(Mixed) Moving Maxima
RPsmith defines a moving maximum process or a mixed moving
maximum process with finite number of shape functions.
RPsmith(shape, tcf, xi, mu, s)
shape |
an |
tcf |
an |
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.
It simulates max-stable processes Z that are referred to as “Smith model”.
Z(x) = max_{i=1, 2, ...} X_i * Y_i(x - W_i),
where (W_i, X_i) are the points of a Poisson point process on R^d x (0, ∞) with intensity dw * c/x^2 dx and Y_i ~ Y are iid measurable random functions with E[int max(0, Y(x)) dx ] < ∞. The constant c is chosen such that Z has standard Frechet margins.
IMPORTANT: For consistency reasons with the geostatistical definitions in this package the scale argument differs froms the original definition of the Smith model! See the example below.
RPsmith depends on RRrectangular
and its arguments.
Advanced options
are maxpoints and max_gauss, see
RFoptions.
Martin Schlather, schlather@math.uni-mannheim.de, https://www.wim.uni-mannheim.de/schlather/
Haan, L. (1984) A spectral representation for max-stable processes. Ann. Probab., 12, 1194-1204.
Smith, R.L. (1990) Max-stable processes and spatial extremes Unpublished Manuscript.
RFoptions(seed=0) ## *ANY* simulation will have the random seed 0; set ## RFoptions(seed=NA) to make them all random again model <- RMball() x <- seq(0, 1000, 0.2) z <- RFsimulate(RPsmith(model, xi=0), x) plot(z) hist(z@data$variable1, 50, freq=FALSE) curve(exp(-x) * exp(-exp(-x)), from=-3, to=8, add=TRUE) ## 2-dim x <- seq(0, 10, 0.1) z <- RFsimulate(RPsmith(model, xi=0), x, x) plot(z) ## original Smith model x <- seq(0, 10, 0.05) model <- RMgauss(scale = sqrt(2)) # !! cf. definition of RMgauss z <- RFsimulate(RPsmith(model, xi=0), x, x) plot(z) ## for some more sophisticated models see 'maxstableAdvanced'
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