Package 'GeoModels'

Annual Precipitation Anomalies in the U.S.

Description

A numerical matrix of dimension $7252 \times 3$ containing longitude, latitude, and yearly total precipitation anomalies registered at 7,352 location sites in the USA.

Usage

data(anomalies)

Format

A numeric matrix with 7,252 rows and 3 columns:

Column 1

Longitude

Column 2

Latitude

Column 3

Annual precipitation anomaly

Source

Kaufman, C.G., Schervish, M.J., Nychka, D.W. (2008). Covariance tapering for likelihood-based estimation in large spatial data sets. Journal of the American Statistical Association, Theory & Methods, 103, 1545–1555.

---

Checking Bivariate Covariance Models

Description

Checks whether the correlation model is bivariate.

Usage

CheckBiv(numbermodel)

Arguments

numbermodel

A numeric value; the number associated with a given correlation model.

Details

This function checks whether the correlation model is bivariate.

Value

A logical value: TRUE if the correlation model is bivariate, and FALSE otherwise.

Author(s)

Moreno Bevilacqua [email protected]
https://sites.google.com/view/moreno-bevilacqua/home
Víctor Morales Oñate [email protected]
https://sites.google.com/site/moralesonatevictor/
Christian Caamaño-Carrillo [email protected]
https://www.researchgate.net/profile/Christian-Caamano

Examples

library(GeoModels)
CheckBiv(CkCorrModel("Bi_matern_sep"))

---

Checking Distance Type

Description

Checks the validity and type of the specified distance.

Usage

CheckDistance(distance)

Arguments

distance

A character string indicating the type of distance. Available options are: "Eucl" (Euclidean), "Geod" (Geodesic), and "Chor" (Chordal). See also GeoCovmatrix.

Details

This function checks whether the specified distance type is valid.

Value

An integer:

• 0 for Euclidean distance

• 1 for Geodesic distance

• 2 for Chordal distance

If the input is not recognized, the function returns NULL.

Author(s)

Moreno Bevilacqua [email protected]
https://sites.google.com/view/moreno-bevilacqua/home
Víctor Morales Oñate [email protected]
https://sites.google.com/site/moralesonatevictor/
Christian Caamaño-Carrillo [email protected]
https://www.researchgate.net/profile/Christian-Caamano

---

Checking if a covariance is valid only on the sphere

Description

Subroutine called by InitParam. The procedure controls if a covariance model is valid only on the sphere.

Usage

CheckSph(numbermodel)

Arguments

numbermodel

Numeric; the code number for the covariance model.

Details

The function checks if a covariance is valid only on the sphere

Value

Returns TRUE or FALSE

Author(s)

Moreno Bevilacqua, [email protected],https://sites.google.com/view/moreno-bevilacqua/home, Víctor Morales Oñate, [email protected], https://sites.google.com/site/moralesonatevictor/, Christian", Caamaño-Carrillo, [email protected],https://www.researchgate.net/profile/Christian-Caamano

---

Checking SpaceTime covariance models

Description

The procedure control if the correlation model is spacetime.

Usage

CheckST(numbermodel)

Arguments

numbermodel

numeric; the number associated to a given correlation model.

Details

The function check if the correlation model is spacetime.

Value

Returns TRUE or FALSE depending if the correlation model is spacetime or not.

Author(s)

Moreno Bevilacqua, [email protected],https://sites.google.com/view/moreno-bevilacqua/home, Víctor Morales Oñate, [email protected], https://sites.google.com/site/moralesonatevictor/, Christian", Caamaño-Carrillo, [email protected],https://www.researchgate.net/profile/Christian-Caamano

Examples

library(GeoModels)
CheckST(CkCorrModel("gneiting"))

---

Checking Correlation Model

Description

The procedure controls if the correlation model inserted is correct.

Usage

CkCorrModel(corrmodel)

Arguments

corrmodel

String; the name of a correlation model, for the description see GeoCovmatrix.

Details

The procedure controls if the correlation model is correct

Value

Return a number associated to a given correlation model if the model is considered in the package. Otherwise return NULL.

Author(s)

Moreno Bevilacqua, [email protected],https://sites.google.com/view/moreno-bevilacqua/home, Víctor Morales Oñate, [email protected], https://sites.google.com/site/moralesonatevictor/, Christian", Caamaño-Carrillo, [email protected],https://www.researchgate.net/profile/Christian-Caamano

---

Checking Input

Description

Subroutine called by the fitting procedures. The procedure controls the the validity of the input inserted by the users.

Usage

CkInput(coordx, coordy, coordz,coordt, coordx_dyn,corrmodel, data, distance, 
           fcall, fixed, grid,likelihood, maxdist, maxtime, 
            model, n,  optimizer, param, radius,
           start, taper, tapsep,  type, varest, 
           weighted,copula,X)

Arguments

coordx	A numeric ($d \times 2$)-matrix or ($d \times 3$)-matrix Coordinates on a sphere for a fixed radius radius are passed in lon/lat format expressed in decimal degrees.
coordy	A numeric vector giving 1-dimension of spatial coordinates; Optional argument, the default is NULL.
coordz	A numeric vector giving 1-dimension of spatial coordinates; Optional argument, the default is NULL.
coordt	A numeric vector assigning 1-dimension of temporal coordinates.
corrmodel	String; the name of a correlation model, for the description see GeoFit.
coordx_dyn	A list of $m$ numeric ($d_t \times 2$)-matrices containing dynamical (in time) spatial coordinates. Optional argument, the default is NULL
data	A numeric vector or a ($n \times d$)-matrix or ($d \times d \times n$)-matrix of observations.
distance	String; the name of the spatial distance. The default is Eucl, the euclidean distance. See the Section Details.
fcall	String; Fitting to call the fitting procedure and simulation to call the simulation.
fixed	A named list giving the values of the parameters that will be considered as known values. The listed parameters for a given correlation function will be not estimated, i.e. if list(nugget=0) the nugget effect is ignored.
grid	Logical; if FALSE (the default) the data are interpreted as a vector or a ($n \times d$)-matrix, instead if TRUE then ($d \times d \times n$)-matrix is considered.
likelihood	String; the configuration of the composite likelihood. Marginal is the default.
maxdist	Numeric; an optional positive value indicating the maximum spatial distance considered in the composite-likelihood computation.
maxtime	Numeric; an optional positive value indicating the maximum temporal lag separation in the composite-likelihood.
radius	Numeric; the radius of the sphere in the case of lon-lat coordinates. The default is 6371, the radius of the earth.
model	String; the density associated to the likelihood objects. Gaussian is the default.
n	Numeric; the number of trials in a binomial random fields. Default is $1$.
optimizer	String; the optimization algorithm (see optim for details). 'Nelder-Mead' is the default.
param	A numeric vector of parameters, needed only in simulation. See GeoSim.
start	A named list with the initial values of the parameters that are used by the numerical routines in maximization procedure. NULL is the default.
taper	String; the name of the tapered correlation function.
tapsep	Numeric; an optional value indicating the separabe parameter in the space time quasi taper (see Details).
type	String; the type of the likelihood objects. If Pairwise (the default) then the marginal composite likelihood is formed by pairwise marginal likelihoods.
varest	Logical; if TRUE the estimate' variances and standard errors are returned. FALSE is the default.
weighted	Logical; if TRUE the likelihood objects are weighted. If FALSE (the default) the composite likelihood is not weighted.
copula	String; the type of copula. It can be "Clayton" or "Gaussian"
X	Numeric; Matrix of space-time covariates in the linear mean specification.

Details

Subroutine called by the fitting procedures. The procedure controls the the validity of the input inserted by the users.

Value

A list with the type of error associated with the input parameters.

Author(s)

Moreno Bevilacqua, [email protected],https://sites.google.com/view/moreno-bevilacqua/home, Víctor Morales Oñate, [email protected], https://sites.google.com/site/moralesonatevictor/, Christian", Caamaño-Carrillo, [email protected],https://www.researchgate.net/profile/Christian-Caamano

See Also

GeoFit

---

Checking Composite-likelihood Type

Description

Subroutine called by InitParam. The procedure controls the type of the composite-likelihood inserted by the users.

Usage

CkLikelihood(likelihood)

Arguments

likelihood

String; the configuration of the composite likelihood. Marginal is the default.

Details

The function controls the type of the composite-likelihood inserted by the users.

Value

The function returns a numeric positive integer, or NULL if the likelihood is invalid.

Author(s)

Moreno Bevilacqua, [email protected],https://sites.google.com/view/moreno-bevilacqua/home, Víctor Morales Oñate, [email protected], https://sites.google.com/site/moralesonatevictor/, Christian", Caamaño-Carrillo, [email protected],https://www.researchgate.net/profile/Christian-Caamano

See Also

GeoFit

---

Checking Random Field Type

Description

Subroutine called by InitParam.\ The procedure controls the type of random field inserted by the users.

Usage

CkModel(model)

Arguments

model

String; the density associated with the likelihood objects. Gaussian is the default.

Details

The function controls the type of random field inserted by the users.

Value

The function returns a numeric positive integer, or NULL if the model is invalid.

Author(s)

Moreno Bevilacqua [email protected] https://sites.google.com/view/moreno-bevilacqua/home \ Víctor Morales Oñate [email protected] https://sites.google.com/site/moralesonatevictor/ \ Christian Caamaño-Carrillo [email protected] https://www.researchgate.net/profile/Christian-Caamano

See Also

GeoFit

---

Checking Likelihood Objects

Description

Subroutine called by InitParam. \ The procedure controls the type of likelihood objects inserted by the users.

