FitComposite: Max-Likelihood-Based Fitting of Gaussian, Binary and...
In CompRandFld: Composite-Likelihood Based Analysis of Random Fields

Description Usage Arguments Details Value Author(s) References See Also Examples

Maximum weighted composite-likelihood fitting of spatio-temporal Gaussian, binary and spatial max-stable random fields. For the spatio-temporal Gaussian random field, (restricted) maximum likelihood and tapered likelihood fitting is also avalable. The function returns the model parameters' estimates and the estimates' variances and allows to fix any of the parameters.

FitComposite(data, coordx, coordy=NULL, coordt=NULL, corrmodel,
             distance='Eucl', fixed=NULL, grid=FALSE,
             likelihood='Marginal', margins='Gev', maxdist=NULL,
             maxtime=NULL, model='Gaussian', optimizer='Nelder-Mead',
             replicates=1, start=NULL, taper=NULL, tapsep=NULL,
             threshold=NULL,type='Pairwise', varest=FALSE,
             vartype='SubSamp', weighted=FALSE, winconst, winstp)

`data`	A d-dimensional vector (a single spatial realisation) or a (n x d)-matrix (n iid spatial realisations) or a (d x d)-matrix (a single spatial realisation on regular grid) or an (d x d x n)-array (n iid spatial realisations on regular grid) or a (t x d)-matrix (a single spatial-temporal realisation) or an (t x d x n)-array (n iid spatial-temporal realisations) or or an (d x d x t)-array (a single spatial-temporal realisation on regular grid) or an (d x d x t x n)-array (n iid spatial-temporal realisations on regular grid). For the description see the Section Details.
`coordx`	A numeric (d x 2)-matrix (where `d` is the number of spatial sites) assigning 2-dimensions of spatial coordinates or a numeric d-dimensional vector assigning 1-dimension of spatial coordinates.
`coordy`	A numeric vector assigning 1-dimension of spatial coordinates; `coordy` is interpreted only if `coordx` is a numeric vector or `grid=TRUE` otherwise it will be ignored. Optional argument, the default is `NULL` then `coordx` is expected to be numeric a (d x 2)-matrix.
`coordt`	A numeric vector assigning 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.
`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 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 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 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.
`margins`	String; the type of the marginal distribution of the max-stable field. `Gev` is the default, see the Section Details.
`maxdist`	Numeric; an optional positive value indicating the maximum spatial distance considered in the composite or tapered likelihood computation. See the Section Details for more information.
`maxtime`	Numeric; an optional positive value indicating the maximum temporal separation considered in the composite or tapered likelihood computation (see Details).
`model`	String; the type of random field and therefore the densities associated to the likelihood objects. `Gaussian` is the default, see the Section Details.
`optimizer`	String; the optimization algorithm (see `optim` for details). 'Nelder-Mead' is the default.
`replicates`	Numeric; a positive integer denoting the number of independent and identically distributed (iid) replications of a spatial or spatial-temporal random field. Optional argument, the default value is 1 then a single realisation is considered.
`start`	An optional named list with the initial values of the parameters that are used by the numerical routines in maximization procedure. `NULL` is the default (see Details).
`taper`	String; the name of the type of covariance matrix. It can be `Standard` (the default value) or `Tapering` for taperd covariance matrix.
`tapsep`	Numeric; an optional value indicating the separabe parameter in the space time quasi taper (see Details).
`threshold`	Numeric; a value indicating a threshold for the binary random field. Optional in the case that `model` is `BinaryGauss`, see the Section Details.
`type`	String; the type of the likelihood objects. If `Pairwise` (the default) then the marginal composite likelihood is formed by pairwise marginal likelihoods (see Details).
`varest`	Logical; if `TRUE` the estimates' variances and standard errors are returned. `FALSE` is the default.
`vartype`	String; (`SubSamp` the default) the type of method used for computing the estimates' variances, see the Section Details.
`weighted`	Logical; if `TRUE` the likelihood objects are weighted, see the Section Details. If `FALSE` (the default) the composite likelihood is not weighted.
`winconst`	Numeric; a positive value for computing the sub-window size where observations are sampled in the sub-sampling procedure (if `vartype=SubSamp`). For increasing `winconst` increasing sub-window sizes are obtained. Optional argument, the default is 1. See Details for more information.
`winstp`	Numeric; a value in (0,1] for computing the sub-window step (in the sub-sampling procedure). This value denote the proportion of the sub-window size. Optional argument, the default is 0.5. See Details for more information.

