GeoFit: Maximum-Likelihood-Based Fitting of Gaussian and non-Gaussian...

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

Maximum-Likelihood-Based Fitting of Gaussian and non-Gaussian random fields.

Description

Maximum weighted (stochastic) composite-likelihood fitting for Gaussian and some non-Gaussian univariate spatial, spatio-temporal and bivariate spatial random fields. The function allows fixing any of the parameters and setting upper/lower bounds in the optimization. Different optimization methods can be used.

Usage

GeoFit(data, coordx, coordy=NULL, coordz=NULL, coordt=NULL,
 coordx_dyn=NULL, copula=NULL, corrmodel=NULL, 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="bernoulli", type="Pairwise", upper=NULL, 
 varest=FALSE, weighted=FALSE,
 X=NULL, spobj=NULL, spdata=NULL, check.duplicates=FALSE)

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 spatio-temporal realisation) or a (d \times d \times t \times n)-array (a single spatio-temporal realisation on regular grid). See Details for the accepted data layouts.
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 one dimension of the observation-time coordinates. Optional argument, default is NULL; if NULL, a spatial random field is expected. Temporal coordinates may be irregularly spaced; temporal lags are computed from the supplied coordinate values.
coordx_dyn	For dynamic locations, a list with one numeric coordinate matrix per temporal instant. If T is length(coordt), the list must have length T; coordx_dyn[[t]] must have N_t rows and two or three columns. Its row order must match data[[t]] and, when supplied as a list, X[[t]]. See GeoModels-spacetime-ordering.
copula	String; the copula used by pairwise copula likelihoods. Supported values are "Gaussian", "Clayton" (the constructive Clayton-like spatial copula), and "SkewGaussian".
corrmodel	String; the name of a correlation model; see GeoCovmatrix for the list of implemented correlation models.
distance	String; the name of the spatial distance. Default is "Eucl" (Euclidean distance). See Details for the accepted options.
fixed	An optional named list giving the values of the parameters that will be considered as known values. Parameter names must match exactly the nuisance/marginal or correlation parameters supported by the selected model and correlation model; invalid names are reported explicitly. The listed parameters for a given model/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. Geometric anisotropy is available only with distance="Eucl".
est.aniso	A bivariate logical vector providing which anisotropy parameters must be estimated.
grid	Logical; if FALSE (default) the data are interpreted as spatial or spatio-temporal realisations on a set of non-equispaced spatial sites. If TRUE, coordx, coordy (and optionally coordz) are interpreted as grid axes and the data dimensions must match those axes; grid inputs are internally converted to the canonical explicit-coordinate layout before fitting.
likelihood	String; the configuration of the composite likelihood. "Marginal" is the default; see Details for the accepted options.
lower	An optional named list giving lower bounds for parameters when the optimizer is L-BFGS-B, nlminb, bobyqa or optimize. Names must match parameters that are being estimated.
maxdist	Numeric; an optional positive value indicating the maximum spatial distance considered in the composite computation. See Details for more information.
neighb	Numeric; an optional positive integer indicating the order of neighborhood in the composite likelihood computation. See Details for more information.
p_neighb	Numeric scalar in (0,1]. If 1 (default), no thinning is applied. For thin_method="bernoulli", it is the candidate-pair inclusion probability in the constant-weight design and controls the retained size in expectation. For thin_method="FixedBudget", it defines the exact global retained-pair budget K=\mathrm{round}(p_{neighb}d), where d is the candidate-pair count.
maxtime	Numeric; an optional non-negative maximum temporal-distance threshold, expressed in the same units as coordt, used to select pairwise composite-likelihood contributions.
memdist	Deprecated logical argument retained for backward compatibility. The selected pair structure is always precomputed and reused during composite-likelihood optimization. Supplying FALSE produces a warning and is treated as TRUE.
method	String; the type of matrix decomposition/linear algebra backend used in likelihood computations. Default is "cholesky". Another possible choice is "svd" (when available).
model	String; the type of random field (and associated density) used in the likelihood objects. Default is "Gaussian"; see Details for the accepted options.
n	Positive integer size parameter. For direct Binomial models it may be scalar or contain one value per observation. For direct Negative-Binomial models it is the common number r of successes and must be scalar; Geometric corresponds to r=1. Non-integer or non-positive values are rejected.
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). Default is "Nelder-Mead". Other possible choices are "nlm", "BFGS", "SANN", "L-BFGS-B", "nlminb", "bobyqa". For "L-BFGS-B", "nlminb" and "bobyqa" bounds can be passed via lower and upper. In the one-dimensional case, optimize is used.
radius	Numeric; the radius of the sphere in the case of lon-lat coordinates. Default is 1.
score	Logical; if TRUE the score function is computed. Default is FALSE.
sensitivity	Logical; if TRUE the sensitivity matrix is computed. Default is FALSE.
sparse	Logical; if TRUE then maximum likelihood / composite likelihood may exploit sparse-matrix algorithms (e.g., spam). Typically used with compactly supported covariance models. Default is FALSE.
start	An optional named list with initial values for parameters to be estimated. Default is NULL. Parameter names must match exactly the nuisance/marginal or correlation parameters supported by the selected model and correlation model; invalid names are reported explicitly. Parameters omitted from start receive conservative internal initial values; explicitly supplied values are preserved (see Details).
thin_method	String; thinning scheme in stochastic weighted pairwise likelihood (used when p_neighb < 1). Default is "bernoulli" (independent Bernoulli thinning). An alternative thinning scheme is "FixedBudget". The legacy name "TargetBalanced" is accepted as an alias for "FixedBudget".
type	String; the type of likelihood objects. If "Pairwise" (default) the composite likelihood is formed by pairwise components (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 parameters that are being estimated.
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 GeoVarest. Default is FALSE.
weighted	Logical; if TRUE the likelihood objects are weighted; see Details for the accepted options. Default is FALSE.
X	Numeric design matrix for the linear mean X\beta. Its columns correspond in order to mean, mean1, .... For fixed-location space-time data, rows follow c(t(data)): all sites at the first time, then all sites at the second time, and so on. For dynamic locations, use either a stacked matrix in time-wise order or a list with X[[t]] aligned with coordx_dyn[[t]]. See GeoModels-spacetime-ordering.
spobj	An object of class sp or spacetime. For space-time objects, the current sp2Geo() conversion uses sequential temporal indices and does not preserve irregular original time spacing; to retain irregular temporal distances, use the explicit-coordinate interface with numeric coordt.
spdata	Character; the name of the data component in the sp or spacetime object.
check.duplicates	Logical. If TRUE, perform a fast exact scan for duplicated observation locations at this user entry point. The default is FALSE, so no duplicate-location scan is imposed. Internal bootstrap, cross-validation, and refitting calls do not repeat the scan.