Usage

CkType(type)

Arguments

type	String; the type of the likelihood objects. If Pairwise (the default) then the marginal composite likelihood is formed by pairwise marginal likelihoods.

Details

The procedure checks the likelihood object.

Value

The function returns a numeric positive integer, or NULL if the type of likelihood is invalid.

Author(s)

Moreno Bevilacqua [email protected] https://sites.google.com/view/moreno-bevilacqua/home \ Víctor Morales Oñate [email protected] https://sites.google.com/site/moralesonatevictor/ \ Christian Caamaño-Carrillo [email protected] https://www.researchgate.net/profile/Christian-Caamano

See Also

GeoFit

---

Optimizes the Composite indipendence log-likelihood

Description

Subroutine called by GeoFit. The procedure estimates the model parameters by maximisation of the indipendence composite log-likelihood.

Usage

CompIndLik2(bivariate, coordx, coordy ,coordz,coordt,
coordx_dyn, data, flagcorr, flagnuis, fixed,grid,
              lower, model, n, namescorr, namesnuis, 
              namesparam,
              numparam, optimizer, onlyvar, 
              param, spacetime, type,
              upper, namesupper, varest, ns, X,
              sensitivity,copula,MM,score)

Arguments

bivariate	Logical; if TRUE then the data come froma a bivariate random field. Otherwise from a univariate random field.
coordx	A numeric ($d \times 2$)-matrix or ($d \times 3$)-matrix Coordinates on a sphere for a fixed radius radius are passed in lon/lat format expressed in decimal degrees.
coordy	A numeric vector giving 1-dimension of spatial coordinates; Optional argument, the default is NULL.
coordz	A numeric vector giving 1-dimension of spatial coordinates; Optional argument, the default is NULL.
coordt	A numeric vector assigning 1-dimension of temporal coordinates. Optional argument, the default is NULL then a spatial random field is expected.
coordx_dyn	A list of $m$ numeric ($d_t \times 2$)-matrices containing dynamical (in time) spatial coordinates. Optional argument, the default is NULL
data	A numeric vector or a ($n \times d$)-matrix or ($d \times d \times n$)-matrix of observations.
flagcorr	A numeric vector of binary values denoting which paramerters of the correlation function will be estimated.
flagnuis	A numeric vector of binary values denoting which nuisance paramerters will be estimated.
fixed	A numeric vector of parameters that will be considered as known values.
grid	Logical; if FALSE (the default) the data are interpreted as a vector or a ($n \times d$)-matrix, instead if TRUE then ($d \times d \times n$)-matrix is considered.
lower	An optional named list giving the values for the lower bound of the space parameter when the optimizer is L-BFGS-B or nlminb or optimize. The names of the list must be the same of the names in the start list.
model	Numeric; the id value of the density associated to the likelihood objects.
n	Numeric; number of trials in a binomial random fields.
namescorr	String; the names of the correlation parameters.
namesnuis	String; the names of the nuisance parameters.
namesparam	String; the names of the parameters to be maximised.
numparam	Numeric; the number of parameters to be maximised.
optimizer	String; the optimization algorithm (see optim for details). Nelder-Mead is the default. Other possible choices are nlm, BFGS L-BFGS-B and nlminb. In these last two cases upper and lower bounds can be passed by the user. In the case of one-dimensional optimization, the function optimize is used.
onlyvar	Logical; if TRUE (and varest is TRUE) only the variance covariance matrix is computed without optimizing. FALSE is the default.
param	A numeric vector of parameters values.
spacetime	Logical; if TRUE the random field is spatial-temporal otherwise is a spatial field.
type	String; the type of the likelihood objects. If Pairwise (the default) then the marginal composite likelihood is formed by pairwise marginal likelihoods.
upper	An optional named list giving the values for the upper bound of the space parameter when the optimizer is or L-BFGS-B or nlminb or optimize. The names of the list must be the same of the names in the start list.
namesupper	String; the names of the upper limit of the parameters.
varest	Logical; if TRUE the estimate variances and standard errors are returned. FALSE is the default.
ns	Numeric; Number of (dynamical) temporal instants.
X	Numeric; Matrix of space-time covariates in the linear mean specification.
sensitivity	Logical; if TRUE then the sensitivy matrix is computed
copula	String; the type of copula. It can be "Clayton" or "Gaussian"
MM	Numeric;a non constant fixed mean
score	Logical; should score function be computed?

Value

Return a list from an optim call.

Author(s)

Moreno Bevilacqua, [email protected],https://sites.google.com/view/moreno-bevilacqua/home, Víctor Morales Oñate, [email protected], https://sites.google.com/site/moralesonatevictor/, Christian", Caamaño-Carrillo, [email protected],https://www.researchgate.net/profile/Christian-Caamano

See Also

GeoFit

---

Optimizes the Composite Log-likelihood

Description

Subroutine called by GeoFit. The procedure estimates the model parameters by maximization of the composite log-likelihood.

Usage

CompLik(copula, bivariate, coordx, coordy, coordz, coordt, 
        coordx_dyn, corrmodel, data, distance, flagcorr, 
        flagnuis, fixed, grid, likelihood,  lower, 
        model, n, namescorr, namesnuis, namesparam,
        numparam, numparamcorr, optimizer, 
        onlyvar,  param,
        spacetime, type, upper, varest,
        weigthed, ns, X, sensitivity, MM, aniso,score)

Arguments

copula	String; the type of copula. It can be "Clayton" or "Gaussian".
bivariate	Logical; if TRUE then the data come from a bivariate random field, otherwise from a univariate random field.
coordx	A numeric $d \times 2$ or $d \times 3$ matrix. Coordinates on a sphere for a fixed radius radius are passed in lon/lat format expressed in decimal degrees.
coordy	A numeric vector giving one dimension of spatial coordinates; optional, default is NULL.
coordz	A numeric vector giving one dimension of spatial coordinates; optional, default is NULL.
coordt	A numeric vector giving one dimension of temporal coordinates; optional, default is NULL (in which case a spatial random field is assumed).
coordx_dyn	A list of $m$ numeric $d_t \times 2$ matrices containing dynamic (in time) spatial coordinates; optional, default is NULL.
corrmodel	Numeric; the ID of the correlation model.
data	A numeric vector, or a $n \times d$ matrix, or a $d \times d \times n$ array of observations.
distance	String; the name of the spatial distance. Default is "Eucl" (Euclidean distance). See Details.
flagcorr	Numeric vector of binary values indicating which parameters of the correlation function will be estimated.
flagnuis	Numeric vector of binary values indicating which nuisance parameters will be estimated.
fixed	Numeric vector of parameters considered as known values.
grid	Logical; if FALSE (default), data are interpreted as vector or $n \times d$ matrix; if TRUE, then a $d \times d \times n$ array is considered.
likelihood	String; configuration of the composite likelihood (see GeoFit).
lower	Named list; optional lower bounds for parameters when using optimizers L-BFGS-B, nlminb, or optimize. Names must match those in the start list.
model	Numeric; ID of the density associated with the likelihood objects.
n	Numeric; number of trials in binomial random fields.
namescorr	Character vector; names of the correlation parameters.
namesnuis	Character vector; names of the nuisance parameters.
namesparam	Character vector; names of the parameters to be maximized.
numparam	Numeric; number of parameters to be maximized.
numparamcorr	Numeric; number of correlation parameters.
optimizer	String; optimization algorithm (see optim). Default is "Nelder-Mead". Other options: "nlm", "BFGS", "L-BFGS-B", "nlminb". For "L-BFGS-B" and "nlminb" bounds can be provided. For 1D optimization, optimize is used.
onlyvar	Logical; if TRUE (and varest is TRUE), only the variance-covariance matrix is computed without optimizing. Default is FALSE.
param	Numeric vector of parameter values.
spacetime	Logical; if TRUE, the random field is spatio-temporal, otherwise spatial.
type	String; type of likelihood object. Default is "Pairwise" (marginal composite likelihood formed by pairwise marginal likelihoods).
upper	Named list; optional upper bounds for parameters when using optimizers L-BFGS-B, nlminb, or optimize. Names must match those in the start list.
varest	Logical; if TRUE, variance estimates and standard errors are returned. Default is FALSE.
weigthed	Logical; if TRUE, decreasing weights from a compactly supported correlation function with compact support maxdist (maxtime) are used.
ns	Numeric; number of (dynamic) temporal instants.
X	Numeric; matrix of space-time covariates in the linear mean specification.
sensitivity	Logical; if TRUE, the sensitivity matrix is computed.
MM	Numeric; a non-constant fixed mean.
aniso	Logical; whether anisotropy should be considered.
score	Logical; should score function be computed?

Details

Subroutine called by GeoFit. The procedure estimates model parameters by maximization of the composite log-likelihood.

Value

Returns a list from an optim call.

Author(s)

Moreno Bevilacqua, [email protected], https://sites.google.com/view/moreno-bevilacqua/home,
Víctor Morales Oñate, [email protected], https://sites.google.com/site/moralesonatevictor/,
Christian Caamaño-Carrillo, [email protected], https://www.researchgate.net/profile/Christian-Caamano

See Also

GeoFit

---

Optimizes the Composite log-likelihood

Description

Subroutine called by GeoFit. The procedure estimates the model parameters by maximisation of the composite log-likelihood.

