The georob Package

Description

This is a summary of the features and functionality of georob, a package in R for robust geostatistical analyses.

Details

georob is a package for robust analyses of geostatistical data. Such data, say y(s_i), are recorded at a set of locations, s_i, i=1,2, …, n, in a domain G in R^d, d in (1,2,3), along with covariate information x_j(s_i), j=1,2, …, p.

Model

We use the following model for the data y_i=y(s_i):

Y(s_i) = Z(s_i) + ε = x(s_i)^T β + B(s_i) + ε_i,

where Z(s_i) = x(s_i)^T β + B(s_i) is the so-called signal, x(s_i)^Tβ is the external drift, {B(s)} is an unobserved stationary or intrinsic spatial Gaussian random field with zero mean, and ε_i is an i.i.d error from a possibly long-tailed distribution with scale parameter τ (τ^2 is usually called nugget effect). In vector form the model is written as

Y = X β + B + ε,

where X is the model matrix with the rows x^T(s_i).

The (generalized) covariance matrix of the vector of spatial Gaussian random effects B is denoted by

E[B B^T] = Γ_θ = σ_n^2 I+σ^2 V_α = σ_Z^2 V_{α,ξ} = σ_Z^2 (ξ I - (1-ξ) V_α) ,

where σ_n^2 is the variance of seemingly uncorrelated micro-scale variation in B(s) that cannot be resolved with the chosen sampling design, σ^2 is the variance of the captured auto-correlated variation in B(s), σ_Z^2=σ_n^2+σ^2 is the signal variance, and ξ=σ_n^2/σ^_Z^2. To estimate both σ_n^2 and τ^2 (and not only their sum), one needs replicated measurements for some of the s_i.

We define V_{α} to be the matrix with elements

(V_{α})_ij = γ_0 - γ(|A (s_i - s_j )|),

where the constant γ_0 is chosen large enough so that V_{α} is positive definite, v() is a valid stationary or intrinsic variogram, and A = A(α, f_1, f_2; ω, φ, ζ) is a matrix that is used to model geometrically anisotropic auto-correlation. In more detail, A maps an arbitrary point on an ellipsoidal surface with constant semivariance in R^d, centred on the origin, and having lengths of semi-principal axes, p_1, p_2, p_3, equal to |p_1|=α, |p_2|=f_1 α and |p_3|=f_2 α, 0 < f_2 <= f_1 <= 1, respectively, onto the surface of the unit ball centred on the origin.

The orientation of the ellipsoid is defined by the three angles ω, φ and ζ:

ω

is the azimuth of p_1 (= angle between north and the projection of p_1 onto the x-y-plane, measured from north to south positive clockwise in degrees),

φ

is 90 degrees minus the altitude of p_1 (= angle between the zenith and p_1, measured from zenith to nadir positive clockwise in degrees), and

ζ

is the angle between p_2 and the direction of the line, say y', defined by the intersection between the x-y-plane and the plane orthogonal to p_1 running through the origin (ζ is measured from y' positive counterclockwise in degrees).

The transformation matrix is given by

A=diag(1/α, 1/(f_1,α),1/(f_2,α)) (C_1, C_2, C_3)

where

C_1^T=(sinφ sinω, -cosζ cosω + sinζ cosφ sinω, -cosω sinζ - cosζ cosφ sinω)

C_2^T=(-sinφ cosω, cosω sinζ cosφ + cosζ sinω, -cosζ cosω cosφ + sinζ sinω)

C_3^T=(cosφ, -sinφ sinζ, cosζ sinφ)

To model geometrically anisotropic variograms in R^2 one has to set φ=90 and f_2 = 1, and for f_1 = f_2 = 1 one obtains the model for isotropic auto-correlation with range parameter α. Note that for isotropic auto-correlation the software processes data for which d may exceed 3.

Two remarks are in order:

1. Clearly, the (generalized) covariance matrix of the observations Y is given by

   Cov[Y, Y^T] = τ^2 I + Γ_θ.

2. 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. σ^2, …, α, … and f_1, …, ζ are denoted by the term “anisotropy parameters”.

Estimation

The unobserved spatial random effects B at the data locations s_i and the model parameters β and θ^T = (σ^2, σ^2_n, τ^2, α, f_1, f_2, ω, φ, ζ, …) are unknown and are estimated in georob either by Gaussian or robust restricted maximum likelihood (REML) or Gaussian maximum likelihood (ML). Here ... denote further parameters of the variogram such as the smoothness parameter of the Whittle-Mat¬érn model.

