# (Mixed) Moving Maxima

### Description

`RPsmith`

defines a moving maximum process or a mixed moving
maximum process with finite number of shape functions.

### Usage

1 | ```
RPsmith(shape, tcf, xi, mu, s)
``` |

### Arguments

`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. |

### 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 default values of *mu* and *s*
are *1* and *ξ*, respectively.

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.

### Note

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`

.

### Author(s)

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

### References

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 Probab.*,**12**, 1194-1204.Smith, R.L. (1990) Max-stable processes and spatial extremes Unpublished Manuscript.

### See Also

`Advanced RMmodels`

,
`Auxiliary RMmodels`

,
`RMmodel`

,
`RPbernoulli`

,
`RPgauss`

,
maxstable
`maxstableAdvanced`

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 | ```
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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