RandomFields (version 3.1.36)

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 arguments; 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) and Strokorb and Schlather (2014). The standard case reduceddim=1 and fulldim=3. If only one of the arguments are given, then the difference of the two arguments equals 2. For d == n + 2, where n=reduceddim and d==fulldim the original dimension, we have $$ C(r) = \phi(r) + r \phi'(r) / n $$ which, for n=1 reduced to the standard TBM operator $$ C(r) =\frac {d}{d r} r \phi(r) $$ For d == 2 && n == 1 we have $$ C(r) = \frac{d}{dr}\int_0^r \frac{u\phi(u)}{\sqrt{r^2 - u^2}} d u $$

‘Turning layers’ is a generalization of the turning bands method, see Schlather (2011).

References

Turning bands

• 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.
• Strokorb, K., Ballani, F., and Schlather, M. (2014) Tail correlation functions of max-stable processes: Construction principles, recovery and diversity of some mixing max-stable processes with identical TCF. Extremes, Submitted.

Turning layers

• Schlather, M. (2011) Construction of covariance functions and unconditional simulation of random fields. In Porcu, E., Montero, J.M. and Schlather, M., Space-Time Processes and Challenges Related to Environmental Problems. New York: Springer.

See Also

RPtbm, RFsimulate.

Examples

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

x <- seq(0, 10, 0.02)
model <- RMspheric()
plot(model, model.on.the.line=RMtbm(RMspheric()), xlim=c(-1.5, 1.5))

z <- RFsimulate(RPtbm(model), x, x)
plot(z)