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

GeoFit {GeoModels}

R Documentation

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

Description

Maximum weighted composite-likelihood fitting for Gaussian and some Non-Gaussian univariate spatial, spatio-temporal and bivariate spatial random fieldss The function allows to fix any of the parameters and setting upper/lower bound in the optimization.

Usage

GeoFit(data, coordx, coordy=NULL, coordt=NULL, coordx_dyn=NULL,copula=NULL,
      corrmodel=NULL,distance="Eucl",fixed=NULL,anisopars=NULL,
      est.aniso=c(FALSE,FALSE),GPU=NULL, grid=FALSE, likelihood='Marginal',
     local=c(1,1),  lower=NULL,maxdist=Inf,neighb=NULL,
      maxtime=Inf, memdist=TRUE,method="cholesky", 
      model='Gaussian',n=1, onlyvar=FALSE ,
      optimizer='Nelder-Mead', parallel=FALSE, 
      radius=6371,  sensitivity=FALSE,sparse=FALSE, 
      start=NULL, taper=NULL, tapsep=NULL, 
      type='Pairwise', upper=NULL, varest=FALSE, 
      vartype='SubSamp', weighted=FALSE, winconst=NULL, winstp=NULL, 
      winconst_t=NULL, winstp_t=NULL,X=NULL,nosym=FALSE,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 (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. Coordinates on a sphere for a fixed radius `radius` are passed in lon/lat format expressed in decimal degrees.
`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 \times 2`)-matrix.
`coordt`	A numeric vector assigning 1-dimension of temporal coordinates. Optional argument, the default is `NULL` then a spatial random fields 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`
`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. 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.
`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.
`GPU`	Numeric; if `NULL` (the default) no OpenCL computation is performed. The user can choose the device to be used. Use `DeviceInfo()` function to see available devices, only double precision devices are allowed
`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.
`local`	Numeric; number of local work-items of the OpenCL setup
`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.
`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.
`neighb`	Numeric; an optional positive integer indicating the order of neighborhood in the composite likelihood computation. See the Section Details for more information.
`maxtime`	Numeric; an optional positive integer indicating the order of temporal neighborhood in the composite likelihood computation.
`memdist`	Logical; if `TRUE` then all the 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 simulation. Default is cholesky. The other possible choices is `svd`.
`model`	String; the type of random fields 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 random fields; number of successes in a negative binomial random fields
`onlyvar`	Logical; if `TRUE` (and varest is `TRUE`) only the variance covariance matrix is computed without optimizing. `FALSE` is the default.
`optimizer`	String; the optimization algorithm (see `optim` for details). `Nelder-Mead` is the default. Other possible choices are `nlm`, `BFGS`, `SANN`, `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.
`parallel`	Logical; if `TRUE` optmization is performed using optimParallel using the maximum number of cores, when optimizer is `L-BFGS-B`.`FALSE` is the default.
`radius`	Numeric; the radius of the sphere in the case of lon-lat coordinates. The default is 6371, the radius of the earth.
`sensitivity`	Logical; if `TRUE` then the sensitivy matrix is computed
`sparse`	Logical; if `TRUE` then maximum likelihood is computed using sparse matrices algorithms (spam packake).It should be used with compactly supported covariance models.`FALSE` is the default.
`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 adaptive 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 (see Details).
`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 estimates' variances and standard errors are returned. For composite likelihood estimation it is deprecated. Use sensitivity TRUE and update the object using the function `GeoVarestbootstrap` `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 bivariate positive vector for computing the spatial sub-window in the sub-sampling procedure. See Details for more information.
`winstp`	Numeric; a value in `(0,1]` for defining the the proportion of overlapping in the spatial sub-sampling procedure. The case `1` correspond to no overlapping. See Details for more information.
`winconst_t`	Numeric; a positive value for computing the temporal sub-window in the sub-sampling procedure. See Details for more information.
`winstp_t`	Numeric; a value in `(0,1]` for defining the the proportion of overlapping in the temporal sub-sampling procedure. The case `1` correspond to no overlapping. See Details for more information.
`X`	Numeric; Matrix of spatio(temporal)covariates in the linear mean specification.
`nosym`	Logical; if TRUE simmetric weights are not considered. This allows a faster but less efficient CL estimation.
`spobj`	An object of class sp or spacetime
`spdata`	Character:The name of data in the sp or spacetime object