Usage

CompLik2(copula,bivariate, coordx, coordy ,coordz,coordt,
coordx_dyn,corrmodel, data, distance, flagcorr, flagnuis, 
         fixed,grid,likelihood, lower, 
         model, n, namescorr, namesnuis, namesparam,
         numparam, numparamcorr, optimizer, onlyvar,
          param, spacetime, type,
         upper, varest, weigthed, ns, X,sensitivity,
         colidx,rowidx,neighb,MM,aniso,score)

Arguments

copula	String; the type of copula. It can be "Clayton" or "Gaussian"
bivariate	Logical; if TRUE then the data come froma a bivariate random field. Otherwise from a univariate random field.
coordx	A numeric ($d \times 2$)-matrix or ($d \times 3$)-matrix Coordinates on a sphere for a fixed radius radius are passed in lon/lat format expressed in decimal degrees.
coordy	A numeric vector giving 1-dimension of spatial coordinates; Optional argument, the default is NULL.
coordz	A numeric vector giving 1-dimension of spatial coordinates; Optional argument, the default is NULL.
coordt	A numeric vector assigning 1-dimension of temporal coordinates. Optional argument, the default is NULL then a spatial random field is expected.
coordx_dyn	A list of $m$ numeric ($d_t \times 2$)-matrices containing dynamical (in time) spatial coordinates. Optional argument, the default is NULL
corrmodel	Numeric; the id of the correlation model.
data	A numeric vector or a ($n \times d$)-matrix or ($d \times d \times n$)-matrix of observations.
distance	String; the name of the spatial distance. The default is Eucl, the euclidean distance. See the Section Details.
flagcorr	A numeric vector of binary values denoting which paramerters of the correlation function will be estimated.
flagnuis	A numeric vector of binary values denoting which nuisance paramerters will be estimated.
fixed	A numeric vector of parameters that will be considered as known values.
grid	Logical; if FALSE (the default) the data are interpreted as a vector or a ($n \times d$)-matrix, instead if TRUE then ($d \times d \times n$)-matrix is considered.
likelihood	String; the configuration of the compositelikelihood, see GeoFit.
lower	An optional named list giving the values for the lower bound of the space parameter when the optimizer is L-BFGS-B or nlminb or optimize. The names of the list must be the same of the names in the start list.
model	Numeric; the id value of the density associated to the likelihood objects.
n	Numeric; number of trials in a binomial random fields.
namescorr	String; the names of the correlation parameters.
namesnuis	String; the names of the nuisance parameters.
namesparam	String; the names of the parameters to be maximised.
numparam	Numeric; the number of parameters to be maximised.
numparamcorr	Numeric; the number of correlation parameters.
optimizer	String; the optimization algorithm (see optim for details). Nelder-Mead is the default. Other possible choices are nlm, BFGS L-BFGS-B and nlminb. In these last two cases upper and lower bounds can be passed by the user. In the case of one-dimensional optimization, the function optimize is used.
onlyvar	Logical; if TRUE (and varest is TRUE) only the variance covariance matrix is computed without optimizing. FALSE is the default.
param	A numeric vector of parameters' values.
spacetime	Logical; if TRUE the random field is spatial-temporal otherwise is a spatial field.
type	String; the type of the likelihood objects. If Pairwise (the default) then the marginal composite likelihood is formed by pairwise marginal likelihoods.
upper	An optional named list giving the values for the upper bound of the space parameter when the optimizer is or L-BFGS-B or nlminb or optimize. The names of the list must be the same of the names in the start list.
varest	Logical; if TRUE the estimate' variances and standard errors are returned. FALSE is the default.
weigthed	Logical; if TRUE then decreasing weigths coming from a compactly supported correlation function with compact support maxdist (maxtime)are used.
ns	Numeric; Number of (dynamical) temporal instants.
X	Numeric; Matrix of space-time covariates in the linear mean specification.
sensitivity	Logical; if TRUE then the sensitivy matrix is computed
colidx	Numeric; Vector of indexes for spatial distances.
rowidx	Numeric; Vector of indexes for spatial distances.
neighb	Numeric; an optional positive integer indicating the order of neighborhood location.
MM	Numeric;a non constant fixed mean
aniso	Logical; should anisotropy be considered?
score	Logical; should score function be computed?

Value

Return a list from an optim call.

Author(s)

Moreno Bevilacqua, [email protected],https://sites.google.com/view/moreno-bevilacqua/home, Víctor Morales Oñate, [email protected], https://sites.google.com/site/moralesonatevictor/, Christian", Caamaño-Carrillo, [email protected],https://www.researchgate.net/profile/Christian-Caamano

See Also

GeoFit

---

Lists the Parameters of a Correlation Model

Description

Subroutine called by InitParam and other procedures. The procedure returns a list with the parameters of a given correlation model.

Usage

CorrelationPar(corrmodel)

Arguments

corrmodel

Integer; an integer associated to a given correlation model.

Details

The function return a list with the Parameters of a Correlation Model

Value

Return a vector string of correlation parameters.

Author(s)

Moreno Bevilacqua, [email protected],https://sites.google.com/view/moreno-bevilacqua/home, Víctor Morales Oñate, [email protected], https://sites.google.com/site/moralesonatevictor/, Christian", Caamaño-Carrillo, [email protected],https://www.researchgate.net/profile/Christian-Caamano

See Also

GeoFit

---

Lists the Parameters of a Correlation Model

Description

The procedure returns a list with the names of the parameters of a given correlation model.

Usage

CorrParam(corrmodel)

Arguments

corrmodel

String: the name associated to a given correlation model.

Details

The function return a list with the Parameters of a Correlation Model

Value

Return a vector string of correlation parameters.

Author(s)

Moreno Bevilacqua, [email protected],https://sites.google.com/view/moreno-bevilacqua/home, Víctor Morales Oñate, [email protected], https://sites.google.com/site/moralesonatevictor/, Christian", Caamaño-Carrillo, [email protected],https://www.researchgate.net/profile/Christian-Caamano

See Also

GeoCovmatrix

Examples

require(GeoModels)
################################################################
###
### Example 1. Parameters of the  Matern  model
###
###############################################################

CorrParam("Matern")


################################################################
###
### Example 2. Parameters of the  Generalized Wendland  model
###
###############################################################

CorrParam("GenWend")


################################################################
###
### Example 3. Parameters of the  Generalized Cauchy  model
###
###############################################################

CorrParam("GenCauchy")


################################################################
###
### Example 4. Parameters of the  space time Gneiting  model
###
###############################################################

CorrParam("Gneiting")


################################################################
###
### Example 5. Parameters of the bi-Matern separable  model.
###            Note that in the bivariate case variance paramters are
###            included
###############################################################

CorrParam("Bi_Matern_sep")

---

Correlation Function for Sinh-Arcsinh Random Fields

Description

Computes the correlations f for a random field transformed via the sinh-arcsinh (SAS) distribution. This transformation introduces flexible skewness and tail behavior to an underlying Gaussian field. The resulting correlation is derived via an infinite Hermite expansion, as described in Equation (16) of Blasi et al. (2022).

Usage

corrsas(corr, skew, tail, max_coeff = NULL)

Arguments

corr	A numeric vector of correlation values of the underlying standard Gaussian random field.
skew	A numeric value representing the skewness parameter $\alpha$ of the sinh-arcsinh transformation. Positive values induce right-skewness, negative values left-skewness.
tail	A positive numeric value representing the tailweight parameter $\kappa$. Values less than 1 yield heavier tails than Gaussian, while values greater than 1 produce lighter tails.
max_coeff	Optional integer. The maximum number of Hermite coefficients used in the infinite series expansion. If NULL, a default truncation value is used internally.

Details

The correlation of the sinh-arcsinh transformed field is computed as:

$\rho_{SAS}(h) = \sum_{j=1}^{\infty} \frac{\xi_j^2(\alpha, \kappa)}{j!} \rho(h)^j$

where $\rho(h)$ is the correlation function of the underlying Gaussian field and $\xi_j(\alpha, \kappa)$ are Hermite coefficients depending on the skewness and tail parameters. This series is truncated at max_coeff terms for computational feasibility.

See Equation (16) in Blasi et al. (2022) for the full derivation.

Value

A numeric vector of adjusted correlation values corresponding to the SAS-transformed process.

References

Blasi, F., Caamaño-Carrillo, C., Bevilacqua, M., Furrer, R. (2022). A selective view of climatological data and likelihood estimation. Spatial Statistics, 50, 100596. doi:10.1016/j.spasta.2022.100596

Examples

# Example usage:
rho <- seq(0, 1, length.out = 50)
rho_sas <- corrsas(rho, skew = 0.5, tail = 0.8, max_coeff = 20)
plot(rho, rho_sas, type = "l", main = "SAS Correlation", 
         xlab = "Original Correlation", ylab = "Transformed Correlation")

---

Spatial Anisotropy correction

Description

Transforms or back-transforms a set of coordinates according to the geometric anisotropy parameters.

Usage

GeoAniso(coords, anisopars=c(0,1), inverse = FALSE)

Arguments

coords	An n x 2 matrix with the coordinates to be transformed.
anisopars	A bivariate vector with the the anisotropy angle and the anisotropy ratio, respectively. The angle must be given in radians in [0,pi] and the anisotropy ratio must be greater or equal than 1.
inverse	Logical: Default to FALSE. If TRUE the reverse transformation is performed.

