% fields, Tools for spatial data
% Copyright 2004-2013, Institute for Mathematics Applied Geosciences
% University Corporation for Atmospheric Research
% Licensed under the GPL -- www.gpl.org/licenses/gpl.html

\name{sim.Krig}
\alias{sim.Krig}
\alias{sim.Krig.approx}
\alias{sim.mKrig.approx}
\alias{sim.fastTps.approx}

\title{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 = NULL, M = 1, nx = 40, ny = 40,
                 verbose = FALSE, extrap = FALSE)

sim.mKrig.approx(mKrigObject, predictionPoints = NULL,
                 predictionPointsList = NULL, simulationGridList =
                 NULL, gridRefinement = 5, gridExpansion = 1 + 1e-07, M
                 = 1, nx = 40, ny = 40, nxSimulation = NULL,
                 nySimulation = NULL, delta = NULL, verbose = FALSE)
 
sim.fastTps.approx(fastTpsObject, 
                 predictionPointsList, simulationGridList =
                 NULL, gridRefinement = 5, gridExpansion = 1 + 1e-07, M
                 = 1,  delta = NULL, verbose=FALSE)

} 
%- maybe also 'usage' for other objects documented here. 
\arguments{
 
 \item{delta}{If the covariance has compact support the simulation method can 
take advantage of this. This is the amount of buffer added for the simulation domain in the circulant embedding method. 
A minimum size would be \code{theta} for the Wendland but a multiple of this maybe needed to obtain a positive definite 
circulant covariance function.  }

  \item{extrap}{ If FALSE conditional process is not evaluated outside 
     the convex hull of observations. }

  \item{fastTpsObject}{The output object returned by fastTps}

  \item{grid.list}{Grid information for evaluating the conditional 
         surface as a grid.list.}
   \item{gridRefinement}{Amount to increase the number of grid points
  for the simulation grid.}

  \item{gridExpansion}{Amount to increase the size of teh simulation
grid. This is used to increase the simulation domain so that the
circulant embedding algorithm works.}

  \item{mKrigObject}{An mKrig Object}

 
  \item{M}{Number of draws from conditional distribution.}

  \item{nx}{ Number of grid points in prediction locations for x coordinate.}
  \item{ny}{ Number of grid points in  prediction locations for x coordinate.}

  \item{nxSimulation}{ Number of grid points in the circulant embedding simulation x coordinate.}
  \item{nySimulation}{ Number of grid points in the circulant embedding simulation x coordinate.}

  \item{object}{A Krig object.}

  \item{predictionPoints}{A matrix of locations defining the 
          points for evaluating the predictions.}

   \item{predictionPointsList}{ A \code{grid.list} defining the 
rectangular grid for evaluating the predictions.}

  \item{simulationGridList}{ A \code{gridlist} describing grid for
simulation. If missing this is created from the range of the
locations, \code{nx}, \code{ny}, \code{gridRefinement}, and \code{gridExpansion}
or from the range and and \code{nxSimulation}, \code{nySimulation}.}

 
  \item{xp}{Same as predictionPoints above.}

 \item{\dots}{Any other arguments to be passed to the predict function. }

  \item{verbose}{If true prints out intermediate information. }
}

\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.  \code{sim.Krig.standard} samples this conditional process at
the points \code{xp} and is exact for fixed covariance parameters.
\code{sim.Krig.grid} also assumes fixed covariance parameters and does
approximate 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 ].

\code{sim.Krig.standard} follows this algorithm exactly. 

\code{sim.Krig.approx} and \code{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 \code{gridExpansion} can be used to increase the
size of the domain to make the algorithm work.

\code{sim.fastTps.approx} Is optimized for the approximate thin plate
spline estimator in two dimensions and \code{k=2}. For efficiency the
ensemble prediction locations must be on a grid.

}
\value{
\code{sim.Krig}
 a matrix with rowss indexed by the locations in \code{xp} and columns being the 
 \code{M} independent draws.

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

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

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

\code{sim.fastTps.approx} is  a wrapper function that calls \code{sim.mKrig.approx}. 

}

\author{Doug Nychka}
\seealso{ sim.rf, Krig}
\examples{
data( ozone2)

set.seed( 399)

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

# 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: predictSE( 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. 

predictSurface( 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$z[,,k], col=tim.colors(), zlim =zr)
}
\dontrun{
data( ozone2)
y<- ozone2$y[16,]
good<- !is.na( y)
y<-y[good]
x<- ozone2$lon.lat[good,]
O3.fit<- mKrig( x,y, Covariance="Matern", theta=.5,smoothness=1.0, lambda= .01 )
set.seed(122)
O3.sim<- sim.mKrig.approx( O3.fit, nx=100, ny=100, gridRefinement=3, M=5 )
set.panel(3,2)
surface( O3.fit)
for ( k in 1:5){
image.plot( as.surface( O3.sim$predictionPoints, O3.sim$Ensemble[,k]) )
}
# conditional simulation at missing data
xMissing<- ozone2$lon.lat[!good,]
O3.sim2<- sim.mKrig.approx( O3.fit, xMissing, nx=80, ny=80, gridRefinement=3, M=4, verbose=TRUE )
#check of Wendland
O3.fit.Wendland<- mKrig( x,y, cov.function="wendland.cov", theta=.5, lambda= .01 )
O3.sim3<- sim.mKrig.approx( O3.fit.Wendland, xMissing, nx=80, ny=80, gridRefinement=3, M=400,
                               verbose=TRUE )
O3.sim4<- sim.mKrig.approx( O3.fit.Wendland, xMissing, nx=80, ny=80, gridRefinement=3, M=400,
                                 verbose=TRUE, delta=O3.fit.Wendland$args$theta )
}
\dontrun{
#An example for fastTps:
  data(ozone2)
  y<- ozone2$y[16,]
  good<- !is.na( y)
  y<-y[good]
  x<- ozone2$lon.lat[good,]
  O3FitMLE<- fastTps.MLE( x,y, theta=1.5 )
  O3Obj<- fastTps( x,y, theta=1.5, lambda=O3FitMLE$lambda.MLE)
# creating a quick grid list based on ranges of locations
  grid.list<- fields.x.to.grid( O3Obj$x, nx=100, ny=100)
  O3Sim<- sim.fastTps.approx( O3Obj,predictionPointsList=grid.list,M=5)
# controlling the grids
  xR<- range( x[,1], na.rm=TRUE)
  yR<- range( x[,2], na.rm=TRUE)
  simulationGridList<- list( x= seq(xR[1],xR[2],,400), y= seq( yR[1],yR[2], ,400))
# very fine localized prediction grid
    O3GridList<- list( x= seq( -90.5,-88.5,,200), y= seq( 38,40,,200))
    O3Sim<- sim.fastTps.approx( O3Obj, M=5, predictionPointsList=O3GridList,
                  simulationGridList = simulationGridList, verbose=TRUE)
# check 
 plot( O3Obj$x)
 US( add=TRUE)
 image.plot( as.surface( O3GridList,O3Sim$Ensemble[,1] ), add=TRUE)
 points( O3Obj$x, pch=16, col="magenta")
}
}
\keyword{spatial}
% at least one, from doc/KEYWORDS