Details

For a univariate model the mean is specified as

\mu = X\beta.

The entries of \beta are represented by the parameters mean, mean1, and so on. The columns of X correspond to these coefficients in exactly this order. When X=NULL, an intercept-only model is used, which is equivalent to a one-column matrix of ones and the single coefficient mean. Missing starting values for regression coefficients and other estimated parameters are initialized internally. Explicit entries in start always define starting values for parameters that remain estimated, including likelihood="Full", type="Standard"; they are never converted to fixed mean coefficients. A coefficient is fixed only when it is supplied in fixed. Supplied coefficient names must be compatible with ncol(X). For difficult non-Gaussian models, user-supplied starting values can still improve numerical optimization.

For model="Wrapped", omitted automatic starting values use circular statistics. If \bar C=n^{-1}\sum_i\cos(\theta_i), \bar S=n^{-1}\sum_i\sin(\theta_i), and R=(\bar C^2+\bar S^2)^{1/2}, the intercept is initialized from the sample mean direction through the model link \mu=2\arctan(\eta)+\pi, while sill is initialized as -2\log R. Additional automatically initialized mean coefficients are set to zero. Explicit entries in start or fixed retain precedence. The wrapped pairwise and conditional densities use winding indices from -3 through 3.

A site-specific known mean can instead be supplied as a vector in fixed$mean, with one value per observation. This external mean is mutually exclusive with X and with estimated mean coefficients in start. A scalar fixed$mean is not an external vector: it is the fixed intercept coefficient.

