georobObject {georob}

Fitted georob Object

Description

An object of class georob as returned by georob and representing a (robustly) fitted spatial linear model. Objects of this class have methods for model building (see georobModelBuilding) and cross-validation (see cv.georob), for computing (robust) Kriging predictions (see predict.georob), for plotting (see plot.georob) and for common generic functions (see georobMethods).

Value

A georob object is a list with following components:

loglik

the maximized (restricted) Gaussian log-likelihood of a non-robust (RE)ML fit or NA for a robust fit if tuning.psi is less than tuning.psi.nr.

variogram.object

the estimated parameters of a possibly nested variograms model. This is a list that contains for each variogram model structure the following components:

• variogram.model: the name of the fitted parametric variogram model.

• param: a named numeric vector with the (estimated) variogram parameters.

• fit.param: a named logical vector with the flags defining what variogram parameters were estimated.

• isotropic: logical indicating whether an isotropic variogram was fitted.

• aniso: a named numeric vector with the (estimated) anisotropy parameters.

• fit.aniso: a named logical vector with the flags defining what anisotropy parameters were estimated.

• sincos: a list with sin and cos of the angles \omega, \phi and \zeta that define the orientation of the anisotropy ellipsoid (see georobPackage).

• rotmat: the matrix (\boldsymbol{C}_1, \boldsymbol{C}_2, \boldsymbol{C}_3) (see georobPackage).

• sclmat: a vector with the elements 1, 1/f_1, 1/f_2 (see georobPackage).

gradient

a named numeric vector with the estimating equations (robust REML) or the gradient of the maximized (restricted) log-likelihood (Gaussian (RE)ML) evaluated at the solution.

tuning.psi

the value of the tuning constant c of the \psi_c-function.

coefficients

a named vector with the estimated regression coefficients \widehat{\boldsymbol{\beta}}.

fitted.values

a named vector with the fitted values of the external drift \boldsymbol{X}\widehat{\boldsymbol{\beta}}.

bhat

a named vector with the predicted spatial random effects \widehat{\boldsymbol{B}} at the data locations.

residuals

a named vector with the residuals \widehat{\boldsymbol{\varepsilon}}=\boldsymbol{Y} - \boldsymbol{X} \widehat{\boldsymbol{\beta}} - \widehat{\boldsymbol{B}}.

rweights

a named numeric vector with the “robustness weights” \psi_c(\widehat{\varepsilon}_i/\widehat{\tau}) / (\widehat{\varepsilon}_i/\widehat{\tau}).

converged

a logical scalar indicating whether numerical maximization of the (restricted)
log-likelihood by nlminb or optim or root finding by nleqslv converged.

convergence.code

a diagnostic integer issued by nlminb, optim (component convergence) or nleqslv (component termcd) about convergence.

iter

a named integer vector of length two, indicating either

• the number of function and gradient evaluations when maximizing the (restricted) Gaussian log-likelihood by nlminb or optim, or

• the number of function and Jacobian evaluations when solving the robustified estimating equations by nleqslv.

Tmat

the compressed design matrix for replicated observations at coincident locations (integer vector that contains for each observation the row index of the respective unique location).

cov

a list with covariance matrices (or diagonal variance vectors). Covariance matrices are stored in compressed form (see compress) and can be expanded to square matrices by expand. What cov actually contains depends on the flags passed to georob for computing covariances (see control.georob). Possible components are:

• cov.bhat: the covariances of \widehat{\boldsymbol{B}}.

• cov.betahat: the covariances of \widehat{\boldsymbol{\beta}}.

• cov.delta.bhat: the covariances of \boldsymbol{B}- \widehat{\boldsymbol{B}}.

• cov.delta.bhat.betahat: the covariances of \boldsymbol{B}- \widehat{\boldsymbol{B}} and \widehat{\boldsymbol{\beta}}.

• cov.ehat: the covariances of \widehat{\boldsymbol{\varepsilon}}=\boldsymbol{Y} - \boldsymbol{X} \widehat{\boldsymbol{\beta}} - \widehat{\boldsymbol{B}}.

• cov.ehat.p.bhat: the covariances of \widehat{\boldsymbol{\varepsilon}} + \widehat{\boldsymbol{B}} =\boldsymbol{Y} - \boldsymbol{X} \widehat{\boldsymbol{\beta}}.