Details

GeoFit provides weighted composite likelihood estimation based on pairs and independence composite likelihood estimation for Gaussian and non Gaussian random fields. Specifically, marginal and conditional pairwise likelihood are considered for each type of random field (Gaussian and not Gaussian). The optimization method is specified using optimizer. The default method is Nelder-mead and other available methods are ucminf, nlm, BFGS, SANN, L-BFGS-B, and nlminb. In the last two cases upper and lower bounds constraints in the optimization can be specified using lower and upper parameters.

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

If data is a numeric d-dimensional vector, coordx and coordy are two numeric d-dimensional vectors (or coordx is (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, coordx and coordy are two numeric d-dimensional vectors (or coordx is (d \times 2)-matrix and coordy=NULL), coordt is a numeric t-dimensional vector, then the data are interpreted as a single spatial-temporal realisation of a random fields observed on d spatial sites and for t times.
If data is a numeric (2 \times d)-matrix, coordx and coordy are two numeric d-dimensional vectors (or coordx is (d \times 2)-matrix and coordy=NULL), then the data are interpreted as a single spatial realisation of a bivariate random fields observed on d spatial sites.
If data is a list, coordxdyn is a list and coordt is a numeric t-dimensional vector, then the data are interpreted as a single spatial-temporal realisation of a random fields observed on dynamical spatial sites (different locations sites for each temporal instants) and for t times.

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

In the spatial case X must be a (d \times k) covariates matrix associated to data a numeric d-dimensional vector;
In the spatiotemporal case X must be a (N \times k) covariates matrix associated to data a numeric (t \times d)-matrix, where N=t\times d;
In the spatiotemporal case X must be a (N \times k) covariates matrix associated to data a numeric (t \times d)-matrix, where N=2\times d;

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

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 (default value);
Full, the composite-likelihood turns out to be the standard likelihood;

It must be coupled with the type parameter that can be fixed to

Pairwise, the composite-likelihood is based on pairs;
Independence, the composite-likelihood is based on indepedence;
Standard, this is the option for the standard likelihood;

The possible combinations are:

likelihood="Marginal" and type="Pairwise" for maximum marginal pairwise likelihood estimation (the default setting)
likelihood="Conditional" and type="Pairwise" for maximum conditional pairwise likelihood estimation
likelihood="Marginal" and type="Independence" for maximum independence composite likelihood estimation
likelihood="Full" and type="Standard" for maximum stardard likelihood estimation

The first three combinations can be used for any model. The standard likelihood can be used only for some specific model.

The model parameter indicates the type of random field considered. The available options are:

random fields with marginal symmetric distribution:

Gaussian, for a Gaussian random field.
StudentT, for a StudentT random field (see Bevilacqua M., Caamaño C., Arellano Valle R.B., Morales-Oñate V., 2020).
Tukeyh, for a Tukeyh random field.
Tukeyh2, for a Tukeyhh random field. (see Caamaño et al., 2023)
Logistic, for a Logistic random field.

random fields with positive values and right skewed marginal distribution:

Gamma for a Gamma random fields (see Bevilacqua M., Caamano C., Gaetan, 2020)
Weibull for a Weibull random fields (see Bevilacqua M., Caamano C., Gaetan, 2020)
LogGaussian for a LogGaussian random fields (see Bevilacqua M., Caamano C., Gaetan, 2020)
LogLogistic for a LogLogistic random fields.

random fields with with possibly asymmetric marginal distribution:

SkewGaussian for a skew Gaussian random field (see Alegrıa et al. (2017)).
SinhAsinh for a Sinh-arcsinh random field (see Blasi et. al 2022).
TwopieceGaussian for a Twopiece Gaussian random field (see Bevilacqua et. al 2022).
TwopieceTukeyh for a Twopiece Tukeyh random field (see Bevilacqua et. al 2022).

random fields with for directional data

Wrapped for a wrapped Gaussian random field (see Alegria A., Bevilacqua, M., Porcu, E. (2016))

random fields with marginal counts data

Poisson for a Poisson random field (see Morales-Navarrete et. al 2021).
PoissonZIP for a zero inflated Poisson random field (see Morales-Navarrete et. al 2021).
Binomial for a Binomial random field.
BinomialNeg for a negative Binomial random field.
BinomialNegZINB for a zero inflated negative Binomial random field.

random fields using Gaussian and Clayton copula (see Bevilacqua, Alvarado and Caamaño, 2023) with the following marginal distribution:

Gaussian for Gaussian random field.
Beta2 for Beta random field.