In brief, the robust REML method is based on the insight that for given θ the kriging predictions (= BLUP) of B and the generalized least squares (GLS = ML) estimates of β can be obtained simultaneously by maximizing

- ∑_i ( y_i - x^T(s_i)β - B(s_i) )^2 / τ^2 - B^T Γ_θ^-1 B

with respect to B and β e.g. Harville (1977). Hence, the BLUP of B and ML estimates of β and θ are obtained by maximizing

- log(det( τ^2 I + Γ_θ) ) - ∑_i ( y_i - x^T(s_i)β - B(s_i) )^2 / τ^2 - B^T Γ_θ^-1 B

jointly with respect to B, β, θ.

The respective estimating equations can then by robustified by

• replacing the standardized residuals, say ε_i/τ, by a bounded or redescending ψ-function, ψ_c(ε_i/τ), of them (e.g. Marona et al, 2006, chap. 2) and by

• introducting suitable bias correction terms for Fisher consistency at the Gaussian model,

see K¬ünsch et al. (2011) for details. The robustified estimating equations are solved numerically by a combination of iterated re-weighted least squares (IRWLS) to estimate B and β for given θ and nonlinear root finding by the function nleqslv of the R package nleqslv to get θ. The robustness of the procedure is controlled by the tuning parameter c of the ψ_c-function. For c>=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) loglikelihood using nlminb or optim.

georob uses variogram models implemented in the R package RandomFields (see RMmodel). Currently, estimation of the parameters of the following models is implemented:

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

For most variogram parameters, closed-form expressions of dγ/dθ_i are used in the computations. However, for the parameter ν of the models "RMbessel", "RMmatern" and "RMwhittle" dγ/dν is evaluated numerically by the function numericDeriv, and this results in an increase in computing time when ν is estimated.

Prediction

Robust plug-in external-drift point kriging predictions can be computed for an unsampled location s_0 from the covariates x(s_0), the estimated parameters hatβ, hatθ and the predicted random effects hatB by

hatY(s_0) = hatZ(s_0) = x^T(s_0) hatβ + γ^T_hatθ(s_0) Γ_hatθ^-1 hatB,

where Γ_hatθ is the estimated (generalized) covariance matrix of B and γ^T_hatθ(s_0) is the vector with the estimated (generalized) covariances between B and B(s_0). Kriging variances can be computed as well, based on approximated covariances of hatB and hatβ (see K¬ünsch et al., 2011, and appendices of Nussbaum et al., 2012, for details).

The package georob provides in addition software for computing robust external-drift block kriging predictions. The required integrals of the generalized covariance function are computed by functions of the R package constrainedKriging.

Functionality

For the time being, the functionality of georob is limited to robust geostatistical analyses of single response variables. No software is currently available for robust multivariate geostatistical analyses. georob offers functions for:

1. 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 georob — which also fits such models efficiently by Gaussian (RE)ML — profilelogLik and control.georob).

2. Extracting estimated model components (see residuals.georob, rstandard.georob,
   rstudent.georob, ranef.georob).

3. Robustly estimating sample variograms and for fitting variogram model functions to them (see sample.variogram and fit.variogram.model).

4. Model building by forward and backward selection of covariates for the external drift (see waldtest.georob, step.georob, add1.georob, drop1.georob, extractAIC.georob,
   logLik.georob, deviance.georob). For a robust fit, the loglikelihood is not defined. The function then computes the (restricted) loglikelihood of an equivalent Gaussian model with heteroscedastic nugget (see deviance.georob for details).

5. Assessing the goodness-of-fit and predictive power of the model by K-fold cross-validation (see cv.georob and validate.predictions).

6. Computing robust external-drift point and block kriging predictions (see predict.georob, control.predict.georob).

7. Unbiased back-transformation of both point and block kriging predictions of log-transformed data to the original scale of the measurements (see lgnpp).

Author(s)

Andreas Papritz andreas.papritz@env.ethz.ch
http://www.step.ethz.ch/people/scientific-staff/andreas-papritz

References

Nussbaum, M., Papritz, A., Baltensweiler, A. and Walthert, L. (2012) Organic Carbon Stocks of Swiss Forest Soils, Institute of Terrestrial Ecosystems, ETH Zurich and Swiss Federal Institute for Forest, Snow and Landscape Research (WSL), pp. 51. http://e-collection.library.ethz.ch/eserv/eth:6027/eth-6027-01.pdf

KŸnsch, H. R., Papritz, A., Schwierz, C. and Stahel, W. A. (in preparation) Robust Geostatistics.

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

Maronna, R. A., Martin, R. D. and Yohai, V. J. (2006) Robust Statistics Theory and Methods, John Wiley & Sons.

See Also

georob for (robust) fitting of spatial linear models; georobObject for a description of the class georob; plot.georob for display of RE(ML) variogram estimates; control.georob for controlling the behaviour of georob; cv.georob for assessing the goodness of a fit by georob; predict.georob for computing robust kriging predictions; and finally georobModelBuilding for stepwise building models of class georob; georobMethods for further methods for the class georob, sample.variogram and fit.variogram.model for robust estimation and modelling of sample variograms.

Results