GeoWls: WLS of Gaussian random fields

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

WLS of Gaussian random fields

Description

the function returns the parameter estimates of a Gaussian random field obtained by the weighted least squares estimator.

Usage

GeoWLS(data, coordx, coordy=NULL,coordz=NULL, coordt=NULL, coordx_dyn=NULL, corrmodel, 
 distance="Eucl", fixed=NULL, grid=FALSE, maxdist=NULL,neighb=NULL,
 maxtime=NULL, optimizer='Nelder-Mead',
 numbins=NULL, radius=1, start=NULL, weighted=FALSE,optimization=TRUE,
 numbins_t=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 spatio-temporal realisation) or an (d \times d \times t \times n )-array (a single spatio-temporal realisation on regular grid). See GeoFit for details.
coordx	A numeric (d \times 2)-matrix or (d \times 3)-matrix Coordinates on a sphere for a fixed radius radius are passed in lon/lat format expressed in decimal degrees.
coordy	A numeric vector giving 1-dimension of spatial coordinates; Optional argument, the default is NULL.
coordz	A numeric vector giving 1-dimension of spatial coordinates; Optional argument, the default is NULL.
coordt	A numeric vector giving one dimension of temporal coordinates. Optional argument; the default is NULL, in which case a spatial random field is expected. Temporal coordinates may be irregularly spaced; temporal lags are computed from the supplied coordinate values.
coordx_dyn	A list of m numeric (d_t \times 2)-matrices containing dynamical (in time) spatial coordinates. Optional argument, the default is NULL
corrmodel	String; the name of a correlation model, for the description (see GeoFit).
distance	String; the name of the spatial distance. The default is Eucl, the Euclidean distance. See GeoFit for details. of GeoFit.
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 \times d)-matrix, instead if TRUE then (d \times d \times n)-matrix is considered.
maxdist	A numeric value denoting the maximum distance, see GeoFit for details.
neighb	Numeric; an optional positive integer indicating the order of neighborhood. See GeoFit for details
maxtime	Numeric; an optional positive maximum temporal lag, expressed in the same units as coordt. See GeoFit for details.
optimizer	String; the optimization algorithm (see optim for details). 'Nelder-Mead' is the default.
numbins	Historical GeoModels argument giving the number of spatial bin boundaries; numbins - 1 empirical spatial classes are used. See Details.
numbins_t	Optional positive integer giving the number of temporal classes for irregularly spaced coordt. If NULL, numbins - 1 is used. Regularly spaced times use their directly attainable temporal lags.
radius	Numeric; a value indicating the radius of the sphere when using the great circle distance. Default value is 1.
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 GeoFit).
weighted	Logical; if TRUE then the weighted least square estimator is considered. If FALSE (the default) then the classic least square is used.
optimization	Logical; if TRUE then the weighted least square minimization is performed. Otherwise the weighted least square function is evaluated at the starting value.

Details

GeoWLS is defined for Gaussian random fields and therefore does not expose a model argument. Its native least-squares kernels compare the empirical variogram with the Gaussian variogram parameterized by nugget, sill, and the selected correlation model.

The historical numbins parameter gives the number of spatial bin boundaries, so numbins - 1 spatial classes are used. For spatio-temporal data, regular temporal coordinates are handled from their common spacing without constructing all pairwise temporal differences; irregular temporal coordinates use numbins_t temporal classes.

Empty empirical cells are retained in the rectangular spatial-temporal lag grid and are skipped by the least-squares objective through their zero pair count. This preserves the correspondence between empirical cells and their spatial/temporal lags.

The maxdist parameter indicates the positive finite maximum distance below which pairs are considered in the (weighted) least squares.

Value

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;
maxdist	The maximum spatial distance used for the calculation of the variogram used in least square estimation. If no spatial distance is specified then it is NULL;
maxtime	The maximum temporal distance used for the calculation of the variogram used in least square estimation. If no temporal distance is specified then it is NULL;
numbins_t	The requested number of temporal classes for irregular time coordinates, or NULL;
message	Extra message passed from the numerical routines;
numcoord	The number of spatial coordinates;
numtime	The number the temporal realisations of the random field;
param	The vector of parameters' estimates;
variograms	The empirical spatial variogram;
variogramt	The empirical temporal variogram;
variogramst	The empirical spatio-temporal variogram;
weighted	A logical value indicating if its the weighted method;
wls	The value of the least squares at the minimum.

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

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

Gaetan, C. and Guyon, X. (2010) Spatial Statistics and Modelling. Springer-Verlag, New York.

See Also

GeoFit, optim

Examples

library(GeoModels)


# Set the coordinates of the sites:

set.seed(211)
x <- runif(200, 0, 1)
set.seed(98)
y <- runif(200, 0, 1)
coords <- cbind(x,y)

################################################################
###
### Example 1. Least square fitting of a Gaussian random field
### with exponential correlation.
###
###############################################################

# Set the model's parameters:
corrmodel <- "Exponential"
mean <- 0
sill <- 1
nugget <- 0
scale <- 0.15/3
param <- list(mean=0,sill=sill, nugget=nugget, scale=scale)
# Simulation of the Gaussian random field:
set.seed(2)
data <- GeoSim(coordx=coords, corrmodel=corrmodel, param=param)$data

fixed=list(nugget=0,mean=mean)
start=list(scale=scale,sill=sill)
# Least square fitting of the random field:
fit <- GeoWLS(data=data,coordx=coords, corrmodel=corrmodel,
 fixed=fixed,start=start,maxdist=0.5)

# Results:
print(fit)

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