GeoSim {GeoModels}

Simulation of Gaussian and non Gaussian Random Fields.

Description

Simulation of Gaussian and some non Gaussian spatial, spatio-temporal and spatial bivariate random fields. The function return a realization of a random Field for a given covariance model and covariance parameters.Simulation is based on Cholesky decomposition.

Usage

GeoSim(coordx, coordy=NULL, coordt=NULL, coordx_dyn=NULL, corrmodel, distance="Eucl",
      GPU=NULL, grid=FALSE, local=c(1,1),method="cholesky", model='Gaussian', n=1, param,
      anisopars=NULL,radius=6371, sparse=FALSE,X=NULL,spobj=NULL,nrep=1)

Arguments

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. 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; 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 RF 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 the Section Details.

distance

String; the name of the spatial distance. The default is Eucl, the euclidean distance. See the Section Details of GeoFit.

GPU

Numeric; if NULL (the default) no GPU computation is performed.

grid

Logical; if FALSE (the default) the data are interpreted as spatial or spatial-temporal realisations on a set of non-equispaced spatial sites (irregular grid).

local

Numeric; number of local work-items of the GPU

method

String; the type of matrix decomposition used in the simulation. Default is cholesky. The other possible choices is svd.

model

String; the type of RF and therefore the densities associated to the likelihood objects. Gaussian is the default, see the Section Details.

n

Numeric; the number of trials for binomial RFs. The number of successes in the negative Binomial RFs. Default is 1.

param

A list of parameter values required in the simulation procedure of RFs, see Examples.

anisopars

A list of two elements "angle" and "ratio" i.e. the anisotropy angle and the anisotropy ratio, respectively.

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)

sparse

Logical; if TRUE then cholesky decomposition is performed using sparse matrices algorithms (spam packake). It should be used with compactly supported covariance models.FALSE is the default.

X

Numeric; Matrix of space-time covariates.

spobj

An object of class sp or spacetime

nrep

Numeric; Numbers of indipendent replicates.

Value

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

bivariate

Logical:TRUE if the Gaussian RF is bivariate, otherwise FALSE;

coordx

A d-dimensional vector of spatial coordinates;

coordy

A d-dimensional vector of spatial coordinates;

coordt

A t-dimensional vector of temporal coordinates;

coordx_dyn

A list of dynamical (in time) spatial coordinates;

corrmodel

The correlation model; see GeoCovmatrix.

data

The vector or matrix or array of data, see GeoFit;

distance

The type of spatial distance;

method

The method of simulation

model

The type of RF, see GeoFit.

n

The number of trial for Binomial RFs;the number of successes in a negative Binomial RFs;

numcoord

The number of spatial coordinates;

numtime

The number the temporal realisations of the RF;

param

The vector of parameters' estimates;

radius

The radius of the sphere if coordinates are passed in lon/lat format;

spacetime

TRUE if spatio-temporal and FALSE if spatial RF;

nrep

The number of indipendent replicates;

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

Examples

library(GeoModels)
library(mapproj)


################################################################
###
### Example 1. Simulation of a spatial Gaussian RF 
### with Matern and Generalized Wendland correlations
###############################################################

# Define the spatial-coordinates of the points:
x <- runif(500);y <- runif(500)
coords=cbind(x,y)
set.seed(261)
# Simulation of a spatial Gaussian RF with Matern correlation function
data1 <- GeoSim(coordx=coords, corrmodel="Matern", param=list(smooth=0.5,
             mean=0,sill=1,scale=0.4/3,nugget=0))$data

set.seed(261)
data2 <- GeoSim(coordx=coords,  corrmodel="GenWend", param=list(smooth=0,
              power2=4,mean=0,sill=1,scale=0.4,nugget=0))$data
opar=par(no.readonly = TRUE)
par(mfrow=c(1,2))
quilt.plot(coords,data1,main="Matern",xlab="",ylab="")
quilt.plot(coords,data2,main="Wendland",xlab="",ylab="")   
par(opar)
             

################################################################
###
### Example 2. Simulation of a spatial geometric RF 
### with  underlying Wend0 correlation
###
################################################################

# Define the spatial-coordinates of the points:
x <- runif(800);y <- runif(800)
coords <- cbind(x,y)
set.seed(251)
# Simulation of a spatial Binomial RF:
sim <- GeoSim(coordx=coords, corrmodel="Wend0",
             model="BinomialNeg",n=1,sparse=TRUE,
             param=list(nugget=0,mean=0,scale=.2,power2=4))

quilt.plot(coords,sim$data,nlevel=max(sim$data),col=terrain.colors(max(sim$data+1))) 


################################################################
###
### Example 3. Simulation of a spatial Weibull  RF
### with  underlying Matern correlation on a regular grid
###
###############################################################
# Define the spatial-coordinates of the points:
x <- seq(0,1,0.032)
y <- seq(0,1,0.032)
set.seed(261)
# Simulation of a spatial Gaussian RF with Matern correlation function
data1 <- GeoSim(x,y,grid=TRUE, corrmodel="Matern",model="Weibull", 
         param=list(shape=1.2,mean=0,scale=0.3/3,nugget=0,smooth=0.5))$data
image.plot(x,y,data1,main="Weibull RF",xlab="",ylab="")



