RMcov {RandomFields}

R Documentation

Non-stationary covariance model corresponding to a variogram model

Description

This function generalizes the well-known non-stationary covariance function 2\min\{x,y\} of the Brownian motion with variogram \gamma(x,y) = |x-y|, x,y\ge 0 to arbitrary variogram models any spatial processes of any dimension and multivariability.

Furthermore, the standard condition for the Brownian motion W is that variance equals 0 at the origin, i.e., W(x) =^d Z(x) -Z(0) for any zero mean Gaussian process Z with variogram \gamma(x,y) = |x-y| is replaced by W(x) = Z(x) -\sum_{i=1}^n a_i Z(x_i) with \sum_{i=1}^n a_i = 1.

For a given variogram \gamma, a_i and x_i, the model equals C(x, y) = \sum_{i=1}^n a_i (\gamma(x, x_i) + \gamma(x_i, y)) - \gamma(x, y) - \sum_{i=1}^n \sum_{j=1}^n a_i a_j \gamma(x_i, y_i)

Usage

RMcov(gamma, x, y=NULL, z=NULL, T=NULL, grid, a,
       var, scale, Aniso, proj, raw, norm)

Arguments

gamma	a variogram model. Possibly multivariate.
x, y, z, T, grid	The usual arguments as in RFsimulate to define the locations where the covariates are given. Additional x might be set to one of the values "origin", "center", "extremals", or "all". If x is not given, x is set to "origin".
a	vector of weights. The length of a must equal the number of points given by x, y, z and T. The values of a must sum up to 1. If a is not given, equals weights are used.
var, scale, Aniso, proj	optional arguments; same meaning for any RMmodel. If not passed, the above covariance function remains unmodified.
raw	logical. If FALSE then the data are interpolated. This approach is always save, but might be slow. If TRUE then the data may be accessed when covariance matrices are calculated. No rescaling or anisotropy definition is allowed in combination with the model. The use is dangerous, but fast. Default: FALSE.
norm	optional model that gives the norm between locations

Value

RMcov returns an object of class RMmodel

Author(s)

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

See Also

RMmodel, RFsimulate, RFfit.

Examples

RFoptions(seed=0) ## *ANY* simulation will have the random seed 0; set
##                   RFoptions(seed=NA) to make them all random again
bm <- RMfbm(alpha=1)
plot(bm)

x <- seq(0, 6, if (interactive()) 0.125 else 3)
plot(RFsimulate(bm, x))

## standardizing with the random variable at the origin
z1 <- RFsimulate(RMcov(bm), x)
plot(z1)
z1 <- as.vector(z1)
zero <- which(abs(x) == 0)
stopifnot(abs(z1[zero]) < 1e-13)

## standardizing with the random variable at the center of the interval
z2 <- RFsimulate(RMcov(bm, "center"), x)
plot(z2)
z2 <- as.vector(z2)
stopifnot(abs(z2[(length(z2) + 1) / 2]) < 1e-13)


## standardizing with the random variables at the end points of the interval
z3 <- RFsimulate(RMcov(bm, "extremals"), x)
plot(z3)
z3 <- as.vector(z3)
stopifnot(abs(z3[1] + z3[length(z3)]) < 1e-13)

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