For a given model the associated parameters are given by nuisance and correlation parameters. In order to obtain the nuisance parameters associated with a specific model use NuisParam. In order to obtain the correlation parameters associated with a given correlation model use CorrParam.

All the nuisance and covariance parameters must be specified by the user using the start and the fixed parameter. Specifically:

The option start sets the starting values parameters in the optimization process for the parameters to be estimated. The option fixed parameter allows to fix some of the parameters.

Regression parameters in the linear specification must be specified as mean,mean1,..meank (see NuisParam). In this case a matrix of covariates with suitable dimension must be specified using X. In the case of a single mean then X should not be specified and it is interpreted as a vector of ones. It is also possible to fix a vector of spatial or spatio-temporal means (estimated with other methods for instance).

Two types of binary weights can be used in the weighted composite likelihood estimation based on pairs, one based on neighboords and one based on distances.

In the first case binary weights are set to 1 or 0 depending if the pairs are neighboords of a certain order (1, 2, 3, ..) specified by the parameter (neighb). This weighting scheme is effecient for large-data sets since the computation of the 'useful' pairs in based on fast nearest neighbour search (Caamaño et al., 2023).

In the second case, binary weights are set to 1 or 0 depending if the pairs have distance lower than (maxdist). This weighting scheme is less inefficient for large data. The same arguments of neighb applies for maxtime that sets the order (1, 2, 3, ..) of temporal neighboords in spatial-temporal field.

For estimation of the variance-covariance matrix of the weighted composite likelihood estimates the option sensitivity=TRUE must be specified. Then the GeoFit object must be updated using the function GeoVarestbootstrap. This allows to estimate the Godambe Information matrix (see Bevilacqua et. al. (2012) , Bevilacqua and Gaetan (2013)). Then standard error estimation, confidence intervals, pvalues and composite likelihood information critera can be obtained.

The option varest=TRUE is deprecated.

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 random fields is bivariate, otherwise `FALSE`;
`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;
`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 of data;
`distance`	The type of spatial distance;
`fixed`	A list of the 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;
`maxdist`	The maximum spatial distance used in the weigthed composite likelihood. If no spatial distance is specified then it is 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 ued in the composite likelihhod;
`n`	The number of trials in a binominal random fields;the number of successes in a negative Binomial random fieldss;
`neighb`	The order of spatial neighborhood in the composite likelihood computation.
`ns`	The number of (different) location sites in the bivariate case;
`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;
`numtime`	The number of the temporal realisations of the random fields;
`param`	A list of the parameters' estimates;
`radius`	The radius of the sphere in the case of great circle distance;
`stderr`	The vector of standard errors for standard maximum likelihood estimation;
`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;
`type`	The type of the likelihood objects.
`winconst`	The constant used to compute the window size in the spatial sub-sampling;
`winstp`	The step used for moving the window in the spatial sub-sampling;
`winconst_t`	The constant used to compute the window size in the spatio-temporal sub-sampling;
`winstp_`	The step used for moving the window in the spatio-temporal sub-sampling;
`X`	The matrix of covariates;

Author(s)

Moreno Bevilacqua, moreno.bevilacqua89@gmail.com,https://sites.google.com/view/moreno-bevilacqua/home, Víctor Morales Oñate, victor.morales@uv.cl, https://sites.google.com/site/moralesonatevictor/, Christian", Caamaño-Carrillo, chcaaman@ubiobio.cl,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:

Alegrıa A., Caro S., Bevilacqua M., Porcu E., Clarke J. (2017) Estimating covariance functions of multivariate skew-Gaussian random fields on the sphere. Spatial Statistics 22 388-402

Alegria A., Bevilacqua, M., 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., Caamano C., Gaetan C. (2020) On modeling positive continuous data with spatio-temporal dependence. Environmetrics 31(7)

Bevilacqua M., Caamaño C., Arellano Valle R.B., Morales-Oñate V. (2020) Non-Gaussian Geostatistical Modeling using (skew) t Processes. Scandinavian Journal of Statistics.