Note, that the standard likelihood may be seen as particular case of the composite likelihood. In this respect FitComposite provides maximum (restricted) likelihood fitting. Only composite likelihood estimation based on pairs are considered. Specifically marginal pairwise, conditional pairwise and difference pairwise. Covariance tapering is considered only for Gaussian random fields.

With data, coordx, coordy, coordt, grid and replicates parameters:

If data is a numeric d-dimensional vector, coordx and coordy are two numeric d-dimensional vectors (or coordx is (d x 2)-matrix and coordy=NULL), coordt=NULL, grid=FALSE and replicates=1, then the data are interpreted as a single spatial realisation observed on d spatial sites;
If data is a numeric (n x d)-matrix, coordx and coordy are two numeric d-dimensional vectors (or coordx is (d x 2)-matrix and coordy=NULL), coordt=NULL, grid=FALSE and replicates=n, then the data are interpreted as n iid replications of a spatial random field observed on d spatial sites.
If data is a numeric (d x d)-matrix, coordx and coordy are two numeric d-dimensional vectors, coordt=NULL, grid=TRUE and replicates=1, then the data are interpreted as a single spatial random field realisation observed on d equispaced spatial sites (named regular grid).
If data is a numeric (d x d x n)-array, coordx and coordy are two numeric d-dimensional vectors, coordt=NULL, grid=TRUE and replicates=n, then the data are interpreted as n iid realisations of a spatial random field observed on d equispaced spatial sites.
If data is a numeric (t x d)-matrix, coordx and coordy are two numeric d-dimensional vectors (or coordx is (d x 2)-matrix and coordy=NULL), coordt is a numeric t-dimensional vector, grid=FALSE and replicates=1, then the data are interpreted as a single spatial-temporal realisation of a random field observed on d spatial sites and for t times.
If data is a numeric (t x d x n)-array, coordx and coordy are two numeric d-dimensional vectors (or coordx is (d x 2)-matrix and coordy=NULL), coordt is a numeric t-dimensional vector, grid=FALSE and replicates=n, then the data are interpreted as n iid realisations of a spatial-temporal random field observed on d spatial sites and for t times.
If data is a numeric (d x d x t)-array, coordx and coordy are two numeric d-dimensional vectors, coordt is a numeric t-dimensional vector, grid=TRUE and replicates=1, then the data are interpreted as a single spatial-temporal realisation of a random field observed on d equispaced spatial sites and for t times.
If data is a numeric (d x d x t x n)-array, coordx and coordy are two numeric d-dimensional vectors, coordt is a numeric t-dimensional vector, grid=TRUE and replicates=n, then the data are interpreted as n iid realisation of a spatial-temporal random field observed on d equispaced spatial sites and for t times.

The corrmodel parameter allows to select a specific correlation function for the random field. (See Covmatrix ).

The distance parameter allows to consider differents kinds of spatial distances. The settings alternatives are:

Eucl, the euclidean distance (default value);
Chor, the chordal distance;
Geod, the geodesic distance;

The likelihood parameter represents the composite-likelihood configurations. The settings alternatives are:

Conditional, the composite-likelihood is formed by conditionals likelihoods;
Marginal, the composite-likelihood is formed by marginals likelihoods;
Full, the composite-likelihood turns out to be the standard likelihood;

The margins parameter concerns only max-stable fields and indicates how the margins are considered. The options are Gev or Frechet, where in the former case the marginals are supposed generalized extreme value distributed and in the latter case unit Frechet distributed.

The maxdist parameter set the maximum spatial distance below which pairs of sites with inferior distances are considered in the composite-likelihood. This can be inferior of the effective maximum spatial distance. Note that this corresponds to use a weighted composite-likelihood with binary weights. Pairs with distance less than maxdist have weight 1 and are included in the likelihood computation, instead those with greater distance have weight 0 and then excluded. The default is NULL, in this case the effective maximum spatial distance between sites is considered.