Details

Geometric anisotropy is defined by a linear tranformation from the anisotropic space to the isotropic space that is

$Y = X R S$

where X is a matrix with original coordinates (anisotropic space), and Y is a matrix with transformed coordinates (isotropic space). Here R is a rotation matrix with associated anisotropy angle parameter (in $[0,pi]$) and a $S$ is a shrinking matrix with associated anisotropy ratio parameter (greeater or equal than one). The two parameters are specified in the anisopars argument as a bivariate numeric vector. The case $(.,1)$ corresponds to the isotropic case.

Value

Returns a matrix of transformed coordinates

Author(s)

Moreno Bevilacqua, [email protected],https://sites.google.com/view/moreno-bevilacqua/home, Víctor Morales Oñate, [email protected], https://sites.google.com/site/moralesonatevictor/, Christian", Caamaño-Carrillo, [email protected],https://www.researchgate.net/profile/Christian-Caamano

---

Spatial and Spatio-temporal correlation or covariance of (non) Gaussian random fields

Description

The function computes the correlations of a spatial (or spatio-temporal or bivariate spatial) Gaussian or non Gaussian randomm field for a given correlation model and a set of spatial (temporal) distances.

Usage

GeoCorrFct(x,t=NULL,corrmodel, model="Gaussian",
distance="Eucl", param, radius=6371,n=1,
covariance=FALSE,variogram=FALSE)

Arguments

x	A set of spatial distances.
t	A set of (optional) temporal distances.
corrmodel	String; the name of a correlation model, for the description see GeoCovmatrix.
model	String; the type of RF. See GeoFit.
distance	String; the name of the spatial distance. The default is Eucl, the euclidean distance. See GeoFit.
param	A list of parameter values required for the covariance model.
radius	Numeric; a value indicating the radius of the sphere when using covariance models valid using the great circle distance. Default value is the radius of the earth in Km (i.e. 6371)
n	Numeric; the number of trials in a (negative) binomial random fields. Default is $1$.
covariance	Logic; if TRUE then the covariance is returned. Default is FALSE
variogram	Logic; if FALSE then the covariance/correlation is returned. Otherwise the associated semivariogram is returned

Value

Returns correlations or covariances values associated to a given parametric spatial and temporal correlation models.

Author(s)

Moreno Bevilacqua, [email protected],https://sites.google.com/view/moreno-bevilacqua/home, Víctor Morales Oñate, [email protected], https://sites.google.com/site/moralesonatevictor/, Christian, Caamaño-Carrillo, [email protected],https://www.researchgate.net/profile/Christian-Caamano

Examples

library(GeoModels)

################################################################
###
### Example 1. Covariance of a Gaussian random field with underlying 
### Matern correlation model with nugget
###
###############################################################
# Define the spatial distances
x = seq(0,1,0.002)
# Correlation Parameters for Matern model 
CorrParam("Matern")
NuisParam("Gaussian")
# Matern Parameters 
param=list(sill=2,smooth=0.5,scale=0.2/3,nugget=0.2,mean=0)
cc= GeoCorrFct(x=x, corrmodel="Matern", covariance=TRUE,
  param=param,model="Gaussian")
plot(cc,ylab="Corr",lwd=2,main="Matern correlation",type="l")

################################################################
###
### Example 2. Covariance of a Gaussian random field with underlying 
### Generalized Wendland-Matern correlation model
###
###############################################################
CorrParam("GenWend_Matern")
NuisParam("Gaussian")
# GenWend Matern Parameters 
param=list(sill=2,smooth=1,scale=0.1,nugget=0,power2=1/4,mean=0)
cc= GeoCorrFct(x=x, corrmodel="GenWend_Matern", param=param,model="Gaussian",covariance=FALSE)
plot(cc,ylab="Cov",lwd=2,,main="GenWend covariance",type="l")

################################################################
###
### Example 3. Semivariogram of a Tukeyh random field with underlying 
### Generalized Wendland correlation model
###
###############################################################
CorrParam("GenWend")
NuisParam("Tukeyh")
x = seq(0,1,0.005)
param=list(sill=1,smooth=1,scale=0.5,nugget=0,power2=5,tail=0.1,mean=0)
cc= GeoCorrFct(x=x, corrmodel="GenWend", param=param,model="Tukeyh",variogram=TRUE)
plot(cc,ylab="Corr",lwd=2,main="Tukey semivariogram",type="l")

################################################################
###
### Example 4. Semi-Variogram of a LoggGaussian random field with underlying 
### Kummer correlation model
###
###############################################################
CorrParam("Kummer")
NuisParam("LogGaussian")
# GenWend Matern Parameters 
param=list(smooth=1,sill=0.5,scale=0.1,nugget=0,power2=1,mean=0)
cc= GeoCorrFct(x=x, corrmodel="Kummer", param=param,model="LogGaussian",
       ,covariance=TRUE,variogram=TRUE)
plot(cc,ylab="Semivario",lwd=2,
  main="LogGaussian semivariogram",type="l")


################################################################
###
### Example 5. Covariance of Poisson random field with underlying 
### Matern correlation model
###
###############################################################
CorrParam("Matern")
NuisParam("Poisson")
x = seq(0,1,0.005)
param=list(scale=0.6/3,nugget=0,smooth=0.5,mean=2)
cc= GeoCorrFct(x=x, corrmodel="Matern", param=param,model="Poisson",covariance=TRUE)
plot(cc,ylab="Cov",lwd=2,
  main="Poisson covariance",type="l")

################################################################
###
### Example 6.  Space time  semivariogram of a Gaussian random field 
### with  separable Matern correlation model
###
###############################################################

## spatial and temporal distances 
h<-seq(0,3,by=0.04)
times<-seq(0,3,by=0.04)

# Correlation Parameters for the space time separable Matern model 
CorrParam("Matern")
NuisParam("Gaussian")
# Matern Parameters 
param=list(sill=1,scale_s=0.6/3,scale_t=0.5,nugget=0,mean=0,smooth_s=1.5,smooth_t=0.5)
cc= GeoCorrFct(x=h,t=times,corrmodel="Matern_Matern", param=param,
        model="Gaussian",variogram=TRUE)
plot(cc,lwd=2,type="l")


################################################################
###
### Example 7. Correlation of a bivariate Gaussian random field 
### with underlying  separable bivariate  Matern correlation model
###
###############################################################
# Define the spatial distances
x = seq(0,1,0.005)
 #Correlation Parameters for the bivariate sep Matern model 
CorrParam("Bi_Matern")
 #Matern Parameters 
param=list(sill_1=1,sill_2=1,smooth_1=0.5,smooth_2=1,smooth_12=0.75,
           scale_1=0.2/3, scale_2=0.2/3, scale_12=0.2/3,
           mean_1=0,mean_2=0,nugget_1=0,nugget_2=0,pcol=-0.2)
cc= GeoCorrFct(x=x, corrmodel="Bi_Matern", param=param,model="Gaussian")
plot(cc,ylab="corr",lwd=2,type="l")

---

Spatial and Spatio-temporal correlation or covariance of (non) Gaussian random fields (copula models)

Description

The function computes the correlations of a spatial or spatio-temporal or a bivariate spatial Gaussian or non Gaussian copula randomm field with a given covariance model and a set of spatial (temporal) distances.

Usage

GeoCorrFct_Cop(x,t=NULL,corrmodel, 
model="Gaussian",copula="Gaussian",
distance="Eucl", param, radius=6371,
n=1,covariance=FALSE,variogram=FALSE)

Arguments

x	A set of spatial distances.
t	A set of (optional) temporal distances.
corrmodel	String; the name of a correlation model, for the description see the Section Details.
model	String; the type of RF. See GeoFit.
copula	String; the type of copula. The two options are Gaussian and Clayton.
distance	String; the name of the spatial distance. The default is Eucl, the euclidean distance. See GeoFit.
param	A list of parameter values required for the covariance model.
radius	Numeric; a value indicating the radius of the sphere when using covariance models valid using the great circle distance. Default value is the radius of the earth in Km (i.e. 6371)
n	Numeric; the number of trials in a (negative) binomial random fields. Default is $1$.
covariance	Logic; if TRUE then the covariance is returned. Default is FALSE
variogram	Logic; if FALSE then the covariance/coorelation is returned. Otherwise the associated semivariogram is returned

Value

Returns a vector of correlations or covariances values associated to a given parametric spatial and temporal correlation models.

Author(s)

Moreno Bevilacqua, [email protected],https://sites.google.com/view/moreno-bevilacqua/home, Víctor Morales Oñate, [email protected], https://sites.google.com/site/moralesonatevictor/, Christian", Caamaño-Carrillo, [email protected],https://www.researchgate.net/profile/Christian-Caamano

Examples

library(GeoModels)

################################################################
###
### Example 1. Correlation of a (mean reparametrized) beta random field with underlying 
### Matern correlation model using Gaussian and Clayton copulas
###
###############################################################

# Define the spatial distances
x = seq(0,0.4,0.01)

# Correlation Parameters for Matern model 
CorrParam("Matern")
NuisParam("Beta2")
# corr Gaussian copula
param=list(smooth=0.5,sill=1,scale=0.2/3,nugget=0,mean=0,min=0,max=1,shape=0.5)
corr1= GeoCorrFct_Cop(x=x, corrmodel="Matern", param=param,copula="Gaussian",model="Beta2")

plot(corr1,ylab="corr",main="Gauss copula correlation",lwd=2)

# corr Clayton copula
param=list(smooth=0.5,sill=1,scale=0.2/3,nugget=0,mean=0,min=0,max=1,shape=0.5,nu=2)
corr2= GeoCorrFct_Cop(x=x, corrmodel="Matern", param=param,copula="Clayton",model="Beta2")
lines(x,corr2$corr,ylim=c(0,1),lty=2)

plot(corr1,ylab="corr",main="Clayton copula correlation",lwd=2)

---

Computes the fitted variogram model.

