Spatial (bivariate) and spatio temporal optimal linear prediction for Gaussian and non Gaussian random fields.
For a given set of spatial location sites (and temporal instants), the function computes optimal linear prediction and associated mean square error for the Gaussian and non Gaussian case. UTF-8
GeoKrig(estobj=NULL,data, coordx, coordy=NULL, coordz=NULL, coordt=NULL, coordx_dyn=NULL, corrmodel,distance="Eucl", grid=FALSE, loc, maxdist=NULL, maxtime=NULL, method="cholesky", model="Gaussian", n=1,nloc=NULL,mse=FALSE, lin_opt=TRUE, param, anisopars=NULL,radius=6371, sparse=FALSE, taper=NULL,tapsep=NULL, time=NULL, type="Standard",type_mse=NULL, type_krig="Simple",weigthed=TRUE,which=1, copula=NULL, X=NULL,Xloc=NULL,Mloc=NULL,spobj=NULL,spdata=NULL)
estobj: An object of class Geofit that includes information about data, model and estimates.
data: A -dimensional vector (a single spatial realisation) or a ()-matrix (a single spatial realisation on regular grid) or a ()-matrix (a single spatial-temporal realisation) or an ()-array (a single spatial-temporal realisation on regular grid) giving the data used for prediction.
coordx: A numeric ()-matrix or ()-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 1-dimension of temporal coordinates used for prediction; the default is NULL
then a spatial random field is expected.
coordx_dyn: A list of numeric ()-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 the Section Details .
distance: String; the name of the spatial distance. The default is Eucl, the euclidean distance. See the Section Details of GeoFit.
grid: Logical; if FALSE (the default) the data used for prediction are interpreted as spatial or spatial-temporal realisations on a set of non-equispaced spatial sites (irregular grid).
lin_opt: Logical;If TRUE (default) then optimal (pairwise) linear kriging is computed. Otherwise optimal (pairwise) kriging is computed in the mean square sense.
loc: A numeric ()-matrix (where n is the number of spatial sites) giving 2-dimensions of spatial coordinates to be predicted.
maxdist: Numeric; an optional positive value indicating the maximum spatial compact support in the case of covariance tapering kriging.
maxtime: Numeric; an optional positive value indicating the maximum temporal compact support in the case of covasriance tapering kriging.
method: String; the type of matrix decomposition used in the simulation. Default is cholesky. The other possible choices is svd.
n: Numeric; the number of trials in a binomial random fields. Default is .
nloc: Numeric; the number of trials of the locations sites to be predicted in a binomial random fields type II. Default is .
mse: Logical; if TRUE (the default) MSE of the kriging predictor is computed
model: String; the type of RF and therefore the densities associated to the likelihood objects. Gaussian is the default, see the Section Details .
param: A list of parameter values required for the correlation model.See the Section Details .
anisopars: A list of two elements: "angle" and "ratio" i.e. the anisotropy angle and the anisotropy ratio, respectively.
radius: Numeric: the radius of the sphere if coordinates are passed in lon/lat format;
sparse: Logical; if TRUE kriging is computed with sparse matrices algorithms using spam package. Default is FALSE. It should be used with compactly supported covariances.
taper: String; the name of the taper correlation function, see the Section Details .
tapsep: Numeric; an optional value indicating the separabe parameter in the space time quasi taper (see Details ).
time: A numeric () vector (where m is the number of temporal instants) giving the temporal instants to be predicted; the default is NULL
then only spatial prediction is performed.
type: String; if Standard then standard kriging is performed;if Tapering
then kriging with covariance tapering is performed;if Pairwise then pairwise kriging is performed
type_mse: String; if Theoretical then theoretical MSE pairwise kriging is computed. If SubSamp then an estimation based on subsampling is computed.
type_krig: String; the type of kriging. If Simple (the default) then simple kriging is performed. If Optim then optimal kriging is performed for some non-Gaussian RFs
weigthed: Logical; if TRUE then decreasing weigths coming from a compactly supported correlation function with compact support maxdist (maxtime)are used in the pairwise kriging.
which: Numeric; In the case of bivariate (tapered) cokriging it indicates which variable to predict. It can be 1 or 2
copula: String; the type of copula. It can be "Clayton" or "Gaussian"
X: Numeric; Matrix of spatio(temporal)covariates in the linear mean specification.