The same arguments of maxdist are valid for maxtime but here the weigthed composite-likelihood regards the case of spatial-temporal field. At the moment is implemented only for Gaussian random fields. The default is NULL, in this case the effective maximum temporal lag between pairs of observations is considered.

In the case of tapering likelihood maxdist and maxtime describes the spatial and temporal compact support of the taper model (see Covmatrix). If they are not specified then the maximum spatial and temporal distances are considered. In the case of space time quasi taper the tapsep parameter allows to specify the spatio temporal compact support (see Covmatrix).

The model paramter indicates the type of random field considered, for instance model=Gaussian denotes a Gaussian random field. Accordingly, this also determines the analytical expression of the finite dimensional distribution associated with the random field. The available options are:

Gaussian, for a Gaussian random field (see i.e. Wackernagel, H. 1998);
BinaryGauss, for a Binary random field (see Heagerty and Lele 1998)
BrowResn, for a Brown-Resnick max-stable random field (see Kabluchko, Z. et al. 2009);
ExtGauss, for an Extremal Gaussian max-stable random field (known also as Schlather model) (see Schlather, M. 2002);
ExtT, for an Extremal t max-stable random field (see Davison, A. C. et al. 2012);

Note, that only for the Gaussian case the estimation procedure is implemented for spatial and spatial-temporal random fields.

The start parameter allows to specify starting values. If start is omitted the routine is computing the starting values using the weighted moment estimator.

The taper parameter, optional in case that type=Tapering, indicates the type of taper correlation model. (See Covmatrix)

The threshold parameter indicates the value (common for all the spatial sites) above which the values of the underlying Gaussian latent process are considered sucesses events (values below are instead failures). See e.g. Heagerty and Lele (1998) for more details.

The type parameter represents the type of likelihood used in the composite-likelihood definition. The possible alternatives are listed in the following scheme.

If a Gaussian random field is considered (model=Gaussian):
- If the composite is formed by marginal likelihoods (likelihood=Marginal):
  - Pairwise, the composite-likelihood is defined by the pairwise likelihoods;
  - Difference, the composite-likelihood is defined by likelihoods which are obtained as difference of the pairwise likelihoods.
- If the composite is formed by conditional likelihoods (likelihood=Conditional)
  - Pairwise, the composite-likelihood is defined by the pairwise conditional likelihoods.
- If the composite is formed by a full likelihood (likelihood=Full):
  - Standard, the objective function is the classical multivariate likelihood;
  - Restricted, the objective function is the restricted version of the full likelihood (e.g. Harville 1977, see References);
  - Tapering, the objective function is the tapered version of the full likelihood (e.g. Kaufman et al. 2008, see References).

The varest parameter specifies if the standard error estimation of the estimated parameters must be computed. For Gaussian random field and standard (restricted) likelihood estimation, standard errrors are computed as square root of the diagonal elements of the Fisher Information matrix (asymptotic covariance matrix of the estimates under increasing domain). For Gaussian random field and tapered and composite likelihood estimation, standard errors estimate are computed as square root of the diagonal elements of the Godambe Information matrix. (asymptotic covariance matrix of the estimates under increasing domain (see Shaby, B. and D. Ruppert (2012) for tapering and Bevilacqua et. al. (2012) , Bevilacqua and Gaetan (2013) for weighted composite likelihood)). The vartype parameter specifies the method used to compute the estimates' variances in the composite likelihood case. In particular for estimating the variability matrix J in the Godambe expression matrix. This parameter is considered if varest=TRUE. The options are:

SubSamp (the default), indicates the Sub-Sampling method;
Sampling, indicates that the variability matrix is estimated by the sample contro-part (available only for n iid replications of the random field, i.e. replicates=n);

The weighted parameter specifies if the likelihoods forming the composite-likelihood must be weighted. If TRUE the weights are selected by opportune procedures that improve the efficient of the maximum composite-likelihood estimator (not implemented yet). If FALSE the efficient improvement procedure is not used.

