This page documents parameters used to control georob
. It
describes the arguments of the functions control.georob
,
param.transf
, fwd.transf
, dfwd.transf
,
bwd.transf
, control.rq
, control.nleqslv
,
control.nlminb
and control.optim
, which all serve to
control the behaviour of georob
.
control.georob(ml.method = c("REML", "ML"), reparam = TRUE,
maximizer = c("nlminb", "optim"), initial.param = TRUE,
initial.fixef = c("lmrob", "rq", "lm"), bhat = NULL,
min.rweight = 0.25,
param.tf = param.transf(), fwd.tf = fwd.transf(),
deriv.fwd.tf = dfwd.transf(), bwd.tf = bwd.transf(),
psi.func = c("logistic", "t.dist", "huber"),
irwls.maxiter = 50,
irwls.ftol = 1.e-5, force.gradient = FALSE,
min.condnum = 1.e-12, zero.dist = sqrt(.Machine[["double.eps"]]),
error.family.estimation = c("gaussian", "long.tailed"),
error.family.cov.effects = c("gaussian", "long.tailed"),
error.family.cov.residuals = c("gaussian", "long.tailed"),
cov.bhat = TRUE, full.cov.bhat = FALSE, cov.betahat = TRUE,
cov.delta.bhat = TRUE, full.cov.delta.bhat = TRUE,
cov.delta.bhat.betahat = TRUE,
cov.ehat = TRUE, full.cov.ehat = FALSE,
cov.ehat.p.bhat = FALSE, full.cov.ehat.p.bhat = FALSE,
hessian = TRUE,
rq = control.rq(), lmrob = lmrob.control(),
nleqslv = control.nleqslv(),
optim = control.optim(), nlminb = control.nlminb(),
pcmp = control.pcmp(), ...)param.transf(variance = "log", snugget = "log", nugget = "log", scale = "log",
alpha = c(
RMaskey = "log", RMdewijsian = "logit2", RMfbm = "logit2", RMgencauchy = "logit2",
RMgenfbm = "logit2", RMlgd = "identity", RMqexp = "logit1", RMstable = "logit2"
),
beta = c(RMdagum = "logit1", RMgencauchy = "log", RMlgd = "log"),
delta = "logit1", gamma = c(RMcauchy = "log", RMdagum = "logit1"),
kappa = "logit3", lambda = "log", mu = "log", nu = "log",
f1 = "log", f2 ="log", omega = "identity", phi = "identity", zeta = "identity")
fwd.transf(...)
dfwd.transf(...)
bwd.transf(...)
control.rq(tau = 0.5, rq.method = c("br", "fnb", "pfn"),
rq.alpha = 0.1, ci = FALSE, iid = TRUE,
interp = TRUE, tcrit = TRUE, rq.beta = 0.99995, eps = 1e-06,
Mm.factor = 0.8, max.bad.fixup = 3, ...)
control.nleqslv(method = c("Broyden", "Newton"),
global = c("dbldog", "pwldog", "qline", "gline", "none"),
xscalm = c("fixed", "auto"), control = list(ftol = 1e-04), ...)
control.optim(method = c("BFGS", "Nelder-Mead", "CG",
"L-BFGS-B", "SANN", "Brent"), lower = -Inf, upper = Inf,
control = list(reltol = 1e-05), ...)
control.nlminb(control = list(rel.tol = 1.e-5), lower = -Inf,
upper = Inf, ...)
control.georob
, control.rq
, control.nleqslv
,
control.optim
and control.nlminb
all create lists with
control parameters passed to georob
,
rq
, nleqslv
,
optim
or nlminb
, see arguments
above and the help pages of the respective functions for information
about the components of these lists. Note that the list returned by
control.georob
contains some components (irwls.initial
,
tuning.psi.nr
, cov.bhat.betahat
,
aux.cov.pred.target
) that cannot be changed by the user.
param.transf
generates a list with character strings that
define what transformations are used for estimating the variogram
parameters, and fwd.transf
, bwd.transf
and
dfwd.transf
return lists of functions with forward and backward
transformations and the first derivatives of the forward
transformations, see section Parameter transformations above.
