gencorr {georob} | R Documentation |
Variogram Models
Description
The function gencorr
computes intrinsic or
stationary isotropic generalized correlations (= negative semi-variances
computed with sill (variance) parameter equal to 1) for a set of basic
variogram models formerly made available by the function RFfctn
of
the now archived R package RandomFields.
Usage
gencorr(x, variogram.model, param)
Arguments
x |
a numeric vector with scaled lag distances, i.e. lag distances
divided by the range parameter |
variogram.model |
a character keyword defining the variogram model.
Currently, the following models are implemented: |
param |
a named numeric vector with values of the additional
parameters of the variogram models such as the smoothness parameter of
the Whittle-Matérn model, see |
Details
The name and parametrization of the variogram models originate from the
function RFfctn
of RandomFields. The equations and further
information are taken (with minor modifications) from the help pages of
the respective functions of the archived R package RandomFields,
version 3.3.14 (Schlather et al., 2022). Note that the variance
(sill, param["variance"]
) and the range parameters
(param["scale"]
) are assumed to be equal to 1 in the following
formulae, and is the lag distance. The variogram functions are
stationary and are valid for any number of dimensions if not mentioned
otherwise.
The following intrinsic or stationary isotropic variogram
functions are implemented in
gencorr
:
-
RMaskey
is the indicator function equal to 1 for
and 0 otherwise. This variogram function is valid for dimension
if
. For
we get the well-known triangle (or tent) model, which is only valid on the real line.
-
RMbessel
where
,
denotes the gamma function and
is a Bessel function of first kind. This models a hole effect (see Chilès and Delfiner, 1999, p. 92). An important case is
which gives the variogram function
and which is only valid for
(this equals
RMdampedcos
for). A second important case is
with variogram function
which is valid for
. This coincides with
RMwave
. -
RMcauchy
where
. The parameter
determines the asymptotic power law. The smaller
, the longer the long-range dependence. The generalized Cauchy Family (
RMgencauchy
) includes this family for the choiceand
.
-
RMcircular
This variogram function is valid only for dimensions
.
-
RMcubic
The model is only valid for dimensions
. It is a 2 times differentiable variogram function with compact support (see Chilès and Delfiner, 1999, p. 84).
-
RMdagum
The parameters
and
can be varied in the intervals
and
, respectively. Like the generalized Cauchy model (
RMgencauchy
) the Dagum family can be used to model separately fractal dimension and Hurst effect (see Berg et al., 2008). -
RMdampedcos
The model is valid for any dimension
. However, depending on the dimension of the random field the following bound
has to be respected. This variogram function models a hole effect (see Chilès and Delfiner, 1999, p. 92). For
we obtain
which is only valid for
and corresponds to
RMbessel
for.
-
RMdewijsian
where
. This is an intrinsic variogram function. Originally, the logarithmic model
was named after de Wijs and reflects a principle of similarity (see Chilès and Delfiner, 1999, p. 90). But note that
is not a valid variogram function.
-
RMexp
This model is a special case of the Whittle model (
RMwhittle
) ifand of the stable family (
RMstable
) if. Moreover, it is the continuous-time analogue of the first order auto-regressive time series covariance structure.
-
RMfbm
where
. This is an intrinsically stationary variogram function. For
we get a variogram function corresponding to a standard Brownian Motion. For
the quantity
is called Hurst index and determines the fractal dimension
of the corresponding Gaussian sample paths where
is the dimension of the random field (see Chilès and Delfiner, 1999, p. 89).
-
RMgauss
The Gaussian model has an infinitely differentiable variogram function. This smoothness is artificial. Furthermore, this often leads to singular matrices and therefore numerically instable procedures (see Stein, 1999, p. 29). The Gaussian model is included in the stable class (
RMstable
) for the choice.
-
RMgencauchy
where
and
. This model has a smoothness parameter
and a parameter
which determines the asymptotic power law. More precisely, this model admits simulating random fields where fractal dimension D of the Gaussian sample path and Hurst coefficient H can be chosen independently (compare also with
RMlgd
): Here, we haveand
. The smaller
, the longer the long-range dependence. The variogram function is very regular near the origin, because its Taylor expansion only contains even terms and reaches its sill slowly. Note that the Cauchy Family (
RMcauchy
) is included in this family for the choiceand
.
-
RMgenfbm
where
and
. This is an intrinsic variogram function.
-
RMgengneiting
This is a family of stationary models whose elements are specified by the two parametersand
with
being a non-negative integer and
with
denoting the dimension of the random field (the models can be used for any dimension). Let
.
For
the model equals the Askey model (
RMaskey
) and is therefore not implemented.For
the model is given by
If
and for
A special case of this family is
RMgneiting
(withthere) for the choice
.
