or irregular sets of locations
    Some of its functionality is illustrated by the next commands.

    sim1 <- grf(100,, .25))
    points.geodata(sim1, main="simulated locations and values")
    plot(sim1, max.dist=1, main="true and empirical variograms") 

    sim2 <- grf(441, grid="reg",, .25))
    image(sim2, main="a small-ish simulation", col=gray(seq(1, .1, l=30)))

    sim3 <- grf(40401, grid="reg",, .2), met="circ")
    image(sim3, main="simulation on a fine grid", col=gray(seq(1, .1, l=30)))

    NOTE: we recommend the package RandomFields for a more comprehensive implementation for simulation of Gaussian Random Fields.

  14. Citing geoR
  15. The function cite.geoR() shows information on how to cite geoR in publications.
     > cite.geoR()
     To cite geoR in publications, use
     RIBEIRO Jr., P.J. & DIGGLE, P.J. (2001) geoR: A package for
     geostatistical analysis. R-NEWS, Vol 1, No 2, 15-18. ISSN 1609-3631.
     Please cite geoR when using it for data analysis!
     A BibTeX entry for LaTeX users is
         title         = {{geoR}: a package for geostatistical analysis},
         author        = {Ribeiro Jr., P.J. and Diggle, P.J.},
         journal       = {R-NEWS},
         year          = {2001},
         volume        = {1},
         number        = {2},
         pages         = {15--18},
         issn          = {1609-3631},
         url           = {}

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