RandomFields: RMcov – R documentation

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RMcov

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 γ(x,y) = |x-y|, x,y≥ 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 γ(x,y) = |x-y| is replaced by W(x) = Z(x) -∑_{i=1}^n a_i Z(x_i) with ∑_{i=1}^n a_i = 1.

For a given variogram γ, a_i and x_i, the model equals C(x, y) = ∑_{i=1}^n a_i (γ(x, x_i) + γ(x_i, y)) - γ(x, y) - ∑_{i=1}^n ∑_{j=1}^n a_i a_j γ(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

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)

RandomFields

Simulation and Analysis of Random Fields

v3.3.10

GPL (>= 3)

Authors

Martin Schlather [aut, cre], Alexander Malinowski [aut], Marco Oesting [aut], Daphne Boecker [aut], Kirstin Strokorb [aut], Sebastian Engelke [aut], Johannes Martini [aut], Felix Ballani [aut], Olga Moreva [aut], Jonas Auel[ctr], Peter Menck [ctr], Sebastian Gross [ctr], Ulrike Ober [ctb], Paulo Ribeiro [ctb], Brian D. Ripley [ctb], Richard Singleton [ctb], Ben Pfaff [ctb], R Core Team [ctb]

Initial release