Non-stationary covariance model corresponding to a variogram model
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)
RMcov(gamma, x, y=NULL, z=NULL, T=NULL, grid, a, var, scale, Aniso, proj, raw, norm)
gamma |
a variogram model. Possibly multivariate. |
x,y,z,T,grid |
The usual arguments as in |
a |
vector of weights. The length of |
var,scale,Aniso,proj |
optional arguments; same meaning for any
|
raw |
logical. If |
norm |
optional model that gives the norm between locations |
Martin Schlather, schlather@math.uni-mannheim.de
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)
Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.