RandomFields (version 3.1.12)

RMstp: Single temporal process

Description

RMstp is a univariate covariance model which depends on a normal mixture submodel $\phi$. The covariance is given by $$C(x,y) = |S_x|^{1/4} |S_y|^{1/4} |A|^{-1/2} \phi(Q(x,y)^{1/2})$$ where $$Q(x,y) = c^2 - m^2 + h^t (S_x + 2(m + c)M) A^{-1} (S_y + 2 (m-c)M)h,$$ $$c = -z^t h + \xi_2(x) - \xi_2(y),$$ $$A = S_x + S_y + 4 M h h^t M$$ $$m = h^t M h$$ $$h = x - y$$

Usage

RMstp(xi, phi, S, z, M, var, scale, Aniso, proj)

Arguments

xi

arbitrary univariate function on $R^d$

phi

an RMmodel that is a normal mixture model, cf. RFgetModelNames(monotone="normal mixture")

S

functions that returns strictly positive definite $d\times d$

z

arbitrary vector, $z \in R^d$

M

an arbitrary, symmetric $d \times d$ matrix

var,scale,Aniso,proj

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

Value

• RMstp returns an object of class RMmodel

Details

See Schlather (2008) formula (13). The model allows for mimicking cyclonic behaviour.

References

• Paciorek C.J., and Schervish, M.J. (2006) Spatial modelling using a new class of nonstationary covariance functions,Environmetrics17, 483-506.
• Schlather, M. (2010) Some covariance models based on normal scale mixtures.Bernoulli,16, 780-797.

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
StartExample()
model <- RMstp(xi = RMrotat(phi= -2 * pi, speed=1),
               phi = RMwhittle(nu = 1),
               M=matrix(nc=3, rep(0, 9)),
               S=RMetaxxa(E=rep(1, 3), alpha = -2 * pi,
                          A=t(matrix(nc=3, c(2, 0, 0, 1, 1 , 0, 0, 0, 0))))
              )
x <- seq(0, 10, 0.7)
plot(RFsimulate(model, x=x, y=x, z=x))
FinalizeExample()