a character keyword defining whether Gaussian maximum
likelihood (ML
) or restricted maximum likelihood (REML
default) estimates will be computed (ignored if tuning.psi <=
tuning.psi.nr
).
a logical scalar. If TRUE
(default) the
re-parametrized variance parameters \(\sigma_B^2\), \(\eta\) and
\(\xi\) are estimated by Gaussian (RE)ML, otherwise the original
parameters \(\tau^2\), \(\sigma_{\mathrm{n}}^2\) and
\(\sigma^2\) (cf. subsection Estimating variance parameters by
Gaussian (RE)ML, section Details of georob
).
a character keyword defining whether the Gaussian
(restricted) log-likelihood is maximized by nlminb
(default) or optim
.
a logical scalar, controlling whether initial values
of variogram parameters are computed for solving the robustified
estimating equations of the variogram and anisotropy parameters. If
initial.param = TRUE
(default) robust initial values of parameters
are computed by discarding outlying observations based on the
“robustness weights” of the initial fit of the regression model by
lmrob
and fitting the spatial linear model by
Gaussian REML to the pruned data set. For initial.param = FALSE
no initial parameter values are computed and the estimating equations are
solved with the initial values passed by param
and aniso
to
georob
(see Details of georob
).
a character keyword defining whether the function
lmrob
or rq
is used to
compute robust initial estimates of the regression parameters
\(\boldsymbol{\beta}\) (default "lmrob"
).
If the fixed effects model matrix has not full columns rank, then
lm
is used to compute initial values of the
regression coefficients.
a numeric vector with initial values for the spatial random
effects \(\widehat{\boldsymbol{B}}\), with
\(\widehat{\boldsymbol{B}}=\boldsymbol{0}\) if bhat
is
equal to NULL
(default).
a positive numeric with the “robustness
weight” of the initial lmrob
fit that
observations must exceed to be used for computing robust initial
estimates of variogram parameters by setting initial.param = TRUE
(see georob
; default 0.25
).
a function such as param.transf
, which returns a
named list of character strings that define the transformations to be
applied to the variogram parameters for model fitting, see
Details.
a function such as fwd.transf
, which returns a named
list of invertible functions to be used to transform variogram
parameters, see Details.
a function such as dfwd.transf
, which
returns a named list of functions corresponding to the first derivatives
of fwd.tf
, see Details.
a function such as bwd.transf
, which returns the
named list of inverse functions corresponding to fwd.tf
, see
Details.
a character keyword defining what \(\psi_c\)-function
should be used for robust model fitting. Possible values are
"logistic"
(a scaled and shifted logistic CDF, default),
"t.dist"
(re-descending \(\psi_c\)-function associated with
Student \(t\)-distribution with \(c\) degrees of freedom) and
"huber"
(Huber's \(\psi_c\)-function).
a positive integer equal to the maximum number of
IRWLS iterations to solve the estimating equations of
\(\boldsymbol{B}\) and \(\boldsymbol{\beta}\) (default
50
).
a positive numeric with the convergence criterion for
IRWLS. Convergence is assumed if the objective function change of a IRWLS
iteration does not exceed ftol
.
a logical scalar controlling whether the estimating
equations or the gradient of the Gaussian restricted log-likelihood are
evaluated even if all variogram parameters are fixed (default
FALSE
).
a positive numeric with the minimum acceptable
ratio of smallest to largest singular value of the model matrix
\(\boldsymbol{X}\) (default 1.e-12
).
a positive numeric equal to the maximum distance, separating two sampling locations that are still considered as being coincident.
a character keyword, defining the
probability distribution for \(\varepsilon\) (default:
"gaussian"
) that is used to approximate the covariance of
\(\widehat{\boldsymbol{B}}\) when solving the
estimating equations, see Details.
a character keyword, defining the
probability distribution for \(\varepsilon\) (default:
"gaussian"
) that is used to approximate the covariances of
\(\widehat{\boldsymbol{\beta}}\),
\(\widehat{\boldsymbol{B}}\) and
\(\boldsymbol{B}-\widehat{\boldsymbol{B}}\),
see Details.
