RMtbm: Turning Bands Method

Description

RMtbm is a univariate stationary isotropic covariance model in dimension reduceddim which depends on a univariate stationary isotropic covariance $\phi$ in a bigger dimension fulldim. For formulas for the covariance function see details.

Usage

RMtbm(phi, fulldim, reduceddim, layers, var, scale, Aniso, proj)

Arguments

phi, fulldim, reduceddim, layers

see RPtbm.

var,scale,Aniso,proj

optional parameters; same meaning for any RMmodel. If not passed, the above covariance function remains unmodified.

Value

RMtbm returns an object of class RMmodel

Details

The turning bands method stems from the 1:1 correspondence between the isotropic covariance functions of different dimensions. See Gneiting (1999).

The standard case reduceddim=1 and fulldim=3. If only one of the parameters are given, then the difference of two parameters equals 2. For d == n + 2, where n=reduceddim and d==fulldim the original dimension, we have

$C (r) = ϕ (r) + r ϕ^{'} (r) / n$ which, for n=1 reduced to the standard TBM operator $C (r) = \frac{d}{d r} r ϕ (r)$ For d == 2 && n == 1 we have $C (r) = \frac{d}{d r} \int_{0}^{r} \frac{u ϕ (u)}{\sqrt{r^{2} - u^{2}}} d u$

References

Gneiting, T. (1999) On the derivatives of radial positive definite function.J. Math. Anal. Appl,236, 86-99
Matheron, G. (1973). The intrinsic random functions and their applications.Adv . Appl. Probab.,5, 439-468.

Examples

Run this code

RFoptions(seed=0)
x <- seq(0,25, if (interactive()) 0.02 else 5)
model <- RPtbm(RMspheric())
z <- RFsimulate(model, x, x)
plot(z)
RFoptions(seed=NA)

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