For computing the standard errors by the sub-sampling procedure, winconst and winstp parameters represent respectively a positive constant used to determine the sub-window size and the the step with which the sub-window moves.

In the spatial case (subset of R^2), the domain is seen as a rectangle BxH, therefore the size of the sub-window side b is given by b=winconst x sqrt(B) (similar is of h). For a complete description see Lee and Lahiri (2002). By default winconst is set B / (2 x sqrt(B)). The winstp parameter is used to determine the sub-window step. The latter is given by the proportion of the sub-window size, so that when winstp=1 there is not overlapping between contiguous sub-windows. In the spatial case by default winstp=0.5. The sub-window is moved by successive steps in order to cover the entire spatial domain. Observations, that fall in disjoint or overlapping windows are considered indipendent samples.

In the spatio-temporal case the subsampling is meant only in time as described by Li et al. (2007). Thus, winconst represents the lenght of the temporal sub-window. By default the size of the sub-window is computed following the rule established in Li et al. (2007). By default winstp is the time step.

Observe that in the spatio-temporal case, the returned values by srange and trange, represent respectively the minimum and maximum of the marginal spatial distances and those of the temporal separations. Thus, the minimum being not the overall (i.e. considering the spatio-temporal coordinates) is not zero, as one could be expect and the latter can be easily added by the user.

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

`clic`	The composite information criterion, if the full likelihood is considered then it coincides with the Akaike information criterion;
`coordx`	A d-dimensional vector of spatial coordinates;
`coordy`	A d-dimensional vector of spatial coordinates;
`coordt`	A t-dimensional vector of temporal coordinates;
`convergence`	A string that denotes if convergence is reached;
`corrmodel`	The correlation model;
`data`	The vector or matrix or array of data;
`distance`	The type of spatial distance;
`fixed`	The vector of fixed parameters;
`iterations`	The number of iteration used by the numerical routine;
`likelihood`	The configuration of the composite likelihood;
`logCompLik`	The value of the log composite-likelihood at the maximum;
`message`	Extra message passed from the numerical routines;
`model`	The density associated to the likelihood objects;
`nozero`	In the case of tapered likelihood the percentage of non zero values in the covariance matrix. Otherwise is NULL.
`numcoord`	The number of spatial coordinates;
`numrep`	The number of the iid replicatations of the random field;
`numtime`	The number the temporal realisations of the random field;
`param`	The vector of parameters' estimates;
`srange`	The minimum and maximum spatial distance (see Details). The maximum is `maxdist`, if inserted, rather the effective maximum distance;
`stderr`	The vector of standard errors;
`sensmat`	The sensitivity matrix;
`varcov`	The matrix of the variance-covariance of the estimates;
`varimat`	The variability matrix;
`vartype`	The method used to compute the variance of the estimates;
`trange`	The minimum and maximum temporal separation (see Details). The maximum is `maxtime`, if inserted, rather then the effective maximum separation;
`threshold`	The threshold used in the binary random field.
`type`	The type of the likelihood objects.
`winconst`	The constant use to compute the window size in the sub-sampling procedure;
`winstp`	The step used for moving the window in the sub-sampling procedure

Simone Padoan, simone.padoan@unibocconi.it, http://faculty.unibocconi.it/simonepadoan; Moreno Bevilacqua, moreno.bevilacqua@uv.cl, https://sites.google.com/a/uv.cl/moreno-bevilacqua/home.

Padoan, S. A. and Bevilacqua, M. (2015). Analysis of Random Fields Using CompRandFld. Journal of Statistical Software, 63(9), 1–27.

Maximum Restricted Likelihood Estimator:

Harville, D. A. (1977) Maximum Likelihood Approaches to Variance Component Estimation and to Related Problems. Journal of the American Statistical Association, 72, 320–338.

Tapered likelihood:

Kaufman, C. G., Schervish, M. J. and Nychka, D. W. (2008) Covariance Tapering for Likelihood-Based Estimation in Large Spatial Dataset. Journal of the American Statistical Association, 103, 1545–1555.

Shaby, B. and D. Ruppert (2012). Tapered covariance: Bayesian estimation and asymptotics. J. Comp. Graph. Stat., 21-2, 433–452.

