R: Conditional simulation of a spatial process

sim.spatialProcess {fields} | R Documentation

## 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.spatialProcess(object, xp,  M = 1, verbose = FALSE, ...)

    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,...)

### Arguments

`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 `theta` for the Wendland but a multiple of this maybe needed to obtain a positive definite circulant covariance function.
`extrap` | If FALSE conditional process is not evaluated outside the convex hull of observations.
`fastTpsObject` | The output object returned by fastTps
`grid.list` | Grid information for evaluating the conditional surface as a grid.list.
`gridRefinement` | Amount to increase the number of grid points for the simulation grid.
`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.
`mKrigObject` | An mKrig Object
`M` | Number of draws from conditional distribution.
`nx` | Number of grid points in prediction locations for x coordinate.
`ny` | Number of grid points in prediction locations for x coordinate.
`nxSimulation` | Number of grid points in the circulant embedding simulation x coordinate.
`nySimulation` | Number of grid points in the circulant embedding simulation x coordinate.
`object` | A Krig object.
`predictionPoints` | A matrix of locations defining the points for evaluating the predictions.
`predictionPointsList` | A `grid.list` defining the rectangular grid for evaluating the predictions.
`simulationGridList` | A `gridlist` describing grid for simulation. If missing this is created from the range of the locations, `nx`, `ny`, `gridRefinement`, and `gridExpansion` or from the range and and `nxSimulation`, `nySimulation`.
`xp` | Same as predictionPoints above.

`...` | Any other arguments to be passed to the predict function. Usually this is the `Z` or `drop.Z` argument when there are additional covariates in the fixed part of the model. (See example below.)
`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 h(x)= P(x) + g(x) observed at

Y.k = Z(x.k)d + P(x.k) + g(x.k) + e.k

where P(x) is a low order, fixed polynomial and g(x) a Gaussian spatial process and Z(x.k) is a vector of covariates that are also indexed by space (such as elevation). Z(x.k)d is a linear combination of the the covariates with the parameter vector d being a component of the fixed part of the model and estimated in the usual way by generalized least squares.

With Y= Y.1, ..., Y.N, the goal is to sample the conditional distribution of the process.

[h(x) | Y ] or the full prediction Z(x)d + h(x)

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 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 h.hat(x) and this is the conditional mean.

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

`sim.spatialProcess` Follows this algorithm exactly. For the case of an addtional covariate this of course needs to be included. For a model with covariates use `drop.Z=TRUE` for the function to ignore prediction using the covariate and generate conditional samples for just the spatial process and any low order polynomial. Finally, it should be noted that this function will also work with an `mKrig` object because the essential prediction information in the mKrig and spatialProcess objects are the same. The naming is through convenience.

`sim.Krig` Also follows this algorithm exactly but for the older `Krig` object. Note the inclusion of drop.Z=TRUE or FALSE will determine whether the conditional simulation includes the covariates Z or not. (See example below.)

`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.

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

### Value

`sim.Krig and sim.spatialProcess` a matrix with rows 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 `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 `attributes` of `predictionPoints`. `Ensemble` is a matrix where rows index the simulated values of the field and columns are the different draws, `call` is the calling sequence. Not that if `predictionPoints` 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)

`sim.fastTps.approx` is a wrapper function that calls `sim.mKrig.approx`.

### Author(s)

Doug Nychka

### See Also

sim.rf, Krig, spatialProcess

### Examples

    ## Not run:
    # conditional simulation with covariates
    # colorado climate example
      data(COmonthlyMet)
      fit1E<- spatialProcess(CO.loc,CO.tmin.MAM.climate, Z=CO.elev   )
    # conditional simulation at missing data
      good<- !is.na(CO.tmin.MAM.climate )
      infill<- sim.spatialProcess( fit1E, xp=CO.loc[!good,],
                    Z= CO.elev[!good], M= 10)
    # get an elevation grid  ... NGRID<- 50 gives a nicer image but takes longer
     NGRID <- 25
     # get elevations on a grid
       COGrid<- list( x=seq( -109.5, -101, ,NGRID), y= seq(39, 41.5,,NGRID) )
       COGridPoints<- make.surface.grid( COGrid)
     # elevations are a bilinear interpolation from the 4km
     # Rocky Mountain elevation fields data set.
       data( RMelevation)
       COElevGrid<- interp.surface( RMelevation, COGridPoints )
    # NOTE call to sim.Krig treats the grid points as just a matrix
    # of locations the plot has to "reshape" these into a grid
    # to use with image.plot
       SEout<- sim.spatialProcess( fit1E, xp=COGridPoints,  Z= COElevGrid, M= 30)
    # for just the smooth surface in lon/lat
    #  SEout<- sim.spatialProcess( fit1E, xp=COGridPoints,  drop.Z=TRUE, M= 30)
    # in practice M should be larger to reduce Monte Carlo error.
       surSE<- apply( SEout, 2, sd )
       image.plot( as.surface( COGridPoints, surSE))
       points( fit1E$x, col="magenta", pch=16)

    ## End(Not run)

    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)
    }

    ## Not run:
    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 )

    ## End(Not run)
    ## Not run:
    #An example for fastTps:
      data(ozone2)
      y<- ozone2$y[16,]
      good<- !is.na( y)
      y<-y[good]
      x<- ozone2$lon.lat[good,]
      O3FitMLE<- fastTpsMLE( 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)
    # check
     plot( O3Obj$x)
     US( add=TRUE)
     image.plot( as.surface( O3GridList,O3Sim$Ensemble[,1] ), add=TRUE)
     points( O3Obj$x, pch=16, col="magenta")

    ## End(Not run)

[Package fields version 9.6.1 Index]