Description

The procedure computes and plots estimated covariance or semivariogram models of a Gaussian or a non Gaussian spatial (temporal or bivariate spatial) random field. It allows to add the empirical estimates in order to compare them with the fitted model.

Usage

GeoCovariogram(fitted, distance="Eucl",answer.cov=FALSE,
            answer.vario=FALSE, answer.range=FALSE, fix.lags=NULL, 
             fix.lagt=NULL, show.cov=FALSE, show.vario=TRUE, 
            show.range=FALSE, add.cov=FALSE, add.vario=FALSE,
            pract.range=95, vario, invisible=FALSE, ...)

Arguments

fitted	A fitted object obtained from the GeoFit or GeoWLS procedures.
distance	String; the name of the spatial distance. The default is Eucl, the euclidean distance. See GeoFit.
answer.cov	Logical; if TRUE a vector with the estimated covariance function is returned; if FALSE (the default) the covariance is not returned.
answer.vario	Logical; if TRUE a vector with the estimated variogram is returned; if FALSE (the default) the variogram is not returned.
answer.range	Logical; if TRUE the estimated pratical range is returned; if FALSE (the default) the pratical range is not returned.
fix.lags	Integer; a positive value denoting the spatial lag to consider for the plot of the temporal profile.
fix.lagt	Integer; a positive value denoting the temporal lag to consider for the plot of the spatial profile.
show.cov	Logical; if TRUE the estimated covariance function is plotted; if FALSE (the default) the covariance function is not plotted.
show.vario	Logical; if TRUE the estimated variogram is plotted; if FALSE (the default) the variogram is not plotted.
show.range	Logical; if TRUE the estimated pratical range is added on the plot; if FALSE (the default) the pratical range is not added.
add.cov	Logical; if TRUE the vector of the estimated covariance function is added on the current plot; if FALSE (the default) the covariance is not added.
add.vario	Logical; if TRUE the vector with the estimated variogram is added on the current plot; if FALSE (the default) the correlation is not added.
pract.range	Numeric; the percent of the sill to be reached.
vario	A Variogram object obtained from the GeoVariogram procedure.
invisible	Logical;If TRUE then a statistic the (sum of the squared diffeence between the empirical semivariogram and the estimated semivariogram) is computed.
...	other optional parameters which are passed to plot functions.

Details

The function computes the fitted variogram model

Value

Produces a plot. No values are returned.

Author(s)

Moreno Bevilacqua, [email protected],https://sites.google.com/view/moreno-bevilacqua/home, Víctor Morales Oñate, [email protected], https://sites.google.com/site/moralesonatevictor/, Christian", Caamaño-Carrillo, [email protected],https://www.researchgate.net/profile/Christian-Caamano

References

Cressie, N. A. C. (1993) Statistics for Spatial Data. New York: Wiley.

Gaetan, C. and Guyon, X. (2010) Spatial Statistics and Modelling. Spring Verlang, New York.

See Also

GeoFit.

Examples

library(GeoModels)
library(scatterplot3d)

################################################################
###
### Example 1. Plot of fitted covariance and fitted 
### and empirical semivariogram from a  Gaussian RF 
### with Matern correlation. 
###
###############################################################
set.seed(21)
# Set the coordinates of the points:
x = runif(300, 0, 1)
y = runif(300, 0, 1)
coords=cbind(x,y)

# Set the model's parameters:
corrmodel = "Matern"
model = "Gaussian"
mean = 0
sill = 1
nugget = 0
scale = 0.2/3
smooth=0.5

param=list(mean=mean,sill=sill, nugget=nugget, scale=scale, smooth=smooth)
# Simulation of the Gaussian random field:
data = GeoSim(coordx=coords, corrmodel=corrmodel, model=model,param=param)$data
I=Inf
start=list(mean=0,scale=scale,sill=sill)
lower=list(mean=-I,scale=0,sill=0)
upper=list(mean= I,scale=I,sill=I)
fixed=list(nugget=nugget,smooth=smooth)
# Maximum composite-likelihood fitting of the Gaussian random field:
fit = GeoFit(data=data,coordx=coords, corrmodel=corrmodel,model=model,
            likelihood="Marginal",type='Pairwise',start=start,
            lower=lower,upper=upper,
            optimizer="nlminb", fixed=fixed,neighb=3)

# Empirical estimation of the variogram:
vario = GeoVariogram(data=data,coordx=coords,maxdist=0.5)

# Plot of covariance and variogram functions:
GeoCovariogram(fit,show.vario=TRUE, vario=vario,pch=20)

################################################################
###
### Example 2. Plot of fitted covariance and fitted 
### and empirical semivariogram from a  Bernoulli 
###  RF with Genwend  correlation. 
###
###############################################################
set.seed(2111)

model="Binomial";n=1
# Set the coordinates of the points:
x = runif(500, 0, 1)
y = runif(500, 0, 1)
coords=cbind(x,y)

# Set the model's parameters:
corrmodel = "GenWend"
mean = 0
nugget = 0
scale = 0.2
smooth=0
power=4
param=list(mean=mean, nugget=nugget, scale=scale,smooth=0,power2=4)
# Simulation of the Gaussian RF:
data = GeoSim(coordx=coords, corrmodel=corrmodel, model=model,param=param,n=n)$data

start=list(mean=0,scale=scale)
fixed=list(nugget=nugget,power2=4,smooth=0)
# Maximum composite-likelihood fitting of the Binomial random field:
fit = GeoFit(data,coordx=coords, corrmodel=corrmodel,model=model,
             likelihood="Marginal",type='Pairwise',start=start,n=n,
             optimizer="BFGS", fixed=fixed,neighb=4)

# Empirical estimation of the variogram:
vario = GeoVariogram(data,coordx=coords,maxdist=0.5)

# Plot of covariance and variogram functions:
GeoCovariogram(fit, show.vario=TRUE, vario=vario,pch=20,ylim=c(0,0.3))


################################################################
###
### Example 3. Plot of fitted covariance and fitted 
### and empirical semivariogram from a  Weibull  RF
###   with Wend0 correlation. 
###
###############################################################
set.seed(111)

model="Weibull";shape=4
# Set the coordinates of the points:
x = runif(700, 0, 1)
y = runif(700, 0, 1)
coords=cbind(x,y)

# Set the model's parameters:
corrmodel = "Wend0"
mean = 0
nugget = 0
scale = 0.4
power2=4


param=list(mean=mean, nugget=nugget, scale=scale,shape=shape,power2=power2)
# Simulation of the Gaussian RF:
data = GeoSim(coordx=coords, corrmodel=corrmodel, model=model,param=param)$data

start=list(mean=0,scale=scale,shape=shape)
I=Inf
lower=list(mean=-I,scale=0,shape=0)
upper=list(mean= I,scale=I,shape=I)
fixed=list(nugget=nugget,power2=power2)

fit = GeoFit(data,coordx=coords, corrmodel=corrmodel,model=model,
             likelihood="Marginal",type='Pairwise',start=start,
             lower=lower,upper=upper,
             optimizer="nlminb", fixed=fixed,neighb=3)

# Empirical estimation of the variogram:
vario = GeoVariogram(data,coordx=coords,maxdist=0.5)

# Plot of covariance and variogram functions:
GeoCovariogram(fit, show.vario=TRUE, vario=vario,pch=20)

################################################################
###
### Example 4. Plot of  fitted  and empirical semivariogram
### from a space time Gaussian random fields  
### with double Matern correlation. 
###
###############################################################
set.seed(92)
# Define the spatial-coordinates of the points:
x = runif(50, 0, 1)
y = runif(50, 0, 1)
coords=cbind(x,y)
# Define the temporal sequence:
time = seq(0, 10, 1)


param=list(mean=mean,nugget=nugget,
  smooth_s=0.5,smooth_t=0.5,scale_s=0.5/3,scale_t=2/2,sill=sill)
# Simulation of the spatio-temporal Gaussian random field:
data = GeoSim(coordx=coords, coordt=time, corrmodel="Matern_Matern",param=param)$data

fixed=list(nugget=0, mean=0, smooth_s=0.5,smooth_t=0.5)
start=list(scale_s=0.2, scale_t=0.5, sill=1)
# Maximum composite-likelihood fitting of the space-time Gaussian random field:
fit = GeoFit(data, coordx=coords, coordt=time, corrmodel="Matern_Matern", maxtime=1,
             neighb=3, likelihood="Marginal", type="Pairwise",fixed=fixed, start=start)

# Empirical estimation of spatio-temporal covariance:
vario = GeoVariogram(data,coordx=coords, coordt=time, maxtime=5,maxdist=0.5)

# Plot of the fitted space-time variogram
GeoCovariogram(fit,vario=vario,show.vario=TRUE)

# Plot of covariance, variogram and spatio and temporal profiles:
GeoCovariogram(fit,vario=vario,fix.lagt=1,fix.lags=1,show.vario=TRUE,pch=20)


################################################################
###
### Example 5. Plot of  fitted  and empirical semivariogram
### from a bivariate Gaussian random fields  
### with  Matern correlation. 
###
###############################################################
set.seed(92)
# Define the spatial-coordinates of the points:
x <- runif(600, 0, 2)
y <- runif(600, 0, 2)
coords <- cbind(x,y)