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.

Weighted Composite-likelihood for binary random fields:

Patrick, J. H. and Subhash, R. L. (1998) A Composite Likelihood Approach to Binary Spatial Data. Journal of the American Statistical Association, Theory & Methods, 93, 1099–1111.

Weighted Composite-likelihood for max-stable random fields:

Davison, A. C. and Gholamrezaee, M. M. (2012) Geostatistics of extremes. Proceedings of the Royal Society of London, series A, 468, 581–608.

Padoan, S. A. (2008). Computational Methods for Complex Problems in Extreme Value Theory. PhD. Thesis, Department of Statistics, University of Padua.

Padoan, S. A. Ribatet, M. and Sisson, S. A. (2010) Likelihood-Based Inference for Max-Stable Processes. Journal of the American Statistical Association, Theory & Methods, 105, 263–277.

Weighted Composite-likelihood for 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 & Methods, 107, 268–280.

Bevilacqua, M. Gaetan, C. (2014) Comparing composite likelihood methods based on pairs for spatial Gaussian random fields Statistics and Computing.DOI:10.1111/2041-210X.12167

Spatial Extremes:

Davison, A. C., Padoan, S. A., and Ribatet, M. (2012) Statistical Modelling of Spatial Extremes, with discussion. Statistical Science, 27, 161–186.

de Haan, L., and Pereira, T. T. (2006) Spatial Extremes: Models for the Stationary Case. The Annals of Statistics, 34, 146–168.

Kabluchko, Z. (2010) Extremes of Independent Gaussian Processes. Extremes, 14, 285–310.

Kabluchko, Z., Schlather, M., and de Haan, L. (2009) Stationary max-stable fields associated to negative definite functions. The Annals of Probability, 37, 2042–2065.

Schlather, M. (2002) Models for Stationary Max-Stable Random Fields. Extremes, 5, 33–44.

Smith, R. L. (1990) Max-Stable Processes and Spatial Extremes. Unpublished manuscript, University of North California.

Sub-sampling estimation:

Carlstein, E. (1986) The Use of Subseries Values for Estimating the Variance. The Annals of Statistics, 14, 1171–1179.

Heagerty, P. J. and Lumley T. (2000) Window Subsampling of Estimating Functions with Application to Regression Models. Journal of the American Statistical Association, Theory & Methods, 95, 197–211.

Lee, Y. D. and Lahiri S. N. (2002) Variogram Fitting by Spatial Subsampling. Journal of the Royal Statistical Society. Series B, 64, 837–854.

Li, B., Genton, M. G. and Sherman, M. (2007). A nonparametric assessment of properties of space-time covariance functions. Journal of the American Statistical Association, 102, 736–744

Covmatrix, GaussRF, MaxStableRF, WLeastSquare, optim

library(CompRandFld)
library(RandomFields)
library(spam)
set.seed(3132)

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

# Define the spatial-coordinates of the points:
x <- runif(100, 0, 10)
y <- runif(100, 0, 10)

# Set the covariance model's parameters:
corrmodel <- "exponential"
mean <- 0
sill <- 1
nugget <- 0
scale <- 1.5
param<-list(mean=mean,sill=sill,nugget=nugget,scale=scale)
coords<-cbind(x,y)
# Simulation of the spatial Gaussian random field:
data <- RFsim(coordx=coords, corrmodel=corrmodel, param=param)$data

# Fixed parameters
fixed<-list(mean=mean,nugget=nugget)

# Starting value for the estimated parameters
start<-list(scale=scale,sill=sill)


################################################################
###
### Example 1. Maximum likelihood fitting of
### Gaussian random fields with exponential correlation.
### One spatial replication.
### Likelihood setting: composite with
### marginal pairwise likelihood objects.
###
###############################################################


# Maximum composite-likelihood fitting of the random field:
fit <- FitComposite(data, coordx=coords, corrmodel=corrmodel, maxdist=2,
                    likelihood="Marginal",type="Pairwise",varest=TRUE,
                    start=start,fixed=fixed)

