GeoWLS {GeoModels}

WLS of Random Fields

Description

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

Usage

GeoWLS(data, coordx, coordy=NULL, coordt=NULL, coordx_dyn=NULL, corrmodel, 
             distance="Eucl", fixed=NULL, grid=FALSE, maxdist=NULL,neighb=NULL,
             maxtime=NULL,  model='Gaussian', optimizer='Nelder-Mead',
             numbins=NULL, radius=6371, start=NULL, weighted=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 an (d \times d \times n)-array (n iid spatial realisations 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). See GeoFit for details.

coordx

A numeric (d \times 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 \times 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.

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 the Section 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 Details in GeoFit.

neighb

Numeric; an optional positive integer indicating the order of neighborhood. See Details and GeoFit

maxtime

Numeric; an optional positive value indicating the maximum temporal lag considered.See Details and GeoFit.

model

String; the type of random field. Gaussian is the default, see GeoFit 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

radius

Numeric; a value indicating the radius of the sphere when using the great circle distance. Default value is the radius of the earth in Km (i.e. 6371)

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.

Details

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.

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;

message

Extra message passed from the numerical routines;

model

The type of random fields;

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 spatial-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. Spring Verlang, 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)