# 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} 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

 1 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.

RMmodel, RFsimulate, RFfit.

### Examples

  1 2 3 4 5 6 7 8 9 10 11 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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