################################################################
###
### Example 4. Simulation of a spatial t  RF
### with  with  underlying Generalized Wendland correlation 
###
###############################################################
# Define the spatial-coordinates of the points:
x <- seq(0,1,0.03)
y <- seq(0,1,0.03)
set.seed(268)
# Simulation of a spatial Gaussian RF with Matern correlation function
data1 <- GeoSim(x,y,grid=TRUE, corrmodel="GenWend",model="StudentT", sparse=TRUE,
         param=list(df=1/4,mean=0,sill=1,scale=0.3,nugget=0,smooth=1,power2=5))$data
image.plot(x,y,data1,col=terrain.colors(100),main="Student-t RF",xlab="",ylab="")




################################################################
###
### Example 5. Simulation of a sinhasinh RF
###   with  underlying Wend0 correlation.
###
###############################################################

# Define the spatial-coordinates of the points:
x <- runif(500, 0, 2)
y <- runif(500, 0, 2)
coords <- cbind(x,y)
set.seed(261)
corrmodel="Wend0"
# Simulation of a spatial Gaussian RF:
param=list(power2=4,skew=0,tail=1,
             mean=0,sill=1,scale=0.2,nugget=0)  ## gaussian case
data0 <- GeoSim(coordx=coords, corrmodel=corrmodel,
               model="SinhAsinh", param=param,sparse=TRUE)$data
plot(density(data0),xlim=c(-7,7))

param=list(power2=4,skew=0,tail=0.7,
             mean=0,sill=1,scale=0.2,nugget=0) ## heavy tails
data1 <- GeoSim(coordx=coords, corrmodel=corrmodel,
               model="SinhAsinh", param=param,sparse=TRUE)$data
lines(density(data1),lty=2)

param=list(power2=4,skew=0.5,tail=1,
             mean=0,sill=1,scale=0.2,nugget=0)  ## asymmetry
data2 <- GeoSim(coordx=coords, corrmodel=corrmodel,
               model="SinhAsinh", param=param,sparse=TRUE)$data
lines(density(data2),lty=3)


################################################################
###
### Example 6. Simulation of a bivariate Gaussian RF
### with  bivariate Matern correlation model
###
###############################################################

# Define the spatial-coordinates of the points:
x <- runif(500, 0, 2)
y <- runif(500, 0, 2)
coords <- cbind(x,y)

# Simulation of a bivariate spatial Gaussian RF:
# with a separable Bivariate Matern
set.seed(12)
param=list(mean_1=4,mean_2=2,smooth_1=0.5,smooth_2=0.5,smooth_12=0.5,
           scale_1=0.12,scale_2=0.1,scale_12=0.15,
           sill_1=1,sill_2=1,nugget_1=0,nugget_2=0,pcol=0.5)
data <- GeoSim(coordx=coords,corrmodel="Bi_matern",
              param=param)$data
opar=par(no.readonly = TRUE)
par(mfrow=c(1,2))
quilt.plot(coords,data[1,],col=terrain.colors(100),main="1",xlab="",ylab="")
quilt.plot(coords,data[2,],col=terrain.colors(100),main="2",xlab="",ylab="")
par(opar)


################################################################
###
### Example 7. Simulation of a  spatio temporal Gaussian RF.
### observed on  dynamical location sites with double exponential correlation 
###
###############################################################

# Define the dynamical spatial-coordinates of the points:

coordt=1:5
coordx_dyn=list()
maxN=30
set.seed(8)
for(k in 1:length(coordt))
{
NN=sample(1:maxN,size=1)
x <- runif(NN, 0, 1)
y <- runif(NN, 0, 1)
coordx_dyn[[k]]=cbind(x,y)
}
coordx_dyn

param<-list(nugget=0,mean=0,scale_s=0.2/3,scale_t=2/3,sill=1)
data <- GeoSim(coordx_dyn=coordx_dyn, coordt=coordt, corrmodel="Exp_Exp",
                     param=param)$data
## spatial realization at first temporal instants
data[[1]]
## spatial realization at third temporal instants
data[[3]]




################################################################
###
### Example 8. Simulation of a Gaussian RF 
###  with a Wend0 correlation in the north emisphere of the planet earth
### using geodesic distance
###############################################################
distance="Geod";radius=6371

NN=3000 ## total point on the sphere on lon/lat format
set.seed(80)
coords=cbind(runif(NN,-180,180),runif(NN,0,90))
## Set the wendland parameters
corrmodel <- "Wend0"
param<-list(mean=0,sill=1,nugget=0,scale=1000,power2=3)
# Simulation of a spatial Gaussian RF on the sphere
#set.seed(2)
data <- GeoSim(coordx=coords,corrmodel=corrmodel,sparse=TRUE,
               distance=distance,radius=radius,param=param)$data
#require(globe)
#globe::globeearth(eye=place("newyorkcity"))
#globe::globepoints(loc=coords,pch=20,col =  cm.colors(length(data),alpha=0.1)[rank(data)])