Xloc: Numeric; Matrix of spatio(temporal)covariates in the linear mean specification associated to predicted locations.
Mloc: Numeric; Vector of spatio(temporal) estimated means associated to predicted locations.
spobj: An object of class sp or spacetime
spdata: Character:The name of data in the sp or spacetime object
Best linear unbiased predictor and associated mean square error is computed for Gaussian and some non Gaussian cases. Specifically, for a spatial or spatio-temporal or spatial bivariate dataset, given a set of spatial locations and temporal istants and a correlation model corrmodel with some fixed parameters and given the type of RF (model) the function computes simple kriging, for the specified spatial locations loc and temporal instants time, providing also the respective mean square error. For the choice of the spatial or spatio temporal correlation model see details in GeoCovmatrix function. The list param specifies mean and covariance parameters, see CorrParam and GeoCovmatrix for details. The type_krig parameter indicates the type of kriging. In the case of simple kriging, the known mean can be specified by the parameter mean in the list param (See examples).
Returns an object of class Kg. An object of class Kg is a list containing at most the following components: - bivariate: TRUE if spatial bivariate cokriging is performed, otherwise FALSE;
coordx: A -dimensional vector of spatial coordinates used for prediction;
coordy: A -dimensional vector of spatial coordinates used for prediction;
coordz: A -dimensional vector of spatial coordinates used for prediction;
coordt: A -dimensional vector of temporal coordinates used for prediction;
corrmodel: String: the correlation model;
covmatrix: The covariance matrix if type is Standard. An object of class spam if type is Tapering
data: The vector or matrix or array of data used for prediction
distance: String: the type of spatial distance;
grid: TRUE if the spatial data used for prediction are observed in a regular grid, otherwise FALSE;
loc: A ()-matrix of spatial locations to be predicted.
n: The number of trial for Binomial RFs
nozero: In the case of tapered simple kriging the percentage of non zero values in the covariance matrix. Otherwise is NULL.
numcoord: Numeric:he number of spatial coordinates used for prediction;
numloc: Numeric: the number of spatial coordinates to be predicted;
numtime: Numeric: the number of the temporal instants used for prediction;
numt: Numeric: the number of the temporal instants to be predicted;
model: The type of RF, see GeoFit.
param: Numeric: The covariance parameters;
pred: A ()-matrix of spatio or spatio temporal kriging prediction;
radius: Numeric: the radius of the sphere if coordinates are pssed in lon/lat format;
spacetime: TRUE if spatio-temporal kriging and FALSE if spatial kriging;
tapmod: String: the taper model if type is Tapering. Otherwise is NULL.
time: A -dimensional vector of temporal coordinates to be predicted;
type: String: the type of kriging (Standard or Tapering).
type_krig: String: the type of kriging.
mse: A ()-matrix of spatio or spatio temporal mean square error kriging prediction;
Gaetan, C. and Guyon, X. (2010) Spatial Statistics and Modelling. Spring Verlang, New York.
GeoCovmatrix
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
library(GeoModels) ################################################################ ########### Examples of spatial kriging ############ ################################################################ ################################################################ ### ### Example 1. Spatial kriging of a ### 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) # Set the exponential cov parameters: 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, param=param)$data ## estimation with pairwise likelihood fixed=list(nugget=nugget,smooth=0,power2=power2) start=list(mean=0,scale=scale,sill=1) I=Inf lower=list(mean=-I,scale=0,sill=0) upper=list(mean= I,scale=I,sill=I) # Maximum