GeoFit provides weighted and stochastic weighted composite-likelihood estimation based on pairs for Gaussian and non-Gaussian random fields, including nearest-neighbor and stochastic nearest-neighbor pairwise likelihoods; see Caamaño-Carrillo et al. (2024) and Bevilacqua et al. (2026). It also provides independence composite-likelihood estimation. The accepted likelihood/type combinations are checked explicitly: likelihood="Full" uses the full-likelihood types (including "Standard"). Full-likelihood objectives are currently implemented for Gaussian, SinhAsinh, LogGaussian, Tukeyh, Tukeyh2, and the internally supported misspecified-Gaussian full likelihoods. Other margins, including Gamma and Weibull, are rejected before optimization with an explicit capability message; likelihood="Marginal" is used with "Pairwise" or "Independence"; and likelihood="Conditional" is used with "Pairwise". The historical Difference composite likelihood is no longer supported. For space-time pairwise fitting, GeoFit checks that a registered space-time native kernel exists for the requested marginal model and stops before optimization when it does not. Bivariate pairwise fitting is currently implemented only for the Gaussian model with marginal pairwise likelihood.

For pairwise copula fitting, the current implementation is univariate and purely spatial. Continuous margins supported with the Gaussian, Clayton-like, and skew-Gaussian copulas are "Gaussian", "StudentT", "LogGaussian", "Gamma", "Weibull", "Beta", "Beta2", "Kumaraswamy", "Kumaraswamy2", "Logistic", and "SkewLaplace". The Gaussian and skew-Gaussian copulas additionally support the discrete margins "Poisson", "Binomial", and "BinomialNeg". For the skew-Gaussian copula these probabilities are evaluated as copula-rectangle probabilities; when nu=0 the calculation reduces exactly to the Gaussian copula likelihood. For copula="Clayton", nu is the positive-integer parameter of the constructive Clayton-like random field and must be supplied in fixed; it is not a continuously estimable copula parameter. For copula="SkewGaussian", nu is the bounded asymmetry parameter \eta\in(-1,1) and invalid optimizer proposals are rejected by the objective function.

The optimization method is specified using optimizer. The default method is Nelder-Mead; other available methods are nlm, BFGS, SANN, L-BFGS-B, bobyqa, and nlminb. In the last three cases, bounds can be specified using lower and upper.

Depending on the dimension of data and on the name of the correlation model, the observations are assumed to be a realization of a spatial, spatio-temporal or bivariate random field. Specifically, with data, coordx, coordy, coordt:

• If data is a numeric d-dimensional vector and coordx, coordy are two numeric d-dimensional vectors (or coordx is a (d \times 2)-matrix and coordy=NULL), then the data are interpreted as a single spatial realisation observed on d spatial sites;

• If data is a numeric (t \times d)-matrix and coordt is a numeric t-dimensional vector, then the data are interpreted as a single spatio-temporal realisation observed on d sites and t times;

• If data is a numeric (2 \times d)-matrix, then the data are interpreted as a single bivariate spatial realisation observed on d spatial sites;

• If data is a list, coordx_dyn is a list and coordt is a numeric t-dimensional vector, then the data are interpreted as a spatio-temporal realisation observed on dynamical spatial sites (different locations for each time) and for t times.

It is also possible to specify a matrix of covariates using X. Specifically:

• In the spatial case, X must be a (d \times k) matrix associated to data (a d-vector);

• In the spatio-temporal case, X must be a (N \times k) matrix associated to data (a t \times d-matrix), where N=t\times d;

• In the bivariate case, X must be a (N \times k) matrix associated to data (a 2 \times d-matrix), where N=2\times d.

The distance parameter allows different kinds of spatial distances:

1. Eucl, Euclidean distance (default);

2. Chor, chordal distance;

3. Geod, geodesic distance.

The likelihood parameter represents the composite-likelihood configuration:

1. Conditional, composite likelihood formed by conditionals;

2. Marginal, composite likelihood formed by marginals (default);

3. Full, standard likelihood.

It must be coupled with type:

1. Pairwise, composite likelihood based on pairs;

2. Independence, composite likelihood based on independence;

3. Standard, standard likelihood.

Observation coordinates are checked for exact duplicates before fitting. For space-time data, the full space-time point must be unique; the same spatial site observed at different times is allowed. The check is hash-based and is performed once at the user-facing fit, not during bootstrap/refit iterations that reuse an already validated design. If stochastic thinning retains no pairwise contributions, fitting stops with an explicit error rather than optimizing an empty criterion.

