GeoFit2: Max-Likelihood-Based Fitting of Gaussian and non Gaussian...

In GeoModels: Procedures for Gaussian and Non Gaussian Geostatistical (Large) Data Analysis

Max-Likelihood-Based Fitting of Gaussian and non Gaussian RFs.

Description

Maximum weighted composite-likelihood fitting for Gaussian and some Non-Gaussian univariate spatial, spatio-temporal and bivariate spatial RFs. A first preliminary estimation is performed using independence composite-likelihood for the marginal parameters of the model. The estimates are then used as starting values in the second final estimation step. The function allows fixing any of the parameters and setting upper/lower bounds in the optimization.

Usage

GeoFit2(data, coordx, coordy=NULL, coordz=NULL, coordt=NULL,
        coordx_dyn=NULL, copula=NULL, corrmodel, distance="Eucl",
        fixed=NULL, anisopars=NULL, est.aniso=c(FALSE,FALSE),
        grid=FALSE, likelihood="Marginal", lower=NULL, maxdist=Inf,
        neighb=NULL, p_neighb=1, maxtime=Inf, memdist=TRUE,
        method="cholesky", model="Gaussian", n=1, onlyvar=FALSE,
        optimizer="Nelder-Mead", radius=1, score=FALSE,
        sensitivity=FALSE, sparse=FALSE, start=NULL,
        thin_method="iid", type="Pairwise", upper=NULL,
        varest=FALSE, weighted=FALSE, X=NULL, 
        spobj=NULL, spdata=NULL)

Arguments

data	A d-dimensional vector (a single spatial realisation) or a (d \times d)-matrix (a single spatial realisation on regular grid) or a (t \times d)-matrix (a single spatial-temporal realisation) or an (d \times d \times t \times n)-array (a single spatial-temporal realisation on regular grid). For the description see the Section Details.
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, default is NULL.
coordz	A numeric vector giving 1-dimension of spatial coordinates; optional argument, default is NULL.
coordt	A numeric vector assigning 1-dimension of temporal coordinates. Optional argument, default is NULL: if NULL then a spatial RF is expected.
coordx_dyn	A list of m numeric (d_t \times 2)-matrices containing dynamical (in time) spatial coordinates. Optional argument, default is NULL.
copula	String; the type of copula. It can be "Clayton" or "Gaussian".
corrmodel	String; the name of a correlation model, for the description see the Section Details.
distance	String; the name of the spatial distance. Default is "Eucl" (euclidean distance). See the Section Details.
fixed	An optional named list giving the values of the parameters that will be considered as known values. The listed parameters for a given correlation function will not be estimated.
anisopars	A list of two elements: "angle" and "ratio", i.e. the anisotropy angle and the anisotropy ratio, respectively.
est.aniso	A bivariate logical vector providing which anisotropic parameters must be estimated.
grid	Logical; if FALSE (default) the data are interpreted as spatial or spatial-temporal realisations on a set of non-equispaced spatial sites (irregular grid).
likelihood	String; the configuration of the composite likelihood. "Marginal" is the default; see the Section Details.
lower	An optional named list giving lower bounds for parameters when the optimizer is L-BFGS-B, nlminb, bobyqa or optimize. Names must match those in start.
maxdist	Numeric; an optional positive value indicating the maximum spatial distance considered in the composite likelihood computation. See the Section Details for more information.
neighb	Numeric; an optional positive integer indicating the order of neighborhood in the composite likelihood computation. See the Section Details for more information.
p_neighb	Numeric; a value in (0,1] specifying the expected fraction of nearest-neighbor pairs retained through stochastic thinning. If 1 (default), no thinning is applied and all nearest-neighbor pairs are used. If <1, pairs are randomly retained using independent Bernoulli sampling.
maxtime	Numeric; an optional positive integer indicating the order of temporal neighborhood in the composite likelihood computation.
memdist	Logical; if TRUE then all distances useful in the composite likelihood estimation are computed before the optimization. FALSE is deprecated.
method	String; the type of matrix decomposition used in the likelihood computation. Default is "cholesky". Another possible choice is "svd" (when available).
model	String; the type of RF and therefore the densities associated to the likelihood objects. "Gaussian" is the default; see the Section Details.
n	Numeric; number of trials in a binomial RF; number of successes in a negative binomial RF.
onlyvar	Logical; if TRUE (and varest=TRUE) only the variance-covariance matrix is computed without optimizing. Default is FALSE.
optimizer	String; the optimization algorithm (see optim for details). "Nelder-Mead" is the default. Other possible choices are "nlm", "BFGS", "SANN", "L-BFGS-B", "nlminb", "bobyqa". In these last three cases upper and lower bounds can be passed by the user. In the one-dimensional case, optimize is used.
radius	Numeric; the radius of the sphere in the case of lon-lat coordinates. Default value is 1.
score	Logical; should score function be computed? Default is FALSE.
sensitivity	Logical; if TRUE then the sensitivity matrix is computed.
sparse	Logical; if TRUE then maximum likelihood is computed using sparse matrix algorithms (e.g., spam). It should be used with compactly supported covariance models. Default is FALSE.
start	An optional named list with initial values for parameters used by the numerical routines in the maximization procedure. Default is NULL (see Details).
thin_method	String; thinning scheme used when p_neighb<1. Default is "iid" (independent Bernoulli thinning).
type	String; the type of the likelihood objects. If "Pairwise" (default) then the marginal composite likelihood is formed by pairwise marginal likelihoods (see Details).
upper	An optional named list giving upper bounds for parameters when the optimizer is L-BFGS-B, nlminb, bobyqa or optimize. Names must match those in start.
varest	Logical; if TRUE the estimates' variances and standard errors are returned. For composite likelihood estimation it is deprecated. Use sensitivity=TRUE and update the object using GeoVarestbootstrap. Default is FALSE.
weighted	Logical; if TRUE the likelihood objects are weighted; see the Section Details. Default is FALSE.
X	Numeric; matrix of spatio(temporal) covariates in the linear mean specification.
spobj	An object of class sp or spacetime.
spdata	Character; the name of data in the sp or spacetime object.