# Simulation of a bivariate spatial Gaussian RF:
# with a  Bivariate Matern
set.seed(12)
param=list(mean_1=4,mean_2=2,smooth_1=0.5,smooth_2=0.5,smooth_12=0.5,
           scale_1=0.12,scale_2=0.1,scale_12=0.15,
           sill_1=1,sill_2=1,nugget_1=0,nugget_2=0,pcol=-0.5)
data <- GeoSim(coordx=coords,corrmodel="Bi_matern",
              param=param)$data



# selecting fixed and estimated parameters
fixed=list(mean_1=4,mean_2=2,nugget_1=0,nugget_2=0,
      smooth_1=0.5,smooth_2=0.5,smooth_12=0.5)
start=list(sill_1=var(data[1,]),sill_2=var(data[2,]),
           scale_1=0.1,scale_2=0.1,scale_12=0.1,
           pcol=cor(data[1,],data[2,]))


# Maximum marginal pairwise likelihood
fitcl<- GeoFit(data=data, coordx=coords, corrmodel="Bi_Matern",
                     likelihood="Marginal",type="Pairwise",
                     optimizer="BFGS" , start=start,fixed=fixed,
                     neighb=4)
print(fitcl)

# Empirical estimation of spatio-temporal covariance:
vario = GeoVariogram(data,coordx=coords,maxdist=0.4,bivariate=TRUE)
GeoCovariogram(fitcl,vario=vario,show.vario=TRUE,pch=20)

---

Image plot displaying the pattern of the sparsness of a covariance matrix.

Description

Image plot displaying the pattern of the sparsness of a covariance matrix.

Usage

GeoCovDisplay(covmatrix,limits=FALSE,pch=2)

Arguments

covmatrix	An object of class GeoCovmatrix. See the Section Details.
limits	Logical; If TRUE and the covariance matrix is spatiotemporal or spatial bivariate then vertical and horizontal lines are added to the image plot.
pch	Type of symbols to use in the image plot.

Details

For a given covariance matrix object (GeoCovmatrix) the function diplays the pattern of the sparsness of a covariance matrix where the white color represents 0 entries and black color represents non zero entries

Value

Produces a plot. No values are returned.

Author(s)

Moreno Bevilacqua, [email protected],https://sites.google.com/view/moreno-bevilacqua/home, Víctor Morales Oñate, [email protected], https://sites.google.com/site/moralesonatevictor/, Christian", Caamaño-Carrillo, [email protected],https://www.researchgate.net/profile/Christian-Caamano

See Also

GeoCovmatrix

Examples

library(GeoModels)


  # Define the spatial-coordinates of the points:
x <- runif(100, 0, 2)
y <- runif(100, 0, 2)
coords=cbind(x,y)
matrix1 <- GeoCovmatrix(coordx=coords, corrmodel="GenWend", param=list(smooth=0,
                      power2=4,sill=1,scale=0.2,nugget=0))
 
GeoCovDisplay(matrix1)

---

Spatial and Spatio-temporal Covariance Matrix of (non-)Gaussian Random Fields

Description

The function computes the covariance matrix associated to a spatial or spatio-temporal or a bivariate spatial Gaussian or non-Gaussian random field with given underlying covariance model and a set of spatial location sites (and temporal instants).

Usage

GeoCovmatrix(estobj = NULL, coordx, coordy = NULL, coordz = NULL, coordt = NULL,
             coordx_dyn = NULL, corrmodel, distance = "Eucl", grid = FALSE,
             model = "Gaussian", n = 1, param, anisopars = NULL, radius = 1,
             sparse = FALSE, copula = NULL, X = NULL, spobj = NULL)

Arguments

estobj	An object of class Geofit that includes information about data, model and estimates.
coordx	A numeric ($d \times 2$)-matrix or ($d \times 3$)-matrix. Coordinates on a sphere for a fixed radius radius are passed in lon/lat format expressed in decimal degrees.
coordy	A numeric vector giving 1-dimension of spatial coordinates; optional argument, the default is NULL.
coordz	A numeric vector giving 1-dimension of spatial coordinates; optional argument, the default is NULL.
coordt	A numeric vector giving 1-dimension of temporal coordinates. At the moment implemented only for the Gaussian case. Optional argument, the default is NULL then a spatial random field is expected.
coordx_dyn	A list of $T$ numeric ($d_t \times 2$)-matrices containing dynamical (in time) coordinates. Optional argument, the default is NULL.
corrmodel	String; the name of a correlation model, for the description see the Section Details.
distance	String; the name of the spatial distance. The default is "Eucl", the Euclidean distance. See GeoFit.
grid	Logical; if FALSE (the default) the data are interpreted as spatial or spatial-temporal realisations on a set of non-equispaced spatial sites (irregular grid). See GeoFit.
n	Numeric; the number of trials in a binomial random field. Default is $1$.
model	String; the type of RF. See GeoFit.
param	A list of parameter values required for the covariance model.
anisopars	A list of two elements "angle" and "ratio", i.e. the anisotropy angle and the anisotropy ratio, respectively.
radius	Numeric; a value indicating the radius of the sphere when using covariance models valid for the great circle distance. Default value is 1.
sparse	Logical; if TRUE the function returns an object of class spam. This option should be used when a parametric compactly supported covariance is used. Default is FALSE.
copula	String; the type of copula. It can be "Clayton" or "Gaussian".
X	Numeric; matrix of space-time covariates.
spobj	An object of class sp or spacetime.

Details

In the spatial case, the covariance matrix of the random vector

$[Z(s_1),\ldots,Z(s_n)]^T$

with a specific spatial covariance model is computed. Here $n$ is the number of spatial location sites.

In the space-time case, the covariance matrix of the random vector

$[Z(s_1,t_1),Z(s_2,t_1),\ldots,Z(s_n,t_1),\ldots,Z(s_n,t_m)]^T$

with a specific space-time covariance model is computed. Here $m$ is the number of temporal instants.

In the bivariate case, the covariance matrix of the random vector

$[Z_1(s_1),Z_2(s_1),\ldots,Z_1(s_n),Z_2(s_n)]^T$

with a specific spatial bivariate covariance model is computed.

The location site $s_i$ can be a point in the $d$-dimensional Euclidean space with $d=2$ or $d=3$ or a point (given in lon/lat degree format) on a sphere of arbitrary radius.

A list with all implemented spatial, space-time and bivariate correlation models is given below. The argument param is a list including all the parameters of a given correlation model specified by the argument corrmodel. For each correlation model one can check the associated parameters' names using CorrParam. In what follows $\kappa>0$, $\beta>0$, $\alpha, \alpha_s, \alpha_t \in (0,2]$ and $\gamma \in [0,1]$. The associated parameters in the argument param are smooth, power2, power, power_s, power_t and sep respectively. Moreover let $1(A)=1$ when $A$ is true and $0$ otherwise.

• Spatial correlation models:

  1. GenCauchy (generalised Cauchy in Gneiting and Schlather 2004) defined as:

     $R(h) = ( 1+h^{\alpha} )^{-\beta / \alpha}$

     If $h$ is the geodesic distance then $\alpha \in (0,1]$.

  2. Matern defined as:

     $R(h) = 2^{1-\kappa} \Gamma(\kappa)^{-1} h^\kappa K_\kappa(h)$

     If $h$ is the geodesic distance then $\kappa \in (0,0.5]$.

  3. Kummer (Kummer hypergeometric in Ma and Bhadra 2022) defined as:

     $R(h) = \Gamma(\kappa+\alpha) U(\alpha,1-\kappa,0.5 h^2 ) / \Gamma(\kappa+\alpha)$

     where $U(.,.,.)$ is the Kummer hypergeometric function. If $h$ is the geodesic distance then $\kappa \in (0,0.5]$.

  4. Kummer_Matern It is a rescaled version of the Kummer model, i.e. $h$ must be divided by $(2(1+\alpha))^{0.5}$. When $\alpha$ goes to infinity it is the Matern model.

  5. Wave defined as:

     $R(h)=\sin(h)/h$

     This model is valid only for dimensions less than or equal to 3.

  6. GenWend (Generalized Wendland in Bevilacqua et al. 2019) defined as:

     $R(h) = A (1-h^2)^{\beta+\kappa} F(\beta/2,(\beta+1)/2,2\beta+\kappa+1,1-h^2) 1(h \in [0,1])$

     where $\mu \ge 0.5(d+1)+\kappa$ and $A=(\Gamma(\kappa)\Gamma(2\kappa+\beta+1))/(\Gamma(2\kappa)\Gamma(\beta+1-\kappa)2^{\beta+1})$ and $F(.,.,.)$ is the Gaussian hypergeometric function. The cases $\kappa=0,1,2$ correspond to the Wend0, Wend1 and Wend2 models respectively.

  7. GenWend_Matern (Generalized Wendland Matern in Bevilacqua et al. 2022). It is defined as a rescaled version of the Generalized Wendland, i.e. $h$ must be divided by $(\Gamma(\beta+2\kappa+1)/\Gamma(\beta))^{1/(1+2\kappa)}$. When $\beta$ goes to infinity it is the Matern model.

  8. GenWend_Matern2 (Generalized Wendland Matern second parametrisation). It is defined as a rescaled version of the Generalized Wendland, i.e. $h$ must be multiplied by $\beta$ and the smoothness parameter is $\kappa-0.5$. When $\beta$ goes to infinity it is the Matern model.