For model="PoissonGamma" and model="PoissonGammaZIP", pairwise fitting accepts every finite shape>0. This is the continuous Kibble–Gamma pairwise extension used by the marginal moments, correlation function, and bivariate probabilities. The constructive random-field simulation based on a finite sum of squared Gaussian fields is more restrictive and requires 2\,shape to be a positive integer; GeoSim enforces that simulation constraint explicitly.

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

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 random field is bivariate, otherwise FALSE.
clic	The composite information criterion after a GeoVarest call; 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/matrix/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 maximum temporal-distance threshold used in the composite likelihood, expressed in the same units as coordt.
message	Extra message passed from the numerical routines.
model	The density associated to the likelihood objects.
estimation_model	The inferential model/working likelihood supplied to GeoFit. For misspecified fits this differs from model, which records the data-generating response model used by simulation, diagnostics, and prediction routines.
missp	TRUE if a misspecified Gaussian model is used in the composite likelihood.
n	The Binomial number of trials, or the common Negative-Binomial number r of successes.
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.
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 likelihood objects.
X	The matrix of covariates.

Spatio-temporal ordering

For fixed spatial locations, data is a T \times N matrix: row t corresponds to coordt[t] and column i to row i of coordx. Observation-level quantities are ordered as c(t(data)), hence X and a vector fixed$mean use the order time then site.

For dynamic locations, coordx_dyn, data, and optionally a list-valued X have one aligned element per time. Element t contains the coordinates, responses, and covariate rows observed at coordt[t]; internal concatenation is by increasing list index. See GeoModels-spacetime-ordering for the complete convention.

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

References

General Composite-likelihood:

Varin, C., Reid, N. and Firth, D. (2011). An overview of composite likelihood methods. Statistica Sinica, 21, 5–42.

Varin, C. and Vidoni, P. (2005). A note on composite likelihood inference and model selection. Biometrika, 92, 519–528.

Non-Gaussian random fields:

Alegría, A., Caro, S., Bevilacqua, M., Porcu, E. and Clarke, J. (2017). Estimating covariance functions of multivariate skew-Gaussian random fields on the sphere. Spatial Statistics, 22, 388–402. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/j.spasta.2017.05.003")}

Alegría, A., Bevilacqua, M. and Porcu, E. (2016). Likelihood-based inference for multivariate space-time wrapped-Gaussian fields. Journal of Statistical Computation and Simulation, 86(13), 2583–2597.

Bevilacqua, M., Caamaño-Carrillo, C. and Gaetan, C. (2020). On modelling positive continuous data with spatio-temporal dependence. Environmetrics, 31(7), e2628. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1002/env.2628")}

Bevilacqua, M., Caamaño-Carrillo, C., Arellano-Valle, R. B. and Morales-Oñate, V. (2021). Non-Gaussian geostatistical modeling using (skew) t processes. Scandinavian Journal of Statistics, 48(1), 212–245. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1111/sjos.12447")}

Blasi, F., Caamaño-Carrillo, C., Bevilacqua, M. and Furrer, R. (2022). A selective view of climatological data and likelihood estimation. Spatial Statistics, 50, 100596. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/j.spasta.2022.100596")}

Bevilacqua, M., Caamaño-Carrillo, C., Arellano-Valle, R. B. and Gómez, C. (2022). A class of random fields with two-piece marginal distributions for modeling point-referenced data with spatial outliers. TEST, 31(3), 644–674. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/s11749-021-00797-5")}

Morales-Navarrete, D., Bevilacqua, M., Caamaño-Carrillo, C. and Castro, L. M. (2024). Modelling point referenced spatial count data: A Poisson process approach. Journal of the American Statistical Association, 119(545), 664–677. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/01621459.2022.2140053")}

Bevilacqua, M., Alvarado, E. and Caamaño-Carrillo, C. (2024). A flexible Clayton-like spatial copula with application to bounded support data. Journal of Multivariate Analysis, 201, 105277. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/j.jmva.2023.105277")}

Weighted composite-likelihood for (non-)Gaussian random fields:

Bevilacqua, M., Gaetan, C., Mateu, J. and Porcu, E. (2012). Estimating space and space-time covariance functions for large data sets: a weighted composite likelihood approach. Journal of the American Statistical Association, Theory and Methods, 107, 268–280. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/01621459.2011.646928")}