-
RMgneiting
if
and
otherwise. Here,
. This variogram function is valid only for dimensions less than or equal to 3. It is 6 times differentiable and has compact support. This model is an alternative to
RMgauss
as its graph is hardly distinguishable from the graph of the Gaussian model, but possesses neither the mathematical nor the numerical disadvantages of the Gaussian model. It is a special case ofRMgengneiting
for the choice.
-
RMlgd
where
and
, with
denoting the dimension of the random field. The model is only valid for dimension
. This model admits simulating random fields where fractal dimension
of the Gaussian sample and Hurst coefficient
can be chosen independently (compare also
RMgencauchy
): Here, the random field has fractal dimensionand Hurst coefficient
for
.
-
RMmatern
where
and
is the modified Bessel function of second kind. This is one of 3 possible parametrizations (Whittle, Matérn, Handcock-Wallis) of the Whittle-Matérn model. The Whittle-Matérn model is the model of choice if the smoothness of a random field is to be parametrized: the sample paths of a Gaussian random field with this covariance structure are
times differentiable if and only if
(see Gneiting and Guttorp, 2010, p. 24). Furthermore, the fractal dimension
of the Gaussian sample paths is determined by
: We have
and
for
where
is the dimension of the random field (see Stein, 1999, p. 32). If
the Matérn model equals
RMexp
. Fortending to
a rescaled Gaussian model
RMgauss
appears as limit for the Handcock-Wallis parametrization. -
RMpenta
The model is only valid for dimensions
. It has a 4 times differentiable variogram function with compact support (cf. Chilès and Delfiner, 1999, p. 84).
-
RMqexp
where
.
-
RMspheric
This variogram model is valid only for dimensions less than or equal to 3 and has compact support.
-
RMstable
where
. The parameter
determines the fractal dimension
of the Gaussian sample paths:
where
is the dimension of the random field. For
the Gaussian sample paths are not differentiable (see Gelfand et al., 2010, p. 25). The stable family includes the exponential model (
RMexp
) forand the Gaussian model (
RMgauss
) for.
-
RMwave
The model is only valid for dimensions
. It is a special case of
RMbessel
for. This variogram models a hole effect (see Chilès and Delfiner, 1999, p. 92).
-
RMwhittle
where
and
is the modified Bessel function of second kind. This is one of 3 possible parametrizations (Whittle, Matérn, Handcock-Wallis) of the Whittle-Matérn model, for further details, see information for entry
RMmatern
above.
Value
A numeric vector with generalized correlations (= negative semi-variances
computed with variance parameter param["variance"] = 1
).
Author(s)
Andreas Papritz papritz@retired.ethz.ch
References
Berg, C., Mateau, J., Porcu, E. (2008) The dagum family of isotropic correlation functions, Bernoulli, 14, 1134–1149, doi:10.3150/08-BEJ139.
Chilès, J.-P., Delfiner, P. (1999) Geostatistics Modeling Spatial Uncertainty, Wiley, New York, doi:10.1002/9780470316993.
Gneiting, T. (2002) Compactly supported correlation functions. Journal of Multivariate Analysis, 83, 493–508, doi:10.1006/jmva.2001.2056.
Gneiting, T., Schlather, M. (2004) Stochastic models which separate fractal dimension and Hurst effect. SIAM review 46, 269–282, doi:10.1137/S0036144501394387.
Gneiting, T., Guttorp, P. (2010) Continuous Parameter Stochastic Process Theory, In Gelfand, A. E., Diggle, P. J., Fuentes, M., Guttrop, P. (Eds.) Handbook of Spatial Statistics, CRC Press, Boca Raton, p. 17–28, doi:10.1201/9781420072884.
Schlather M., Malinowski A., Oesting M., Boecker D., Strokorb K., Engelke S., Martini J., Ballani F., Moreva O., Auel J., Menck P.J., Gross S., Ober U., Ribeiro P., Ripley B.D., Singleton R., Pfaff B., R Core Team (2022). RandomFields: Simulation and Analysis of Random Fields. R package version 3.3.14, https://cran.r-project.org/src/contrib/Archive/RandomFields/.
Stein, M. L. (1999) Interpolation of Spatial Data: Some Theory for Kriging, Springer, New York, doi:10.1007/978-1-4612-1494-6.
See Also
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;
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.
Examples
## scaled lag distances
x <- seq(0, 3, length.out = 100)
## generalized correlations stable model
y <- gencorr(x, variogram.model = "RMstable", param = c(alpha = 1.5))
plot(x, y)
## generalized correlations circular model
y <- gencorr(x, variogram.model = "RMcircular")
plot(x, y)