Simulation methods for Brown-Resnick processes

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Description

These models define the particular way to simulate Brown-Resnick processes

Usage

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RPbrmixed(phi, tcf, xi, mu, s, meshsize, vertnumber, optim_mixed,
          optim_mixed_tol, optim_mixed_maxpo, lambda, areamat, variobound) 

RPbrorig(phi, tcf, xi, mu, s)

RPbrshifted(phi, tcf, xi, mu, s)

Arguments

phi

object of class RMmodel; specifies the covariance model to be simulated.

tcf

the extremal correlation function; either phi or tcf must be given.

xi, mu, s

the shape parameter, the location parameter and the scale parameter, respectively, of the generalized extreme value distribution. See Details.

lambda

positive constant factor in the intensity of the Poisson point processused in the M3 representation, cf. Thm. 6 and Remark 7 in Oesting et. al (2012); can be estimated by setting optim_mixed if unknown. Default value is 1.

areamat

vector or matrix of values in [0,1] with odd length (odd number of rows and columns, respectively). Each value represents the portion of processes whose maximum is located at a specific location on a grid taken into account for the simulation of the shape function in the M3 representation. The center of areamat represents the value for the origin, the other entries belong to the corresponding locations on a 1D or 2D grid. areamat can be used for dimensions 1 and 2 only; can be optimized by setting optim_mixed if unknown. Default value is 1.

meshsize, vertnumber, optim_mixed, optim_mixed_tol, optim_mixed_maxpo, variobound

further arguments for simulation via the mixed moving maxima (M3) representation; see RFoptions

Details

The argument xi is always a number, i.e. ξ is constant in space. In contrast, μ and s might be constant numerical value or given a RMmodel, in particular by a RMtrend model.

The functions RPbrorig, RPbrshifted and RPbrmixed simulate a Brown-Resnick process, which is defined by

Z(x) = max_{i=1, 2, ...} X_i * exp(W_i(x) - gamma),

where the X_i are the points of a Poisson point process on the positive real half-axis with intensity 1/x^2 dx, W_i ~ Y are iid centered Gaussian processes with stationary increments and variogram gamma given by model. The functions correspond to the following ways of simulation:

RPbrorig

simulation via using the original definition (method 0 in Oesting et al., 2012)

RPbrshifted

simulation using a random shift (similar to method 1 and 2)

RPbrmixed

simulation using M3 representation (method 4)

Value

The functions return an object of class RMmodel

Note

Advanced options for RPbroriginal and RPbrshifted are maxpoints and max_gauss, see RFoptions.

Author(s)

Marco Oesting, oesting@math.uni-mannheim.de, Martin Schlather, schlather@math.uni-mannheim.de http://ms.math.uni-mannheim.de/de/publications/software

References

  • Oesting, M., Kabluchko, Z. and Schlather M. (2012) Simulation of Brown-Resnick Processes, Extremes, 15, 89-107.

See Also

RPbrownresnick, RMmodel, RPgauss, maxstable, maxstableAdvanced

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



## currently does not work

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