Bevilacqua, M. and Gaetan, C. (2015). Comparing composite likelihood methods based on pairs for spatial Gaussian random fields. Statistics and Computing, 25(5), 877–892. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/s11222-014-9471-3")}

Caamaño-Carrillo, C., Bevilacqua, M., López, C. and Morales-Oñate, V. (2024). Nearest neighbours weighted composite likelihood based on pairs for (non-)Gaussian massive spatial data with an application to Tukey-hh random fields estimation. Computational Statistics and Data Analysis, 191, 107887. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/j.csda.2023.107887")}

Bevilacqua, M., Cuevas-Pacheco, F. and Caamaño-Carrillo, C. (2026). Fast stochastic nearest neighbor pairwise composite likelihood for massive spatial datasets. arXiv preprint, arXiv:2607.06142. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.48550/arXiv.2607.06142")}

See Also

GeoCovmatrix for covariance matrix construction, GeoSim for simulation, GeoKrig for prediction, GeoVariogram and GeoWLS for variogram-based tools, GeoVarest for bootstrap variance estimation.

Examples

library(GeoModels)

###############################################################
############ Examples of spatial Gaussian random fields ################
###############################################################


# Define the spatial-coordinates of the points:
set.seed(3)
N=300 # number of location sites
x <- runif(N, 0, 1)
y <- runif(N, 0, 1)
coords <- cbind(x,y)

# Define spatial matrix covariates and regression parameters
X=cbind(rep(1,N),runif(N))
mean <- 0.2
mean1 <- -0.5

# Set the covariance model's parameters:
corrmodel <- "Matern"
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 random field:
data <- GeoSim(coordx=coords,corrmodel=corrmodel, param=param,X=X)$data



################################################################
###
### Example 0. Maximum independence composite likelihood fitting of
### a Gaussian random field (no dependence parameters)
### 
###############################################################
# setting starting parameters to be estimated
start<-list(mean=mean,mean1=mean1,sill=sill)

fit1 <- GeoFit(data=data,coordx=coords,likelihood="Marginal",
 type="Independence", start=start,X=X)
print(fit1)


################################################################
###
### Example 1. Maximum conditional pairwise likelihood fitting of
### a Gaussian random field using Nelder-Mead
### 
###############################################################
# setting fixed and starting parameters to be estimated
fixed<-list(nugget=nugget,smooth=smooth)
start<-list(mean=mean,mean1=mean1,scale=scale,sill=sill)

fit1 <- GeoFit(data=data,coordx=coords,corrmodel=corrmodel, 
 neighb=3,likelihood="Conditional",optimizer="Nelder-Mead",
 type="Pairwise", start=start,fixed=fixed,X=X)
print(fit1)

################################################################
###
### Example 2. Maximum stochastic marginal pairwise likelihood 
### fitting of a Gaussian random field using Nelder-Mead
### 
###############################################################

#N=100000 # number of location sites
#x <- runif(N, 0, 1)
#y <- runif(N, 0, 1)
#coords <- cbind(x,y)
#X=cbind(rep(1,N),runif(N))
#data <- GeoSimapprox(coordx=coords,method="TB",L=20000,parallel=TRUE,
#	  corrmodel=corrmodel, param=param,X=X)$data
#fixed<-list(nugget=nugget)
#start<-list(mean=mean,mean1=mean1,scale=scale,sill=sill,smooth=smooth)

#fit2 <- GeoFit(data=data,coordx=coords,corrmodel=corrmodel,
# neighb=3,likelihood="Marginal",optimizer="Nelder-Mead",
# p_neighb=0.2, thin_method="bernoulli",
# type="Pairwise", start=start,fixed=fixed,X=X)
#print(fit2)

################################################################
###
### Example 3. Standard Maximum likelihood fitting of
### a Gaussian random field using nlminb
###
###############################################################
# Define the spatial-coordinates of the points:
set.seed(3)
N=250 # number of location sites
x <- runif(N, 0, 1)
y <- runif(N, 0, 1)
coords <- cbind(x,y)

param<-list(mean=mean,sill=sill,nugget=nugget,scale=scale,smooth=smooth)

data <- GeoSim(coordx=coords,corrmodel=corrmodel, param=param)$data

# setting fixed and parameters to be estimated
fixed<-list(nugget=nugget,smooth=smooth)
start<-list(mean=mean,scale=scale,sill=sill)