  9. Hypergeometric (Hypergeometric model in Bevilacqua et al. 2025).

  10. Hypergeometric_Matern (Hypergeometric model first parametrisation).

  11. Hypergeometric_Matern2 (Hypergeometric model second parametrisation).

  12. Multiquadric defined as:

      $R(h) = (1-\alpha 0.5)^{2\beta}/(1+(\alpha 0.5)^2-\alpha \cos(h))^{\beta}, \quad h \in [0,\pi]$

      This model is valid on the unit sphere and $h$ is the geodesic distance.

  13. Sinpower defined as:

      $R(h) = 1-(\sin(h/2))^{\alpha},\quad h \in [0,\pi]$

      This model is valid on the unit sphere and $h$ is the geodesic distance.

  14. F_Sphere (F family in Alegria et al. 2021) defined as:

      $R(h) = K F(1/\alpha,1/\alpha+0.5,2/\alpha+0.5+\kappa),\quad h \in [0,\pi]$

      where $K =(\Gamma(a)\Gamma(i))/(\Gamma(i)\Gamma(o))$. This model is valid on the unit sphere and $h$ is the geodesic distance.

• Spatio-temporal correlation models:

  • Non-separable models:

    1. Gneiting defined as:

       $R(h, u) = \exp(-h^{\alpha_s}/((1+u^{\alpha_t})^{0.5 \gamma \alpha_s}))/(1+u^{\alpha_t})$

    2. Gneiting_GC defined as:

       $R(h, u) = \exp(-u^{\alpha_t}/((1+h^{\alpha_s})^{0.5 \gamma \alpha_t}))/(1+h^{\alpha_s})$

       where $h$ can be either Euclidean or geodesic distance.

    3. Iacocesare defined as:

       $R(h, u) = (1+h^{\alpha_s}+u^{\alpha_t})^{-\beta}$

    4. Porcu defined as:

       $R(h, u) = (0.5 (1+h^{\alpha_s})^\gamma + 0.5 (1+u^{\alpha_t})^\gamma)^{-\gamma^{-1}}$

    5. Porcu1 defined as:

       $R(h, u) = \exp(-h^{\alpha_s}(1+u^{\alpha_t})^{0.5 \gamma \alpha_s})/((1+u^{\alpha_t})^{1.5})$

    6. Stein defined as:

       $R(h, u) = (h^{\psi(u)}K_{\psi(u)}(h))/(2^{\psi(u)}\Gamma(\psi(u)+1))$

       where $\psi(u)=\nu+u^{0.5\alpha_t}$.

    7. Gneiting_mat_S defined as:

       $R(h, u) = \phi(u)^{\tau_t} \mathrm{Mat}(h \phi(u)^{-\beta},\nu_s)$

       where $\phi(u)=(1+u^{0.5\alpha_t})$, $\tau_t \ge 3.5+\nu_s$, $\beta \in [0,1]$.

    8. Gneiting_mat_T defined by interchanging $h$ with $u$ in Gneiting_mat_S.

    9. Gneiting_wen_S defined as:

       $R(h, u) = \phi(u)^{\tau_t} \mathrm{GenWend}(h \phi(u)^{\beta},\nu_s,\mu_s)$

       where $\phi(u)=(1+u^{0.5\alpha_t})$, $\tau_t \ge 2.5+2\nu_s$, $\beta \in [0,1]$.

    10. Gneiting_wen_T defined by interchanging $h$ with $u$ in Gneiting_wen_S.

    11. Matern_Matern_nosep defined as:

        $R(h, u) = \frac{\mathrm{Matern}(h;\nu_s)\,\mathrm{Matern}(u;\nu_t)} {1+\lambda\,h^{2}u^{2}/(1+\lambda)}$

        with $\nu_s,\nu_t>0$ and $\lambda\in[0,1]$.

    12. GenWend_GenWend_nosep defined as:

        $R(h, u) = \frac{\mathrm{GenWend}(h;\nu_s,\delta_s)\,\mathrm{GenWend}(u;\nu_t,\delta_t)} {1+\lambda\,h^{2}u^{2}/(1+\lambda)}$

        with $\nu_x \geq -0.5$, $\delta_x>(d+1)/2+\nu_x$ for $x=s,t$, and $\lambda\in[0,1]$.

    13. Multiquadric_st defined as:

        $R(h, u)= ((1-0.5\alpha_s)^2/(1+(0.5\alpha_s)^2-\alpha_s \psi(u) \cos(h)))^{a_s},\quad h \in [0,\pi]$

        where $\psi(u)=(1+(u/a_t)^{\alpha_t})^{-1}$. This model is valid on the unit sphere and $h$ is the geodesic distance.

    14. Sinpower_st defined as:

        $R(h, u)=(\exp(\alpha_s \cos(h) \psi(u)/a_s)(1+\alpha_s \cos(h) \psi(u)/a_s))/k$

        where $\psi(u)=(1+(u/a_t)^{\alpha_t})^{-1}$ and $k=(1+\alpha_s/a_s)\exp(\alpha_s/a_s)$, $h \in [0,\pi]$. This model is valid on the unit sphere and $h$ is the geodesic distance.

  • Separable models:

    Space-time separable correlation models are easily obtained as the product of a spatial and a temporal correlation model, that is

    $R(h,u)=R(h) R(u)$

    Several combinations are possible:

    1. Exp_Exp defined as:

       $R(h, u) = \exp(-h)\exp(-u)$

    2. Matern_Matern defined as:

       $R(h, u) = \mathrm{Matern}(h;\kappa_s)\mathrm{Matern}(u;\kappa_t)$

    3. GenWend_GenWend defined as:

       $R(h, u) = \mathrm{GenWend}(h;\kappa_s,\mu_s)\mathrm{GenWend}(u;\kappa_t,\mu_t)$

    4. Stable_Stable defined as:

       $R(h, u) = \exp(-h^{\alpha_s})\exp(-u^{\alpha_t})$

    Note that some models are nested (e.g. Exp_Exp within Matern_Matern).

• Spatial bivariate correlation models:

  1. Bi_Matern (Bivariate full Matern model)

  2. Bi_Matern_contr (Bivariate Matern model with constraints)

  3. Bi_Matern_sep (Bivariate separable Matern model)

  4. Bi_LMC (Bivariate linear model of coregionalization)

  5. Bi_LMC_contr (Bivariate LMC with constraints)

  6. Bi_Wendx (Bivariate full Wendland model)

  7. Bi_Wendx_contr (Bivariate Wendland model with constraints)

  8. Bi_Wendx_sep (Bivariate separable Wendland model)

  9. Bi_F_Sphere (Bivariate full F model on the unit sphere)

Remarks:
In what follows we assume $\sigma^2,\sigma_1^2,\sigma_2^2,\tau^2,\tau_1^2,\tau_2^2,a,a_s,a_t,a_{11},a_{22},a_{12},\kappa_{11},\kappa_{22},\kappa_{12},f_{11},f_{12},f_{21},f_{22}$ positive.

The associated names of the parameters in param are sill, sill_1, sill_2, nugget, nugget_1, nugget_2, scale, scale_s, scale_t, scale_1, scale_2, scale_12, smooth_1, smooth_2, smooth_12, a_1, a_12, a_21, a_2 respectively.

Let $R(h)$ be a spatial correlation model given in standard notation. Then the covariance model applied with arbitrary variance, nugget and scale equals to $\sigma^2$ if $h=0$ and

$C(h)=\sigma^2(1-\tau^2)R(h/a,\ldots), \quad h>0$

with nugget parameter $\tau^2$ between 0 and 1.

Similarly, if $R(h,u)$ is a spatio-temporal correlation model given in standard notation, then the covariance model is $\sigma^2$ if $h=0$ and $u=0$ and

$C(h,u)=\sigma^2(1-\tau^2)R(h/a_s,u/a_t,\ldots), \quad h>0, u>0$

Here ‘...’ stands for additional parameters.

The bivariate models implemented are the following:

1. Bi_Matern defined as:

   $C_{ij}(h)=\rho_{ij}(\sigma_i\sigma_j+\tau_i^2 1(i=j,h=0))\mathrm{Matern}(h/a_{ij},\kappa_{ij}), \quad i,j=1,2,\; h\ge 0$

   where $\rho=\rho_{12}=\rho_{21}$ is the colocated correlation parameter and $\rho_{ii}=1$. The model Bi_Matern_sep (separable Matern) is a special case when $a=a_{11}=a_{12}=a_{22}$ and $\kappa=\kappa_{11}=\kappa_{12}=\kappa_{22}$. The model Bi_Matern_contr (constrained Matern) is a special case when $a_{12}=0.5(a_{11}+a_{22})$ and $\kappa_{12}=0.5(\kappa_{11}+\kappa_{22})$.

2. Bi_GenWend defined as:

   $C_{ij}(h)=\rho_{ij}(\sigma_i\sigma_j+\tau_i^2 1(i=j,h=0))\mathrm{GenWend}(h/a_{ij},\nu_{ij},\kappa_{ij}), \quad i,j=1,2,\; h\ge 0$

   where $\rho=\rho_{12}=\rho_{21}$ is the colocated correlation parameter and $\rho_{ii}=1$. The model Bi_GenWend_sep (separable GenWendland) is a special case when $a=a_{11}=a_{12}=a_{22}$ and $\mu=\mu_{11}=\mu_{12}=\mu_{22}$. The model Bi_GenWend_contr (constrained GenWendland) is a special case when $a_{12}=0.5(a_{11}+a_{22})$ and $\mu_{12}=0.5(\mu_{11}+\mu_{22})$.

