GeoKrigWeights: Compute kriging weights (and related quantities) for Gaussian / non-Gaussian spatial, spatio-temporal or multivariate random fields

Description

Given a set of spatial locations (and possibly temporal instants) the function returns the vector (or matrix) of kriging weights, the covariance matrix used in the kriging system and, optionally, the inverse of the left-hand-side matrix.

Usage

GeoKrigWeights(
    coordx, coordy = NULL, coordz = NULL, coordt = NULL,
    coordx_dyn = NULL, corrmodel, distance = "Eucl",
    grid = FALSE, loc, method = "cholesky", model = "Gaussian",
    n = 1, nloc = NULL, param, anisopars = NULL, radius = 1,
    sparse = FALSE, time = NULL, which = 1, copula = NULL,
    X = NULL, Xloc = NULL, Mloc=NULL)

Value

An object of class KgWeights, a list containing:

weights: Numeric vector (or matrix) of kriging weights.
covmat: Covariance matrix $\Sigma$ used in the kriging system.
invcov: Inverse of $\Sigma$ (only if sparse = FALSE).
beta: Estimated regression coefficients when covariates are supplied.
type: Character string: "simple" or "universal".
model: Input model.
corrmodel: Input corrmodel.
spacetime: Logical: TRUE for spatio-temporal case, FALSE otherwise.
sparse: Logical: was a sparse algorithm used?
loc: Copy of the loc argument.
time: Copy of the time argument (if any).

Arguments

coordx: Numeric ($d\times 2$) or ($d\times 3$) matrix of spatial coordinates. Coordinates on a sphere are accepted (lon/lat in decimal degrees) when distance = "Sphere".
coordy: Optional numeric vector giving an additional spatial coordinate dimension. Ignored if coordx is already a matrix.
coordz: Optional numeric vector giving a third spatial coordinate dimension.
coordt: Optional numeric vector of temporal coordinates (length $t$). If missing, a purely spatial random field is assumed.
coordx_dyn: List of $m$ matrices ($d_t\times 2$) containing time-varying spatial coordinates. See GeoKrig.
corrmodel: Character string naming a valid correlation model. See GeoCovmatrix for admissible choices.
distance: Character string specifying the spatial distance. Default is "Eucl" (Euclidean). See GeoFit.
grid: Logical. If TRUE, coordinates are interpreted as defining a regular grid (see GeoKrig).
loc: Numeric ($n\times 2$) matrix of locations for which the kriging weights are required.
method: Character string indicating the matrix factorisation used to solve the kriging system: "cholesky" (default) or "svd".
model: Character string specifying the random field type (e.g.\ "Gaussian", "SkewGaussian", "Gamma", …). See GeoFit.
n: Integer. Number of trials for Binomial random fields (default 1).
nloc: Integer. Number of trials for the prediction locations in Binomial random fields (default 1).
param: Named list of covariance and mean parameters. See CorrParam and GeoCovmatrix.
anisopars: List with components angle and ratio defining geometric anisotropy (optional).
radius: Positive numeric value: sphere radius when coordinates are lon/lat (default 1).
sparse: Logical. If TRUE, sparse‐matrix algorithms (package spam) are employed. Only effective with compactly supported covariance functions.
time: Numeric vector of length $m$ giving the temporal instants for which weights are required. Ignored if coordt is missing.
which: Integer (1 or 2) selecting the variable whose weights are returned in the bivariate case.
copula: Character string naming a copula when a non-Gaussian dependence structure is used ("Clayton" or "Gaussian").
X: Numeric matrix of spatio-temporal covariates at data locations.
Xloc: Numeric matrix of spatio-temporal covariates at prediction locations.
Mloc: Numeric; Vector of spatio(temporal) estimated means associated to predicted locations.

Author

Moreno Bevilacqua, moreno.bevilacqua@uai.cl, Víctor Morales-Oñate, victor.morales@uv.cl, Christian Caamaño-Carrillo, chcaaman@ubiobio.cl

Details

The function builds the kriging system $$\Sigma \mathbf{w} = \boldsymbol{\sigma}_0$$ where $\Sigma$ is the covariance matrix between observed locations and $\boldsymbol{\sigma}_0$ the vector of covariances between observed and prediction locations. The solution $\mathbf{w}$ is returned together with $\Sigma$ and, optionally, $\Sigma^{-1}$. Universal kriging with covariates is supported; in that case the generalised least squares estimate of the mean parameters is appended. No actual prediction is carried out; for full kriging use GeoKrig.

References

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

Examples

Run this code

library(GeoModels)
################################################################

################################################################
###
### Example 1. Spatial kriging weights for
### Gaussian random fields with Gen wendland correlation.
###
################################################################

model="Gaussian"
set.seed(79)
x = runif(300, 0, 1)
y = runif(300, 0, 1)
coords=cbind(x,y)

corrmodel = "GenWend"
mean=0; sill=5; nugget=0
scale=0.2;smooth=0;power2=4

param=list(mean=mean,sill=sill,nugget=nugget,scale=scale,smooth=smooth,power2=power2)

# Simulation of the spatial Gaussian random field:
data = GeoSim(coordx=coords, corrmodel=corrmodel,model=model,
              param=param)$data

xx=seq(0,1,0.25)
loc_to_pred=as.matrix(expand.grid(xx,xx))

W=GeoKrigWeights(,coordx=coords,loc=loc_to_pred,corrmodel=corrmodel,
       model=model,param=param)

dim(W$weights)  ### kriging weights

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