Brown-Resnick process
RPbrownresnick defines a Brown-Resnick process.
RPbrownresnick(phi, tcf, xi, mu, s)
phi |
specifies the covariance model or variogram, see RMmodel and RMmodelsAdvanced. |
tcf |
the extremal correlation function; either |
xi, mu, s |
the extreme value index, the location parameter and the scale parameter, respectively, of the generalized extreme value distribution. See Details. |
The argument xi is always a number, i.e. ξ is constant in space. In contrast, μ and s might be constant numerical values or (in future!) be given by an RMmodel, in particular by an RMtrend model.
For xi=0, the default values of mu and s are 0 and 1, respectively. For xi\not=0, the default values of mu and s are 1 and |ξ|, respectively, so that it defaults to the standard Frechet case if ξ > 0.
The functions RPbrorig, RPbrshifted and RPbrmixed
perform the simulation of a Brown-Resnick process, which is defined by
Z(x) = max_{i=1, 2, ...} X_i * exp(W_i(x) - gamma^2),
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
phi.
For simulation, internally, one of the methods
RPbrorig, RPbrshifted and
RPbrmixed is chosen automatically.
Advanced options
are maxpoints and max_gauss, see
RFoptions.
Further advanced options related to the
simulation methods RPbrorig,
RPbrshifted and RPbrmixed can be
found in the paragraph ‘Specific method options for Brown-Resnick
Fields’ in RFoptions.
Brown, B.M. and Resnick, S.I. (1977). Extreme values of independent stochastic processes. J. Appl. Probab. 14, 732-739.
Buishand, T., de Haan , L. and Zhou, C. (2008). On spatial extremes: With application to a rainfall problem. Ann. Appl. Stat. 2, 624-642.
Kabluchko, Z., Schlather, M. and de Haan, L (2009) Stationary max-stable random fields associated to negative definite functions Ann. Probab. 37, 2042-2065.
Oesting, M., Kabluchko, Z. and Schlather M. (2012) Simulation of Brown-Resnick Processes, Extremes, 15, 89-107.
RFoptions(seed=0) ## *ANY* simulation will have the random seed 0; set ## RFoptions(seed=NA) to make them all random again ## for some more sophisticated models see 'maxstableAdvanced'
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