3. Bi_LMC defined as:

   $C_{ij}(h)=\sum_{k=1}^{2}(f_{ik}f_{jk}+\tau_i^2 1(i=j,h=0))R(h/a_k)$

   where $R(h)$ is a correlation model. The model Bi_LMC_contr is a special case when $f=f_{12}=f_{21}$. Bivariate LMC models, in the current version of the package, are obtained with $R(h)$ equal to the exponential correlation model.

Value

Returns an object of class GeoCovmatrix. An object of class GeoCovmatrix is a list containing at most the following components:

bivariate	Logical: TRUE if the Gaussian random field is bivariate, otherwise FALSE.
coordx	A $d$-dimensional vector of spatial coordinates.
coordy	A $d$-dimensional vector of spatial coordinates.
coordt	A $t$-dimensional vector of temporal coordinates.
coordx_dyn	A list of $t$ matrices of spatial coordinates.
covmatrix	The covariance matrix if type is Standard. An object of class spam if type is Tapering or Standard and sparse is TRUE.
corrmodel	String: the correlation model.
distance	String: the type of spatial distance.
grid	Logical: TRUE if the spatial data are on a regular grid, otherwise FALSE.
nozero	In the case of tapered matrix the percentage of non-zero values in the covariance matrix; otherwise NULL.
n	The number of trials for binomial RFs.
namescorr	String: the names of the correlation parameters.
numcoord	Numeric: the number of spatial coordinates.
numtime	Numeric: the number of temporal coordinates.
model	The type of RF, see GeoFit.
param	Numeric: the covariance parameters.
spacetime	TRUE if spatio-temporal and FALSE if spatial covariance model.
sparse	Logical: is the returned object of class spam?

Author(s)

Moreno Bevilacqua, [email protected], https://sites.google.com/view/moreno-bevilacqua/home

Víctor Morales-Oñate, [email protected], https://sites.google.com/site/moralesonatevictor/

Christian Caamaño-Carrillo, [email protected], https://www.researchgate.net/profile/Christian-Caamano

References

Alegria, A., Cuevas-Pacheco, F., Diggle, P. and Porcu, E. (2021). The F-family of covariance functions: A Matérn analogue for modeling random fields on spheres. Spatial Statistics 43, 100512.

Bevilacqua, M., Faouzi, T., Furrer, R. and Porcu, E. (2019). Estimation and prediction using generalized Wendland functions under fixed domain asymptotics. Annals of Statistics 47(2), 828–856.

Bevilacqua, M., Caamaño-Carrillo, C. and Porcu, E. (2022). Unifying compactly supported and Matérn covariance functions in spatial statistics. Journal of Multivariate Analysis 189, 104949.

Daley, D. J., Porcu, E. and Bevilacqua, M. (2015). Classes of compactly supported covariance functions for multivariate random fields. Stochastic Environmental Research and Risk Assessment 29(4), 1249–1263.

Emery, X. and Alegria, A. (2022). The Gauss hypergeometric covariance kernel for modeling second-order stationary random fields in Euclidean spaces: its compact support, properties and spectral representation. Stochastic Environmental Research and Risk Assessment 36, 2819–2834.

Gneiting, T. (2002). Nonseparable, stationary covariance functions for space-time data. Journal of the American Statistical Association 97, 590–600.

Gneiting, T., Kleiber, W. and Schlather, M. (2010). Matérn cross-covariance functions for multivariate random fields. Journal of the American Statistical Association 105, 1167–1177.

Ma, P. and Bhadra, A. (2022). Beyond Matérn: on a class of interpretable confluent hypergeometric covariance functions. Journal of the American Statistical Association, 1–14.

Porcu, E., Bevilacqua, M. and Genton, M. (2015). Spatio-temporal covariance and cross-covariance functions of the great circle distance on a sphere. Journal of the American Statistical Association. DOI: 10.1080/01621459.2015.1072541.

Gneiting, T. and Schlather, M. (2004). Stochastic models that separate fractal dimension and the Hurst effect. SIAM Review 46, 269–282.

See Also

GeoKrig, GeoSim, GeoFit

Examples

library(GeoModels)
################################################################
###
### Example 1. Estimated spatial covariance matrix associated to
### a WCL estimates using the Matern correlation model
###
###############################################################

set.seed(3)
N <- 300  # number of location sites
x <- runif(N, 0, 1)
y <- runif(N, 0, 1)
coords <- cbind(x, y)
# Set the covariance model's parameters:
corrmodel <- "Matern"
mean <- 0.5
sill <- 1
nugget <- 0
scale <- 0.2/3
smooth <- 0.5

param <- list(mean = mean, sill = sill, nugget = nugget, scale = scale, smooth = smooth)
data <- GeoSim(coordx = coords, corrmodel = corrmodel, param = param)$data
fixed <- list(nugget = nugget, smooth = smooth)
start <- list(mean = mean, scale = scale, sill = sill)

fit0 <- GeoFit(data = data, coordx = coords, corrmodel = corrmodel,
               neighb = 3, likelihood = "Conditional", optimizer = "BFGS",
               type = "Pairwise", start = start, fixed = fixed)
print(fit0)

# estimated covariance matrix using Geofit object
mm <- GeoCovmatrix(fit0)$covmatrix
# estimated covariance matrix
mm1 <- GeoCovmatrix(coordx = coords, corrmodel = corrmodel,
                    param = c(fit0$param, fit0$fixed))$covmatrix
sum(mm - mm1)

################################################################
###
### Example 2. Spatial covariance matrix associated to
### the Generalized Wendland-Matern correlation model
###
###############################################################

# Correlation Parameters for Gen Wendland model
CorrParam("GenWend_Matern")
# Gen Wendland Parameters
param <- list(sill = 1, scale = 0.04, nugget = 0, smooth = 0, power2 = 1/1.5)

matrix2 <- GeoCovmatrix(coordx = coords, corrmodel = "GenWend_Matern",
                        param = param, sparse = TRUE)

# Percentage of non-zero values
matrix2$nozero

################################################################
###
### Example 3. Spatial covariance matrix associated to
### the Kummer correlation model
###
###############################################################

# Correlation Parameters for Kummer model
CorrParam("Kummer")
param <- list(sill = 1, scale = 0.2, nugget = 0, smooth = 0.5, power2 = 1)

matrix3 <- GeoCovmatrix(coordx = coords, corrmodel = "Kummer", param = param)

matrix3$covmatrix[1:4, 1:4]

################################################################
###
### Example 4. Covariance matrix associated to
### the space-time double Matern correlation model
###
###############################################################

# Define the temporal coordinates:
times <- seq(1, 4, 1)

# Correlation Parameters for double Matern model
CorrParam("Matern_Matern")

# Define covariance parameters
param <- list(scale_s = 0.3, scale_t = 0.5, sill = 1, smooth_s = 0.5, smooth_t = 0.5)

# Simulation of a spatial Gaussian random field:
matrix4 <- GeoCovmatrix(coordx = coords, coordt = times,
                        corrmodel = "Matern_Matern", param = param)

dim(matrix4$covmatrix)

################################################################
###
### Example 5. Spatial covariance matrix associated to
### a skew Gaussian RF with Matern correlation model
###
###############################################################

param <- list(sill = 1, scale = 0.3/3, nugget = 0, skew = 4, smooth = 0.5)
# Simulation of a spatial Gaussian random field:
matrix5 <- GeoCovmatrix(coordx = coords, corrmodel = "Matern", param = param,
                        model = "SkewGaussian")

# covariance matrix
matrix5$covmatrix[1:4, 1:4]

################################################################
###
### Example 6. Spatial covariance matrix associated to
### a Weibull RF with GenWend correlation model
###
###############################################################

param <- list(scale = 0.3, nugget = 0, shape = 4, mean = 0, smooth = 1, power2 = 5)
# Simulation of a spatial Gaussian random field:
matrix6 <- GeoCovmatrix(coordx = coords, corrmodel = "GenWend", param = param,
                        sparse = TRUE, model = "Weibull")

# Percentage of non-zero values
matrix6$nozero

################################################################
###
### Example 7. Spatial covariance matrix associated to
### a binomial Gaussian RF with Generalized Wendland correlation model
###
###############################################################

param <- list(mean = 0.2, scale = 0.2, nugget = 0, power2 = 4, smooth = 0)
# Simulation of a spatial Gaussian random field:
matrix7 <- GeoCovmatrix(coordx = coords, corrmodel = "GenWend", param = param,
                        n = 5, sparse = TRUE, model = "Binomial")

as.matrix(matrix7$covmatrix)[1:4, 1:4]

################################################################
###
### Example 8. Covariance matrix associated to
### a bivariate Matern exponential correlation model
###
###############################################################

set.seed(8)
# Define the spatial coordinates of the points:
x <- runif(4, 0, 1)
y <- runif(4, 0, 1)
coords <- cbind(x, y)

# Parameters
param <- list(mean_1 = 0, mean_2 = 0, sill_1 = 1, sill_2 = 2,
              scale_1 = 0.1, scale_2 = 0.1, scale_12 = 0.1,
              smooth_1 = 0.5, smooth_2 = 0.5, smooth_12 = 0.5,
              nugget_1 = 0, nugget_2 = 0, pcol = -0.25)

# Covariance matrix
matrix8 <- GeoCovmatrix(coordx = coords, corrmodel = "Bi_matern", param = param)$covmatrix

matrix8

---