I=Inf
lower<-list(mean=-I,scale=0,sill=0)
upper<-list(mean=I,scale=I,sill=I)
fit2 <- GeoFit(data=data,coordx=coords,corrmodel=corrmodel,
 optimizer="nlminb",upper=upper,lower=lower,
 likelihood="Full",type="Standard", 
 start=start,fixed=fixed)
print(fit2)


###############################################################
############ Examples of spatial non-Gaussian random fields #############
###############################################################


################################################################
###
### Example 4. Maximum pairwise likelihood fitting of a Weibull random field 
### with Generalized Wendland correlation with Nelder-Mead
### 
###############################################################
set.seed(524)
# Define the spatial-coordinates of the points:
N=300
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
shape=2
scale=0.2
smooth=0

model="Weibull"
corrmodel="GenWend"
param=list(mean=mean,mean1=mean1,scale=scale,
 shape=shape,nugget=nugget,power2=4,smooth=smooth)
# Simulation of a non-stationary Weibull random field:
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,shape=shape)

# Maximum independence likelihood:
fit <- GeoFit(data=data, coordx=coords, X=X,
 likelihood="Marginal", type="Independence", corrmodel=corrmodel,
 model=model, start=start, fixed=fixed)
print(unlist(fit$param))

## estimating dependence parameter fixing vector mean parameter
Xb <- as.numeric(X %*% unlist(fit$param)[1:2])
fixed<-list(nugget=nugget,power2=4,smooth=smooth,mean=Xb)
start<-list(scale=scale,shape=shape)

# Maximum conditional composite-likelihood fitting of the random fields:
fit1 <- GeoFit(data=data,coordx=coords,corrmodel=corrmodel, model=model,
 neighb=3,likelihood="Conditional",type="Pairwise",
 optimizer="Nelder-Mead",
 start=start,fixed=fixed)
print(unlist(fit1$param))



### joint estimation of the dependence parameter and mean parameters
fixed<-list(nugget=nugget,power2=4,smooth=smooth)
start<-list(mean=mean,mean1=mean1,scale=scale,shape=shape)
fit2 <- GeoFit(data=data,coordx=coords,corrmodel=corrmodel, model=model,
 neighb=3,likelihood="Conditional",type="Pairwise",X=X,
 optimizer="Nelder-Mead",
 start=start,fixed=fixed)
print(unlist(fit2$param))



################################################################
###
### Example 5. Maximum pairwise likelihood fitting of
### a Skew-Gaussian spatial random fields with Wendland correlation
###
###############################################################
set.seed(261)
model="SkewGaussian"
# 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=-4.5
power2=4
c_supp=0.2

# model parameters
param=list(power2=power2,skew=skew,
 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,mean=mean,sill=sill)
lower=list(scale=0,skew=-I,mean=-I,sill=0)
upper=list(scale=I,skew=I,mean=I,sill=I)
# Maximum marginal pairwise likelihood:
fit1 <- GeoFit(data=data,coordx=coords,corrmodel=corrmodel, model=model,
 neighb=3,likelihood="Marginal",type="Pairwise",
 optimizer="bobyqa",lower=lower,upper=upper,
 start=start,fixed=fixed)
print(unlist(fit1$param))


################################################################
###
### Example 6. Maximum pairwise likelihood fitting of 
### a Bernoulli random field with exponential correlation 
###
###############################################################

set.seed(422)
N=250
x <- runif(N, 0, 1)
y <- runif(N, 0, 1)
coords <- cbind(x,y)
mean=0.1; mean1=0.8; mean2=-0.5 # regression parameters
X=cbind(rep(1,N),runif(N),runif(N)) # matrix covariates
corrmodel <- "Wend0"
param=list(mean=mean,mean1=mean1,mean2=mean2,nugget=0,scale=0.2,power2=4)
# Simulation of the spatial Binomial-Gaussian random field:
data <- GeoSim(coordx=coords, corrmodel=corrmodel, model="Binomial", n=1,X=X,
 param=param)$data


## estimating the marginal parameters using independence cl
fixed <- list(power2=4,scale=0.2,nugget=0)
start <- list(mean=mean,mean1=mean1,mean2=mean2)