a character keyword, defining the
probability distribution for \(\varepsilon\) (default:
"long.tailed"
) that is used to approximate the covariances of
\(\widehat{\boldsymbol{\varepsilon}}=\boldsymbol{Y}
- \boldsymbol{X}
\widehat{\boldsymbol{\beta}} -
\widehat{\boldsymbol{B}}\) and \(\widehat{\boldsymbol{\varepsilon}} +
\widehat{\boldsymbol{B}} =\boldsymbol{Y}
- \boldsymbol{X}
\widehat{\boldsymbol{\beta}}\), see Details.
a logical scalar controlling whether the covariances of
\(\widehat{\boldsymbol{B}}\) are returned by georob
(default FALSE
).
a logical scalar controlling whether the full
covariance matrix (TRUE
) or only the variance vector of
\(\widehat{\boldsymbol{B}}\) is returned (default FALSE
).
a logical scalar controlling whether the covariance
matrix of \(\widehat{\boldsymbol{\beta}}\) is returned
(default TRUE
).
a logical scalar controlling whether the covariances of
\(\boldsymbol{B}-
\widehat{\boldsymbol{B}}\) are returned (default TRUE
).
a logical scalar controlling whether the full covariance
matrix (TRUE
) or only the variance vector of
\(\boldsymbol{B}-
\widehat{\boldsymbol{B}}\) is returned (default TRUE
).
a logical scalar controlling whether the covariance
matrix of \(\boldsymbol{B}-
\widehat{\boldsymbol{B}}\) and
\(\widehat{\boldsymbol{\beta}}\) is returned
(default TRUE
).
a logical scalar controlling whether the covariances of
\(\widehat{\boldsymbol{\varepsilon}}=\boldsymbol{Y}
- \boldsymbol{X}
\widehat{\boldsymbol{\beta}} -
\widehat{\boldsymbol{B}}\) are returned (default TRUE
).
a logical scalar controlling whether the full covariance
matrix (TRUE
) or only the variance vector of
\(\widehat{\boldsymbol{\varepsilon}}=\boldsymbol{Y}
- \boldsymbol{X}
\widehat{\boldsymbol{\beta}} -
\widehat{\boldsymbol{B}}\) is returned (default FALSE
).
a logical scalar controlling whether the covariances of
\(\widehat{\boldsymbol{\varepsilon}} +
\widehat{\boldsymbol{B}} =\boldsymbol{Y}
- \boldsymbol{X}
\widehat{\boldsymbol{\beta}}\) are returned (default FALSE
).
a logical scalar controlling whether the full
covariance matrix (TRUE
) or only the variance vector
of \(\widehat{\boldsymbol{\varepsilon}}
+ \widehat{\boldsymbol{B}}
=\boldsymbol{Y} - \boldsymbol{X}
\widehat{\boldsymbol{\beta}}\) is returned (default FALSE
).
a logical scalar controlling whether for Gaussian (RE)ML the Hessian should be computed at the MLEs.
a list of arguments passed to rq
or a function such as
control.rq
that generates such a list (see
rq
for allowed arguments).
a list of arguments passed to the control
argument of
lmrob
or a function such as
lmrob.control
that generates such a list (see
lmrob.control
for allowed arguments).
a list of arguments passed to
nleqslv
or a function such as
control.nleqslv
that generates such a list (see
nleqslv
for allowed arguments).
a list of arguments passed to nlminb
or a function such as control.nlminb
that generates such a list
(see nlminb
for allowed arguments).
a list of arguments passed to optim
or a function
such as control.optim
that generates such a list (see
optim
for allowed arguments).
a list of arguments, passed e.g. to pmm
or a
function such as control.pcmp
that generates such a list
(see control.pcmp
for allowed arguments).
for fwd.transf
, dfwd.transf
and
bwd.transf
a named vector of functions, extending the definition
of transformations for variogram parameters (see Details).
character strings with names of transformation functions of the variogram parameters.
character strings with names of transformation functions of the anisotropy variogram parameters.
arguments passed
as ...
to rq
. Note that only "br"
,
"fnb"
and "pfn"
methods of rq()
are currently
supported.
arguments passed as
...
to rq
.
arguments passed to related arguments of
nleqslv
, nlminb
and
optim
, respectively.