# Results:
print(fit)

################################################################
###
### Example 2. Maximum likelihood fitting of
### Gaussian random fields with exponential correlation.
### One spatial replication.
### Likelihood setting: standard full likelihood.
###
###############################################################

# Maximum composite-likelihood fitting of the random field:
fit <- FitComposite(data, coordx=coords, corrmodel=corrmodel,likelihood="Full",
                    type="Standard",varest=TRUE,start=start,fixed=fixed)
# Results:
print(fit)

################################################################
###
### Example 3. Maximum likelihood fitting of
### Gaussian random fields with exponetial correlation.
### One spatial replication.
### Likelihood setting: tapered full likelihood.
###
###############################################################

# Maximum tapered likelihood fitting of the random field:
fit <- FitComposite(data, coordx=coords, corrmodel=corrmodel,likelihood="Full",
                    type="Tapering",taper="Wendland1",maxdist=1.5,
                    varest=TRUE,start=start,fixed=fixed)

# Results:
print(fit)


################################################################
###
### Example 4. Maximum composite-likelihood fitting of
### max-stable random fields. Extremal Gaussian model with
### exponential correlation. n iid spatial replications.
### Likelihood setting: composite with marginal pairwise
### likelihood objects.
###
###############################################################

# Simulation of a max-stable random field in the specified points:
data <- RFsim(x, y, corrmodel=corrmodel, model="ExtGauss", replicates=30,
              param=list(mean=mean,sill=sill,nugget=nugget,scale=scale))$data

# Maximum composite-likelihood fitting of the random field:
fit <- FitComposite(data, x, y, corrmodel=corrmodel, model="ExtGauss",
                    replicates=30, varest=TRUE, vartype="Sampling",
                    margins="Frechet",start=list(sill=sill,scale=scale))

# Results:
print(fit)

################################################################
###
### Example 5. Maximum likelihood fitting of
### Binary-Gaussian random fields with exponential correlation.
### One spatial replication.
### Likelihood setting: composite with marginal pairwise
### likelihood objects.
###
###############################################################

#set.seed(3128)

#x <- runif(200, 0, 10)
#y <- runif(200, 0, 10)

# Simulation of the spatial Binary-Gaussian random field:
#data <- RFsim(coordx=coords, corrmodel=corrmodel, model="BinaryGauss",
#              threshold=0, param=list(mean=mean,sill=.8,
#              nugget=nugget,scale=scale))$data

# Maximum composite-likelihood fitting of the random field:
#fit <- FitComposite(data, coordx=coords, corrmodel=corrmodel, threshold=0,
#                    model="BinaryGauss", fixed=list(nugget=nugget,
#                    mean=0),start=list(scale=.1,sill=.1))

# Results:
#print(fit)


###############################################################
######### Examples of spatio-temporal random fields ###########
###############################################################

# Define the temporal sequence:
#time <- seq(1, 80, 1)

# Define the spatial-coordinates of the points:
#x <- runif(10, 0, 10)
#y <- runif(10, 0, 10)
#coords=cbind(x,y)

# Set the covariance model's parameters:
#corrmodel="exp_exp"
#scale_s=0.5
#scale_t=1
#sill=1
#nugget=0
#mean=0

#param<-list(mean=0,scale_s=1,scale_t=1,sill=sill,nugget=nugget)

# Simulation of the spatial-temporal Gaussian random field:
#data <- RFsim(coordx=coords,coordt=time,corrmodel=corrmodel,
#              param=param)$data

# Fixed parameters
#fixed<-list(mean=mean,nugget=nugget)

# Starting value for the estimated parameters
#start<-list(scale_s=scale_s,scale_t=scale_t,sill=sill)

################################################################
###
### Example 6. Maximum likelihood fitting of
### Gaussian random field with double-exponential correlation.
### One spatio-temporal replication.
### Likelihood setting: composite with conditional pairwise
### likelihood objects.
###
###############################################################

# Maximum composite-likelihood fitting of the random field:
#fit <- FitComposite(data=data,coordx=coords,coordt=time,corrmodel="exp_exp",
#                    maxtime=2,maxdist=1,likelihood="Marginal",type="Pairwise",
#                    start=start,fixed=fixed)

# Results:
#print(fit)