# Maximum independence likelihood:
fit <- GeoFit(data=data, coordx=coords, n=1, X=X,
 likelihood="Marginal", type="Independence", corrmodel=corrmodel,
 model="Binomial", start=start, fixed=fixed)
 
print(fit)


## estimating dependence parameter fixing vector mean parameter
Xb <- as.numeric(X %*% unlist(fit$param))
fixed <- list(nugget=0,power2=4,mean=Xb)
start <- list(scale=0.2)
lower <- list(scale=0)
upper <- list(scale=2)
# Maximum Marginal pairwise likelihood:
fit1 <- GeoFit(data=data, coordx=coords, corrmodel=corrmodel, n=1,
 likelihood="Marginal", type="Pairwise", neighb=3,
 model="Binomial", start=start, fixed=fixed,
 lower=list(scale=0.001), upper=list(scale=1))
 
print(fit1)


## estimating jointly marginal and dependence parameters
fixed <- list(nugget=0,power2=4)
start <- list(mean=mean,mean1=mean1,mean2=mean2,scale=0.2)

# Maximum conditional pairwise likelihood:
fit2 <- GeoFit(data=data, coordx=coords, corrmodel=corrmodel, n=1, X=X,
 likelihood="Marginal", type="Pairwise", neighb=3,
 model="Binomial", start=start, fixed=fixed)
 
print(fit2)


###############################################################
######### Examples of Gaussian spatio-temporal random fields ###########
###############################################################
set.seed(52)
# Define the temporal sequence:
time <- seq(1, 9, 1)

# Define the spatial-coordinates of the points:
x <- runif(20, 0, 1)
y <- runif(20, 0, 1)
coords=cbind(x,y)

# Set the covariance model's parameters:
scale_s=0.2/3;scale_t=1
smooth_s=0.5;smooth_t=0.5
sill=1
nugget=0
mean=0

param<-list(mean=0,scale_s=scale_s,scale_t=scale_t,
 smooth_t=smooth_t, smooth_s=smooth_s ,sill=sill,nugget=nugget)

# Simulation of the spatio-temporal Gaussian random field:
data <- GeoSim(coordx=coords,coordt=time,corrmodel="Matern_Matern",
 param=param)$data

################################################################
###
### Example 7. Maximum pairwise likelihood fitting of a
### space time Gaussian random fields with double-exponential correlation
###
###############################################################
# Fixed parameters
fixed<-list(nugget=nugget,smooth_s=smooth_s,smooth_t=smooth_t)
# Starting value for the estimated parameters
start<-list(mean=mean,scale_s=scale_s,scale_t=scale_t,sill=sill)

# Maximum composite-likelihood fitting of the random fields:
fit <- GeoFit(data=data,coordx=coords,coordt=time,
 corrmodel="Matern_Matern",maxtime=1,neighb=3,
 likelihood="Marginal",type="Pairwise",
 start=start,fixed=fixed)
print(fit)



###############################################################
######### Examples of a bivariate Gaussian random field ###########
###############################################################

################################################################
### Example 8. Maximum pairwise likelihood fitting of a
### bivariate Gaussian random fields with separable Bivariate matern 
### (cross) correlation model 
###############################################################

# Define the spatial-coordinates of the points:
set.seed(89)
x <- runif(300, 0, 1)
y <- runif(300, 0, 1)
coords=cbind(x,y)
# parameters
param=list(mean_1=0,mean_2=0,scale=0.1,smooth=0.5,sill_1=1,sill_2=1,
 nugget_1=0,nugget_2=0,pcol=0.2)

# Simulation of a spatial bivariate Gaussian random field:
data <- GeoSim(coordx=coords, corrmodel="Bi_Matern_sep", 
 param=param)$data

# selecting fixed and estimated parameters
fixed=list(mean_1=0,mean_2=0,nugget_1=0,nugget_2=0,smooth=0.5)
start=list(sill_1=var(data[1,]),sill_2=var(data[2,]),
 scale=0.1,pcol=cor(data[1,],data[2,]))


# Maximum marginal pairwise likelihood
fitcl<- GeoFit(data=data, coordx=coords, corrmodel="Bi_Matern_sep",
 likelihood="Marginal",type="Pairwise",
 start=start,fixed=fixed,
 neighb=3)
print(fitcl)

GeoModels documentation built on Sept. 23, 2026, 5:07 p.m.