R: Space-time moving average model

RMave {RandomFields} | R Documentation

## Space-time moving average model

### Description

`RMave` is a univariate stationary covariance model which depends on a normal scale mixture covariance model phi.

The corresponding covariance function only depends on the difference between two points in the d-dimensional space and is given by

C(h, u) = |E + 2 A h h^t A|^{-1/2} phi(sqrt[|h|^2/2 + (z^t h + u)^2 (1 - 2h^t A (E + 2 A h h^t A)^{-1} A h)])

where E is the identity matrix. The spatial dimension is d-1 and h is real-valued.

### Usage

    RMave(phi, A, z, spacetime, var, scale, Aniso, proj)

### Arguments

`phi` | a covariance model which is a normal mixture, that means an `RMmodel` whose `monotone` property equals `'normal mixture'`, see `RFgetModelNames(monotone="normal mixture")`
`A` | a symmetric d-1 x d-1-matrix if the corresponding random field is in the d-dimensional space
`z` | a d-1 dimensional vector if the corresponding random field is on d-dimensional space
`spacetime` | logical. If FALSE then the model is interpreted as if h=0, i.e. the spatial dimension is d. Default is `TRUE`
`var,scale,Aniso,proj` | optional arguments; same meaning for any `RMmodel`. If not passed, the above covariance function remains unmodified.

### Details

See Schlather, M. (2010), Example 13 with l=1)

### Value

`RMave` returns an object of class `RMmodel`

### Author(s)

Martin Schlather, schlather@math.uni-mannheim.de

### References

- Schlather, M. (2010) Some covariance models based on normal scale mixtures. Bernoulli, 16, 780-797.

### See Also

`RFfit`. `RFsimulate`, `RMmodel`, `RMstp`

### Examples

    RFoptions(seed=0) ## *ANY* simulation will have the random seed 0; set
    ##                   RFoptions(seed=NA) to make them all random again

    ## Example of an evaluation of the ave2-covariance function
    ## in three different ways
    ## ---------------------------------------------------------
    ## some parameters A and z
    A <- matrix(c(2,1,1,2),ncol=2)
    z <- c(1,2)
    ## h for evalutation
    h <- c(1,2)
    ## some abbreviations
    E <- matrix(c(1,0,0,1),ncol=2)
    B <- A %*% h %*% t(h) %*% A
    phi <- function(t){return(RFcov(RMwhittle(1), t))}
    ## ---------------------------------------------------------
    ## the following should yield the same value 3 times
    ## (also for other choices of A,z and h)
    z1 <- RFcov( model=RMave(RMwhittle(1),A=A,z=z) , x=t(c(h,0)) )
    z2 <- RFcov( model=RMave(RMwhittle(1),A=A,z=z,spacetime=FALSE) , x=t(h) )
    z3 <- ( (det(E+2*B))^(-1/2) ) *
     phi( sqrt( sum(h*h)/2 + (t(z) %*% h)^2 *
     ( 1-2*t(h) %*% A %*% solve(E+2*B) %*% A %*% h) ) )
    ##
    stopifnot(abs(z1-z2)<1e-12, abs(z2-z3)<1e-12)

[Package RandomFields version 3.1.50 Index]