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

Bevilacqua M., Caamaño C., Arellano-Valle R. B., Camilo Gomez C. (2022) A class of random fields with two-piece marginal distributions for modeling point-referenced data with spatial outliers. Test 10.1007/s11749-021-00797-5

Morales-Navarrete D., Bevilacqua M., Caamaño C., Castro L.M. (2022) Modelling Point Referenced Spatial Count Data: A Poisson Process Approach TJournal of the American Statistical Association To appear

Caamaño C., Bevilacqua M., López C., Morales-Oñate V. (2023) 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 To appear

Bevilacqua M., Alvarado E., Caamaño C., (2023) A flexible Clayton-like spatial copula with application to bounded support data Journal of Multivariate Analysis To appear

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 & Methods, 107, 268–280.

Bevilacqua, M., Gaetan, C. (2015) Comparing composite likelihood methods based on pairs for spatial Gaussian random fields. Statistics and Computing, 25(5), 877-892.

Sub-sampling estimation:

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.

Examples

library(GeoModels)

###############################################################
############ Examples of spatial Gaussian random fieldss ################
###############################################################


# 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 fields:
data <- GeoSim(coordx=coords,corrmodel=corrmodel, param=param,X=X)$data



################################################################
###
### Example 0. Maximum independence composite likelihood fitting of
### a Gaussian random fields (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 fields using BFGS
### 
###############################################################
# 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="BFGS",
                    type="Pairwise", start=start,fixed=fixed,X=X)
print(fit1)

################################################################
###
### Example 2. Standard Maximum likelihood fitting of
### a Gaussian random fields 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 fieldss #############
###############################################################


################################################################
###
### Example 3. Maximum pairwise likelihood fitting of a Weibull  random fields 
### 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 fields:
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 %*% c(mean,mean1))
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 4. Maximum pairwise likelihood fitting of
### a SinhAsinh-Gaussian spatial  random fields with Wendland 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 marginal pairwise likelihood:
fit1 <- GeoFit(data=data,coordx=coords,corrmodel=corrmodel, model=model,
                    neighb=3,likelihood="Marginal",type="Pairwise",
                    start=start,fixed=fixed)
print(unlist(fit1$param))


################################################################
###
### Example 5. Maximum pairwise likelihood fitting of 
### a Binomial random fields 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)) # marix 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 fields:
data <- GeoSim(coordx=coords, corrmodel=corrmodel, model="Binomial", n=10,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=10, 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 %*% c(mean,mean1,mean2))
fixed <- list(nugget=0,power2=4,mean=Xb)
start <- list(scale=0.2)

# Maximum conditional pairwise likelihood:
fit1 <- GeoFit(data=data, coordx=coords, corrmodel=corrmodel,n=10, 
          likelihood="Conditional",type="Pairwise",  neighb=3
         ,model="Binomial", start=start, fixed=fixed)
                  
print(fit1)


## estimating jointly marginal   and dependnce 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=10, X=X, 
          likelihood="Conditional",type="Pairwise",  neighb=3
         ,model="Binomial", start=start, fixed=fixed)
                  
print(fit2)


###############################################################
######### Examples of Gaussian spatio-temporal random fieldss ###########
###############################################################
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 spatial-temporal Gaussian random fields:
data <- GeoSim(coordx=coords,coordt=time,corrmodel="Matern_Matern",
              param=param)$data

################################################################
###
### Example 6. 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 fields ###########
###############################################################

################################################################
### Example 7. 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 fields:
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",
                     optimizer="BFGS" , start=start,fixed=fixed,
                     neighb=c(4,4,4))
print(fitcl)

[Package GeoModels version 2.0.4 Index]