Details

The function GeoFit2 is similar to GeoFit. However, GeoFit2 performs a preliminary estimation using maximum independence composite likelihood for the marginal parameters of the model and then uses the obtained estimates as starting values in the final weighted composite likelihood estimation (that includes both marginal and dependence parameters). This provides robust starting values for the marginal parameters in the optimization algorithm.

Stochastic thinning of nearest-neighbor pairs can be enabled via p_neighb<1. The argument thin_method controls the thinning scheme (default "iid").

Value

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

bivariate	Logical: TRUE if the Gaussian RF is bivariate, otherwise FALSE.
clic	The composite information criterion; if the full likelihood is considered then it coincides with AIC.
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 dynamical (in time) spatial coordinates.
conf.int	Confidence intervals for standard maximum likelihood estimation.
convergence	A string that denotes if convergence is reached.
copula	The type of copula.
corrmodel	The correlation model.
data	The vector or matrix or array (or list) of data.
distance	The type of spatial distance.
fixed	A list of fixed parameters.
iterations	The number of iterations used by the numerical routine.
likelihood	The configuration of the composite likelihood.
logCompLik	The value of the log composite-likelihood at the maximum.
maxdist	The maximum spatial distance used in the weighted composite likelihood (or NULL).
maxtime	The order of temporal neighborhood in the composite likelihood computation.
message	Extra message passed from the numerical routines.
model	The density associated to the likelihood objects.
missp	TRUE if a misspecified Gaussian model is used in the composite likelihood.
n	The number of trials in a binomial RF; the number of successes in a negative binomial RF.
neighb	The order of spatial neighborhood in the composite likelihood computation.
ns	The number of (different) location sites in the bivariate case.
numcoord	The number of spatial coordinates.
numtime	The number of temporal realisations of the RF.
param	A list of parameter estimates.
radius	The radius of the sphere in the case of great-circle distance.
stderr	Standard errors for standard maximum likelihood estimation.
sensmat	The sensitivity matrix.
varcov	The variance-covariance matrix of the estimates.
type	The type of the likelihood objects.
X	The matrix of covariates.

