R: Empirical (Cross-)Madogram

RFempiricalmadogram {RandomFields} | R Documentation

## Empirical (Cross-)Madogram

### Description

Calculates the empirical (cross-)madogram. The empirical (cross-)madogram of two random fields X and Y is given by

γ(r):=1/N(r) ∑_{(t_{i},t_{j})|t_{i,j}=r} |(X(t_{i})-X(t_{j}))||(Y(t_{i})-Y(t_{j}))|

where t_{i,j}:=t_{i}-t_{j}, and where N(r) denotes the number of pairs of data points with distancevector t_{i,j}=r.

### Usage

    RFempiricalmadogram(x, y = NULL, z = NULL, T = NULL, data, grid,
     bin=NULL, phi=NULL, theta=NULL, deltaT=NULL, distances, vdim, ...)

### Arguments

`x` | matrix of coordinates, or vector of x coordinates, or object of class `GridTopology` or `raster`. If matrix, `ncol(x)` is the dimension of the index space. Matrix notation is required in case of more than 3 spatial dimensions; in this case, if `grid=FALSE`, `x_ij` is the i-th coordinate in the j-th dimension. Otherwise, if `grid=TRUE`, the columns of `x` are interpreted as gridtriples (see `grid`). if of class `GridTopology` , `x` is interpreted as grid definition and `grid` is automatically set to `TRUE`. Coordinates are not required if the data is an object of class `RFsp`, as these objects already contain its coordinates
`y` | optional vector of y coordinates, ignored if `x` is a matrix
`z` | optional vector of z coordinates, ignored if `x` is a matrix
`T` | optional vector of time coordinates, `T` must always be an equidistant vector or given in a gridtriple format (see `grid`); for each component of `T`, the random field is simulated at all location points; the argument `T` is in an experimental stage.
`grid` | logical; determines whether the vectors `x`, `y`, and `z` or the columns of `x` should be interpreted as a grid definition (see Details). If `grid=TRUE`, either `x`, `y`, and `z` must be equidistant vectors in ascending order or the columns of `x` must be given in the gridtriple format `c(from, stepsize, len)` (see Details); Not required if data is of class `RFsp`
`data` | matrix, data.frame or object of class `RFsp`;
`bin` | a vector giving the borders of the bins; If not specified an array describing the empirical (pseudo-)(cross-) madogram in every direction is returned.
`phi` | an integer defining the number of sectors one half of the X/Y plane shall be devided into. If not specified, either an array is returned (if bin missing) or isotropy is assumed (if bin specified)
`theta` | an integer defining the number of sectors one half of the X/Z plane shall be devided into. Use only for dimension d=3 if phi is already specified
`deltaT` | vector of length 2, specifying the temporal bins. The internal bin vector becomes `seq(from=0, to=deltaT[1], by=deltaT[2])`
`distances` | object of class `dist` representing the upper trianguar part of the matrix of Euclidean distances between the points at which the field is to be simulated; only applicable for stationary and isotropic models; if not `NULL`, `dim` must be given and `x`, `y`, `z` and `T` must be missing or `NULL`.
`vdim` | the number of variables of a multivariate data set. If not given and `data` is an `RFsp` object created by RandomFields, the information there is taken from there. Otherwise `vdim` is assumed to be one. NOTE: still the argument `vdim` is an experimental stage.
`...` | further options and control arguments for the simulation that are passed to and processed by `RFoptions`.

### Details

`RFempiricalmadogram` computes the empirical cross-madogram for given (multivariate) spatial data.

The spatial coordinates `x`, `y`, `z` should be vectors. For random fields of spatial dimension d > 3 write all vectors as colums of matrix x. In this case do neither use y, nor z and write the colums in `gridtriple` notation.

If the data is spatially located on a grid a fast algorithm based on the fast Fourier transformed (fft) will be used. As advanced option the calculation method can also be changed for grid data (see `RFoptions`.)

It is also possible to use `RFempiricalmadogram` to calulate the pseudomadogram (see `RFoptions`).

### Value

`RFempiricalmadogram` returns objects of class `RFempVariog`.

### Author(s)

Jonas Auel, jauel@mail.uni-mannheim.de Sebastian Engelke, sebastian.engelke@unil.ch Johannes Martini, jmartin2@uni-goettingen.de Martin Schlather, schlather@math.uni-mannheim.de http://ms.math.uni-mannheim.de/de/publications/software

### References

Gelfand, A. E., Diggle, P., Fuentes, M. and Guttorp, P. (eds.) (2010) Handbook of Spatial Statistics. Boca Raton: Chapman & Hall/CRL.

Stein, M. L. (1999) Interpolation of Spatial Data. New York: Springer-Verlag

### See Also

`RMstable`, `RMmodel`, `RFsimulate`, `RFfit`. `RFempiricalcovariance` `RFempiricalvariogram`

### Examples

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

    n <- 1 ## use n <- 2 for better results

    ## isotropic model
    model <- RMexp()
    x <- seq(0, 10, 0.02)
    z <- RFsimulate(model, x=x, n=n)
    emp.vario <- RFempiricalmadogram(data=z)
    plot(emp.vario)

    ## anisotropic model
    model <- RMexp(Aniso=cbind(c(2,1), c(1,1)))
    x <- seq(0, 10, 0.05)
    z <- RFsimulate(model, x=x, y=x, n=n)
    emp.vario <- RFempiricalmadogram(data=z, phi=4)
    plot(emp.vario)

    ## space-time model
    model <- RMnsst(phi=RMexp(), psi=RMfbm(alpha=1), delta=2)
    x <- seq(0, 10, 0.05)
    T <- c(0, 0.1, 100)
    z <- RFsimulate(x=x, T=T, model=model, n=n)
    emp.vario <- RFempiricalmadogram(data=z, deltaT=c(10, 1))
    plot(emp.vario, nmax.T=3)

    ## multivariate model
    model <- RMbiwm(nudiag=c(1, 2), nured=1, rhored=1, cdiag=c(1, 5),
                    s=c(1, 1, 2))
    x <- seq(0, 20, 0.1)
    z <- RFsimulate(model, x=x, y=x, n=n)
    emp.vario <- RFempiricalmadogram(data=z)
    plot(emp.vario)

    ## multivariate and anisotropic model
    model <- RMbiwm(A=matrix(c(1,1,1,2), nc=2),
                    nudiag=c(0.5,2), s=c(3, 1, 2), c=c(1, 0, 1))
    x <- seq(0, 20, 0.1)
    data <- RFsimulate(model, x, x, n=n)
    ev <- RFempiricalmadogram(data=data, phi=4)
    plot(ev, boundaries=FALSE)

[Package RandomFields version 3.1.50 Index]