Andreas Papritz papritz@retired.ethz.ch.
The arguments param.tf
, fwd.tf
, deriv.fwd.tf
,
bwd.tf
define the transformations of the variogram parameters for
RE(ML) estimation. Implemented are currently "log"
,
"logit1"
, "logit2"
, "logit3"
(various variants of
logit-transformation, see code of function fwd.transf
) and "identity"
(= no)
transformations. These are the possible values that the many arguments
of the function param.transf
accept (as quoted character strings)
and these are the names of the list components returned by
fwd.transf
, dfwd.transf
and bwd.transf
. Additional
transformations can be implemented by:
Extending the function definitions by arguments like
fwd.tf = fwd.transf(my.fun = function(x) your transformation)
,
deriv.fwd.tf = dfwd.transf(my.fun = function(x) your derivative)
,
bwd.tf = bwd.transf(my.fun = function(x) your back-transformation)
,
Assigning to a given argument of param.transf
the name of
the new function, e.g.
variance = "my.fun"
.
Note that the values given for the arguments of param.transf
must match the names of the functions returned by fwd.transf
,
dfwd.transf
and bwd.transf
.
The robustified estimating equations of robust REML depend on the
covariances of \(\widehat{\boldsymbol{B}}\).
These covariances (and the covariances of
\(\boldsymbol{B}-\widehat{\boldsymbol{B}}\),
\(\widehat{\boldsymbol{\beta}}\),
\(\widehat{\boldsymbol{\varepsilon}}\),
\(\widehat{\boldsymbol{\varepsilon}} +
\widehat{\boldsymbol{B}}\)) are
approximated by expressions that in turn depend on the variances of
\(\varepsilon\),
\(\psi(\varepsilon/\tau)\) and the expectation
of \(\psi'(\varepsilon/\tau) (= \partial / \partial \varepsilon \,
\psi(\varepsilon/\tau))\). The arguments
error.family.estimation
, error.family.cov.effects
and
error.family.cov.residuals
control what parametric distribution
for \(\varepsilon\) is used to compute the variance of
\(\varepsilon\),
\(\psi(\varepsilon/\tau)\) and the expectation
of \(\psi'(\varepsilon/\tau)\) when
solving the estimating equations (error.family.estimation
),
computing the covariances of
\(\widehat{\boldsymbol{\beta}}\),
\(\widehat{\boldsymbol{B}}\) and
\(\boldsymbol{B}-\widehat{\boldsymbol{B}}\)
(error.family.cov.effects
) and
computing the covariances of
\(\widehat{\boldsymbol{\varepsilon}}=\boldsymbol{Y}
- \boldsymbol{X}
\widehat{\boldsymbol{\beta}} -
\widehat{\boldsymbol{B}}\) and \(\widehat{\boldsymbol{\varepsilon}} +
\widehat{\boldsymbol{B}}
=\boldsymbol{Y} - \boldsymbol{X}
\widehat{\boldsymbol{\beta}}\)
(error.family.cov.residuals
).
Possible options are: "gaussian"
or "long.tailed"
. In
the latter case the probability density function of
\(\varepsilon\) is assumed to be proportional to
\(1/\tau \, \exp(-\rho_c(\varepsilon/\tau))\), where
\(\psi_c(x)=\rho_c^\prime(x)\).
georobPackage
for a description of the model and a brief summary of the algorithms;
georob
for (robust) fitting of spatial linear models;
georobObject
for a description of the class georob
;
profilelogLik
for computing profiles of Gaussian likelihoods;
plot.georob
for display of RE(ML) variogram estimates;
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.
data(meuse)
r.logzn.rob <- georob(log(zinc) ~ sqrt(dist), data = meuse, locations = ~ x + y,
variogram.model = "RMexp",
param = c(variance = 0.15, nugget = 0.05, scale = 200),
tuning.psi = 1, control = control.georob(cov.bhat = TRUE,
cov.ehat.p.bhat = TRUE, initial.fixef = "rq"), verbose = 2)
qqnorm(rstandard(r.logzn.rob, level = 0)); abline(0, 1)
qqnorm(ranef(r.logzn.rob, standard = TRUE)); abline(0, 1)
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