Single temporal process

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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} φ(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 + ξ_2(x) - ξ_2(y),

A = S_x + S_y + 4 M h h^t M

m = h^t M h

h = x - y

Usage

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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 x d

z

arbitrary vector, z \in R^d

M

an arbitrary, symmetric d x d matrix

var,scale,Aniso,proj

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

Details

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

Value

RMstp returns an object of class RMmodel

Author(s)

Martin Schlather, schlather@math.uni-mannheim.de

References

  • Paciorek C.J., and Schervish, M.J. (2006) Spatial modelling using a new class of nonstationary covariance functions, Environmetrics 17, 483-506.

  • Schlather, M. (2010) Some covariance models based on normal scale mixtures. Bernoulli, 16, 780-797.

See Also

RMmodel, RFsimulate, RFfit.

Examples

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

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))

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