Author(s)

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

Examples

library(GeoModels)


###############################################################
############ Examples of spatial Gaussian RFs ################
###############################################################

################################################################
###
### Example 1 : Maximum pairwise conditional likelihood fitting 
###  of a Gaussian RF with Matern correlation
###
###############################################################
model="Gaussian"
# Define the spatial-coordinates of the points:
set.seed(3)
N=400  # number of location sites
x <- runif(N, 0, 1)
set.seed(6)
y <- runif(N, 0, 1)
coords <- cbind(x,y)

# Define spatial matrix covariates
X=cbind(rep(1,N),runif(N))

# Set the covariance model's parameters:
corrmodel <- "Matern"
mean <- 0.2
mean1 <- -0.5
sill <- 1
nugget <- 0
scale <- 0.2/3
smooth=0.5
param<-list(mean=mean,mean1=mean1,sill=sill,nugget=nugget,scale=scale,smooth=smooth)

# Simulation of the spatial Gaussian RF:
data <- GeoSim(coordx=coords,model=model,corrmodel=corrmodel, param=param,X=X)$data

fixed<-list(nugget=nugget,smooth=smooth)
start<-list(mean=mean,mean1=mean1,scale=scale,sill=sill)

################################################################
###
### Maximum pairwise likelihood fitting of
### Gaussian RFs with exponential correlation.
### 
###############################################################
fit1 <- GeoFit2(data=data,coordx=coords,corrmodel=corrmodel, 
                    optimizer="BFGS",neighb=3,likelihood="Conditional",
                    type="Pairwise", start=start,fixed=fixed,X=X)
print(fit1)




###############################################################
############ Examples of spatial non-Gaussian RFs #############
###############################################################


################################################################
###
### Example 2. Maximum pairwise likelihood fitting of 
### a LogGaussian  RF with Generalized Wendland correlation
### 
###############################################################
set.seed(524)
# Define the spatial-coordinates of the points:
N=500
x <- runif(N, 0, 1)
y <- runif(N, 0, 1)
coords <- cbind(x,y)
X=cbind(rep(1,N),runif(N))
mean=1; mean1=2 # regression parameters
nugget=0
sill=0.5
scale=0.2
smooth=0

model="LogGaussian"
corrmodel="GenWend"
param=list(mean=mean,mean1=mean1,sill=sill,scale=scale,
                    nugget=nugget,power2=4,smooth=smooth)
# Simulation of a  non stationary LogGaussian RF:
data <- GeoSim(coordx=coords, corrmodel=corrmodel,model=model,X=X,
           param=param)$data

fixed<-list(nugget=nugget,power2=4,smooth=smooth)
start<-list(mean=mean,mean1=mean1,scale=scale,sill=sill)
I=Inf
lower<-list(mean=-I,mean1=-I,scale=0,sill=0)
upper<-list(mean= I,mean1= I,scale=I,sill=I)

# Maximum pairwise composite-likelihood fitting of the RF:
fit <- GeoFit2(data=data,coordx=coords,corrmodel=corrmodel, model=model,
                    neighb=3,likelihood="Conditional",type="Pairwise",X=X,
                    optimizer="nlminb",lower=lower,upper=upper,
                    start=start,fixed=fixed)
print(unlist(fit$param))


################################################################
###
### Example 3. Maximum pairwise likelihood fitting of
### SinhAsinh   RFs with Wendland0 correlation
###
###############################################################
set.seed(261)
model="SinhAsinh"
# Define the spatial-coordinates of the points:
x <- runif(500, 0, 1)
y <- runif(500, 0, 1)
coords <- cbind(x,y)

corrmodel="Wend0"
mean=0;nugget=0
sill=1
skew=-0.5
tail=1.5
power2=4
c_supp=0.2

# model parameters
param=list(power2=power2,skew=skew,tail=tail,
             mean=mean,sill=sill,scale=c_supp,nugget=nugget)
data <- GeoSim(coordx=coords, corrmodel=corrmodel,model=model, param=param)$data

plot(density(data))
fixed=list(power2=power2,nugget=nugget)
start=list(scale=c_supp,skew=skew,tail=tail,mean=mean,sill=sill)
# Maximum pairwise likelihood:
fit1 <- GeoFit2(data=data,coordx=coords,corrmodel=corrmodel, model=model,
                    neighb=3,likelihood="Marginal",type="Pairwise",
                    start=start,fixed=fixed)
print(unlist(fit1$param))

GeoModels documentation built on April 6, 2026, 5:08 p.m.