sim.Krig: Conditional simulation of a spatial process

Description

Generates exact (or approximate) random draws from the conditional distribution of a spatial process given specific observations. This is a useful way to characterize the uncertainty in the predicted process from data. This is known as conditional simulation in geostatistics or generating an ensemble prediction in the geosciences. sim.Krig.grid can generate a conditional sample for a large regular grid but is restricted to stationary correlation functions.

Usage

sim.Krig(object, xp, M = 1, verbose = FALSE)
sim.Krig.approx(object, grid.list = NA, M = 1, nx = 40, ny = 40, verbose = 
FALSE, 
 extrap = FALSE)
sim.mKrig.approx(mKrigObject, predictionGrid = NULL,
                            simulationGridlist=NULL, gridRefinement=5, gridExpansion=1,
    M = 1, nx = 40, ny = 40,  verbose = FALSE )

Arguments

Value

sim.Krig a matrix with rowss indexed by the locations in xp and columns being the M independent draws.

sim.Krig.approx a list with components x, y and z. x and y define the grid for the simulated field and z is a three dimensional array with dimensions c(nx, ny, M) where the first two dimensions index the field and the last dimension indexes the draws.

sim.mKrig.approx a list with predictionGrid being the locations where the field has been simulated. Ensemble is a matrix where rows index the simulated vlaues of the field and columns are the different draws, call is the calling sequence. Not that if predictionGrid has been omitted in the call or is created beforehand using make.surface.grid it is easy to reformat the results into an image format for ploting using as.surface. e.g. if simOut is the output object then to plot the 3rd draw:

imageObject<- as.surface(simOut$PredictionGrid, simOut$Ensemble[,3] ) image.plot( imageObject)

Details

These functions generate samples from a conditional multivariate distribution, or an approximate one, that describes the uncertainty in the estimated spatial process under Gaussian assumptions. An important assumption throughout these functions is that all covariance parameters are fixed at their estimated or prescribed values from the passed object.

Given a spatial process Z(x)= P(x) + h(x) observed at

Y.k = P(x.k) + h(x.k) + e.k

where P(x) is a low order, fixed polynomial and h(x) a Gaussian spatial process. With Y= Y.1, ..., Y.N, the goal is to sample the conditional distribution of the process. [Z(x) | Y ]

For fixed a covariance this is just a multivariate normal sampling problem. sim.Krig.standard samples this conditional process at the points xp and is exact for fixed covariance parameters. sim.Krig.grid also assumes fixed covariance parameters and does approxiamte sampling on a grid.

The outline of the algorithm is

0) Find the spatial prediction at the unobserved locations based on the actual data. Call this Z.hat(x).

1) Generate an unconditional spatial process and from this process simluate synthetic observations. At this point the approximation is introduced where the field at the observation locations is approximated using interpolation from the nearest grid points.

2) Use the spatial prediction model ( using the true covariance) to estimate the spatial process at unobserved locations.

3) Find the difference between the simulated process and its prediction based on synthetic observations. Call this e(x).

4) Z.hat(x) + e(x) is a draw from [Z(x) | Y ].

sim.Krig.standard follows this algorithm exactly.

sim.Krig.approx and sim.mKrig.approx evaluate the conditional surface on grid and simulates the values of h(x) off the grid using bilinear interpolation of the four nearest grid points. Because of this approximation it is important to choose the grid to be fine relative to the spacing of the observations. The advantage of this approximation is that one can consider conditional simulation for large grids -- beyond the size possible with exact methods. Here the method for simulation is circulant embedding and so is restricted to stationary fields. The circulant embedding method is known to fail if the domain is small relative to the correlation range. The argument gridExpansion can be used to increase the size of the domain to make the algorithm work.

Examples

Run this code

data( ozone2)

set.seed( 399)

# fit to day 16 from Midwest ozone data set.
Krig( ozone2$lon.lat, ozone2$y[16,], Covariance="Matern", 
theta=1.0,smoothness=1.0, na.rm=TRUE)-> out

# NOTE theta =1.0 is not the best choice but 
# allows the sim.rf circulant embedding algorithm to 
# work without increasing the domain.

#six missing data locations
 xp<-  ozone2$lon.lat[ is.na(ozone2$y[16,]),]

# 5 draws from process at xp given the data 
# this is an exact calculation
 sim.Krig( out,xp, M=5)-> sim.out

# Compare: stats(sim.out)[3,] to  Exact: predict.se( out, xp)

# simulations on a grid
# NOTE this is approximate due to the bilinear interpolation
# for simulating the unconditional random field. 
# also more  grids points ( nx and  ny) should be used  

sim.Krig.approx(out,M=5, nx=20,ny=20)-> sim.out

# take a look at the ensemble members. 

predict.surface( out, grid= list( x=sim.out$x, y=sim.out$y))-> look

zr<- c( 40, 200)

set.panel( 3,2)
image.plot( look, zlim=zr)
title("mean surface")
for ( k in 1:5){
image( sim.out$x, sim.out$y, sim.out$z[,,k], col=tim.colors(), zlim =zr)
}
data( ozone2)
y<- ozone2$y[16,]
good<- !is.na( y)
y<-y[good]
x<- ozone2$lon.lat[good,]
O3.fit<- mKrig( x,y, Covariance="Matern", theta=1.0,smoothness=1.0, lambda= .01 )
set.seed(122)
O3.sim<- sim.mKrig.approx( O3.fit, nx=100, ny=100, gridRefinement=4, M=5 )
set.panel(3,2)
surface( O3.fit)
for ( k in 1:5){
image.plot( as.surface( O3.sim$predictionGrid, O3.sim$Ensemble[,k]) )
}