• cov.pred.target: a covariance term required for the back-trans-
  formation of Kriging predictions of log-transformed data.

expectations

a named numeric vector with the expectations of \partial \psi_c(x)/\partial x and \psi_c^2(x) with respect to a standard normal distribution (exp.gauss.dpsi, var.gauss.psi) and the expectations of \varepsilon^2 and \psi_c^2(x) with respect to the long-tailed distribution of \varepsilon (var.f0.eps, var.f0.psi).

Valphaxi.objects

a list of matrices in compressed form with (among others) the following components:

• Valpha: a list with the (generalized) correlation matrices (Valpha) of the nested variogram models structures along with the constants (gcr.constant) added to the respective semi-variance matrices.

• Valphaxi: the (generalized) correlation matrix \boldsymbol{V}_{\alpha,\xi} = \boldsymbol{\Gamma}_{\alpha,\xi} / (\sigma_{\mathrm{n}}^2+\sigma^2 ) that includes the spatial nugget effect.

• Valphaxi.inverse: the inverse of \boldsymbol{V}_{\alpha,\xi}.

• log.det.Valphaxi: \log(\det(\boldsymbol{V}_{\alpha,\xi})).

zhat.objects

a list of matrices in (partly) compressed form with the following components:

• Aalphaxi: the matrix (\boldsymbol{X}^T \boldsymbol{V}_{\alpha,\xi}^{-1}\boldsymbol{X})^{-1} \boldsymbol{X}^T\boldsymbol{V}_{\alpha,\xi}^{-1} .

• Palphaxi: the matrix \boldsymbol{I}- \boldsymbol{X} \boldsymbol{A}_{\alpha,\xi} .

• Valphaxi.inverse.Palphaxi: the matrix \boldsymbol{V}^{-1}_{\alpha,\xi} \boldsymbol{P}_{\alpha,\xi} .

locations.object

a list with 3 components:

• locations: a formula indicating the coordinates of the measurement locations.

• coordinates: a numeric matrix with the coordinates of the measurement locations.

• lag.vectors: a numeric matrix with the lag vectors between any distinct pairs of measurement locations.

initial.objects

a list with 3 components:

• coefficients: initial estimates of \boldsymbol{\beta} computed either by lmrob or rq.

• bhat: initial predictions of \boldsymbol{B}.

• variogram.object: the initial values of the parameters of a possibly nested variograms model. This is a list with the same structure as described above for the component variogram.object.

hessian.tfpa

a symmetric matrix with the Hessian (observed Fisher information) at the solution with respect to the transformed variogram and anisotropy parameters if the model was fitted non-robustly with the argument hessian = TRUE (see control.georob). Missing otherwise. This Hessian is used by summary.georob to compute confidence intervals for the estimated parameters.

hessian.ntfpa

a symmetric matrix with the Hessian (observed Fisher information) at the solution with respect to the non-transformed variogram and anisotropy parameters if the model was fitted non-robustly with the argument hessian = TRUE (see control.georob). Missing otherwise.

control

a list with control parameters generated by control.georob.

MD

optionally a matrix of robust distances in the space spanned by \boldsymbol{X} (see argument compute.rd of lmrob.control and control.georob).

model, x, y

if requested the model frame, the model matrix and the response, respectively.

na.action, offset, contrasts, xlevels, rank, df.residual, call, terms

further components of the fit as described for an object of class lm.

Author(s)

Andreas Papritz [e-mail].

See Also

georobPackage for a description of the model and a brief summary of the algorithms;

georob for (robust) fitting of spatial linear models;

profilelogLik for computing profiles of Gaussian likelihoods;

plot.georob for display of RE(ML) variogram estimates;

control.georob for controlling the behaviour of georob;

georobModelBuilding for stepwise building models of class georob;

cv.georob for assessing the goodness of a fit by georob;

georobMethods for further methods for the class georob;

predict.georob for computing robust Kriging predictions;

lgnpp for unbiased back-transformation of Kriging prediction of log-transformed data;

georobSimulation for simulating realizations of a Gaussian process from model fitted by georob; and finally

sample.variogram and fit.variogram.model for robust estimation and modelling of sample variograms.