pairwise likelihood fitting : fit = GeoFit(data, coordx=coords, corrmodel=corrmodel,model=model, likelihood='Marginal', type='Pairwise',neighb=3, optimizer="nlminb", lower=lower,upper=upper, start=start,fixed=fixed) # locations to predict xx=seq(0,1,0.03) loc_to_pred=as.matrix(expand.grid(xx,xx)) ## first option #param=append(fit$param,fit$fixed) #pr=GeoKrig(loc=loc_to_pred,coordx=coords,corrmodel=corrmodel, # model=model,param=param,data=data,mse=TRUE) ## second option using object GeoFit pr=GeoKrig(fit,loc=loc_to_pred,mse=TRUE) colour = rainbow(100) opar=par(no.readonly = TRUE) par(mfrow=c(1,3)) quilt.plot(coords,data,col=colour) # simple kriging map prediction image.plot(xx, xx, matrix(pr$pred,ncol=length(xx)),col=colour, xlab="",ylab="", main=" Kriging ") # simple kriging MSE map prediction variance image.plot(xx, xx, matrix(pr$mse,ncol=length(xx)),col=colour, xlab="",ylab="",main="Std error") par(opar) ################################################################ ### ### Example 2. Spatial kriging of a Skew ### Gaussian random fields with Matern correlation. ### ################################################################ model="SkewGaussian" set.seed(79) x = runif(300, 0, 1) y = runif(300, 0, 1) coords=cbind(x,y) # Set the exponential cov parameters: corrmodel = "Matern" mean=0 sill=2 nugget=0 scale=0.1 smooth=0.5 skew=3 param=list(mean=mean,sill=sill,nugget=nugget,scale=scale,smooth=smooth,skew=skew) # Simulation of the spatial skew Gaussian random field: data = GeoSim(coordx=coords, corrmodel=corrmodel,model=model, param=param)$data fixed=list(nugget=nugget,smooth=smooth) start=list(mean=0,scale=scale,sill=1,skew=skew) I=Inf lower=list(mean=-I,scale=0,sill=0,skew=-I) upper=list(mean= I,scale=I,sill=I,skew=I) # Maximum pairwise likelihood fitting : fit = GeoFit2(data, coordx=coords, corrmodel=corrmodel,model=model, likelihood='Marginal', type='Pairwise',neighb=3, optimizer="nlminb", lower=lower,upper=upper, start=start,fixed=fixed) # locations to predict xx=seq(0,1,0.03) loc_to_pred=as.matrix(expand.grid(xx,xx)) ## optimal linear kriging pr=GeoKrig(fit,loc=loc_to_pred,mse=TRUE) colour = rainbow(100) opar=par(no.readonly = TRUE) par(mfrow=c(1,3)) quilt.plot(coords,data,col=colour) # simple kriging map prediction image.plot(xx, xx, matrix(pr$pred,ncol=length(xx)),col=colour, xlab="",ylab="", main=" Kriging ") # simple kriging MSE map prediction variance image.plot(xx, xx, matrix(pr$mse,ncol=length(xx)),col=colour, xlab="",ylab="",main="Std error") par(opar) ################################################################ ### ### Example 3. Spatial kriging of a ### Gamma random field with mean spatial regression ### ############################################################### set.seed(312) model="Gamma" corrmodel = "GenWend" # Define the spatial-coordinates of the points: NN=300 coords=cbind(runif(NN),runif(NN)) ## matrix covariates a0=rep(1,NN) a1=runif(NN,0,1) X=cbind(a0,a1) ##Set model parameters shape=2 ## regression parameters mean = 1;mean1= -0.2 # correlation parameters nugget = 0;power2=4 scale = 0.3;smooth=0 ## simulation param=list(shape=shape,nugget=nugget,mean=mean,mean1=mean1, scale=scale,power2=power2,smooth=smooth) data = GeoSim(coordx=coords,corrmodel=corrmodel, param=param, model=model,X=X)$data #####starting and fixed parameters fixed=list(nugget=nugget,power2=power2,smooth=smooth) start=list(mean=mean,mean1=mean1, scale=scale,shape=shape) ## estimation with pairwise likelihood fit2 = GeoFit(data=data,coordx=coords,corrmodel=corrmodel,X=X, neighb=3,likelihood="Conditional",type="Pairwise", start=start,fixed=fixed, model = model) # locations to predict with associated covariates xx=seq(0,1,0.03) loc_to_pred=as.matrix(expand.grid(xx,xx)) NP=nrow(loc_to_pred) a0=rep(1,NP) a1=runif(NP,0,1) Xloc=cbind(a0,a1) #optimal linear kriging pr=GeoKrig(fit2,loc=loc_to_pred,Xloc=Xloc,sparse=TRUE,mse=TRUE) ## map opar=par(no.readonly = TRUE) par(mfrow=c(1,3)) quilt.plot(coords,data,main="Data") map=matrix(pr$pred,ncol=length(xx)) mapmse=matrix(pr$mse,ncol=length(xx)) image.plot(xx, xx, map, xlab="",ylab="",main="Kriging ") image.plot(xx, xx, mapmse, xlab="",ylab="",main="MSE") par(opar) ################################################################ ########### Examples of spatio temporal kriging ############ ################################################################ ################################################################ ### ### Example 4. Spatio temporal simple kriging of n locations ### sites and m temporal instants for a Gaussian random fields ### with estimated double Wendland correlation. ### ############################################################### model="Gaussian" # Define the spatial-coordinates of the points: x = runif(300, 0, 1) y = runif(300, 0, 1) coords=cbind(x,y) times=1:4 # Define model correlation modek and associated parameters corrmodel="Wend0_Wend0" param=list(nugget=0,mean=0,power2_s=4,power2_t=4, scale_s=0.2,scale_t=2,sill=1) # Simulation of the space time Gaussian random field: set.seed(31) data=GeoSim(coordx=coords,coordt=times,corrmodel=corrmodel,sparse=TRUE, param=param)$data # Maximum pairwise likelihood fitting of the space time random field: start = list(scale_s=0.15,scale_t=2,sill=1,mean=0) fixed = list(nugget=0,power2_s=4,power2_t=4) fit = GeoFit(data, coordx=coords, coordt=times, model=model, corrmodel=corrmodel, likelihood='Conditional', type='Pairwise',start=start,fixed=fixed, neighb=3,maxtime=1) # locations to predict xx=seq(0,1,0.04) loc_to_pred=as.matrix(expand.grid(xx,xx)) # Define the times to predict times_to_pred=2 pr=GeoKrig(fit,loc=loc_to_pred,time=times_to_pred,sparse=TRUE,mse=TRUE) opar=par(no.readonly = TRUE) par(mfrow=c(1,3)) zlim=c(-2.5,2.5) colour = rainbow(100) quilt.plot(coords,data[2,] ,col=colour,main = paste(" data at Time 2")) image.plot(xx, xx, matrix(pr$pred,ncol=length(xx)),col=colour, main = paste(" Kriging at Time 2"),ylab="") image.plot(xx, xx, matrix(pr$mse,ncol=length(xx)),col=colour, main = paste("Std err Time at time 2"),ylab="") par(opar) ################################################################ ### ### Example r. Spatial bivariate simple cokriging of n locations ### sites for a bivariate Gaussian random fields ### with estimated Matern correlation. ### ############################################################### #set.seed(6) #NN=1500 # number of spatial locations #x = runif(NN, 0, 1); #y = runif(NN, 0, 1) #coords=cbind(x,y) ## setting parameters #mean_1 = 2; mean_2= -1 #nugget_1 =0;nugget_2=0 #sill_1 =0.5; sill_2 =1; ### correlation parameters #CorrParam("Bi_Matern") #scale_1=0.2/3; scale_2=0.15/3; scale_12=0.5*(scale_2+scale_1) #smooth_1=smooth_2=smooth_12=0.5 #pcol = -0.4 #param= list(nugget_1=nugget_1,nugget_2=nugget_2, # sill_1=sill_1,sill_2=sill_2, # mean_1=mean_1,mean_2=mean_2, # smooth_1=smooth_1, smooth_2=smooth_2,smooth_12=smooth_12, # scale_1=scale_1, scale_2=scale_2,scale_12=scale_12, # pcol=pcol) ## simulation #data = GeoSim(coordx=coords, corrmodel="Bi_Matern",model=model,param=param)$data #fixed=list(mean_1=mean_1,mean_2=mean_2, nugget_1=nugget_1,nugget_2=nugget_2, # smooth_1=smooth_1, smooth_2=smooth_2,smooth_12=smooth_12) #start=list( sill_1=sill_1,sill_2=sill_2, # scale_1=scale_1,scale_2=scale_2,scale_12=scale_12, pcol=pcol) ## estimation with maximum likelihood #fit = GeoFit(data=data,coordx=coords, corrmodel="Bi_Matern", #likelihood="Marginal",type="Pairwise",optimizer="BFGS",neighb=5, #start=start,fixed=fixed) ###### co-kriging for the fist component ############## #xx=seq(0,1,0.022) #loc_to_pred=as.matrix(expand.grid(xx,xx)) #pr1 = GeoKrig(fit,which=1,mse=TRUE,loc=loc_to_pred) #opar=par(no.readonly = TRUE) #par(mfrow=c(1,2)) #zlim=c(-2.5,2.5) #colour = rainbow(100) #quilt.plot(coords,data[1,] ,col=colour,main = paste(" Fist component")) #quilt.plot(loc_to_pred,pr1$pred,col=colour, # main = paste(" Kriging first component"),ylab="") #par(opar)