WLeastSquare: WLS of Gaussian and Max-Stable Random Fields
In CompRandFld: Composite-Likelihood Based Analysis of Random Fields

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

the function returns the parameters' estimates and the estimates' variances of a random field obtained by the weigthed least squares estimator.

WLeastSquare(data, coordx, coordy=NULL, coordt=NULL, corrmodel,
             distance="Eucl", fixed=NULL, grid=FALSE, maxdist=NULL,
             maxtime=NULL,  model='Gaussian', optimizer='Nelder-Mead',
             numbins=NULL, replicates=1, start=NULL, weighted=FALSE)

`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). See `FitComposite` for details.
`coordx`	A numeric (d x 2)-matrix (where `d` is the number of spatial sites) giving 2-dimensions of spatial coordinates or a numeric d-dimensional vector giving 1-dimension of spatial coordinates.
`coordy`	A numeric vector giving 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 giving 1-dimension of temporal coordinates. 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 `FitComposite`).
`distance`	String; the name of the spatial distance. The default is `Eucl`, the euclidean distance. See the Section Details of `FitComposite`.
`fixed`	A named list giving the values of the parameters that will be considered as known values. The listed parameters for a given correlation function will be not estimated, i.e. if `list(nugget=0)` the nugget effect is ignored.
`grid`	Logical; if `FALSE` (the default) the data are interpreted as a vector or a (n x d)-matrix, instead if `TRUE` then (d x d x n)-matrix is considered.
`maxdist`	A numeric value denoting the maximum distance, see Details and `FitComposite`.
`maxtime`	Numeric; an optional positive value indicating the maximum temporal lag considered in the composite-likelihood computation (see `FitComposite`.
`model`	String; the type of random field. `Gaussian` is the default, see `FitComposite` for the different types.
`optimizer`	String; the optimization algorithm (see `optim` for details). 'Nelder-Mead' is the default.
`numbins`	A numeric value denoting the numbers of bins, see the Section Details
`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`	A named list with the initial values of the parameters that are used by the numerical routines in maximization procedure. `NULL` is the default (see `FitComposite`).
`weighted`	Logical; if `TRUE` then the weighted least square estimator is considered. If `FALSE` (the default) then the classic least square is used.

The numbins parameter indicates the number of adjacent intervals to consider in order to grouped distances with which to compute the (weighted) lest squares.

The maxdist parameter indicates the maximum distance below which the shorter distances will be considered in the calculation of the (weigthed) least squares.

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

`bins`	Adjacent intervals of grouped distances;
`bint`	Adjacent intervals of grouped temporal separations
`centers`	The centers of the bins;
`coordx`	The vector or matrix of spatial coordinates;
`coordy`	The vector of spatial coordinates;
`coordt`	The vector of temporal coordinates;
`convergence`	A string that denotes if convergence is reached;
`corrmodel`	The correlation model;
`data`	The vector or matrix of data;
`distance`	The type of spatial distance;
`fixed`	The vector of fixed parameters;
`iterations`	The number of iteration used by the numerical routine;
`message`	Extra message passed from the numerical routines;
`model`	The type of random fields;
`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;
`trange`	The minimum and maximum temporal separations;
`variograms`	The empirical spatial variogram;
`variogramt`	The empirical temporal variogram;
`variogramst`	The empirical spatial-temporal variogram;
`weighted`	A logical value indicating if its the weighted method;
`wls`	The value of the least squares at the minimum.

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.

Barry, J. T., Crowder, M. J. and Diggle, P. J. (1997) Parametric estimation of the variogram. Tech. Report, Dept Maths & Stats, Lancaster University.

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

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

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

FitComposite, optim

library(CompRandFld)
library(RandomFields)
set.seed(2111)

# Set the coordinates of the sites:
x <- runif(100, 0, 10)
y <- runif(100, 0, 10)


################################################################
###
### Example 1. Least square fitting of a Gaussian random field
### with exponential correlation.
### One spatial replication is simulated.
### Unweighted version (all weights equals to 1).
###
###############################################################

# Set the model's parameters:
corrmodel <- "exponential"
mean <- 0
sill <- 1
nugget <- 0
scale <- 2

# Simulation of the Gaussian random field:
data <- RFsim(x, y, corrmodel=corrmodel, param=list(mean=mean,
              sill=sill, nugget=nugget, scale=scale))$data

fix<-list(nugget=0)
ini<-list(scale=scale,sill=sill)
# Least square fitting of the random field:
fit <- WLeastSquare(data, x, y, corrmodel=corrmodel,fixed=fix,start=ini)

# Results:
print(fit)


################################################################
###
### Example 2. Least square fitting of a max-stable random field
### (Extremal Gaussian model) with exponential correlation
### n iid spatial replications.
### Unweighted version (all weights equals to 1).
###
###############################################################


# Simulation of the max-stable random field:
data <- RFsim(x, y, corrmodel=corrmodel, model="ExtGauss",
              param=list(mean=mean, sill=sill, nugget=nugget,
              scale=scale), replicates=40)$data

# Least square fitting of the random field:
fit <- WLeastSquare(data, x, y, corrmodel=corrmodel, model="ExtGauss",
                    replicates=40)

# Results:
print(fit)

################################################################
###
### Example 3. Least square fitting of a spatio-temporal
### Gaussian random field with double exponential correlation.
### One replication is simulated.
### Weighted version (all weights equals to 1).
###
###############################################################

# Define the temporal sequence:
#time <- seq(1, 25, 1)

# Simulation of the Gaussian random field:
#data <- RFsim(x, y, time, corrmodel="exp_exp", param=list(mean=mean,
#             scale_s=scale,scale_t=1,sill=sill,nugget=nugget))$data

#fix<-list(nugget=nugget)
#ini<-list(scale_s=scale,scale_t=1,sill=1)
# Weighted least square estimation:
#fit <- WLeastSquare(data, x, y, time, corrmodel="exp_exp", maxdist=5,
#                    maxtime=5,fixed=fix,start=ini)

# Results
#print(fit)